Advances in Laser Slowing, Cooling, and Trapping using
Narrow Linewidth Optical Transitions
byJohn Patrick Bartolotta
B.S., University of Connecticut, 2014
M.S., University of Connecticut, 2016

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2021

ii
Bartolotta, John Patrick (Ph.D., Physics)
Advances in Laser Slowing, Cooling, and Trapping using Narrow Linewidth Optical Transitions
Thesis directed by Prof. Murray J. Holland

Laser light in combination with electromagnetic traps has been used ubiquitously throughout
the last three decades to control atoms and molecules at the quantum level. Many approacheswith varying degrees of reliance on dissipative processes have been discovered, investigated, andimplemented in both academic and industrial settings. In this thesis, we theoretically investigate
and propose three practical techniques which have the capacity to slow, cool, and trap particles withnarrow linewidth optical transitions. Each of these methods emphasizes a high degree of coherent
control over quantum states, which lends their operation a reduced reliance on spontaneous emission
and thus makes them appealing candidates for application to systems that lack closed cyclingtransitions. In a more general setting, we also study the transfer of entropy from a quantum
system to a classically-behaved coherent light field, and therefore advance our understanding of therole of dissipative processes in optical pumping and laser cooling.
First, we describe theoretically a cooling technique named “sawtooth wave adiabatic passage
cooling,” or “SWAP cooling,” which was discovered experimentally at JILA and has since been
implemented elsewhere with various atomic species. Second, we discuss some of the results ofan experimental realization of SWAP cooling and connect them to the theoretical model. Third,
we propose the extension of the coherent mechanism underlying SWAP cooling to the additional
task of trapping in a configuration named the “SWAP magneto-optical trap,” or “SWAP MOT.”
Fourth, we propose a method to fundamentally speed up the adiabatic transfer process required forthese techniques by using a shortcut-to-adiabaticity protocol based on Lewis-Riesenfeld invariants.
Lastly, we propose and theoretically analyze a simple Gedankenexperiment which can be used to
study the potential transfer of entropy from an ensemble of particles to a coherent light source.

Dedication

To my fiancée Kelsey, who knows me more than I know myself.

To my parents, Robert and Karen. Thanks for always believing in me.

To my siblings, Mary and Sarah. I wouldn’t be who I am today without you.

To the friends and family who helped to shape me along the way.

iv

Acknowledgements

There has never been a time when I recognized the importance and impact of others on mywork more than the years I spent as a graduate student at CU Boulder. My challenging journey
toward doctorship would certainly not have been possible without the support, reassurance, andwisdom of my mentors and colleagues.
I would first like to thank Murray for granting me the opportunity to grow as a person
and physicist under his advisement. I always felt supported by him in both my studies and extracurricular activities. Thanks to Jinx for teaching me the importance of physical intuition overmathematical formalism. Thanks to Athreya Shankar, Simon Jäger, Jarrod Reilly, Haonan Liu,
Liang-Ying Chih, and Victor Colussi for being supportive group mates. Thanks to Graeme Smith,
John Bohn, Dana Anderson, Jun Ye, James Thompson, Ana Maria Rey, Shiqian Ding, Yewei Wu,
Matthew Norcia, and Julia Cline for countless stimulating and productive discussions.
I am indebted to many people and organizations for balancing my lifestyle with activities
separate from my academic pursuits. Thanks to the many local climbing gyms and outdoor cragsfor challenging me both physically and mentally with fantastic climbing routes. Thanks to William
Lewis and Ben Galloway for memorable game nights and career advice. Thanks to the members of
CU-Prime for working toward an inclusive and equitable physics community. And many thanks tothe staff at JILA and the CU Physics Department.
I would not have continued to pursue a physics career without the mentorship of several
motivating physicists at the University of Connecticut. Many thanks to Vasili Kharchenko, Richard
T. Jones, and Philip Mannheim for these experiences.

v

Contents

Chapter
1 Context and outline

1

2 Introduction3

2.1

Coherent models of light-matter interaction . . . . . . . . . . . . . . . . . . . . . . .

3

2.1.1

First quantization: quantum treatment of matter . . . . . . . . . . . . . . . .

5

2.1.2

The two-level particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.3

Interaction of a moving two-level quantum system with a classical light field . 10

2.1.4

Second quantization: interaction of motionless two-level quantum systemwith a quantized light field in a cavity . . . . . . . . . . . . . . . . . . . . . . 12

2.1.5
2.2

2.3

The spontaneous emission process . . . . . . . . . . . . . . . . . . . . . . . . 14

Theory of open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1

The density matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2

The quantum master equation . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3

The Lindblad superoperator in laser slowing, cooling, and trapping . . . . . . 21

2.2.4

The Monte Carlo wave function method . . . . . . . . . . . . . . . . . . . . . 23

Doppler-based cooling and trapping processes . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1

Approximation of classical particle motion . . . . . . . . . . . . . . . . . . . . 25

2.3.2

Cooling force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.3

Minimum temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.4

The magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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3 SWAP cooling: theoretical investigation

35

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2

Basic mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3

System dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4

Dynamics in the high-velocity regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5

3.6

3.4.1

Resonance times and intervals in the adiabatic limit . . . . . . . . . . . . . . 42

3.4.2

Dressed state picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.3

Coherent dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.4

High-velocity regime dynamics including dissipation . . . . . . . . . . . . . . 46

Forces and capture range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1

Conservative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5.2

Forces including dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Effect of higher-order processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6.1

Doppleron resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6.2

Bragg oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.7

Temperature limit as γ → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.8

Cooling efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.9

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 SWAP cooling: experimental results and comparison to theory

62

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2

Basic mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3

Numerical demonstration of basic mechanism . . . . . . . . . . . . . . . . . . . . . . 65

4.4

Demonstration of steady-state cooling and phase space compression

4.5

Comments on entropy removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.6

Cooling forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.7

Temperature measurement and comparison to simulation . . . . . . . . . . . . . . . 71

. . . . . . . . . 66

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4.8

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 SWAP MOT

75

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2

SWAP MOT mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3

Model and semiclassical approximations . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4

Variant of quantum trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.5

Validity of three-state basis in SWAP MOT simulation . . . . . . . . . . . . . . . . . 88

5.6

Phase space dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7

5.6.1

Choosing appropriate experimental parameters . . . . . . . . . . . . . . . . . 91

5.6.2

Cooling and trapping of the molecule YO . . . . . . . . . . . . . . . . . . . . 94

5.6.3

Characteristic dynamics in different phase space regions . . . . . . . . . . . . 96

5.6.4

Phase space compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6.5

MOT loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Equilibrium temperatures and cloud sizes . . . . . . . . . . . . . . . . . . . . . . . . 102
5.7.1

5.8

SWAP MOT effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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

6 Speeding up particle slowing using shortcuts to adiabaticity

108

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2

Motivation for the speed-up protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3

Lewis-Riesenfeld Invariant-based inverse engineering . . . . . . . . . . . . . . . . . . 113

6.4

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5

6.4.1

System dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4.2

Eigensystem in a momentum subspace . . . . . . . . . . . . . . . . . . . . . . 118

6.4.3

Deriving detuning profiles from the auxiliary equations . . . . . . . . . . . . . 119

6.4.4

Bloch sphere trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Slowing dynamics in classical phase space . . . . . . . . . . . . . . . . . . . . . . . . 124

viii
6.6

Slowing Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.7

Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.8

Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7 Entropy transfer from a quantum particle to a classical coherent light field

139

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.2

Motivation

7.3

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.4

Calculating the final cavity field state . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.5

Q-function for the final cavity field state . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.6

Calculating the fidelity between the initial and final cavity field states . . . . . . . . 148
7.6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Fidelity in the thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . 148

7.7

Analyzing the fidelity result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.8

Bayesian analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.9

Calculating entropies and mutual information . . . . . . . . . . . . . . . . . . . . . . 154
7.9.1

Mutual information from cavity measurements . . . . . . . . . . . . . . . . . 156

7.9.2

Quantum mutual information . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.10 Analyzing the entropies and mutual information . . . . . . . . . . . . . . . . . . . . 159
7.10.1 Entropy of the spontaneous photon . . . . . . . . . . . . . . . . . . . . . . . . 162
7.11 Additional calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.11.1 Displaced final cavity field photon distribution . . . . . . . . . . . . . . . . . 163
7.11.2 Fidelity in the infinite coupling limit m → 0 . . . . . . . . . . . . . . . . . . . 167
7.12 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8 Conclusion

170

Bibliography

172

ix

TablesTable
2.1

Vector operator notation conventions . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Comparison of slowing dynamics between the shortcut slowing scheme based on

5

Lewis-Riesenfeld invariants and radiation-pressure slowing . . . . . . . . . . . . . . . 125
6.2

Properties of candidates for slowing proposal . . . . . . . . . . . . . . . . . . . . . . 129

6.3

Slowing results for candidates listed in Table 6.2 . . . . . . . . . . . . . . . . . . . . 130

xFiguresFigure2.1

Spatial diagram of the light-matter interaction . . . . . . . . . . . . . . . . . . . . .

4

2.2

Energy diagram of two-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

Optical cavity containing a light field and quantum particle (Jaynes-Cummings model) 12

2.4

Diagram of a bipartite quantum system . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5(a) Laser setup for Doppler cooling (b) Energy level diagram with laser frequency . . 27

2.6

Relevant states of a Jg → Je = Jg + 1 transition for a conventional MOT . . . . . . . 31

2.7

Laser and anti-Helmholtz coil configuration for conventional MOT . . . . . . . . . . 32

2.8

Energy level and laser scheme for conventional MOT . . . . . . . . . . . . . . . . . . 33

3.1(a) Spatial diagram of 1D SWAP cooling. (b) Energy diagram for a two-level system 38

3.2

Laser frequency scheme and associated excited state dynamics . . . . . . . . . . . . . 38

3.3

Definition of high-velocity regime time intervals . . . . . . . . . . . . . . . . . . . . . 43

3.4

Energy vs. detuning for bare eigenstates in high-velocity regime . . . . . . . . . . . . 44

3.5

Demonstration of slowing: root-mean-square (rms) momentum prms vs. sweep number 46

3.6

Coherent dynamics: change in rms momentum ∆prms and resultant excited state
fraction Pe vs. initial momentum pi after a single sweep for a system that begins in
the ground state. Diabatic and adiabatic results included . . . . . . . . . . . . . . . 49

3.7

Coherent dynamics: change in rms momentum ∆prms and resultant excited state
fraction Pe vs. initial momentum pi after a single sweep for a system that begins in
the excited state. Diabatic and adiabatic results included . . . . . . . . . . . . . . . 49

xi
3.8

Various quantities for a particle that is subject to a single sweep with γ 6= 0 as afunction of its initial momentum, pi

3.9. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Doppleron and Bragg resonances: energy E vs. momentum p . . . . . . . . . . . . . 53

3.10 Role of Bragg oscillations: sum and difference impulses ∆p± (see text) vs. initialmomentum pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.11 Frequency profile for the simplified “sweep-wait” cooling scheme. . . . . . . . . . . . 57
3.12 Demonstration of recoil-limited cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.13 Stationary values of kinetic energy hp2 /2mi vs. Rabi frequency Ω0 in the sweep-wait
sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.14 Kinetic energy hp̂2 i/2m vs. the average number of scattered photons for SWAP
cooling and Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1(a) Schematic of setup (b) Two level atom model (c) Laser frequency ω and excitedstate fraction P (e) vs. time (d) Simulation trajectories in momentum space vs. time

4.2

65

(a) Temperature of atomic ensemble vs. time (b) Relative phase-space density ρ/ρ0in one and three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3

Cooling into a moving reference frame. (a) Laser frequency offset (b) Atomic velocity vs. time for various sweep directions (c) Acceleration vs. relative laser beampolarization angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4

Temperature vs. Rabi frequency (experimental and theoretical data) . . . . . . . . . 71

5.1

Schematic of SWAP MOT experimental setup . . . . . . . . . . . . . . . . . . . . . . 77

5.2(a) Minimum internal state structure required for the SWAP MOT process. (b)Laser detuning δ(t) and excited state fraction Pe over a single period . . . . . . . . . 78

5.3

Diagrammatic description of (a) trapping and (b) slowing forces in a SWAP MOT . 81

5.4

Phase space contour plots for testing the validity of the three-state basis simulation
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5

Phase space dynamics in a SWAP MOT . . . . . . . . . . . . . . . . . . . . . . . . . 95

xii
5.6

Evolution of the phase space density metric G for the YO system subject to the
SWAP MOT and traditional MOT (Doppler) . . . . . . . . . . . . . . . . . . . . . . 98

5.7

Simulated phase space trajectories of particles being loaded into a SWAP MOT andtraditional MOT (Doppler) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.8

Simulated equilibrium properties of a SWAP MOT . . . . . . . . . . . . . . . . . . . 104

6.1(a) Schematic of the normalized velocity distributions P (v) of the particle beam
before and after the slowing process. (b) Schematic of experimental setup . . . . . . 109

6.2(a) Frequency diagram of the transfer processes. (b) Experimental parameters andparticle dynamics over a |p̄i → |p̄ − 2~ki sequence of period 2T .

. . . . . . . . . . . 115

6.3

Bloch sphere trajectories between the initial and final states . . . . . . . . . . . . . . 123

6.4

Slowing dynamics in momentum space . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.5

Robustness to laser intensity and frequency errors

6.6

Robustness to pulse overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.1

Experimental schematic and energy level diagram . . . . . . . . . . . . . . . . . . . . 141

7.2

Q-function for the final cavity state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.3

Fidelity F between the initial and final cavity states ρ̂L (0) and ρ̂L (∞) . . . . . . . . 151

7.4

Entropic quantities S and I scaled by the initial particle entropy . . . . . . . . . . . 160. . . . . . . . . . . . . . . . . . . 133

Chapter 1

Context and outline

It is only within the last century that we have discovered ways to create light sources witha high degree of coherence, i.e., light comprised of a multitude of photons with a narrow range of
frequencies in lockstep with one another. One such source, the laser, has enabled us to control
the dynamics of an ever-growing number of atomic and molecular species with unprecedented
precision. By using lasers in parallel with other electromagnetic fields, we have learned how to
drastically reduce the speed, temperature, and size of dilute gases consisting of these quantum
particles. These techniques are essential to countless studies in both applied and basic science,from Bose-Einstein condensation to condensed matter and quantum computing [9, 43, 97, 102].
The diverse quantum nature of atoms and molecules often requires us to devise laser slowing,cooling, and trapping protocols that are unique to the species. This is especially true for heavier
atoms and most molecules since they contain a relatively rich internal quantum structure due to theinteractions among their constituent protons, neutrons, and electrons. Therefore, an obvious way tosimplify the complexity of a slowing, cooling, or trapping process is to limit the number of internal
states the system can occupy. However, this is a difficult task because optical pumping supplies
energy to the system, which nature actively dissipates in the form of spontaneous photons. These
spontaneous emission events irreversibly transfer the system to a potentially undesired quantum
state, which can ultimately lead to a loss of control over its motion. Therefore, methods which
reduce these events could allow for previously unachievable experiments on quantum gases.
At the same time, some type of irreversible process is a fundamental ingredient of a cooling2
or trapping process, so we can not rely on purely coherent (non-dissipative) techniques. This isbecause entropy must be removed from a system in order for it to become colder and denser. An
ideal mechanism then must have a minimal (but non-negligible) reliance on an irreversible process,which is most often spontaneous emission. As we will show, the recently discovered “sawtooth
wave adiabatic passage” (SWAP) cooling technique [13, 141] exhibits reduced reliance on spontaneous emission compared to the oldest and most commonly implemented laser cooling mechanism:
Doppler cooling. Starting from this discovery, we dedicate our focus to answering the following
questions in this thesis:(1) How exactly does the SWAP protocol lead to cooling if its underlying mechanism seems tobe purely coherent in nature, and therefore not capable of entropy reduction?
(2) Can the SWAP protocol be extended to trap particles in space, as is done with Doppler
cooling in a conventional magneto-optical trap?
(3) What are the lowest achievable temperatures and cloud sizes with SWAP-based methods?
(4) Can we speed up these techniques in time, although they are fundamentally adiabatic andconsequently slow in nature?
(5) Is there some irreversible process other than spontaneous emission which could be responsible for the reduction in entropy of a quantum gas?
We begin with a survey of relevant background information in Chapter 2 which further
contextualizes our contributions. We then answer (1) and the first part of (3) in Chapter 3, in
which we present our experimental and theoretical studies of SWAP cooling. Subsequently, we
study (2) and (3) in Chapter 5 by combining the SWAP protocol with a magnetic trap to produce
an additional trapping force. Next, we apply an adiabatic speedup protocol to the state-transfer
process in Chapter 6, thereby addressing (4). From a more general perspective, we then address(5) in Chapter 7 by studying the entropy absorption properties of a classically-behaved coherent
light field. We conclude with a brief summary in Chapter 8.

Chapter 2

Introduction

Now that we have laid out the general framework of this thesis, we provide a review of
the topics that are essential to understanding its contents. We begin with highlighting the mainconcepts behind the interaction of matter with light in first and second quantization in Section 2.1.
We then turn to a quick overview of the theory of open quantum systems in Section 2.2, whichallows us to efficiently study a quantum system that is interacting with its surrounding environment.
In our description of open quantum systems, we will introduce the density matrix formulation ofquantum mechanics, the quantum master equation, and a stochastic computational tool called
the Monte Carlo wave function method, or quantum jump method. We will use these techniques
extensively throughout our work. Next, we introduce the semiclassical theories of Doppler coolingand the conventional magneto-optical trap (MOT) in Section 2.3. These widely-used cooling andtrapping processes will serve as a benchmark to which we will compare the results of SWAP coolingand the SWAP MOT in Chapters 3 and 5, respectively.

2.1

Coherent models of light-matter interaction
The interaction of quantum matter with light forms the very core of each study we present in

the following chapters. There exist many excellent textbooks [71, 127, 174] that comprehensively
cover various models developed by the scientific community for this type of interaction. In all cases,
these models can be sorted into one of two categories: they treat the light as either a classical or
quantum mechanical electromagnetic field. While the latter is unequivocally closer to reality, it

4comes with the additional cost of complexity, and in some cases introduces unnecessary detail.
Because we are interested in physics involving both models, we derive the Hamiltonian for a twolevel quantum particle interacting with light in either case in Sections 2.1.3 and 2.1.4. In theclassical case, we include the motional degrees of freedom of the particles, which will be essential
to our studies on the mechanical effects of laser light in Chapters 3-6. In the quantum case, we
limit our discussion to motionless particles, as done in our study of quantum entropy transfer in
Chapter 7. On a related note, we also briefly discuss the manifestation of spontaneous emission
and its various models in Section 2.1.5.
Of course, we also need a quantum mechanical description of the matter, which is comprisedof individual particles such as atoms or molecules. The light interaction alters both the quantum mechanical “internal” states of the electrons bound to these particles and the motional, or“external,” state of the entire particle, so we need to choose a model that incorporates both ofthese effects. With this in mind, in Section 2.1.1 we derive an interaction Hamiltonian in the casewhen the particle is explicitly treated quantum mechanically (first quantization), whereas the exact
nature of the light field is not yet specified. We then simplify this Hamiltonian for the case of a
two-level particle and introduce some common notation in Section 2.1.2.

Figure 2.1: Spatial diagram of the light-matter interaction. A neutral quantum particle consistingof a single electron at position r e and a nucleus with center-of-mass position r n interacts with alight field with wave vector k, wavelength λ = 2π/|k|, electric field E, and magnetic field B.

5
2.1.1

First quantization: quantum treatment of matterWe consider the neutral, single-electron particle depicted in Fig. 2.1. The electron has chargeq = −e, where e is the elementary charge, which means that the rest of the particle, which we
simply call the “nucleus,” has a net charge q = e. We loosely follow the derivations for the lightparticle interaction described in Refs. [71] and [127], although we explicitly factor in the nucleus
subspace and therefore describe the physics in detail. For clarity, we first provide our conventions
for labeling various vectors and operators in Table 2.1.
Table 2.1: Conventions for vector operator quantities. Here we use the short hand “COM” forcenter-of-mass.
quantity

symbolelectron positionr̂ e

nucleus COM positionr̂ nparticle COM positionr̂

relative electron-nucleus position∆r̂

arbitrary positionR̂electron kinetic momentump̂enucleus kinetic momentump̂ntotal particle momentump̂

relative electron-nucleus momentum∆p̂

The Hamiltonian Ĥ0 of the particle in the absence of light is
Ĥ0 =

p̂2e
p̂2
+ n + V̂ (r̂ e − r̂ n ).
2me 2mn(2.1)

Here, we have labeled quantities related to the electron (nucleus) with the subscript e (n). For
j ∈ {e, n}, the quantities p̂j = −i~∇j = −i~∂r̂j are kinetic momentum operators, mj are masses,
and r̂ j are position operators. The potential energy V̂ encapsulates all electron-nucleus interactions. Notice that Eq. (2.1) looks identical to the classical Hamiltonian of the electron-nucleus
system except the position and momentum variables have been replaced with quantum mechanical6
operators. These operators satisfy the commutation relations[(r̂ j )a , (p̂k )b ] = i~δab δjk ,[(r̂ j )a , (r̂ k )b ] = 0,[(p̂j )a , (p̂k )b ] = 0(2.2)

for particles j, k ∈ {e, n} and spatial dimensions a, b ∈ {x, y, z}.
In the presence of light, the Hamiltonian becomesĤ =[p̂e + eA(r̂ e , t)]2 [p̂n − eA(r̂ n , t)]2+
+ e[Φ(r̂ n , t) − Φ(r̂ e , t)] + V̂ (r̂ e − r̂ n ),
2me
2mn

(2.3)

where t is time and A(r̂, t) and Φ(r̂, t) are the vector and scalar potentials associated with thelight field evaluated at the appropriate positions. Note that these potentials are functions of theelectron and nucleus position operators and thus can affect their quantum states, but we have
not yet specified whether the light itself is classical or quantum mechanical. These potentials are
related to the light’s (physical) electric and magnetic fields E(R̂, t) and B(R̂, t) by
E(R̂, t) = −∇Φ(R̂, t) −∂A(R̂, t)
,
∂t

(2.4)

B(R̂, t) = ∇ × A(R̂, t),where R̂ is any position vector operator. (We reserve the symbol r̂ for the center-of-mass positionof the particle, which is introduced later.)
The form of the particle Hamiltonian given in Eq. (2.3) can be simplified through a series ofgauge transformations. First, we choose the radiation gauge so that Φ = 0 and ∇ · A = 0, whichremoves the electron-light and nucleus-light electric potential energy terms in Eq. (2.3). Next, wetransform the Hamiltonian under the gauge transformation Û = exp [−ieχ(r̂ n , r̂ e , t)], whereχ(r̂ n , r̂ e , t) = χe (r̂ e , t) − χn (r̂ n , t),(2.5)

and the χj ’s are undetermined gauge functions. (See Refs. [44] and [45] for a detailed discussion ofgauge choice in quantum electrodynamics.) After this operation, the Hamiltonian readsĤ ={p̂e + e[A(r̂ e , t) + ∇e χe (r̂ e , t)]}2 {p̂n − e[A(r̂ n , t) + ∇n χ(r̂ n , t)]}2
+
2me
2mn


∂χe (r̂ e , t) ∂χn (r̂ n , t)+e−
+ V̂ (r̂ e − r̂ n ).
∂t
∂t

(2.6)

7
Before we choose the gauge functions χj , we apply the “electric dipole approximation,” which
will drastically simplify the form of the Hamiltonian. From electromagnetic theory, the light field’s
vector potential A(R̂, t) satisfies the wave equation, which has solutionA(R̂, t) = A0 exp[i(k · R̂ − ωt)] + c.c.

(2.7)

Here, c.c. denotes the complex (or Hermitian) conjugate, A0 is the amplitude of the vector potential, and k and ω are the wave vector and frequency of the light. Because typical atomic
dimensions (|r|atom / 10−9 m) are much smaller than optical wavelengths (λ ' 10−7 m), it follows
that k ·r atom ∝ |r|atom /λ  1, and thus that the vector potential is approximately uniform over thedimensions of the particle. It is therefore valid to approximate the vector potential as a spatially
uniform vector, which we evaluate for convenience at the center-of-mass r of the particle (definedin Eq. (2.11)):A(r̂ j , t) ≈ A(r̂, t) = A0 exp[i(k · r̂ − ωt)] + c.c.,|∇A|∝ |k|  1/|r atom |.
|A|

(2.8)

Eqns. (2.8) are what is meant by the electric dipole approximation in the field of light-matterinteractions, although the exact position at which A is evaluated depends on the context. Equippedwith this useful approximation, we now make the clever choice χj (r̂ j , t) = −A(r̂, t) · r̂ j which
satisfies∇j χj (r̂ j , t) ≈ −A(r̂, t),∂χj (r̂ j , t)
= E(r̂, t) · r̂ j ,
∂t

(2.9)

in which we used Eq. (2.4). With these results, the Hamiltonian reduces to
Ĥ =

p̂2e
p̂2
+ n + e(r̂ e − r̂ n ) · E(r̂, t) + V̂ (r̂ e − r̂ n ).
2me 2mn(2.10)

Our next alteration is to describe the physics in terms of the center-of-mass of the particle andrelative motion of the electron and nucleus. This is motivated by the relative position dependenceof the last two terms in Eq. (2.10). The center-of-mass r̂ and relative ∆r̂ coordinates arer̂ = (me r̂ e + mn r̂ n )/(me + mn ),∆r̂ = r̂ e − r̂ n ,(2.11)

8and the total p̂ and relative ∆p̂ momenta arep̂ = p̂e + p̂n ,∆p̂ = (me p̂e − mn p̂e )/(me + mn ).

(2.12)(These coordinates also satisfy the canonical commutation relations [73].) In terms of these coordinates, Eq. (2.10) becomes
Ĥ =

p̂2
(∆p̂)2+− dˆ · E(r̂, t) + V̂ (∆r̂),2m
2µ(2.13)

in which dˆ ≡ e∆r̂ is the particle’s dipole moment operator. We shall refer to the term in Eq. (2.13)
involving dˆ as the “dipole term.” By inspecting this result, we notice that two of the four terms
determine the internal bound states of the electron-nucleus interaction in terms of their relative
positions and momenta. This part of the Hamiltonian,Ĥinternal =(∆p̂)2+ V̂ (∆r̂),
2µ(2.14)

has been characterized to high precision in previous theories and experiments for each atomic andmolecular species we study. As such, its spectrum and states are well defined, so we consider it tobe “solved” and thus separate it from the rest of our Hamiltonian. Substituting this notation into
Eq. (2.13), we at last arrive at the desired form of the Hamiltonian:Ĥ = Ĥinternal +p̂2− dˆ · E(r̂, t).
2m(2.15)

Eq. (2.15) is the form of the Hamiltonian we will use to describe all particle-light interactions.

2.1.2

The two-level particle

In the previous section, we did not specify the internal quantum structure of the particle
as given by Eq. (2.14). Here, we briefly describe the simplifications made to Eq. (2.15) when theparticle can be treated as a two-level system. That is, we suppose that only two of the quantum
internal states determined by Ĥinternal are relevant during the interaction, and thereby isolate ourattention to the idealized, yet rich model of the two-level particle.

9

Figure 2.2: Energy level diagram of the internal states of a two-level system, separated by energy∆E = ~ωa .

As is customary, we refer to the two internal states of the particle as the “excited” and“ground” state with associated vectors |ei and |gi in the Hilbert space of Ĥinternal . This terminology reflects the former state’s higher relative energy by an amount ~ωa , where ωa is the angularfrequency of a photon containing the same amount of energy (see Fig. 2.2). We refer to these states
as the “bare eigenstates” or “bare basis” of Ĥinternal because the Hamiltonian is diagonal in thisbasis in the absence of light. In fact, defining the baseline energy at ~ωa /2 above the ground state,
we may write Ĥinternal , often called the “free particle Hamiltonian,” asĤinternal =

~ωa
~ωa
~ωa z
|ei he| −|gi hg| =
σ̂ ,
2
2
2

(2.16)

in which σ̂ z is a Pauli matrix. For this reason, it is said that the excited and ground states |ei and|gi form a pseudo-spin 1/2 system. More generally, the usual raising (σ̂ + = |ei hg|) and lowering(σ̂ − = |gi he|) operators, along with the usual Pauli spin operators, σ̂ x,y,z , whose coordinate labels
are implicitly understood to refer to the pseudo-spin space, operate on the internal states.
We now shift our attention to the other operator acting on the internal Hilbert space: theparticle dipole moment operator d̂ [see Eq. (2.13)]. This operator induces transitions between theinternal states, which physically corresponds to the absorption and emission of photons from thelight field. The polarization ˆ of the light field is chosen based on the magnetic quantum numbers
me and mg of the states |ei and |gi (linear if me = mg and circular if me = mg ± 1), so we shouldˆalso include it in when calculating the matrix elements of d:dˆ · ˆ = hg|dˆ · ˆ|ei |gi he| + he|dˆ · ˆ|gi |ei hg| = hg|dˆ · ˆ|ei (σ̂ − + σ̂ + ).

(2.17)In the second step, we (without loss of generality) chose the phase of the off-diagonal dipole matrix

10
elements to be real. (See Ref. [174] for a derivation of this result from a parity argument.) We
note that the extension of these methods to a three (or more) level system, as done in Chapter 5,
is straightforward.

2.1.3

Interaction of a moving two-level quantum system with a classical light field

For many purposes in the field of light-matter interactions, a classical treatment of the lightis sufficient. In particular, the effects of a light field containing many photons (such as a laser) isoften treated as an unperturbed, time-dependent electric potential. That is, the amplitude E0 of
the light’s electric field E(r̂, t) is unchanged by the particle interaction, even though the particle
absorbs photons. (We address the validity of this treatment to some extent in Chapter 7.) In thiscase, we can write the electric field as
E(r̂, t) = ˆ E0 f (r̂, t),(2.18)

where ˆ is the polarization direction, E0 is the (constant) field amplitude, and the spatiotemporalprofile f (r̂, t) depends on the laser setup.
We restrict our studies to laser light propagating along one spatial dimension, which we callthe z-direction so that the wave vector is k = ±k êz . Here, we have used the notation êj to denotea unit vector pointing along the j-direction. In Chapters 5 and 6, we will consider the effects oftraveling waves, for which the function f is [126]ftraveling (r̂, t) = cos(kẑ − ωL t) =1  i(kẑ−ωl t)e+ e−i(kẑ−ωL t) ,2

(2.19)

where ωL is the laser frequency. In Chapter 3, we will consider the effects of a standing wave (thesum of counterpropagating traveling waves), for which the function f isfstanding (r̂, t) = cos(kẑ − ωL t) + cos(kẑ + ωL t) = 2 cos(kẑ) cos(ωL t).

(2.20)

Notice that the amplitude of a standing wave is twice as large as that of a traveling wave. For thisreason, care needs to be taken when defining the coupling strength, often called the Rabi frequency,
between the light field and the particle.

11
The next obvious step is to substitute the electric field E [Eq. (2.18)] for the laser setup ofinterest into the Hamiltonian Ĥ [Eq. (2.15)]. For the case of the traveling wave [Eq. (2.19)], thedipole term is−dˆ · E(r̂, t) = −hg|d̂ · ˆ|ei E0 −(σ̂ + σ̂ + ) ei(kẑ−ωL t) + e−i(kẑ−ωL t) ,2

(2.21)

in which we used Eq. (2.17) for the dipole moment operator. This expression can be simplified
by invoking the “rotating wave approximation,” which removes terms that are unlikely to affectthe dynamics due to their required energy when the light is quasi-resonant. Inspecting the relative
energies of the bare eigenstates [see Fig. 2.2], we learn that the raising σ̂ + = |ei hg| and loweringσ̂ − = |gi he| operators supply and remove energy from the particle, respectively. This means that
their expectation values have a time-dependence that satisfies σ̂ ± ∝ e±iωa t , and hence that eachterm in Eq. (2.21) (when distributed) oscillates in time with frequencies of magnitude |ωa ± ωL |.
Under the rotating wave approximation, we remove terms oscillating with the sum frequency, as
only the terms oscillating with the difference frequency |ωa − ωL | govern the observable dynamics.
Simultaneously defining the Rabi frequency asΩ≡−

hg|dˆ · ˆ|ei E0,~(2.22)

Eq. (2.21) reduces to−dˆ · E(r̂, t) =~Ω  − −i(kẑ−ωL t)σ̂ e+ σ̂ + ei(kẑ−ωL t) ,2

(2.23)

which is the form often used in the literature. This process is very similar to the standing wave
case. We will extensively use Eq. (2.23) and its equivalent standing wave form in our studies oflaser slowing, cooling, and trapping.
If the frequency of the light changes in time, we replace the accumulated temporal phase of
the light ωL t in all above equations with its equivalent integral form:ωL t → η(t) ≡Z t
t0

ωL (t0 ) dt0 .

(2.24)

12
This replacement is important if the range of the applied light frequencies δωL = max[ωL (t)] −min[ωL (t)] is at least on the order of the other relevant frequencies in the interaction, such as the
Rabi frequency Ω [Eq. (2.22)], recoil frequency ωr [Eq. (2.66)], and spontaneous emission rate γ[Eq. (2.35)]. In the following chapters, δωL is on the order of MHz, which is often larger than theseother frequencies, and hence we make the adjustment. In principle, the wave vector k should also
be adjusted in the spatial phase:k=ωLωL (t)→ k(t) ≡.
cc(2.25)

However, the change in wave vector δk = δωL /c is much smaller than the only other relevant
quantity: the particle “wave vector,” p/~. This is true because the spread in the particle momentum
is fundamentally bounded from below by the photon momentum ~k in the processes we propose,as we show in Chapter 3. Therefore, we approximate k to remain fixed in all cases.

2.1.4

Second quantization: interaction of motionless two-level quantum systemwith a quantized light field in a cavity

Figure 2.3: A quantum particle is placed in an optical cavity that contains a quantized light field.

When the quantum nature of the light is essential to understanding the physics of interest,
then one must quantize the light field through the process of second quantization. This procedure
promotes the light’s electric and magnetic fields to operators (related to individual quanta, or
photons) that evolve according the laws of quantum mechanics. For brevity, we refer the reader

13
to the second chapter of Ref. [71] for an in-depth derivation of quantized light fields and only
derive the interaction Hamiltonian, called the Jaynes-Cummings Hamiltonian, for a single-mode
quantum electromagnetic field and a quantum two-level particle in an optical cavity [see Fig. 2.3].
Although it is not strictly necessary, we also remove the atomic motion for simplicity. The resulting
interaction is related to what we study in Chapter (7).
Because the boundary conditions of electrodynamics and quantum mechanics play a role inthe derivation, it is necessary to quantize the electromagnetic field within a “quantization volume”
V . In the case of Chapter (7), this volume will correspond to the optical cavity which contains
the quantized field. The single-mode quantized electric field of light with frequency ω propagating
along the z-direction in a cavity is found to beÊ(r̂, t) = ˆ

~ω
0 V

1/2

sin(kẑ)(â + â† ) = ê E0 f (r̂) (â + â† ),(2.26)

in which E0 = (~ω/0 V )1/2 characterizes the maximum vacuum electic field amplitude, and f (r̂) =sin(kẑ) is the spatial profile. (We have written this operator in the Schrödinger picture, so it is timeindependent.) Notice that Eq. (2.26) looks very similar to its classical counterpart in Eq. (2.18),
except we now include the creation â† and annihilation â operators of the quantum harmonic
oscillator, which have the basic propertiesâ† |ni =√n + 1 |n + 1i ,â |ni =√n |n − 1i ,[â, â† ] = 1.

(2.27)In this case, the operators act on a Fock space {|ni}∞
n=0 which describes the photon occupancy ofthe cavity. Their inclusion in Eq. (2.26) reflects the dependence of the electric field amplitude on
the number of cavity photons.
If the particle is motionless, then we can demote the particle position operator ẑ to a classicalvalue z, which is the (fixed) position of the particle in the cavity. With this result, the dipole termin Eq. (2.15) becomes−dˆ · Ê(r̂, t) = − hg|dˆ · ˆ|ei E0 sin(kz) (σ̂ − + σ̂ + ) (â + â† ),(2.28)

14
where we have again used Eq. (2.17). Invoking the rotating wave approximation and defining thevacuum Rabi frequency g as
g(z) ≡ −2 hg|dˆ · ˆ|ei E0 sin(kz),~(2.29)~g(z) − †(σ̂ â + σ̂ + â),2(2.30)

the dipole term becomes−dˆ · Ê(r̂, t) =which is the typical form of the Jaynes-Cummings interaction Hamiltonian. Combining this resultwith the free particle Hamiltonian [Eq. (2.16)] and the free field HamiltonianĤ0,field = ~ωâ† â

(2.31)(for which we dropped the trivial vacuum energy term), we arrive at the full Jaynes-Cummings
Hamiltonian:
ĤJC =~ωa z~gσ̂ + ~ωâ† â + (σ̂ − â† + σ̂ + â).
22(2.32)

Notice that we have absorbed the position dependence into the definition of g. We will study asystem very closely related to this result in Chapter 7.

2.1.5

The spontaneous emission process

The last fundamental physical process we need to introduce is spontaneous emission, whichcauses a quantum system in a state other than the lowest-energy state to eventually “relax” toa lower-energy state. The scientific community has developed many different physical theories to
explain this phenomenon over the last century [131, 132]. From a classical perspective, this process
is induced by the radiation reaction, i.e., the interaction of the electron with its own electromagnetic
field. The theory of quantum electrodynamics, however, elicits a different origin: it is caused by theinteraction of the particle with the ground, or “vacuum” states of the universe’s electromagnetic
field. In other words, the quantum perspective argues that the non-zero energy of the ground stateof the free-space electromagnetic field (or “zero-point energy”), which contains no photons, causes
an irreversible exponential decay of excited state populations. We provide the reader with thefollowing insightful quote, which connects these two perspectives:

15“Of course we always get the same expression γ for the spontaneous emission rate
[when deriving it with different orderings of quantum mechanical operators], because the operators we are reordering do in fact commute, but our interpretationof the origin of spontaneous emission may change. Again this is a consequence of
the intimate, fluctuation-dissipation connection between radiation reaction and thezero-point, vacuum electromagnetic field.”
P. W. Milonni, Why spontaneous emission? [130]Because the classical perspective underestimates the value of γ by a factor of 2, modern
authors adopt the quantum perspective [71, 127]. This derivation involves coupling a quantum
system to the free-space (“infinite”-volume) quantized electromagnetic field, or “background radiation field,” which has a continuum of frequency modes. Generalizing the single-mode formalism
discussed in Section 2.1.4, the multi-mode (m.m.) Jaynes Cummings Hamiltonian isĤJC,m.m. =

X
X ~gs
~ωa z
σ̂ + ~
ωs â†s âs +(σ̂ − â†s + σ̂ + âs ),22ss(2.33)

in which s is a mode index. To gain intuition for why we expect dynamics to occur even in a spacedevoid of photons, consider the following matrix elements of ĤJC,m.m. :hg, 1s |ĤJC,m.m. |e, 0s i =

~gs
.
2

(2.34)

Physically, the fact that the matrix element in Eq. (2.34) is non-zero tells us that a quantum particle
can still transition from its excited state to a ground state. More specifically, we should expect aquantum particle prepared in the excited state |ei to undergo vacuum Rabi oscillations with eachmode s.
We now introduce an important approximation: the “Weisskopf-Wigner approximation” [127,189]. Under the assumption that the bandwidth of the background radiation field is large, thecorrelation time τcorr between any two different modes is correspondingly small. If this correlation
time is small compared to the timescales associated with the particle evolution, then we can “coarsegrain” over the evolution of the particle state. (We will revisit this approximation in the derivationof the master equation in Section 2.2.2.) This approximation leads to destructive interference ofthe vacuum Rabi oscillations, and the net effect is irreversible exponential decay into the ground16
state |gi. It can be shown that this decay rate, called the free-space spontaneous emission rate γ,
is
γ=

ˆ |2
ωa3 | hg|d|ei.
3π0 ~c3(2.35)

In principle, we could accurately evolve the dynamics of our quantum particle by incorporating the background radiation field into all of our calculations. However, the detailed evolutionof the this field is generally not of interest, and we have shown that it can be effectively modeledby the inclusion of γ. Therefore, it would be desirable to construct a general theory that accounts
for the effects of the background radiation field without its explicit inclusion. This leads us to thetheory of open quantum systems, which is discussed in the next section.

2.2

Theory of open quantum systems
In Section 2.1, we derived a mathematical formalism for the interaction of a two-level quantum

particle with light under “coherent,” i.e., Hamiltonian dynamics. In the following chapters, we will
not be interested in the evolution of the background radiation field introduced in Section 2.1.5, orfor that matter, any other presence in the universe that could possibly affect the system of interest.
However, we have no choice but to incorporate the background field’s pervasive effects if we hopeto accurately capture the system’s dynamics. We therefore study the system through the theoryof open quantum systems [25]. In this framework, the universe is split into two subspaces: the“system,” which we are interested in studying, and the “environment” (the rest of the universe),
which potentially affects the system but is not of interest.
Before we can introduce this theory, we must introduce the density operator ρ̂, also called thedensity matrix. This matrix will take the place of the state vector |ψi in describing the state of oursystem, and is described in Section 2.2.1. We will then show how the background radiation field canbe formally separated from the dynamics, yielding a quantum master equation that describes thesystem in Section 2.2.2. We then present a convenient way to incorporate the effects of dissipation
on particle motion in Section 2.2.3. In Section 2.2.4, we discuss a useful and experimentally-

17
motivated calculational tool for simulating the master equation: the Monte Carlo wave function
method.

2.2.1

The density matrix formalism

We often find ourselves in situations in which the exact quantum state of a system is unknown,
but we do know that it may be in one of several states |ψi i, each with a probability pi . The systemis then said to be in a statistical ensemble of the states |ψi i. This is a fundamentally separate
notion from a quantum superposition
|Ψi =

X
i

αi |ψi i ,(2.36)

in which αi is a probability amplitude and |Ψi is the quantum state of the system, because Eq. (2.36)
assumes that the state |Ψi is a specific vector sum of the states |ψi i. In other words, if we were tocreate many copies of our system, then a randomly chosen copy would be in |ψii with probability
pi . In this sense, the system exists as a classical distribution of quantum states, which has themathematical representation [127]
ρ̂ =

X
i

pi |ψi i hψi | ,(2.37)

in which ρ̂ is the “density matrix.” This type of state may arise due to experimental limitations instate preparation. However, even an initial pure state will evolve into a mixed state if it undergoes
some decoherence process, in which the system irreversibly interacts with its environment. In thefollowing chapters, we extensively study systems which may undergo the decoherent spontaneousemission process, and hence we must adopt the density matrix perspective.
We now point out some important properties and terminology for the density matrix formalism. We refer the reader to Ref. [139] for a much more detailed and complete presentation.
Unit trace: We immediately see from Eq. (2.37) that the trace of the density matrix satisfies
Tr[ρ̂] =Xjhj|ρ̂|ji =Xi,jpi hj|ψi i hψi |ji =X
i,j,k,k0

pi c∗k0 ck hj|ki hk 0 |ji =

X
i

pi

X
j

|cj |2 = 1,

(2.38)

in which the sets {|ji}, {|ki}, and {|k 0 i} form a basis for the Hilbert space. This is the equivalentcondition for state normalization hΨ|Ψi = 1.

18
Pure vs. mixed states: Notice that the density matrix reduces to a single outer product stateρ̂ = |ψi i hψi | if the system can only be in the state |ψi i, i.e., pi = 1 and therefore pj = 0 for j 6= i.
In this case, there is no classical uncertainty in the state of the system, and the density matrix istherefore said to be in a “pure state.” If ρ̂ is not pure, then it is said to be in a “mixed state.” One
way to measure whether a density matrix is pure or mixed (without explicit diagonalization, which
may be difficult) is to calculate its “purity” Tr[ρ̂2 ]. It can be shown that1
≤ Tr[ρ̂2 ] ≤ 1,d(2.39)

where d is the dimension of the Hilbert space, and Tr[ρ̂2 ] = 1 if and only if ρ̂ is pure.
Operator expectation values: For a state |Ψi, the expectation value of an observable Â is given by
the familiar calculationhÂi = hΨ|Â|Ψi .

(2.40)

Generalizing this notion to the density matrix case, we see from Eq. (2.37) that


h i
X
X
X
X
hÂi =
pi hψi |Â|ψi i =pi hψi |Â |ji hj| |ψi i =hj| ρ̂Â |ji = Tr ρ̂Â .
iij(2.41)

j

Evolution: For a closed quantum system (no interactions with an external entity), the state ρ̂(t) is
evolved in time according to the von-Neumann equation
i
dρ̂
1 h
Ĥ, ρ̂ .
=dti~(2.42)

This result can be easily derived from the fact that each state |ψi i evolves under the Schrödinger
equationi~d |ψi i= Ĥ |ψi i .
dt(2.43)

Reduced density matrices: If the density matrix ρ̂ describes a bipartite system, i.e, a system comprised of two distinct parts a and b, each of which live in a subspace of the total Hilbert space
Ha , Hb ⊆ H respectively, then the “reduced density matrix” ρ̂i of each subsystem is given byρ̂i = Trj [ρ̂],i 6= j.

(2.44)

19
Equipped with these properties, we can now derive the quantum master equation which isused to describe open quantum systems.

2.2.2

The quantum master equation

Figure 2.4: Diagram of a bipartite quantum system. The total Hilbert space HT is divided into
the “system” subspace H and the “environment” subspace HE . Image from Ref. [113].
As shown in the previous section, a closed quantum system living in a Hilbert space HT ,where T represents the “total” system, described by a density matrix ρ̂T is evolved in time by
the von-Neumann equation [Eq. (2.42)], which is the density matrix version of the Schrödinger
equation. In this section, we discuss the key concepts behind the derivation of the quantum masterequation, which evolves states in a subspace of interest H ⊆ HT of HT , which we call the “system”
subspace, potentially interacting with the (uninteresting) remainder of the total system, which wecall the “environment” subspace HE . The various spaces are shown diagrammatically in Fig. 2.4.
We refer the reader to Ref. [70] for an excellent formulation of this derivation.
The steps for deriving the master equation is as follows:(1) Solve the von-Neumann equation [Eq.(2.42)] for the total density matrix ρ̂T under theeffects of the total Hamiltonian ĤT to second order in perturbation theory (the Born

20
approximation)(2) Trace over the environment subspace HE using Eq. (2.44)(3) Simplify the result based on the timescale hierarchy
τcorr  τsys  τband ,(2.45)

where τcorr is the (vanishingly small) correlation time of the background field modes, τsys
is the characteristic timescale of the system of interest, and τband is the (very large) inverse
bandwidth of the background radiation field (the Weisskopf-Wigner, or Markov approximation)
The resulting equation for the system density matrix ρ̂ reads [113]dρ̂1= [Ĥ, ρ̂] + L̂ρ̂,dt
i~(2.46)

where Ĥ is the system Hamiltonian, and
n
o
X
1
†
†
Jˆ Jˆi , ρ̂L̂ρ̂ =Jˆi ρ̂Jˆi −2 i(2.47)iis the “Lindblad superoperator.” The operators Jˆi , called “jump operators,” incorporate the dissipative effects of the system’s interaction with the environment by projecting higher-energy state
populations to lower-energy states at the appropriate dissipative rate. In other words, the Jˆi ’s
describe the various decay channels of the system. Eq. (2.46) is the workhorse of our studies in thefollowing chapters.
There are important consequences to introducing a dissipative process into a quantum system.
First, it allows for an arbitrarily initialized system to reach a steady-state configuration. This canbe seen by setting the time derivative of ρ̂ in Eq. (2.46) to zero:0=1[Ĥ, ρ̂ss ] + L̂ρ̂ssi~⇒[Ĥ, ρ̂ss ] = −i~L̂ρ̂ss .

(steady-state)(2.48)

For L̂ρ̂ 6= 0, this equation can be solved for the steady-state density matrix ρ̂ss . However, if there
were no dissipative processes (L̂ρ̂ = 0), then Eq. (2.48) states that ρ̂ must commute with Ĥ to reach21
steady-state, which is not true for an arbitrary initial state. Second, a dissipative interaction allows
for an irreversible change of the system’s entropy. Of course, there would be a corresponding change
in the environment’s entropy by the second law of thermodynamics. In the following chapters, wewill often choose the system Hamiltonian Ĥ to evolve the system into a relatively low-entropy
steady-state configuration.

2.2.3

The Lindblad superoperator in laser slowing, cooling, and trapping

In Chapters 3-6, we will be studying two- and three- level quantum particles that are free
to move in one spatial dimension. To determine the Lindblad superoperator L̂ρ̂ for this type ofsystem, we need to characterize the decay channels Jˆ associated with the spontaneous emission
process. For simplicity, we demonstrate the derivation of L̂ρ̂ for a moving two-level particle that
may emit linearly polarized light (Chapters 3, 4, and 6) or circularly polarized light (Chapter 5).
The extension to the three-level case is straightforward.
Because the particle exists in three-dimensional space, it may emit spontaneous photons
with angular momentum ~q and wave vector ksp according to a (normalized) dipole radiation
pattern [135]N± (u) =3
1 + u2 ,8

N0 (u) =3
1 − u2 ,4(2.49)

in which u = cos φ, and 0 ≤ φ ≤ π is the angle between ksp and the axis of motion, which we call thez-axis here. The subscripts denote the angular momentum of the emitted photons, which depends
on the internal level structure of the system of interest. Notice that Eq. (2.49) is continuous,
which means that there is a continuum of decay channels corresponding to the different emission
directions. The jump operator corresponding to the variable u is thenJˆu =pγN (u)e−ikẑu σ̂ − ,(2.50)

where we have used |ksp | = k, and the exponential operator accounts for the associated momentum
recoil∆pu = ~ku

(2.51)

22of the particle along the z-axis. Generalizing the sum in Eq. (2.47) to an integral, we find thatL̂ρ̂ = γ

Z 1
du N (u)e−ikẑu −−1+ ikẑuσ̂ ρ̂σ̂ e1 + −−σ̂ σ̂ , ρ̂2

.

(2.52)

We emphasize that we have integrated over u in the anticommutator terms in Eq. (2.52).
For computational simplicity, we now discuss how to perform a coarse discretization of thedipole radiation pattern such that the particle remains on a momentum grid with a spacing definedby the photon momentum ~k, but the distribution still sufficiently models the momentum diffusioncaused by the continuous pattern [Eq. (2.49)]. More specifically, let us define the discretizedradiation pattern
Nqdisc (u) = Aq [δ(u − 1) + δ(u + 1)] + Bq δ(u),(2.53)

where δ is the Dirac delta function, which allows for the emission of a spontaneous photon with
momentum ~q along the ±ẑ-directions (with equal probability Aq by symmetry) and spontaneousemission perpendicular to the ẑ-direction with probability Bq . Then, the Lindblad superoperator
in Eq. (2.52) can be approximated by
1 + −−ikẑ −+ ikẑ
ikẑ −+ −ikẑ−+L̂q ρ̂ = γ Aq eσ̂ ρ̂σ̂ e + Aq e σ̂ ρ̂σ̂ e+ Bq σ̂ ρ̂σ̂ −σ̂ σ̂ , ρ̂ .
2

(2.54)We choose Aq and Bq by enforcing that the resulting first and second moments of the momentum
recoil [see Eq. (2.51)] match that of the continuous pattern. The first moments are h∆pi = 0 in allcases by symmetry. For the case of linear polarization (q = 0), the second moments are
Z 1
1
2
2
h∆p i = (~k)
du u2 N0 (u) = (~k)2 ,
5
1
8
−1
⇒
A0 = , B0 = .
Z 1
10
10
disc
h∆p2 i= (~k)2
du u2 N0disc (u) = 2A0 (~k)2 .

(2.55)−1

For circular polarization (q = ±1), the second moments are
Z 12
h∆p2 i = (~k)2
du u2 N± (u) = (~k)2 ,
5
−1
⇒
Z 1
2 disc2
2 disc2h∆p i= (~k)
du u N± (u) = 2A± (~k) .

1
A± = ,
5

3
B± = .
5(2.56)−1

We will use Eq. (2.54) with the Aq ’s and Bq ’s determined in Eqns. (2.55) and (2.56) in each of thefollowing chapters that involve particle motion.

23
2.2.4

The Monte Carlo wave function method

Armed with the quantum master equation [Eq. (2.46)], we are in principle ready to begin investigating the physics of interest in each of our proposed projects. Our investigations in
Chapters 3-6 involve the usage of time-dependent light fields, which makes Eq. (2.46) analyticallyintractable. We are therefore compelled to adopt a numerical approach. While the dimension of
the internal state Hilbert spaces dint of the two- and three-level quantum particles we consider are
relatively small and therefore numerically tractable, their external Hilbert space dimension dext is
potentially very large. This means that calculating the density matrix ρ̂(t), which has (dint dext )2 −1
degrees of freedom, is computationally expensive. Fortunately, there exist various methods of approximating the solution to the quantum master equation, such as mean field theory, quantum andc-number Langevin theory [127], truncated cumulant theory, and the Monte Carlo wave function
method [63, 70, 135, 151].
We adopt the last procedure, which involves the stochastic evolution of many state vectors|ψi, which have 2(dint + dext ) − 2 degrees of freedom. This lower scaling makes their evolution fastercomputationally at the expense of only approximating the dynamics. This method is based on theassumption that the information gained by continuously monitoring the system for signatures ofdissipative events, which yield “quantum jumps” [133] to lower energy states in the system, canbe used to infer the state of the system at any given time for a particular experiment iteration.
In all of the cases we consider, a quantum jump corresponds to the emission of a spontaneous
photon, although there may be multiple “decay channels” depending on the emission pattern andthe number of internal states. These experimental iterations can be either physical (performed in alaboratory) or numerical (simulated on a computer); we consider the latter scenario. Each iteration
can be roughly understood as a particular (and potentially stochastic) run of the experiment and isaccordingly called a “quantum trajectory.” It follows that the density matrix, which represents theaverage state of the system over all possible outcomes, can then be approximated as the average ofthese individual trajectories [see Eq. (2.61)].

24
We follow the Monte Carlo wave function approach outlined in Ref. [63]. The algorithm forthe evolution of each quantum trajectory, which can performed in parallel, is as follows:(1) Initialization: If the initial state of the system is pure (ρ̂(0) = |φi hφ|), then initialize thewave function as the corresponding state vector: |ψ(0)i = |φi. If the initial state is mixed,
then randomly initialize the wave function as one of the |ψi i’s [Eq. (2.37)] weighted by theirprobabilities. Generate and store a random number 0 ≤ r ≤ 1. Proceed to step (2).
(2) Evolution: Evolve the wave function under the non-Hermitian HamiltonianĤeff = Ĥ −i~ X ˆ† ˆ
Ji Ji ,
2

(2.57)

iin which the Jˆi ’s are the jump operators introduced in the previous section, until either (i)the final desired time is reached, or (ii) the norm of the wave function satisfieshψ(t)|ψ(t)i = r.

(2.58)(Note that the norm of the wave function |ψ(t)i may have decreased due to the nonHermiticity of Ĥeff .) In case (i), proceed directly to step (4). In case (ii), proceed to step(3).
(3) Quantum jump: A quantum jump has occurred. Simulate the jump by randomly choosing
a decay channel Jˆi weighted by the decay channel probabilitieshψ(t)|Jˆi† Jˆi |ψ(t)i.
Pi = Pˆ† ˆj hψ(t)|Jj Jj |ψ(t)i(2.59)

Once the decay channel has been determined, set the (normalized) state after the jump tobe|ψ 0 (t)i =Jˆi |ψ(t)i.
hψ(t)|Jˆ† Jˆi |ψ(t)i
i

Repeat step (2).
(4) Repeat the entire process if another quantum trajectory is desired.

(2.60)

25
At any time during the evolution, the approximate density matrix can be calculated according
to the equationN

1 X
ρ̂(t) ≈
|ψj (t)i hψj (t)| ,N(2.61)

j=1

where j indexes a particular quantum trajectory, and N  1 is the total number of quantum
trajectories. We would arrive at the exact form of the density matrix in the limit N → ∞.
Operator expectation values can then be approximated through Eq. (2.41), although it generally
requires less memory to calculate the expectation value on a trajectory-by-trajectory basis with
Eq. (2.40) and then average the results.
Eq. (2.61) demonstrates why decoherence processes tend to change an initially pure state
to a mixed state. Namely, if there were no decoherence processes, then Ĥeff would be Hermitian.
Consequently, the norm of each wave function wouldn’t change, and no stochasticity would be
introduced into the quantum trajectories. Each quantum trajectory would hence be identical, andEq. (2.61) would then be pure.

2.3

Doppler-based cooling and trapping processes
At this point, we have presented the essential mathematical and computational tools neces-sary for calculating the quantities we discuss in the following chapters. In this section, we describe
the commonly-implemented methods of two-level Doppler cooling and the conventional magnetooptical trap (MOT), which provide us with a benchmark to which we will compare the laser coolingand trapping techniques we describe in Chapters 3 and 5.

2.3.1

Approximation of classical particle motion

In most theoretical treatments of Doppler cooling and MOTs, the particle is approximatedto be a classical point particle, i.e., the position and momentum operators r̂ and p̂ are replacedwith the classical variables r and p which are evaluated at the center-of-mass of the particle. (Inthe context of laser cooling and trapping, this is often called the “semiclassical approximation,”
which is not to be confused with the approximation of light as a classical electromagnetic field,

26
although these approximations are often invoked in tandem.) We showed in Section 2.1 that theelectric field E(r, t) of the applied light has a characteristic length determined by the wavelength
λ = 2π/k. To treat the motion of the particle classically, then, the spatial width σr̂ of the particle’s
wave function (or wave packet) must be small compared to the wavelength of the light:
σr̂  λ.

(2.62)

Moreover, the wave packet’s momentum spread σp̂ must correspond to a spread of Doppler shiftsthat is small compared to the linewidth γ of the cooling transition [53]:
kσp̂
 γ,
m

(2.63)

where m is once again the particle mass. Together with the Heisenberg uncertainty principleσr̂ σp̂ ≥ ~,(2.64)

ωr
 1,γ(2.65)~k 2
2m(2.66)

these localization conditions imply that
χ≡in whichωr =is the “recoil frequency” of the transition. Notice that ~ωr corresponds to the kinetic energy of aparticle with a momentum equal to that of a single photon p = ~k and hence sets a frequency scale
for the particle motion.
Eq. (2.65) is equivalent to the condition that the timescale of the internal state dynamics(determined by γ) is much faster than the timescale of the particle motion (determined by ωr ),and therefore that we can approximate the internal states to be in steady state at all times. For
most laser cooling transitions, ωr is on the order of kHz and γ is on the order of MHz, so thisapproximation is valid. However, the systems we consider in the following chapters correspond toχ ≈ 1, so we must preserve the quantum mechanical nature of the center-of-mass position and27
momentum. Nevertheless, it is still useful to understand the effects of Doppler cooling in thisregime for the sake of context and comparison. For the remainder of this chapter, we will consider
systems obeying Eq. (2.65). Moreover, we will only study motion along one dimension, which wecall the z-direction.

2.3.2

Cooling force

In Section 2.1.1, we derived the following expression [Eq. (2.15)]
Ĥ = Ĥinternal +p̂2− dˆ · E(r̂, t).
2m(2.67)

for the interaction of a quantum particle with a light field. We now wish to investigate the resulting
mechanical effects on the particle, i.e., the evolution of its center-of-mass motion due to the forcesinduced by the light. Of course, this is described by Newton’s second law of motion [127]F̂ (t) =


dp̂
i
∂ Ĥ
∂ ˆ
d · E(r̂, t) .
= [Ĥ, p̂] = −
=
dt
~
∂r̂
∂r̂

(2.68)

Notice that only the dipole term results in a force because Ĥinternal involves the particle’s relative
coordinates (and not the center-of-mass) and the kinetic energy term commutes with p̂.

Figure 2.5: (a) Laser setup for Doppler cooling. Counterpropagating beams interact with particles
of velocity v. (b) Energy level diagram of the two-level particle with the laser frequency ωL . Noticethat the laser is red detuned, i.e., ωL < ωa so that δ < 0 [see Eq. (2.70)].

Under the assumptions introduced in Section 2.65 (classical, one-dimensional motion) andthe electric dipole approximation of Section 2.1.1 (electric field is uniform over the spatial extent
of the particle), the expectation value of the light force operator F̂ (t) defined in Eq. (2.68) is∂f (z, t)
F (t) = hF̂ (t)i = hdˆ · ˆi E0,∂z(2.69)

28
in which we used Eq. (2.18). It is necessary to use a standing wave [two counterpropagating laser
beams, Fig. 2.5(a)] to realize cooling, which means that function f is given by Eq. (2.20). After therotating wave approximation, incorporating the steady-state solution of the internal state variables(as given by the optical Bloch equations [126]), and averaging over a wavelength, the resulting forceisFDoppler = −βv = ~k 2

8s0 (δ/γ)(1 + (2δ/γ)2 )2v = ~k 2

8s(δ/γ)v,
1 + (2δ/γ)2(2.70)

in which v = p/m is the particle velocity,δ = ωL − ωa(2.71)

is the laser “detuning” from the transition frequency of the two-level particle,
s0 ≡ 2|Ω|2 /γ 2

(2.72)

is the on-resonance saturation parameter, ands≡s0
1 + (2δ/γ)2(2.73)

is the saturation parameter. We emphasize that Eq. (2.70) is only valid in the limit of low saturation,s  1, so that the steady state excited state population
ρee =

s
,
2(1 + s)(2.74)

is low. This is because a low excited state fraction ensures that spontaneous emission returns
particles to the ground state. Otherwise, stimulated emission would occur, which in this casewould cancel the average force. Additionally, some nontrivial, velocity-dependent, multiphoton
forces (see Sections 3.6.1 and 3.6.2) begin to affect the dynamics if the saturation is not low.
Notice that Eq. (2.70) has the form of a damping force, so it will tend to reduce the speedsof moving particles if the friction coefficient satisfies β > 0. The reduction of velocities to zero,
regardless of the initial velocity, is exactly what is meant by laser “cooling.” The cooling condition
β > 0 is equivalent to δ < 0, or ωL < ωa [see Fig. 2.5(b)]. So, this procedure will tend to cool the29particles if the laser frequency is designed to be lower than the transition frequency of the two-levelparticle. In analogy to the red color of lower frequency light, this is referred to as a “red detuned”
configuration.
Let us now provide an intuitive, microscopic description of the mechanism behind this“Doppler cooling” process. Because both lasers are red detuned in the laboratory frame (by design), they are not “on resonance” with a motionless particle, i.e., the magnitudes of the frequencyof the light ωL and the transition frequency ωa do not exactly match. However, if the particle ismoving along the z-direction, then it will perceive a Doppler shift kv for each laser in its frame ofreference. More specifically, the light which is propagating in the opposite direction of its motion(counterpropagating) will be “blue shifted,” and the light which is propagating along the same
direction as its motion (copropagating) will be further “red shifted.” Therefore, the counterpropagating laser is closer to resonance with the particle, and is then more likely to cause the particle toabsorb a photon. Because photons have momentum, this absorption will impart an impulse to theparticle in the opposite direction of its motion. Soon after, the particle will release a spontaneous
photon. (This is much more likely than stimulated emission because s  1.) Many iterations of
this absorption - spontaneous emission process will result in the cooling of the particle to a zeroaverage velocity, as the absorbed photons always push against its motion, and the effects of the
spontaneous photons average to zero due to the random emission direction. This photon picture
will prove to be invaluable in understanding the laser cooling and slowing protocols we propose in
the following chapters.

2.3.3

Minimum temperature

The previous section elucidated the circumstances under which the laser configuration depicted in Fig. 2.5(a) would apply a viscous optical force to a two-level particle exhibiting classicalmotion. It seems from what we presented so far that all particles would eventually reach zero
velocity and remain motionless. This is of course not what happens because the particles also
undergo momentum diffusion due to their continuous interaction with the laser [126]. For example,

30
near zero velocity, the particles no longer preferentially interact with a single beam, which causes
them to randomly absorb a photon from either beam and potentially gain speed. Additionally,
the particles have a chance of emitting spontaneous photons if they have a nonzero excited state
fraction, which yields a force in a random direction.
It can be shown that the balance between the cooling force and the momentum diffusion
results in a Gaussian spread of velocities. This suggests that we can characterize the ensemble ofparticles by a “temperature” T in a similar fashion to the theory of statistical mechanics. Morespecifically, it can be shown by modelling the motion as a random walk (or a Fokker-Planck equation) that the momentum distribution W (p) of the particle ensemble equilibrates to the form2

W (p) ∝ e−βp /2mD ,(2.75)

in which β is the friction coefficient derived in the previous section [Eq. (2.70)] and D is the diffusion
coefficient. While D can be derived in a rigorous way from these models, we instead provide an
intuitive, quick derivation [126]. Consider a particle locked to a momentum grid with spacing~k. As a consequence, the particle can emit spontaneous photons only along the z-axis. For asingle beam, the probability per unit time that the particle emits a spontaneous photon is givenby the scattering rate γp = γρee . Multiplying this rate by two (since there are two beams) andincorporating the magnitude of the momentum recoil results in the diffusion coefficientD=2(∆p)2= 4γp (~k)2 .
∆t(2.76)

Comparing the form of Eq. (2.75) to that of a 1D Maxwell-Boltzmann momentum distribution, we
arrive at the Einstein relation
kB T =

D0.
β(2.77)

Substituting our results from Eqns. (2.70) and (2.76) into Eq. (2.77) and minimizing the temperature with respect to detuning δ, we arrive at the “Doppler temperature” formula
TD =

~γ
.
2kB

(2.78)

31
This corresponds to a temperature of roughly 25 µK for a γ = 2π × 1 MHz linewidth transition.
We will compare the Doppler temperature TD to the temperatures resulting from the methods we
propose in the following chapters.

2.3.4

The magneto-optical trap

Figure 2.6: Relevant states of a Jg → Je = Jg + 1 transition for a conventional MOT. Using thenotation |J, mj i, the labeled states are |0i = |0, 0i, |±i = |1, ±1i. We also provide the Zeeman
shifts ±bz in the case of a nonzero magnetic field.

Now that we have characterized the forces and temperatures achieved with Doppler cooling, we explain its combination with a quadrupole magnetic trap to form a magneto-optical trap
(MOT) [126]. Such a configuration is relatively easy and inexpensive to create, robust to experimental errors, and exhibits both cooling and trapping capabilities. We will replace the coolingmechanism of this apparatus to generate higher coherent forces in Chapter 5. We first specify
the necessary state structure, magnetic field, and laser configuration, and then we discuss theunderlying physics of the MOT.
The trapping force is a consequence of the energy of a magnetic dipole moment µ̂ in an
external magnetic field B. In analogy with the classical case,ĤB = −µ̂ · B.

(2.79)The magnetic dipole moment of a quantum system is related to its total angular momentum Jˆ32through the relationshipµ̂ = −

gj µB ˆJ,~

(2.80)

in which µB is the Bohr magneton, and gj is the Landé g-factor. We consider the simple three levelsystem depicted in Fig. 2.6, which has two excited Zeeman sub-levels |Je , Me = ±1i = |±i and one
ground state |Jg , Mg = 0i = |0i. Such a system exists as part of a Jg → Je = Jg + 1 transition,
which occurs in many atoms. Notice that we have labeled the states by their magnetic quantum
numbers mj = 0, ±1. The ground and excited state manifolds are separated by an energy ~ω0 ,and the magnetic sub-levels decay to the ground at a rate γ. We have omitted the |Je , me i = |1, 0i
state from the model because it will not be optically pumped, and hence does not participate in
the dynamics.

Figure 2.7: The laser (σ − − σ + ) and current (I) distribution for a 1D conventional MOT. The
anti-Helmholtz coils generate a magnetic field B(z) = |∇B|z, where |∇B| is the magnetic field
gradient [see Eq. (2.82)].

The magnetic field induced by the anti-Helmholtz coils of a magnetic quadrupole trap [seeFig. 2.7] is given byp
B(r) = |∇B| y 2 + z 2 + 4x2(2.81)

in which |∇B| is the magnetic field gradient, z and y form a plane parallel to the current loops, andx is along the axial direction. Because we consider motion only along the z-direction, the magnetic
field becomesB(z) = |∇B|z êz .

(2.82)

33
Using Eqns. (2.82) and 2.80 in Eq. (2.79), we find thatĤB = ~bz (|+i h+| − |−i h−|) ,(2.83)

in which b = gµB |∇B|/~, and g is the g-factor of the transition. The laser configuration consistsof counterpropagating lasers of opposite circular polarization σ − and σ + , as shown in Fig. 2.7.
These lasers can drive the |0i → |−i and |0i → |+i transitions, respectively. Care must be taken
to properly align the currents and laser polarizations into the exact configuration shown in Fig. 2.7(or the equivalent configuration flipped about an axis pointing out of the paper). Otherwise, theparticles will be driven out of the trap.

Figure 2.8: Energy diagram and laser configuration for a conventional MOT.

The laser frequency ωL is tuned below the zero-magnetic field transition frequency ω0 (δ < 0)so that the Doppler cooling process still operates in a MOT. We now demonstrate the microscopic
mechanism for realizing a trapping force. Consider a particle with zero momentum on the right side

34of the trap at position z 0 , as shown in Figure 2.8. Because the laser is red detuned and the excitedstates have a Zeeman shift [Eq. (2.83)], it is closer to resonance with the |0i → |−i transition.
Therefore, the particle will preferentially absorb from the σ − beam, which is coming from the rightside of the trap. The particle will then experience an impulse that drives it toward the center of the
trap. The same process occurs for particles on the left side of the trap with the |0i → |+i transition
and the σ + laser. Just as in Doppler cooling, we operate in the low-saturation limit s  1 so theparticles emit a spontaneous photon soon after the absorption. (This explains why the coherent
Zeeman force F̂B = −∂ ĤB /∂z, which tends to pull particles away from the center of the trap withthis absorption procedure, does not play a major role in the dynamics.) This cycle is repeatedmany times, which ultimately leads to both cooling and trapping of the particle ensemble.
The trapping force can also be demonstrated from a macroscopic force perspective. The force
imparted to the particles by each of the traveling waves is given by [126]
F± = ±~kγ
s0,
2 1 + s0 + (2δ± /γ)2(2.84)

in which
δ± = δ ∓ k · v ± bz.

(2.85)(We will find that this is very similar to an important, motion-based detuning we define in Chapter 5.) We can find the total force F on the particles through the vector sum of the two forces F ±due to the circularly polarized traveling waves:
F = F + + F − = −βv − κr.

(2.86)

Notice that the first term on the rightmost side of Eq. (2.86) is the same friction force we foundin Doppler cooling. This verifies that particles in a MOT are still cooled by the Doppler coolingmechanism. The second term is a restoring force with
κ=

b
β,
k

(2.87)

which immediately implies that particles will also experience a conservative trapping force toward
the center of the trap.

Chapter 3

SWAP cooling: theoretical investigation

In this chapter, we present our theoretical contributions toward understanding the sawtooth
wave adiabatic (SWAP) cooling method. This work resulted in our Physical Review A article(Ref. [13]), which was published in August 2018 and largely constitutes the material presented
in this chapter. The experimental realization of SWAP cooling is the subject of Chapter 4. In
this cooling method, two-level particles undergo successive adiabatic transfers upon interaction
with counter-propagating laser beams that are repeatedly swept over the transition frequency.
We explain the basic mechanism behind its operation and characterize the parameter regime inwhich it is most efficient and practical. We show that particles with narrow linewidth transitions
can be cooled to near the recoil limit. We also demonstrate that SWAP cooling has an inherently
reduced reliance on spontaneous emission compared to Doppler cooling, which makes it particularly
applicable to systems lacking closed cycling transitions, such as molecules.

3.1

Introduction
The use of coherent light to cool the motion of particles such as atoms to sub-Kelvin tem-

peratures has greatly expanded the capabilities of atomic and molecular physics [123]. The earliest
and simplest mechanism demonstrated and understood was Doppler cooling [77, 192], which relies
on preferential absorption between counterpropagating lasers, followed by spontaneous emission.
While Doppler cooling is simple and robust, the large number of spontaneous emissions involved
has several drawbacks: spontaneously emitted photons impart random momentum kicks to the36particle that cause diffusion and limit the achievable temperatures. These spontaneous emission
events can also result in the particle falling into internal states that are no longer near resonancewith the applied light so that the cooling ceases. In the case of atoms, the latter can be mitigatedby adding a small number of additional lasers to “repump” into cooled states, but in the case ofmolecules the number of uncooled states may be so large that this approach becomes a significant
challenge [31]. Doppler cooling can reach the recoil temperature (set by the recoil energy from asingle emitted photon) for narrow linewidth transitions such that γ < ωr , where γ and ωr are theoptical transition linewidth and recoil frequency respectively. However, the cooling time scale inthis parameter regime is long, scaling inversely with γ.
Several approaches to laser cooling have been developed to mitigate the negative effects of
spontaneous emission [3]. In sub-Doppler cooling mechanisms such as Sisyphus cooling [54], theenergy removed per spontaneously emitted photon is large, allowing lower temperatures to bereached. In Raman sideband cooling [78], tight confinement of the atom can suppress the effectof momentum recoil associated with spontaneous emission, enabling cooling to nearly the groundstate of the external potential well. Cavity-cooling techniques [84] can be used to reduce free-spacespontaneous emission by causing preferential decay to a desired state via the output coupler of an
optical resonator.
Recently, we have experimentally demonstrated a new cooling mechanism, named “SWAP
cooling” (sawtooth wave adiabatic passage cooling) in which particles are coherently driven between ground and excited states of a narrow-linewidth optical transition by counterpropagating,frequency-swept lasers. Spontaneous emission is still critical in order to remove entropy from thesystem to achieve steady-state cooling, but by coherently driving a particle between its ground andexcited state multiple times, large amounts of energy can be removed per spontaneous emission.
This provides a way to generate significant forces and to reach low temperatures while maintaining
a large velocity capture range. The approach is largely insensitive to perturbations such as laser frequency drifts, magnetic fields, and AC Stark shifts. These factors have made SWAP cooling a usefulexperimental technique for cooling atomic strontium using its 7.5 kHz linewidth, dipole-forbidden

37
1S

0 →

3P

1 transition [141].

A related procedure has been used in the generation of a new formof magneto-optical trap [137]. Sub-Doppler cooling of 87 Rb was recently observed using SWAP
cooling with two-photon Raman transitions between ground hyperfine states [74]. In general, theapproach is applicable to any atomic species, especially alkaline-earth-like atoms that possess intercombination transitions. Further, the reduced reliance on spontaneous emission may make SWAP
cooling a useful tool for cooling molecules that have narrow linewidth optical transitions, such as
the 160 kHz linewidth X 2 Σ → A0 2 ∆3/2 transition in YO [49].
Here, we present a detailed theoretical analysis of SWAP cooling. We explore the minimum
achievable temperature as well as the various laser-particle interactions that affect particle dynamics. We investigate its capture range and the forces involved in the cooling process. We show thatfor appropriate parameters, SWAP cooling can be used to cool to near the recoil limit. We also
simulate the rate at which spontaneous photons are emitted during cooling, confirming that theamount of energy removed per spontaneous emission event can greatly exceed the limits of Doppler
cooling.

3.2

Basic mechanism
The main mechanism for momentum removal in SWAP cooling is the coherent transfer ofa particle toward zero momentum via adiabatic passage. Momentum is removed by time-ordered
stimulated absorption and emission of photons caused by interaction with a standing wave formedby counter-propagating laser beams. To describe the cooling, we consider a two-level particle withinternal states |ei and |gi, separated in energy by ~ωa , and one dimension of motional freedom
along the ẑ-direction (see Figure 3.1). We choose to represent the external degree of freedom in the
momentum basis and with label |pi, where p denotes a momentum eigenstate such that p̂ |pi = p |pi.
As shown in Figure 2, the laser frequencies, ωL (t), follow an asymmetric sawtooth waveformwith full range ∆s and period Ts so that the slope of the sawtooth ramp is α ≡ ∆s /Ts . The
sawtooth frequency ramp is centered on the transition frequency, ωa , and the frequency is linearlyramped from below to above ωa . We approximate the wavenumber, k, to be fixed throughout the38

Figure 3.1: (a) Spatial setup of the standing wave (two counterpropagating lasers with frequencyωL (t)) and particle (circle) with velocity v in the lab frame. (b) Energy level diagram of the bareinternal states, separated by energy ∆E = ~ωa .

sweeping sequence. This also implies that the recoil frequency, ωr = ~k 2 /2m, is fixed, where m is
the mass.

Figure 3.2: Top: The laser frequency, ωL (t), as a function of time. The approximate resonancefrequencies for a particle with velocity v are labeled. The sweep range is chosen to be large enoughsuch that both beams will become resonant with the particle at some time during the sweep.
Bottom: The ideal excited-state fraction, Pe , in the adiabatic regime. The particle remains in the
excited state for a time interval τe .

The direction of particle motion matters since Doppler shifts set the time-ordering of whichbeam first interacts with the particle. To understand this time-ordering, consider a particle initiallyin the state |g, pi i. The Doppler shift ensures that the counter-propagating beam is the first to
sweep across the transition frequency. If we first focus on the case with pi > 0, this adiabaticallytransfers the particle to the state |e, pi − ~ki via stimulated absorption. After some time, the co-39
propagating beam will achieve resonance, adiabatically transferring the particle to |g, pi − 2~ki via
stimulated emission. For pi < 0, the mapping is |g, pi i → |e, pi + ~ki → |g, pi + 2~ki. In either
case, the particle is transferred closer to zero momentum. The net result is the removal 2~k ofmomentum and transfer back to the particle’s initial internal state without spontaneous emission.
Subsequent sweeps would then continue to remove momentum in units of 2~k. Figure 3.2 shows
this process by illustrating the ideal excited state fraction, Pe , over two sweeps.
The particle’s internal state at the beginning of a sweep is a crucial factor in determining
the impulse it receives from each laser. If the particle instead begins a sweep in the excited state,
it is transferred away from zero momentum. Hence, one of the roles of spontaneous emission isto ensure that the particle begins a sweep in the ground state. This is achieved by requiring an
imbalance between the times spent inside and outside of the two resonances, which corresponds to∆s > 4|kv|.

(3.1)

This condition also enforces the requirement that the sweep range, ∆s , is large enough for bothlaser beams to achieve resonance with the particle.
There are several other conditions that must be satisfied in order to realize SWAP cooling.
In order to ensure a low probability of decay during the time interval the particle is in the excitedstate, τe , we require the condition
τe 

1
.
γ

(3.2)

Importantly, the Rabi frequency of each laser beam, Ω0 , and the laser frequency sweep rate, α,
must satisfy the condition
κ≡

Ω20
≥ 1,
α

(3.3)

so that there is a substantial probability, Pa , for an adiabatic transition at each resonance [194]:π Ω20Pa = 1 − exp −
.
2 α


(3.4)

We shall refer to κ as the adiabaticity parameter. Any κ that satisfies Eq. (3.3) is said to be within
the adiabatic regime, and any κ that does not satisfy Eq. (3.3) is said to be within the diabatic

40
regime.

3.3

System dynamics
The quantum master equationdρ̂
1= [Ĥ, ρ̂] + L̂ρ̂
dt
i~

(3.5)

governs the time evolution of the density matrix, ρ̂, whose Hilbert space includes both the particle’s
internal and external degrees of freedom. The Hamiltonian, Ĥ, captures the coherent dynamics,
while the Lindblad superoperator, L̂ρ̂, captures the incoherent dynamics due to spontaneous emission and the associated recoil.
The operators ẑ and p̂ describe the particle’s position and momentum, respectively. The
excited and ground states, |ei and |gi, form a pseudo-spin 1/2 system, as discussed in Section 2.1.2.
The electric field of the applied lasers is polarized along the quantization axis, x̂, (see Figure 3.1)
and is described in first quantization by the operatorhiÊ(ẑ, t) = x̂E0 cos (kẑ + η(t)) + cos (kẑ − η(t)) .

(3.6)

Here, the amplitude of each standing wave is E0 , and the time-dependent accumulated phase of
the laser field, η(t), from initial time t0 isη(t) ≡Z t

ωL (t0 ) dt0 .

(3.7)

t0

In the interaction picture defined by the free HamiltonianĤ0 (t) =~p̂2+ ωa σ̂ z ,
2m 2

(3.8)

the particle’s non-dissipative dynamics, under the dipole and rotating wave approximations, isdescribed by the interaction Hamiltonian~k p̂  + −iθ(t)Ĥ = Ωs cos kẑ + tσ̂ e+ h.c. ,
2
m

(3.9)

41
where
θ(t) ≡Z t
t0

ωL (t0 ) − ωa dt0 = η(t) − ωa t

(3.10)

is the time-dependent phase of the laser field’s detuning from resonance. The standing wave’s peak
Rabi frequency,Ωs ≡ 2Ω0 = −2 he|dˆ · x̂|gi E0,~(3.11)

characterizes the interaction strength of the electric field with the particle’s electric dipole operator
ˆ We refer the reader to Sections 2.1 and 2.2 for more model details.
d.
In order to keep the momentum distribution on a discretized grid for computational simplicity,
we approximate the dipole radiation pattern to produce recoil of magnitudes −~k, 0, and ~k along ẑ
with probabilities 15 : 35 : 51 , respectively [135]. As discussed in Section 2.2.3, the resulting Lindblad
superoperator isL̂ρ̂ = γ

1 ikẑ − + −ikẑ 1 −ikẑ − + ikẑ 3 − + 1  + −e σ̂ ρ̂σ̂ e+ eσ̂ ρ̂σ̂ e + σ̂ ρ̂σ̂ −σ̂ σ̂ , ρ̂
5
5
5
2


.

(3.12)

We emphasize that this form of L̂ρ̂ is only valid for states |ei and |gi with magnetic quantum
numbers m satisfying ∆m = ±1. For states with the selection rule ∆m = 0, such as the dipole
forbidden 1 S0 →3 P1 transition |gi ≡ |1 S0 , m = 0i and |ei ≡ |3 P1 , m = 0i studied in Chapter 4,
1
8
1
: 10
: 10
.
one should use recoil probabilities 10

In summary, we consider numerical and theoretical results of Eq. (3.5) with Ĥ given by
Eq. (3.9) and L̂ρ̂ given by Eq. (3.12) in the following sections.

3.4

Dynamics in the high-velocity regime
The core mechanism whereby SWAP cooling removes momentum and energy from a particle’s

motion is most easily understood in a regime in which one can consider that the particle interacts
sequentially with one traveling wave and then the other. We shall henceforth refer to this as the“high-velocity regime.” To define it, we must consider the time it takes to adiabatically transfer aparticle with initial velocity vi between its internal states, which we call τjump , as well as the time42
interval separating the two resonances, which we denote by τres . In the adiabatic regime, it can beshown that τjump = 2Ω0 /α (see [185]). We derive τres in the following subsection.

3.4.1

Resonance times and intervals in the adiabatic limit

In the adiabatic limit (Ω20  α), most of the population transfer for a motionless particle
occurs around t = 0 for a detuning profile δ(t) = αt, as this is when the laser is in resonance withthe particle. By energy conservation (see Eq. (3.24) with n = 0), motional degrees of freedom
translate this “resonance period” for a particle in the state |g, pi interacting with a right-travelingwave to the time that satisfies
αtright − ωr (2β + 1) = αtright − kv − ωr = 0,

(3.13)

where β ≡ p/~k. For high-velocity particles (kv  ωr ), the resonance period agrees with theintuitive Doppler shift result. However, the additional recoil term is important for low velocity(kv ' ωr ) particles. A similar result can be found for the interaction of a ground-state particle
with a left traveling wave (β → −β), which translates the resonance period toαtleft − ωr (−2β + 1) = αtleft + kv − ωr = 0.

(3.14)The resonance periods for a particle starting in the excited state can be found with the substitution
ωr → −ωr on the additional recoil term. It is easily shown that adiabatic transfer with motion has
the same probability as the motionless case [194].
In the high-velocity regime [see Eq. (3.16)], there is one stimulated absorption and one stimulated emission per sweep in SWAP cooling. From Eqs. (3.13) and (3.14), the time separating theseresonant phenomena for a particle with initial velocity vi , labeled τres , isτres =

2(kvi − 2ωr ).
α(3.15)It should be noted that (3.15) is only valid in the adiabatic limit, since the time associated withthe stimulated absorption recoil has been included.

43
Figure 3.3 shows the time-ordering of these processes, as well as a measure of the total excitedtime, τe , defined as the sum of τjump and τres in this regime. Roughly half of each τjump overlaps
1

Pe

τe

τres

0.5

τjump

τjump

0

Time (one sweep)
Figure 3.3: The excited state fraction, Pe , of a particle prepared in the state |g, 10~ki over one
sweep. Values are: ∆s = 200ωr , Ts = 22/ωr , and Ω0 = 5ωr .

with τres . Therefore, to keep the resonances separated, we define the high-velocity regime to be therange of velocities that satisfy τjump < τres , or|Ω0 | < |kvi − 2ωr |.

(high-velocity regime)(3.16)

It is important to emphasize that particles outside of the high-velocity regime may still be cooled
under the SWAP cooling procedure. However, their dynamics are more difficult to describe andanalyze qualitatively.

3.4.2

Dressed state picture

A convenient and intuitive way to understand adiabatic transfer is the dressed state formalism. Working in the laser frame, we diagonalize the HamiltonianĤ(t) =p̂2~~
− δ(t)σ̂ z + Ωs cos(kẑ)σ̂ x ,2m 2
2(3.17)

44
at each instant in time. Here, δ(t) ≡ αt is the laser detuning from resonance, since the detuning
is linearly ramped from −∆s /2 to ∆s /2. We track the evolution of the minimal set of eigenstatesnecessary to demonstrate the evolution of a particle that begins a sweep in the state |g, pi in thehigh-velocity regime. This set maps to the bare eigenstates{|g, pi , |e, p − ~ki , |g, p − 2~ki , |e, p − 3~ki}(3.18)

in the limit of large detuning (|δ(t)|  |kv|). The avoided crossing of a multi-photon process known
as a Doppleron resonance, which does not affect the dynamics of a particle in the high-velocity
regime, is also present (see Section 3.6.1 for details).

n=1
Doppleronresonance|g, pi

Energy|e, p − h̄ki|g, p − 2h̄ki|e, p − 3h̄ki|g, pi|e, p − h̄ki|g, p − 2h̄ki|e, p − 3h̄ki

Detuning
Figure 3.4: Eigenvalue energy vs. detuning for the four coupled states given in Eq. (3.18). The inset
shows the splitting of an n = 1 Doppleron resonance, which is small in the high-velocity regime.
The red arrows identify the physical path being considered. Dashed lines show the evolution of theuncoupled states. Values are: Ω0 = 2ωr , α = ωr2 , Ts = 50/ωr . p = 4~k.

Figure 3.4 shows the instantaneous eigenvalues as the detuning is linearly ramped. Startingin the state |g, pi, the particle diabatically crosses the higher-order Doppleron resonance, thus beingtransferred into a different eigenstate. It then undergoes two adiabatic crossings, which correspond45
to the two resonances previously discussed, and ends up in the state |g, p − 2~ki, signifying theremoval of 2~k of momenta. It is important to note that Figure 3.4 only depicts the correct
evolution of a particle that follows the red arrows; in reality, the surrounding states would couple
to states of higher and lower momenta.

3.4.3

Coherent dynamics

In order to further illustrate the dynamics strictly due to coherent evolution, we numericallycalculated the time evolution of the root-mean-square (rms) momentum of the particle, prms =p
hp̂2 i, as it underwent SWAP cooling starting from the initial state |ψi i = |g, 10~ki and without
spontaneous emission (γ = 0). The Rabi frequency was chosen such that the simulation operated
in the high-velocity regime for all momentum states |pi | > ~k.
Figure 3.5 shows the rms momentum, prms , vs. time, expressed in terms of the number ofsweeps. The probability of finding the particle in the excited state, Pe = |he|ψi|2 , is also shownon the right hand axis. One sees that during each sweep the particle was adiabatically transferredto the excited state and then back to the ground state. Each transition was accompanied by areduction in the particle’s momentum by ~k for a total of 2~k per sweep. As the momentum of
the particle was reduced, the time between transitions became shorter (i.e., the width of the pulses
became smaller). This is what one would expect since the velocity of the particle and therefore theaccompanying Doppler shift is decreased.
By symmetry, and as confirmed by calculation, the rms momentum decreased in an identicalmanner for the state with opposite initial momentum, |g, −10~ki. Course graining over the individual sweeps, the effective force exerted on the particle drives it toward zero velocity, independentof its initial direction of motion. As a result, one should draw an important distinction between theforce exhibited here and a “slowing force” that can be understood as applying a uniform translation to the momentum of all particles. Such slowing forces are often implemented in the context ofslowing a molecular beam with chirped-frequency light [196] or using rapidly varying electrostatic
potentials as is done in Stark decelerators [21], but do not lead to steady-state cooling.

46

prms

1

8

6

Pe

(units of h̄k)

10

4

2

0

0
0

1

2

3

4

5

Number of Sweeps
Figure 3.5: Root-mean-square momentum, prms (step-like curve; magenta), for a particle startingin the state |g, 10~ki over five sweeps. The curve exhibiting rising and falling pulses shows theexcited state fraction, Pe (cyan). The particle experiences a significant reduction in prms and is leftin a superposition of the internal states. Values are: ∆s = 120ωr , Ts = 1000/ωr , and Ω0 = ωr .

On the final sweep shown in Figure 3.5, the particle approached zero momentum where
the dynamics are modified. At the end of the final sweep, the particle had a 50% probabilityof occupying the |g, 0i state, and a 25% probability of occupying each of the |e, ~ki and |e, −~ki√states, resulting in a final rms momentum of prms = ~k/ 2 and a final excited state probability of
Pe = 1/2. We will see that being left in the excited state at the end of the sweep is an importantconsideration for understanding the final equilibrium temperature.

3.4.4

High-velocity regime dynamics including dissipation

We have shown that under the influence of purely coherent dynamics, a particle prepared in
its internal ground state can be sufficiently transferred to low momentum. In a realistic system,
however, the presence of spontaneous emission restricts the amount of time a particle may remainin the excited state, Eq. (3.2), which increases for higher initial momentum. Nevertheless, inthe context of the high-velocity regime in the adiabatic limit, it can be shown that there is no47
upper bound on the momentum states that can be transferred to lower momentum via the SWAP
cooling procedure, contingent that arbitrarily high Rabi frequencies and sweep rates are accessible.
However, there exists a fundamental lower bound:
p
2κγ
.
'1+
~k
ωr

(3.19)

This motivates the use of SWAP cooling on a transition for which κγ/ωr is small. It is a requirementof adiabatic transfer to have κ at least on the order of unity, so the experimentalist only has the
freedom to vary the ramp slope α and Rabi frequency Ω0 accordingly. Regardless of experimental
laser parameters, narrow linewidth transitions (on the scale of the recoil frequency) are preferable
if the goal is to maximally cool the system.

3.5

Forces and capture range
For analyzing the cooling dynamics, it is useful to investigate the equivalent classical force

exerted on a particle as a function of its velocity or momentum. We have chosen to describe thisrelationship by defining various quantities that provide information about the impulse imparted tothe particle over a single sweep.

3.5.1

Conservative forces

One way to describe impulsive momentum kicks applied to the particle is the change in its
rms momentum due to a single sweep:∆prms ≡qphψf | p̂2 |ψf i − hψi | p̂2 |ψi i,(3.20)

where |ψi i is the state of the particle prior to the sweep and |ψf i is the state of the particle after
the sweep. We describe the impulse in this way (rather than the average momentum) because thesystem exhibits Bragg oscillations, which are transitions between resonantly coupled momentum
states |pi ↔ |−pi (see Section 3.6.2). These oscillations yield an additional momentum change thatis qualitatively different to the adiabatic transfer dynamics that we are interested in and does not

48
contribute to the cooling process. The rms momentum is a measure that by construction excludes
the effect of such Bragg oscillations.
Figures 3.6 and 3.7 show the computed rms impulse ∆prms vs. the initial momentum pi fora particle initially prepared in |ψi i = |g, pi i and |ψi i = |e, pi i, respectively. The values chosen for
the adiabaticity parameter, κ = 0.5 (top rows) and κ = 4 (bottom rows), demonstrate the system’s
behavior in the diabatic and adiabatic regimes, respectively, and were varied by changing only the
Rabi frequency, Ω0 . The probability of being left in the excited state at the end of the sweep, Pe ,is also provided.
To aid in the description of the dynamics, we have labeled specific regions of momentum
space with the symbols (0), (1), and (2), where (i) labels the maximum number of lasers theparticle substantially interacts with at any time during the sweep. More specifically, a particle
with initial velocity vi lies within the region defined by:∆s2∆s|Ω0 | < |kvi | <2|kvi | >|kvi | < |Ω0 |region (0)region (1)(3.21)

region (2).

Note that region (1) roughly corresponds to the high-velocity regime, Eq (3.16). Uninterestingly,
particles in region (0) do not significantly interact with the laser field, so we restrict our discussion
to regions (1) and (2).
For most states in region (1), the resulting ∆prms and Pe after the sweep are roughly constant.
The general results of a diabatic sweep in region (1), as seen in both Figures, may be interpreted
as giving a low imparted impulse and a failure to return the particle to its initial internal state. In
contrast, the general results of an adiabatic sweep in region (1) are an impulse of |∆prms | ≈ 2~kand significant return to the initial internal state; these are the ideal coherent dynamics of thehigh-velocity regime as previously discussed. The highest momentum states within region (1) do
not quite undergo this ideal behavior, even in the adiabatic regime, because the particle does not
begin the sweep in an eigenstate of the Hamiltonian, Eq. (3.9).

(c)

2 (0)

(1)

(2) (1)

(0)

1

(b)

Pe

0
-2

2

(d)

0

49

0.5
0
-120 -60 0
60 120pi (units of h̄k)
1-120 -60 0
60 120pi (units of h̄k)Pe∆prms (units of h̄k)∆prms (units of h̄k)

(a)

-2

0.5
0
-120 -60 0
60 120pi (units of h̄k)-120 -60 0
60 120pi (units of h̄k)∆prms (units of h̄k)

(c)

4 (0)

(1)

(2) (1)

(0)

1

(b)

Pe

(a)

2

4
2
0
-120 -60 0
60 120pi (units of h̄k)

0.5
0-120 -60 0
60 120pi (units of h̄k)
10-120 -60 0
60 120pi (units of h̄k)

(d)

Pe

∆prms (units of h̄k)

Figure 3.6: ∆prms vs. pi ((a) and (c)) and Pe vs. pi ((b) and (d)) for a particle that started in thestate |g, pi i at the beginning of a sweep. The various momentum regions in Eqs. (3.21) are labeledin Figure 3.6(a) and are presented in all subplots by vertical, dashed lines. The horizontal, dashedline corresponds to ∆prms = 0. The adiabaticity parameter κ lies in the diabatic regime for plots (a)and (b) (κ = 0.5, Ω0 = 9.5ωr ) and in the adiabatic regime for plots (c) and (d) (κ = 4, Ω0 = 26.8ωr ).
Values are: ∆s = 360ωr , Ts = 2/ωr .

0.5
0-120 -60 0
60 120pi (units of h̄k)

Figure 3.7: ∆prms vs. pi ((a) and (c)) and Pe vs. pi ((b) and (d)) for a particle that started in state|e, pi i at the beginning of a sweep. The various momentum regions in Eqs. (3.21) are labeled in
Figure 3.7(a) and are presented in all subplots by vertical, dashed lines. The adiabaticity parameterκ lies in the diabatic regime for plots (a) and (b) (κ = 0.5, Ω0 = 9.5ωr ) and in the adiabatic regime
for plots (c) and (d) (κ = 4, Ω0 = 26.8ωr ). Values common to all plots are: ∆s = 360ωr , Ts = 2/ωr .

50
As previously mentioned, Dopplerons and the ambiguous time-ordering of the two laser interactions can significantly modify the force in region (2) such that the physics is more complex.
The effect of this complex behavior is clearly visible in Figures 3.6 and 3.7 as |pi | approaches zero.
In particular, a particle within region (2) initially in the internal ground state will be transferredto the excited state after a sweep, which would then send it on a trajectory toward increasingly
higher momentum. This motivates the requirement described by Eq. (3.1), which will on average
reset the particle to the ground state for the next sweep via spontaneous emission.

3.5.2

Forces including dissipation

While Section 3.5.1 provides insight into the conservative forces in SWAP cooling, it does notinclude the dissipative features that ultimately lead to phase-space compression and equilibration.
Moreover, we enforced specific state preparation at the beginning of each sweep. In order to explore
the forces one would expect in the laboratory, i.e., with γ 6= 0 and no specific state preparation, we
define an average impulse as∆pavg ≡ Tr [p̂ρ̂f ] − Tr [p̂ρ̂i ](3.22)

with the constraint that the internal state populations are the same at the beginning and endof the sweep. The quantities ρ̂i and ρ̂f are the density operators associated with the initial andfinal particle states, respectively. We shall call these internal state populations the “steady-state”
populations, Pess for each pi . Note that steady-state here refers only to the internal state populations
being equal before and after the sweep cycle; the momentum in general will change. This choiceof representing the impulse is motivated by the desire to compare to other cooling methods where
the internal populations reach a stationary situation, and to thereby allow investigation of therelationship between force and particle velocity in a more general context.
Figure 3.8(a) displays ∆pavg for a large range of initial momentum states pi . The parameters
were chosen such that the time between the two laser interactions, τres , obeyed τres ≤ 1/γ for all|kvi | < ∆s /2. We see that the overall effect of SWAP cooling yielded an impulse toward zero

(b)

∆pavg

(a)

2
0
-2
1

Pess

51

0.5

(c)

ξ

0
0.6
0.3
0
-500 -375 -250 -125 0 125
pi (units of h̄k)

250

375

500

Figure 3.8: Various quantities for a particle that is subject to a single sweep with γ 6= 0 as afunction of its initial momentum, pi . (a) ∆pavg (average impulse, in units of ~k). (b) Pess (steadystate population). (c) ξ (number of scattered photons). The vertical, dashed lines correspondto the four conditions |kv| < ∆s /2 (where ∆prms falls to zero) and |kv| < ∆s /4 (the capture
range). Each point was averaged over 1000 trajectories. Values are: ∆s = 1800ωr , Ts = 1.0/ωr ,Ω0 = 60ωr , γ = ωr .

momentum for |kvi | < ∆s /4 and an impulse away from zero momentum for ∆s /4 < |kvi | < ∆s /2.
This motivates Eq. (3.1) as a characterization of the momentum capture range of SWAP cooling.
For low momentum, the effects of Bragg oscillations, Dopplerons, and the ambiguous time-orderingof laser interactions result in momentum dynamics that differ from those in the high-velocity regime.
Figures 3.8(b) and 3.8(c) present the steady-state excited state fraction, Pess , and the average
number of incoherent scattering events per sweep, which we call ξ, for the same parameters. We
see that impulses with magnitudes of nearly 2~k are imparted for |pi | near 25~k with only ∼ 0.2
scattering events per sweep. Moreover, the momentum states around |pi | = 25~k experienced anaverage force of |∆pavg /Ts | ≈ 2~kγ, which is roughly four times the cooling force that one expects
from a radiation pressure force that fully saturates the atomic transition. This means that SWAP
cooling can provide large cooling forces with a relatively low scattering rate. We investigate thisuseful feature in more detail in Section 3.8.

52

3.6

Effect of higher-order processes

3.6.1

Doppleron resonances

As the laser frequencies are swept, multi-photon transitions often called Dopplerons cometo resonance even when single-photon transitions do not. As shown in Figure 3.9(a), an nth order Doppleron process is characterized by the absorption of n + 1 photons from one beam and theemission of n photons into the other, resulting in a (2n+1)~k net momentum transfer to the particle,
and it being left in the opposite internal state from which it started [134]. In general, Doppleron
resonances can occur both before and after the single-photon resonances, and their existence has
the potential to substantially affect particle dynamics outside the high-velocity regime.
Doppleron resonances only occur for specific values of laser detuning. For a particle withinitial momentum pi = βi ~k, the time of an nth -order Doppleron resonance can be found fromenergy conservation:
βi2 ~ωr + (n + 1)~ωL (t) = n~ωL (t) + ~ωa + βf2 ~ωr .

(3.23)We have neglected the AC Stark shift here for simplicity. In Eq. (3.23), the terms proportional ton are the energies of the photons being absorbed and emitted, ~ωa is the transition energy, and theterms proportional to ωr are the initial and final kinetic energies of the particle.
Paired with the momentum condition βf = βi −(2n+1), where βf is the particle’s momentum
after the Doppleron process, the time, tn , of an nth -order Doppleron resonance for a lab-frame laser
detuning profile δn (t) = αtn obeys1
αtn = −(2n + 1)kvi + (2n + 1) ωr = − n +(kvi + kvf ).
2
2(3.24)

Regardless of the Rabi frequency, multiphoton effects become relevant for low momentum
states. From numerical methods, the 1st -order splitting in a dressed state picture (see Figure 3.4)
behaves asΩ(1) ≈Ω30,
16(kvi − 3ωr )2(3.25)

53

(a)

(b)

Figure 3.9: (a) A 1st -order Doppleron resonance, characterized by the condition βf = βi − 3. (b)A Bragg resonance between the states |g, ±2~ki.

which suggests that the probability of adiabatically passing through a first-order Doppleron resonance is [194]Pa(1) = 1 − exp




π
Ω60
−
.
512 α(kvi − 3ωr )4(3.26)The size of the argument in this exponential can be used to define the conditions for which
Dopplerons are relevant for a particle with velocity vi . Because SWAP cooling requires Eq. (3.3)
to be satisfied, we choose to define the Doppleron regime as follows:|Ω0 | > |kvi − 3ωr |(Doppleron regime).

(3.27)

Comparing this result with Eq. (3.16), we see that the Doppleron regime describes nearly all

54remaining states outside the high-velocity regime. Of course, there exist similar conditions forhigher-order Doppleron resonances.
Increasing Ω0 , i.e., allowing for more Doppleron resonances, may decrease the amount of time
it takes to reach a steady-state temperature, as they allow for more momentum transfer. However,the final temperature increases with Ω0 (see Figure 3.13). Thus, minimum temperatures with ahigh capture range may be achieved by dynamically changing the Rabi frequency as the particle iscooled over time.

3.6.2

Bragg oscillations

Momentum states experience nontrivial effects due to Bragg oscillations, i.e., particle scattering from a light grating (the standing wave). These transitions, which are between states with
±β~k momenta where β is an integer, occur at a rate [114]ΩB,β (t) ≈|Ω0 |2β,4β (8ωr )β−1 [(β − 1)!]2 δ(t)β(3.28)

provided that |δ(t)| > |Ω0 | > γ. Figure 3.9(b) displays the energy diagram for a Bragg resonancebetween the |g, ±2~ki states. If there are many oscillations, it is likely that system noise will cause
the oscillations for different particles to become out of phase, leaving, on average, half of the atoms
in either momentum state. This phenomenon is an important consideration in the analysis of theimpulses in Figure 3.10.
Integrating Eq. (3.28) and dividing by 2π gives the total number of β th -order Bragg oscillations between times ti and tf :1
Nβ (ti , tf ) =
2π

Z tf
ti

0

0

ΩB,β (t ) dt ∝

Z tf
ti

1
dt0 .
δ(t0 )β

(3.29)The quantities ti and tf are related to the times of stimulated absorption/emission during thesweep process, as these will carry the particle away from the states that are experiencing Bragg
oscillations.
To provide a few interesting examples, 1st -order Bragg transitions affect the dynamics of the55
initial states |ψi0 = |g, ~ki , |g, 3~ki. From Eq. (3.29) and using Eqns. (3.13) and (3.14),
κ
2ωr
ln
,
8π
∆s
κ
∆s
N1 ≈
,
ln
8π
6ωr
N1 ≈

|ψi0 = |g, ~ki ;(3.30)|ψi0 = |g, 3~ki .

The Bragg oscillations occur before the SWAP resonances for |g, ~ki and after the SWAP resonancesfor (|g, 3~ki). Although N1 scales logarithmically with ∆s in both cases, a sufficiently large sweep
range will promote multiple Bragg oscillations.
The effect of Bragg oscillations at low pi , which manifests as a sharp, linear feature in Figure
3.8(a), is elucidated by considering the effect of switching the sweep direction. Bragg oscillations
can mix the particle between the |±pi i states before the particle resonates with the lasers, so the net∆pavg is independent of the sweep direction. To compare ∆pavg with and without these oscillations,
we define impulses ∆p± in the following way:∆p± ≡(∆pavg )pos ± (∆pavg )neg
2

(3.31)

in which the subscripts “pos” and “neg” refer to the sign of the ramp, i.e., red to blue or blue to
red detuning, respectively. By symmetry, we expect the effect of Bragg oscillations to be present
in ∆p+ and to cancel in ∆p− . Figure 3.10 displays ∆p± as a function of the initial momentum,
pi . The substantial effect of Bragg oscillations were seen to be present in ∆p+ for |pi | < 7~k.
For pi nearer to the high-velocity regime, which is |pi | > 30.5~k for this set of parameters, Bragg
oscillations do not play a significant role, and we observe ∆p+ tending to zero, and ∆p− tending
to ±1.5~k.

3.7

Temperature limit as γ → 0
We now provide the results of a SWAP cooling simulation that includes the effects of dissi-

pation, but in the limit γ → 0. To accomplish this, we modifed the frequency profile to be a seriesof single sawtooth ramps with period Ts  1/γ, each separated by a time Twait  1/γ, as shownin Figure 3.11. This “sweep-wait” scheme effectively mimics the cooling process for a particle with

56

Impulse (units of h̄k)

4

∆p+
∆p−

2

0

-2

-4
-50

-25

0

25

50

pi (units of h̄k)
Figure 3.10: ∆p± vs. pi . The sharp, linear feature in ∆p+ is due to Bragg oscillations, which mix
the particle between the |±pi i states before it reaches resonance with the lasers. Each point isaveraged over 1000 trajectories. Values are: ∆s = 1800ωr , Ts = 1.0/ωr , Ω0 = 60ωr , γ = ωr .

an ultranarrow linewidth. It also reduced the required computation time by allowing (i) ∆s to besmall and (ii) the use of the analytical expression for free-space spontaneous decay. We modeled thelimit Twait → ∞ and γ → 0 by setting γ = 0 and projecting any remaining excited state population
to the ground state, along with simulating any accompanying momentum recoil.
Figure 3.12 presents the evolution of a particle that started in the state |g, 10~ki under thiscooling scheme. At the end of this simulation, the temperature of the particle was observed toasymptote to a value near the recoil limit, 2Tr , wherekB Tr ≡(~k)2
= ~ωr ,
2m

(3.32)and kB is Boltzmann’s constant. We use the variance in p as a measurement of the 1D temperature,
T , i.e.,
σp2
hp2 i
1
≈
= kB T.
2m
2m
2(3.33)It should be noted that not all steady state solutions are Gaussian in nature, but are centeredon zero momentum. Thus, the first equality in Eq. (3.33) universally holds once the system has
equilibrated.

57

Figure 3.11: Frequency profile for the simplified “sweep-wait” cooling scheme.

120
P (p)

0.4

hp̂2 /2mi (units of h̄ωr )

100

0.2
0

hp̂2 /2mi (units of h̄ωr )

80
60
40
20

−3
0
3
p (units of h̄k)

20Recoil limit

6

7

8

9

10

Number of sweeps

0
0

2

4

6

8

10

Number of sweeps
Figure 3.12: Cooling of a particle that begins in the state |g, 10~ki to the recoil limit after 5 sweepwait cycles. Top inset: A snapshot of the momentum distribution P (p) halfway through the 7th
sweep. Bottom inset: A closer look at the cooling trajectory once the system reached equilibration.
The recoil limit, 2Tr , is included as a horizontal, dashed line. This curve is the average of 100
trajectories. Values are: ∆s = 100ωr , Ts = 60/ωr , and Ω0 = 2ωr .

58
It is also interesting to consider how the final temperature scales with Ω0 . Figure 3.13 displays
this relationship in both the diabatic and adiabatic regimes. A minimum temperature was foundjust within the adiabatic regime. From the numerics, the temperature was observed to follow a
linear relationship1
kB T = ~Ω02(3.34)

in the adiabatic regime, which we attribute to decreased time ordering between adiabatic transfers
and the particle spending more time in the excited state, hence more scattering events. This linear
relationship held for at least twice the domain of Figure 3.13. An increase in temperature wasobserved in the diabatic regime, which we attribute to both a reduction in conservative forces as
described in Section 3.5.1, and the presence of a significant excited state fraction at the end of each
sweep, leading to more diffusion from spontaneous emission.
8

hp2 /2mi (units of h̄ωr )76Ω20 = α5kB T = 21 h̄Ω0

4
3
2
1
0

Recoil Limit

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Ω0 (units of ωr )Figure 3.13: Stationary values of hp2 /2mi as a function of Ω0 in the sweep-wait sequence. The
vertical, solid line labels the value of Ω0 that divides the diabatic and adiabatic regimes. The
recoil limit is represented as a horizontal, dashed line. Deep within the adiabatic regime, thetemperature scales linearly with Ω0 . Values are: ∆s = 100ωr , Ts = 60/ωr . Each point is averaged
over 500 trajectories.

In Chapter 4, we present a plot which shows the same quantities as in Fig. 3.13, but with

59
experimental and simulation data for the γ = 2π × 7.5 kHz linewidth 1 S0 →3 P1 transition in 88 Sr.
The simulation data was created by employing the method of Monte Carlo wave functions (seeSection 2.2.4) with the SWAP cooling quantum master equation [Eq. (3.5)].

3.8

Cooling efficiency
We now compare the efficiency of SWAP cooling with Doppler cooling. We define “coolingefficiency” here as the energy carried away from the system per scattering event. This choice ismotivated in part by the potential application of SWAP cooling to systems where closed cyclingtransitions may not be accessible and therefore a large number of spontaneous emission events are
undesirable. We calculate the cooling efficiency by preparing an ensemble of particles in a specificinitial state, |g, 20~ki, and subsequently applying both Doppler and SWAP cooling for comparison.
Here, we used the SWAP cooling scheme presented in Figure 3.2, in which dissipation is included
at all times.
Clearly, many fewer scattered photons are required for SWAP cooling to reduce the energy
and to bring the system close to equilibrium. In fact, in the simulation, SWAP cooling was able
to remove up to an average of 5~k of momentum per scattered photon. As a consequence, incomparison to Doppler cooling’s ideal cooling efficiency, we deduce that SWAP cooling promises tobe well-suited to cooling particles where the adverse affects of spontaneous emission are significant,such as those that lack closed cycling transitions.

3.9

Conclusion
The SWAP (sawtooth wave adiabatic passage) procedure proves to be a robust and simple

cooling mechanism. Our analysis of both the coherent and dissipative dynamics of two-level particles suggests that it has numerous applications, such as to systems with narrow linewidth or no
closed cycling transitions. We have shown its ability to cool particles to the recoil limit while simultaneously inducing larger cooling forces and maintaining a higher cooling efficiency than Doppler
cooling.

60
400

hp̂2 /2mi (units of h̄ωr )SWAP
Doppler, Ω0 = kvi
Idealized Doppler

300

200

100

0
0

3

6

9

12

15

Scattered Photons
Figure 3.14: hp̂2 i/2m vs. the average number of scattered photons for SWAP cooling and Doppler
cooling. For both methods, ωr = γ. The SWAP parameters are: Ω0 = 28ωr , ∆s = 391ωr , Ts = 1/ωr ,and Doppler parameters are: Ω = 40ωr , δ = −40ωr . The SWAP cooling data is over 38 sweeps.
Each curve is averaged over 1000 trajectories. We extract an average cooling efficiency for SWAP
cooling of up to 5~k per scattered photon.

The scope of the calculations presented in this paper were limited to two-level atoms moving along one-dimension. It will be interesting to consider more general atomic systems and moregeneral geometries of laser and trapping fields. In particular, SWAP cooling may be applied tomagneto-optical trapping [12, 137], to optical lattices, and to general multilevel laser cooling strategies [74, 171]. The realization of the SWAP protocol in a magneto-optical trap is in fact the topic
of the next chapter. Furthermore, while it is important to recognize that while our calculations
are fully quantum mechanical (consisting of a complete description of the internal and externalvariables) with the inherent advantages of not having to make approximations, it will also be useful
to develop and validate semiclassical methods where many-sweep cycles can be treated with higher
efficiency. This will be important to treat more massive systems such as complex molecules.
In the future, it will be interesting to further elaborate on the concept of a cooling efficiency,
i.e., the removal of a system’s energy and entropy per scattering event, in the general context oflaser cooling theory. For example, the information associated with the momentum of a particle

61
initially in the state |g, pi i may be encoded in the time record of spontaneously emitted photons.
This provides significant motivation for a systematic analysis of entropy dynamics in the various
implementations of laser cooling that may prove to be an insightful and useful endeavor for thelaser cooling and atomic physics community. We discuss this topic to some extent in Chapter 7.
The protocol we have proposed is inherently limited by the requirement of adiabatic transfer
between quantum states. We discuss the relaxation of this requirement by use of an adiabatic
speedup protocol in Chapter 6.

Chapter 4

SWAP cooling: experimental results and comparison to theory

Now that we have developed and studied SWAP cooling from a theoretical perspective, we
present the results of its first experimental realization as well as a comparison of the experimental
results to theory. While we also discuss its basic mechanism, forces, and achievable temperatures inthis chapter, we refer the reader to Chapter 3 for a more detailed analysis. The content shown hereis mainly from our New Journal of Physics article (Ref. [141]), which was published in February
2018. To put the discovery of SWAP cooling into context, we quote the wise and humbling wordsof Matthew A. Norcia, who first witnessed its cooling potential in the laboratory as a graduate
research assistant in James Thompson’s lab at JILA:“Basically, I pressed a button on the function generator.”
Dr. Matthew Norcia, A Little Less Spontaneous [98]The content shared in the chapter demonstrates that SWAP cooling is a robust coolingmechanism capable of exerting large forces via a weak transition and with reduced reliance on
spontaneous emission. We present a simple intuitive model for the resulting frictional force, anddirectly demonstrate its efficacy for increasing the total phase-space density of an atomic ensemble.
We rely on both simulation and experimental studies using the 7.5 kHz linewidth 1 S0 →3 P1 transition in 88 Sr. The reduced reliance on spontaneous emission may allow this adiabatic sweep method
to be a useful tool for cooling particles that lack closed cycling transitions, such as molecules.

63

4.1

Introduction
The development of Doppler cooling and sub-Doppler cooling in the 1980’s has revolutionized

our ability to control neutral atoms, ions, and mechanical resonators [34, 42, 59]. Since then,Doppler cooling techniques have been extended to narrow-linewidth optical transitions to achieve
lower temperatures and high phase-space density [62, 86, 91, 110, 176, 186]. Additionally, there
is an ongoing effort to use Doppler cooling for molecules [85, 170]. Each of these pursuits facecertain persistent limitations. The use of weak transitions limits the forces achievable with Doppler
cooling, and the narrow transition linewidth makes the cooling very sensitive to perturbations of
the cooling transition frequency [92] and to drifts in the cooling laser frequency. In the case ofmolecules, the large number of spontaneous emissions required for Doppler cooling is a key obstacle
due to the high probability of spontaneous Raman transitions to undesired states.
Here we present a new form of laser cooling that mitigates these issues. Our technique relieson the adiabatic transfer of atoms to and from a long-lived optically excited state to both slow
and cool the atoms. Because the role of spontaneous emission is reduced (though not eliminated)
relative to standard Doppler cooling techniques, our technique enables large forces to be generated
even on weak transitions, and may facilitate the extension of laser cooling techniques to systems
that lack closed cycling transitions. Of particular interest, this technique may be applied to theslowing and cooling of molecules with narrow linewidth optical transitions [49, 99].
From a fundamental perspective, this work has important implications for the ongoing discussion of the role of spontaneous emission in dissipating entropy during laser cooling [51, 124]. Ourview is that spontaneous emission (or another form of dissipation) is necessary to achieve phasespace compression, but that the total number of spontaneously scattered photons required can bequite low (of order one). Similar conclusions have been reached in the study of optical pumping inhigh angular momentum states of atoms [159] and cooling with single spontaneous emission events[152].

64

4.2

Basic mechanism
To understand our cooling mechanism in its simplest form, we consider a two-level atomwith a long-lived optically excited state |ei and ground state |gi moving in one dimension withvelocity v (Fig. 4.1a). Two counterpropagating laser beams with wave-number k and frequency ωare linearly ramped in frequency from below to above the atomic transition frequency ωa , with fullsweep range ∆s . This ramp is repeated to form a sawtooth pattern in time (Fig. 4.1c). Each laserinteracts with the two-level atom with a Rabi frequency Ω, which is larger than the spontaneous
decay rate γ from |ei to |gi. The sweep range is adjusted such that ∆s > Ω, 4kv. The frequency
sweep rate α = (dω/dt) fulfills the Landau-Zener condition α  Ω2 for adiabatic transfer of
population between ground and excited states. Lastly, the jump in laser frequency at the end ofeach ramp is treated as perfectly diabatic.
In the reference frame of the atom, both cooling beams start below resonance with the atomic
transition. Doppler shifts due to the atomic motion cause the beam counter-propagating to theatom’s velocity to appear kv higher in frequency and the co-propagating beam to appear kv lower
in frequency. As the laser sweep upward in frequency, the counter-propagating beam sweeps overresonance first, and adiabatically transfers the atom from |gi to |ei. Because of the long lifetimeof the excited state, the atom remains in |ei until the co-propagating beam sweeps over resonanceand adiabatically transfers it back to the ground state. In this process, the atom has absorbed
one photon from the beam propagating against its motion, and emitted a photon into the beam
propagating along its motion, resulting in a net momentum transfer of 2~k against its motion. The
laser frequency is then diabatically jumped back to its start frequency such that the atom remains
in |gi, and then the process is repeated.
Similar principles have been explored [87, 129, 142, 172, 173] to generate large forces. In
these works, the time-ordering of the left and right-going pulses is fixed by parameters external tothe atom. Here, because the time-ordering of the interaction with the two beams is determinedby the atomic velocity, the force always opposes the atom’s velocity, and thus causes slowing for

a.

b.

v

65

P(e)

laser
frequency

c.

1
0

d.

time

Figure 4.1: (a) A moving atom is illuminated by two counter-propagating laser beams, whose
frequencies are modulated with a saw-tooth wave-form. (b) The atom is treated as a two-levelsystem with transition frequency ωa . (c) Representation of frequencies in lab frame. Becauseof Doppler shifts resulting from atomic motion, the co-propagating (counter-propagating) laser
beams are resonant with |gi to |ei transition when the laser frequency ω = ωa − kv (ωa + kv).
Because ω increases with time, the counter-propagating laser sweeps over the transition before theco-propagating laser. If the atom starts in |gi, the counter-propagating laser transfers the atomfrom |gi to |ei as it sweeps over resonance, and the co-propagating laser transfers the atom backto |gi. This results in the transfer of two photon recoils of momenta to the atom in the correct
direction to slow the atom. (d) Simulation of trajectories in momentum space. When the atoms
have large momenta, each sweep lowers the momentum by 2~k. At lower momenta, multi-photon
processes can transfer larger amounts of momentum.

atoms moving in either direction.

4.3

Numerical demonstration of basic mechanism
Fig. 4.1d shows simulated trajectories for atoms that begin at different initial momenta usinga Monte Carlo wave function trajectory method [63], which is further discussed in Section 4.7.
Atoms are prepared in a pure quantum state specified by their internal ground state and fixed
momentum. The average momentum over 50 trajectories is calculated at each time step. At high66
momentum, each sweep lowers the momentum of the atom by roughly 2~k. At lower momentum,
we find that the change in momentum per sweep can greatly exceed 2~k, as multiple photons aretransferred between the two cooling beams.
At low momentum, the role of spontaneous emission becomes more important. As the atom
is slowed to near-zero velocity, the Doppler shift becomes small compared to the Rabi frequency,
kv . Ω, and the condition for deterministic time ordering of adiabatic transfers from the two beams
no longer holds. When this occurs, the simple picture of sequential adiabatic transfers to and from|ei becomes invalid, and the probability that the atom is found in |ei at the end of the sweep, which
ideally should be 0, becomes appreciable. While the sawtooth frequency sweep leads to slowing for
an atom that is in the ground state at the beginning of a sweep, it causes an atom that starts inthe excited state to increase in velocity. In the absence of spontaneous emission, the atom wouldfind itself on a heating trajectory after initial cooling. The presence of spontaneous emission istherefore critical, as it ensures that the atom preferentially starts in |gi at the beginning of each
sweep. This breaks time reversibility, enabling cooling and phase space compression to occur.
We can extract the rate of spontaneous emission events from the simulation. For a trajectorythat begins at 100 photon momenta and realistic sweep and atomic properties for 88 Sr, the atom’s
momentum is initially reduced by an average of five photon momenta for each spontaneous emission
event. This can be compared to Doppler cooling, which requires at least one scattered photon perphoton momentum removed, and in practice typically much more. Once the atom has reached its
equilibrium temperature, the average scattering rate is slightly less than one photon per sweep forthe same parameters. If applied to a species for which spontaneous emission is problematic (likemolecules), it may be useful to apply velocity-selective optical pumping to a dark state to avoid
the continued scattering of photons from already cooled particles.

4.4

Demonstration of steady-state cooling and phase space compression
We experimentally verify that our mechanism leads to cooling by applying two counter-

propagating beams to an ensemble of 88 Sr atoms pre-cooled to roughly 600 µK (Fig. 4.2a). The

K

67

Figure 4.2: Experimental study of an atomic ensemble using one-dimensional cooling by adiabatic
transfer. (a) An ensemble of atoms, precooled to around 600 µK, is illuminated by frequencyswept counterpropagating beams for a variable amount of time. The one-dimensional temperature
is decreased to a steady-state value of 45 µK by the cooling lasers. (b) The phase-space density
in one dimension (grey trace) and three dimensions (black trace) is increased during the coolingprocess.

frequency of the two beams is swept upwards by ∆s = 2π × 6.6 MHz every 50 µs. The frequency sweep is centered on the resonance of the dipole-forbidden 1 S0 →3 P1 transition, which has
a linewidth of γ = 2π ×7.5 kHz. The two beams are linearly polarized in the direction of a 3.3 gaussmagnetic field, such that the light-atom interaction can be described as a two-level system with|gi ≡ |1 S0 , mj = 0i and |ei ≡ |3 P1 , mj = 0i, where mj labels the magnetic Zeeman sub-level.
The cooling beams are applied for a time of 50 µs to 1200 µs (i.e. application of 1 to 24
sweeps) after which the beams are turned off. The temperature of the atomic cloud is determinedfrom a fluorescence image of the cloud after free ballistic expansion for 10 ms. In the directionof beam propagation, we find that for these parameters the atoms are cooled from their initial
temperature of roughly 600 µK to 45 µK in of order 300 µs or 6 sweeps.
Importantly, we directly observe an increase in phase-space density during the cooling process,and not simply velocity reduction. We measure atom loss during cooling to be negligible and therelative increase in phase space density is ρ/ρ◦ = ∆x◦ ∆v◦ /(∆x∆v), where ∆x and ∆v (∆x◦ and68∆v◦ ) are the measured cloud size and velocity spread after (before) cooling. In Fig. 4.2b, thisquantity is shown as the grey line. The black line in Fig. 4.2b accounts for measured heating in
the orthogonal directions (which for simplicity we have assumed to be equal in the two directions).
Note that because the atoms are not confined and the cooling is only applied along one dimension,
this increase of phase-space density is much smaller than one would obtain with the same decreasein temperature for three-dimensional cooling in a harmonic trap.

4.5

Comments on entropy removal
The fact that phase-space density increases indicates that entropy is removed during the cool-ing process. This may seem problematic for a mechanism that relies heavily on unitary dynamics
and stimulated emission. However, as discussed above, the presence of even small amounts of spontaneous emission is critical for breaking time reversibility and enabling phase-space compression.
In related works [152, 159], it was found that the number of scattered photons required to remove
entropy from a system can be quite low (of order one).
In these protocols, the timing of the emitted photons encodes key information about the initialstate of the quantum system. In an idealized version of our cooling mechanism, the dynamics arecompletely unitary as the atom is slowed from a high initial momentum. When it reaches somelower momentum threshold, the simple picture of subsequent adiabatic transfer breaks down andthe atom begins to scatter photons. In this way, the initial velocity of the atom is encoded in thetime delay before the atom began to scatter photons, and the atom’s initial entropy in momentum
space is mapped onto entropy contained in the measurement record of when photons are and are
not spontaneously emitted. While a detailed analysis of the entropy transfer is beyond the scopeof the work here and will be explored more fully in future work, we believe that the above picture
captures the essential physics of how phase-space compression is achieved.

a.

frequency

69

left beamtimeright beam

m/s

b.

m/s2

c.

Figure 4.3: Cooling into a moving reference frame. (a) We apply a frequency offset in the lab
reference frame between the two beams, which creates a moving reference frame in which thebeams have equal frequency. (b) If the laser frequency is swept upwards, (green points) the atoms
settle into the moving reference frame, which has a velocity indicated by the dashed black line. If
the laser frequency is swept downwards (orange points), the atoms accelerate in the other direction,
without cooling. If the frequency sweep is symmetric (purple points), atoms experience a small
acceleration due to radiation-pressure imbalance. (c) Acceleration vs. polarization angle between
two beams. Acceleration is enhanced when the polarizations of the two beams are aligned. The
solid line represents the maximum acceleration for a saturated two-level atom without stimulated
emission. The dashed red line represents the measured acceleration with a single beam. Black points
represent acceleration with both beams applied. The dashed black line represents the accelerationthat two photon recoils per sweep of the laser frequency would produce.

4.6

Cooling forces
We now turn to a more detailed characterization of the forces involved in the cooling process.

Previously, we have considered the case where the instantaneous frequencies of the two beams areequal in the lab frame. If, however, we introduce a fixed relative offset frequency between the70
counter-propagating beams (as shown in Fig. 4.3a), then atoms that are stationary in the lab frame
will be accelerated until their velocity matches that of the frame F moving at velocity vF = ∆/2k,
in which the two beams appear to have equal instantaneous frequencies.
We apply the frequency-swept beams for a variable amount of time Ta , then measure theresulting velocity of the atoms. This is shown in Fig. 4.3b for several sweep configurations. The
green points represent the cooling configuration described above, with the laser frequency adiabatically swept from low to high and diabatically jumped from high to low. The two beams are offset
by a detuning ∆/2π = 400 kHz, and are swept by ∆s /2π = 8 MHz every 33 µs. We observe thatthe atoms undergo initial acceleration until they reach equilibrium with the velocity of the moving
frame vF = 0.14 m/s.
To rule out the interpretation that the atoms are merely being dragged by the moving standing
wave formed by the detuned lasers, we reverse the direction of the sawtooth sweep without changing
the relative detuning ∆ of the two beams. The standing wave still moves in the same direction
as before, but now the atoms accelerate in the other direction. While the upwards sweep causes
saturation of the atomic velocity at vF , no such saturation is apparent for a downwards sweep.
This confirms that the downwards sweep configuration does not lead to cooling.
Finally, when we apply a symmetric triangle-ramped frequency sweep to the lasers, the atoms
undergo a much smaller acceleration, likely due to incidental residual radiation pressure from anintensity imbalance between the two beams.
Because our mechanism relies on stimulated emission, much larger accelerations can beachieved than would be possible with Doppler cooling on such a narrow transition. To quantify this acceleration, we apply the beams for a time much shorter than it takes the atoms to reachequilibrium velocity, and measure the resulting change in velocity. If we apply only one of the twocooling beams, the atoms experience an acceleration consistent with (γ/2)(~k/m) = 155 m/s2 , theexpected value for a maximally saturated atom.
We see a far more dramatic effect when we apply both beams at the same time (Fig. 4.3c).
When the polarizations of the two beams are aligned, we observe a maximum acceleration of around71
600 m/s2 for a sweep period of 20 µs, a factor of almost 4 above both the observed accelerationfrom the leftward beam alone and the maximum expected acceleration for a two-level atom without
stimulated emission. This measured acceleration is within ten percent of the value we would expect
if each sweep of the laser frequency led to two photon recoils of momentum transfer, though thisagreement may partly be due to a cancellation between the effects of imperfect adiabatic transfersand multi-photon processes. When the polarizations are made orthogonal, the acceleration returns
to near the single-beam value, as only one of the beams interacts with |ei.

Temperature measurement and comparison to simulation

Temperature (µK)

4.7

60
40
20
0
0.0

0.2

0.4

0.6 0.8 1.0
Rabi frequency Ω/2π (MHz)Figure 4.4: Temperature vs. Rabi frequency for a sweep range of 7 MHz and repetition rate of
15 kHz. Red line represents the results of our simulations, and black points are experimental
data. The red band indicates uncertainty of prediction due to statistical uncertainty in simulation
results, experimental calibration of Ω, and variations in Rabi frequency during the sweep. The
vertical line at 130 kHz represents the approximate point at which we expect adiabaticity to break
down: Ω2 = α. We discuss possible sources for the disagreement at high Rabi frequency in themain text.

We now compare our experimentally measured temperature to simulation in Fig. 4.4. This
plot presents the same quantities as the idealized γ → 0 study in Section 3.7, but for γ 6= 0.
The simulation used is a Monte Carlo wave function approach, as introduced in Section 2.2.4,
which evolves the quantum master equationdρ̂
1= [Ĥ, ρ̂] + L̂ρ̂
dt
i~

(4.1)

72
described in detail in Section 3.3. While the external electric field associated with the coolinglasers is treated classically, the internal and external atomic degrees of freedom are treated fully
quantum mechanically. The wave function contains ground and excited internal states in addition
to a discrete family of external momentum states, which are equally spaced in units of ~k up to a
maximum cutoff. The unitary dynamics are generated by the single-particle Hamiltonian
Ĥ =



p̂2
1
+ ~ωa σ̂ z + ~Ω cos(kẑ) σ̂ + e−iη(t) + h.c. ,
2m 2

(4.2)

where the instantaneous phase of the applied coherent field η(t) is the time integrated instantaneous
frequency, with the frequency ramped in the sawtooth pattern previously described. Because thefull atom and field system is an open quantum system due to the presence of spontaneous emission,
the Lindblad superoperator proportional to the spontaneous emission rateL̂ρ̂ = γ

1 ikẑ − + −ikẑ 1 −ikẑ − + ikẑ 3 − + 1  + −e σ̂ ρ̂σ̂ e+ eσ̂ ρ̂σ̂ e + σ̂ ρ̂σ̂ −σ̂ σ̂ , ρ̂
5
5
5
2


(4.3)

is also included to fully simulate the associated quantum master equation.
Both experiment and simulation show a minimum temperature as a function of Rabi frequency. At low Rabi frequency, the Landau-Zener condition α  Ω2 breaks down, leading to inefficient adiabatic transfers. For the parameters used in (Fig. 4.4), α = Ω2 when Ω = 2π × 130 kHz,roughly the point at which temperature is minimized in both the simulation and experimental
results.
At larger Ω, both the experimentally measured and simulated temperatures rise with Ω,
though the predicted and observed values appear to disagree at large Ω. This may be explained by
the use of an approximate dipole radiation pattern which is better suited for a ∆mj = ±1 transitionin the Lindblad superoperator [Eq. (4.3)], which introduces additional momentum diffusion andtherefore results in a higher temperature. Moreover, several experimental factors could lead tothis discrepancy. Because an acousto-optic modulator (AOM) is used to sweep the frequency ofthe cooling lasers, there are significant power variations during the sweep. Including this effect inour simulation resulted in lower temperatures at high Rabi frequency, and this effect can explain

73
much of the disagreement (as indicated by the red band in Fig. 4.4). Furthermore, imperfect laser
polarization and finite bias field may lead to cooling on the transitions to 3 P1 , mj = ±1, especiallyat high Rabi frequency. Nevertheless, the data roughly obeys the linear relationshipkB T ∝ ~Ω,(4.4)

for values of Ω in the adiabatic limit, which we discussed in Section 3.7 in the limit γ → 0.
These results highlight that obtaining low temperatures requires low Rabi frequencies, and a
sufficiently slow sweep such that the Landau-Zener condition α . Ω2 is maintained. These conditions indirectly constrain the values of the atomic decay rate γ that are compatible with achieving
low temperatures. In order to be in a regime where adiabatic transfer can be performed efficiently,
we require that Ω  γ, which motivates the use of narrow linewidth transitions in order to utilize
a low Rabi frequency and obtain a low temperature. The sweep rate α is further constrained by
the condition that the atoms should not decay between subsequent adiabatic transfers to and fromthe excited state, which requires α/2kv & γ. If we take v to be the initial velocity of the particles
to be cooled, this condition represents a tradeoff between low final temperatures (which necessitatea small sweep rate α), and a large velocity capture range (which requires large α). This tradeoff
becomes more favorable for long-lived excited states, though in principle can be mitigated in certain
applications by dynamically adjusting the sweep rate and range as the particles are cooled.
By optimizing the sweep parameters in our simulation to achieve low equilibrium temperatures for 88 Sr atoms that begin at low temperature (making capture range a more minor concern),
we achieve an equilibrium temperature of 2.3 µK. We find that colder temperatures can be achieved
in simulation by using a longer-lived excited state. For comparison, this temperature is roughly anorder of magnitude above the Doppler cooling limit for the same transition. Clearly, the key meritof our cooling mechanism is not its ability to achieve the lowest possible temperature, but rather toprovide relatively strong forces on weak transitions, a large capture range, and a reduced relianceon spontaneous emission.

74

4.8

Conclusion
We have demonstrated a novel cooling mechanism in which large accelerations may beachieved even on optical transitions with low scattering rates. This mechanism may find application to the cooling of molecules, and can improve the performance of atomic cooling using
dipole-forbidden optical transitions. From a technical perspective, this cooling method has theadvantage of low sensitivity to long-term laser frequency drifts and to perturbations of the atomic
transition frequency, for example to those encountered while loading into a deep optical dipole trap
[92].
After the publication of this study, various extensions and modifications to this cooling mechanism have been realized and proposed, including the theoretical investigation [13] described in theprevious chapter, the use of circularly polarized beams and magnetic field gradients to form amagneto-optical trap [12, 137, 171] (which is the subject of the next chapter), the application to87 Rb (which has no narrow-line optical transition) by using Raman transitions [74], and the decel-

eration of a beam of 163 Dy atoms [146]. We also expect that this cooling mechanism can be used
to achieve near ground state cooling of tightly confined atoms. To speed up the adiabatic transfers
and therefore increase the stimulated forces, we also consider the incorporation of an adiabatic
speedup protocol [14] in Chapter 6.

Chapter 5

SWAP MOTIn the previous two chapters, we demonstrated how the SWAP protocol can be used tocool atoms and molecules with narrow linewidth optical transitions. In this chapter, we show
theoretically how it can be combined with a magnetic trap to supply an additional trapping force.
This work resulted in a publication in Physical Review A (Ref. [12]) in May 2020. As opposed
to related methods that have been previously reported [137], our approach utilizes repeated cycles
of stimulated absorption and emission processes to achieve both trapping and cooling, thereby
reducing the adverse effects that arise from photon scattering. Specifically, we demonstrate thismethod’s ability to cool, slow, and trap particles with fewer spontaneously emitted photons, higher
forces and in less time when compared to a traditional MOT scheme that utilizes the same narrow
linewidth transition (see Section 2.3.4). We calculate the phase space compression that is achievableand characterize the resulting system equilibrium cloud size and temperature.

5.1

Introduction
Trapping and slowing devices based on the removal of momentum, energy, and entropy via

light are ubiquitous and essential in most experiments on quantum gases. Techniques such as theslowing and cooling of particles by preferential absorption from a counterpropagating laser [68,77], the magneto-optical trap [156], Zeeman decelerators [172], the bichromatic force [191], andsawtooth-wave adiabatic passage (SWAP) cooling [13, 141] all rely on many cycles of an engineered
light-matter interaction for this purpose.

76
It is widely understood that spontaneous emission is a fundamental requirement for the laser
cooling and trapping of atoms and molecules. This is due to the fact that the scattered photons
irreversibly remove entropy, allowing the system to violate the conditions for Liouville’s theorem toapply and to thereby undergo compression in terms of the occupied volume in phase space. While
incorporating repeated scattering events is acceptable for systems with a closed cycling transition,
this can lead to significant loss for systems with many degrees of freedom, such as molecules, which
may have a large number of dynamically decoupled internal states, or “dark states.” Protocols suchas the optical repumping from uncoupled states to cooled states can mitigate this issue [90, 170],but a more absolute solution would be to significantly reduce the number of spontaneous emissions
necessary to achieve slowing, cooling, and trapping.
SWAP cooling can cool a system to equilibrium with fewer spontaneous emissions than
Doppler cooling and slow a distribution of particles using purely coherent dynamics [13]. We
present the results of combining SWAP cooling with a quadrupole magnetic trap, which we calla SWAP MOT [140]. We show that a SWAP MOT is able to demonstrate slowing, cooling, andtrapping with fewer scattered photons, higher conservative forces, and in less time when comparedwith a traditional MOT scheme.
Our approach differs from other methods of implementing the SWAP procedure in a MOT[137, 171] because we formulate a method to incorporate the stimulated emission process originallyenvisioned in the first proposals of SWAP cooling [13, 141]. This addition promotes both coolingand trapping and reduces the number of scattered photons required for equilibration. In orderto achieve the desired coherent dynamics, we propose that the cooling laser polarizations andmagnetic field directions (or, equivalently, the direction of the electron’s magnetic moment) shouldbe abruptly switched at the center and end of each sweep (see Fig. 5.1).
In Section 5.2 we explicate the particle dynamics achieved with the SWAP MOT protocol.
In Section 5.3 we develop a semiclassical model in which the internal states are treated quantum
mechanically and the external states are treated classically. Section 5.4 explains the details of ournumerical algorithm. In Section 5.6 we define regimes in phase space that exhibit different types of

77

Figure 5.1: A schematic of the two experimental setups for the 1D SWAP MOT (motion only
along the x-direction) that operate in alternating periods of the cooling cycle. Anti-Helmholtz coilswith current I and counterpropagating lasers of opposite circular polarization σ + and σ − create amagnetic-optical trap for neutral particles (pink circle). The laser polarizations and current directions are (ideally) instantaneously, periodically switched between the two setups as the detuning ofeach laser, which is linearly ramped from below to above the resonance of the cooling transition,passes through zero.

dynamic behavior under SWAP MOT evolution, define capture range conditions, and demonstratephase space compression and MOT loading. In Section 5.7 we provide various scaling properties ofthe procedure over the range of interesting system parameters.

5.2

SWAP MOT mechanism
The SWAP procedure relies on the coherent transfer of a particle between quantum states via

adiabatic rapid passage. The internal structure of the simplest quantum system that demonstrates
the desired dynamics consists of two excited states with a common ground state, which we label|+i , |−i , and |0i, respectively. The labels correspond to the values m = ±1 and 0, where m is themagnetic quantum number, such as that occurs in a system with a J = 0 → J = 1 transition, where
J is an angular momentum quantum number. Due to our choice of laser polarization, omission of

78

Figure 5.2: (a) The minimum internal state structure necessary for demonstrating SWAP MOT
dynamics. The transition frequency ω0 , natural linewidth γ, laser frequency ωL (t), detuning δ(t),and Zeeman shifts ±bx of the excited states are included. (b) The laser detuning δ(t) (dashed) andexcited state fraction Pe (t) (solid) vs time t over a sweep of period Ts . The particle resonates withthe lasers at the times t1 and t2 when δ(t) = ±δ m (dot-dashed), where δ m is the motional detuning[see Eq. (5.2)].

the |J, mi = |1, 0i state is valid, as it is not optically pumped. As shown in Fig. 5.2(a), the excitedstates are separated in energy from the ground state by ~ω0 , where ω0 is the transition frequency,
assumed to be in the optical domain. In the presence of a magnetic field, the excited states are
shifted in energy according to the Zeeman shift ±~bx, where b = gµB (∇B)/~, g is the g-factor of
the transition, µB is the Bohr magneton, and x is the displacement of the particle from the trap
center. Both excited states decay to the ground state with a rate given by the linewidth γ. For
simplicity, we limit our discussion to one dimension.
As displayed in Fig. 5.1, the experimental setup is nearly identical to that of a type I 1D

79
MOT [126]. However, instead of fixing the laser detunings
δ(t) = ωL (t) − ω0

(5.1)

below the transition frequency, they are repeatedly swept from below to above the cooling transition
in a sawtooth pattern (see the dashed curve in Fig. 5.2(b)) with full period Ts . In Eq. (5.1), ωL (t)is the instantaneous laser frequency. Additionally, the directions of the currents I in the antiHelmholtz coils and the polarizations of the cooling lasers are switched at a rate 2/Ts . The current
directions are chosen such that the magnetic field along the radial direction at the center of thetrap, which we call the x-direction (see Fig. 5.1), has the form B(x) = (∇B)x during the first halfof the sweep, and B(x) = −(∇B)x during the second half, where ∇B > 0 is the magnetic field
gradient. The laser traveling along the +x (−x)-direction has circular polarization σ + (σ − ) during
the first half, and then these polarizations are exchanged for the second half.
The general desired coherent dynamics over a single cycle of the SWAP procedure is asfollows. Let us assume that the particle begins in |0i, which is a good assumption due to the effectsof spontaneous emission for appropriate system parameters. The particle resonates with one of thelasers when the laser detuning δ(t) is equal in magnitude to the particle’s motional detuningδ m ≡ bx + kv,(5.2)

which is the sum of its Doppler and Zeeman shifts, as shown in Fig. 5.2(b). Here, k is the wavenumber of the transition and v is the velocity of the particle. More specifically, at the time t1 during
the first half of the sweep defined by δ(t1 ) = −|δ m |, the particle absorbs a photon from one of
the cooling lasers and is transferred into whichever excited state |ei ∈ {|+i , |−i} first comes into
resonance. Then, at the time t2 in the second half of the sweep defined by δ(t2 ) = |δ m |, the particle
emits a photon into the other laser by stimulated emission and is transferred coherently back to|0i [see Pe (t) in Fig. 5.2(b)] with the corresponding momentum shift. It is essential to our methodthat both the laser polarizations and magnetic field direction are switched between the times t1
and t2 so that the particle resonates with the correct laser as to achieve stimulated emission and80
the consequent second momentum recoil. This second stimulated process in a sweep is exactly
what differentiates our method from other SWAP MOT protocols [137, 171]. Very importantly,
this protocol replaces the scattering event required after every absorption in Doppler cooling witha stimulated emission, mitigating the adverse effects of momentum diffusion that would otherwise
occur. This feature allows the particle to experience the impulse of many photon momenta while
avoiding spontaneous emission.
It is necessary to demonstrate that this protocol inherently generates a force toward thecenter of the trap (trapping), a force that opposes the particle’s motion (slowing), and an overall
frictional force (cooling). In order to illustrate its trapping capability, consider a motionless particle
with position x > 0 [see Fig. 5.3(a)]. The Zeeman shift causes the particle to absorb a σ − photon
from the left-traveling laser during the first half of the cycle and to emit a photon into the righttraveling σ − laser during the second half, transferring the particle back to |0i with a net impulseof two photon momenta toward the center of the trap. Its slowing capability is elucidated byconsidering a particle with momentum p > 0 near the center of the trap [see Fig. 5.3(b)]. In
this case, the Doppler shift causes the particle to absorb a σ − photon from the left-traveling laser
during the first half and to emit a photon into the right-traveling σ − laser during the second half,transferring the particle back to |0i also with a net impulse of two photon momenta that opposes
the particle’s motion. Examples with negative x or p similarly would cause trapping and slowing via
interaction with the σ + laser. The cooling effect is more subtle. To be precise, we define coolingas compression of the particle’s classical phase space volume, which is achieved through irreversible
entropy flow from the particle to free space via the process of spontaneous emission. This mostfrequently occurs when the Doppler and Zeeman shifts are equal in magnitude but opposite in sign,
as the particle resonates with both lasers simultaneously and is left with a significant excited state
fraction at the end of a sweep. Spontaneous emission then resets the particle to the ground state
for the next sweep, yielding a net drift in phase space toward the phase space origin. Although
the discussion so far has been descriptive, we we will elucidate the details of the full dynamics in
Section 5.6 through the numerical solutions to follow.

81

Figure 5.3: A qualitative demonstration of the trapping force (a) and slowing force (b) in a SWAP
MOT. The cooling lasers (with circular polarizations σ ± ) and sign of the magnetic field B(x) setup
are included. (a) Top: The energy eigenvalues E(x) of Ĥ(t) with Ω = 0 of a motionless particle(p = 0) as a function of position x during the first half of the sweep. An example system (pink circle)
with x > 0 in the state |0i absorbs a σ − photon, yielding an impulse toward x = 0 and transferring
the particle into the |−i state. This occurs at the time t1 when the swept laser frequency satisfies
δ(t1 ) = −bx. Bottom: E(x) vs. x and laser and magnetic field setup during the second half ofthe sweep. The example system emits a σ − photon, yielding another impulse toward x = 0 andtransferring the particle back to the |0i state. This occurs at the time t2 > t1 when the swept laserp2
frequency satisfies δ(t2 ) = bx. (b) Top: The relative energy eigenvalues ∆E(p) ≡ E(p) − 2mof ofĤ(t) with Ω = 0 for the states W (p) [see Eqn. (5.11)] for a particle at the center of the trap (x = 0)
as a function of momentum p in the first half (top) and second half (bottom) of the sweep. An
example system experiences an impulse toward p = 0 by undergoing similar dynamics to Fig. 5.3(a)with the substitutions x → p and bx → kv.

5.3

Model and semiclassical approximations
In order to capture the intricate features of the SWAP procedure, we first develop a fully

quantum mechanical model for the particle and then make appropriate semiclassical approximations
to create a computationally tractable simulation. We refer the reader to Sections 2.1 and 2.2 forgeneral details on modeling light-matter interactions. As previously described, Fig. 5.2(a) displays

82
the internal state structure. We track motion along one dimension, for which the particle has
momentum and position operators p̂ and x̂.
The system is subject to counter-propagating lasers of opposite circular polarization (which
we denote as σ + and σ − ) of instantaneous frequency ωL (t) and a magnetic field that depends
linearly on the coordinate as B̂(x̂) = (∇B)x̂. The detunings of the lasers δ(t) [see Eq. (5.1)] are
set to follow a sawtooth waveform pattern centered at zero with range ∆s and period Ts . In the
Schrödinger picture, the coherent dynamics is described by the HamiltonianĤ(t) =

p̂2
+ ~ω0 (|+i h+| + |−i h−|) + ~bx̂ (|+i h+| − |−i h−|)2m
 ~Ω 
~Ω  i[kx̂−η(t)] ++e
σ̂+ + h.c. +e−i[kx̂+η(t)] σ̂−+ h.c.
+
2
2

(5.3)

during the first half of the sweep, with similar form but substitutions k → −k and b → −b during
the second half of the sweep. The particle mass is m, the wavenumber of the laser light (which
we approximate to be constant) is k, and the magnetic field gradient is characterized by b, which
was defined in Section 5.2. The (equal) Rabi frequencies of both of the counterpropagating lasersare Ω, andη(t) ≡Z t

ωL (t0 ) dt0

(5.4)

0

is the time-dependent accumulated phase of the laser field. The operators σ̂i+ ≡ |ii h0| and σ̂i− ≡(σ̂i+ )† = |0i hi| are the raising and lowing operators corresponding to transitions between the groundstate and the two excited states.
The quantum master equation
i
dρ̂
1 h
=
Ĥ, ρ̂ + L̂ρ̂
dt
i~

(5.5)

governs the time evolution of the density matrix, ρ̂, which fully describes the particle’s internal andexternal states. The Lindblad superoperator captures the incoherent dynamics due to spontaneousemission and its associated recoil. As discussed in Section 2.2.3, we approximate the continuous
dipole radiation pattern to produce discrete recoils of magnitudes −~k, 0, and ~k along the xdirection with probabilities of 15 : 53 : 15 , respectively [63, 135]. Although it is not absolutely

83
necessary to make the simplification that there are only three allowed impulses arising from thedirection of the photon emission, the resulting numerical implementation is more straightforward
due to the fact that one may take advantage of the presence of uncoupled momentum families.
This approximation yields a Lindblad superoperator of the formX 3
1 ikx̂ − + −ikx̂ 1 −ikx̂ − + ikx̂  + −−+L̂ρ̂ = γσ̂ ρ̂σ̂i + e σ̂i ρ̂σ̂i e
σ̂i ρ̂σ̂i e − σ̂i σ̂i , ρ̂ .
+ e
5 i
5
5

(5.6)

i∈{+,−}In the interaction picture defined by the free HamiltonianĤ0 (t) = ~ωL (t) (|+i h+| + |−i h−|) ,(5.7)

the interaction Hamiltonian isĤI (t) =p̂2− ~δ(t) (|+i h+| + |−i h−|) + ~bx̂ (|+i h+| − |−i h−|)2m
 ~Ω 
~Ω  ikx̂ +++e σ̂+ + h.c. +e−ikx̂ σ̂−+ h.c. .
2
2

(5.8)

The corresponding force operator that describes the coherent evolution of the external variables isthenF̂ ≡ −

 i~Ωk ∂ ĤIi~Ωk  ikx̂ ++e σ̂+ − h.c. +e−ikx̂ σ̂−= ~b (|−i h−| − |+i h+|) −− h.c. .
∂ x̂
2
2

(5.9)

We now make several approximations to form a tractable, semiclassical model. By the term
“semiclassical,” we refer to reducing the number of motional degrees of freedom of the particle thatneed to be tracked, while the laser field is treated classically. The parameter that characterizes theseparation of timescales required to treat the motion classically is [53]
χ≡

ωr
 1,
γ

(5.10)

where ωr ≡ ~k 2 /2m is the recoil frequency of the transition (see Section 2.3). However, the SWAP
procedure causes fewer spontaneous emissions as χ increases [13], so Eq. (5.10) may not necessarily
hold. Moreover, the rapid adiabatic passage protocol generates coherences between quantum states,
so tracking a single value for the semiclassical momentum p is insufficient. Instead, we simplify our

84formalism by limiting the number of quantum states we track at any given time. We retain theterm “semiclassical” for this simplification and our consequent treatment of the particle’s externalvariables, and we consider its validity in Section 5.5.
The σ + − σ − polarization scheme prevents multiphoton processes by angular momentum
conservation [53], so a particle in an eigenstate |e0(p)i ≡ |0, pi can only be transferred to the statesee|−(p)i≡ |−, p − ~ki and |+(p)i≡ |+, p + ~ki (and vice versa) during the first half of the sweep.
The same statement holds true for the second half of the sweep with k → −k. Therefore, we only
track the particle’s evolution in the subset of basis statese , |−i}eW (p) ≡ {|e0i , |+i(5.11)of the composite Hilbert space, then update p appropriately as the particle absorbs and emits
photons.
Next, we treat the particle’s external variables in the Hamiltonian ĤI (t) semiclassically bymaking the substitutions p̂ → p and x̂ → x, where p is a momentum eigenvalue satisfying p̂ |pi =p |pi [which also parameterizes W (p)], and x is a c-number that characterizes the particle’s spatial
position. These classical phase space coordinates will be updated according to a procedure that wedevelop in Section 5.4. With the state |e0i defining zero energy, the semiclassical Hamiltonian thatevolves W (p) ise h+|e − ~(δ(t) + δ m ) |−i
e h−|
e +
Ĥs (t) = −~(δ(t) − δ m ) |+i~Ω  xσ̂ e + σ̂ xe ,+−2(5.12)

where σ̂ex ≡ σ̂e+ + σ̂e− . Note that the motional detuning δ m determines the condition for resonanceiiiin Eq. (5.12) as previously discussed. Additionally, we have omitted recoil energy terms ~ωr under
the approximation that they are small in comparison to ~δ m .
The coherent evolution of x and p is typically accounted for via Newton’s laws of motion
under the classical force


D
E
2
2
F (t) = F̂ (t) = ~b c−e (t) − c+e (t)

 h
i
h
i
+
+ ~Ωk Re ihσ̂ +
(t)i
−
Re
ihσ̂
(t)i
,
e
e
−

+

(5.13)

85
where cei (t)2≡ hei|Ψ(t)i2

are the instantaneous populations of state |eii determined from theparticle’s wave function |Ψ(t)i evolved under Ĥs (t). As opposed to other semiclassical treatmentsof laser cooling [126], the effects of F (t) on the particle momentum are intrinsically incorporated
into our model due to the p-dependence of the quantum states in W (p). Nevertheless, we provide
this force analysis as a consistency check. As we will motivate in Section 5.6, we make the physicallyreasonable assumption that the coherent force due to photon absorption and emission is much larger
than the coherent Zeeman force that arises from the gradient of the magnetic field, i.e., b  Ωk,which reduces Eq. (5.13) to
 h
i
h
i
+
(t)i
.
F (t) ≈ ~Ωk Re ihσ̂ +
(t)i
−
Re
ihσ̂
e
e
−

+

(5.14)In this limit, the impulse experienced by the particle at any time in the first half of the sweep isZ t∆p(t) =t0

F (t0 ) dt0 ≈ ~k [∆P+ (t0 , t) − ∆P− (t0 , t)] ,(5.15)

e between thewhere ∆P± (t0 , t) ≡ P± (t) − P± (t0 ) are the changes in population of the states |±itimes t0 and t. Eq. (5.15) agrees with the result obtained from calculating∆p(t) = hΨ(t)|p̂|Ψ(t)i − hΨ(t0 )|p̂|Ψ(t0 )i .

(5.16)

Eqns. (5.12) and (5.15) uniquely determine the coherent evolution of a particle during thefirst half of a sweep. The corresponding equations for the second half of the sweep are found by
the substitutions k → −k and b → −b. The incoherent dynamics are described by the Lindblad
superoperator Lρ̂ in Eq. (5.6) with the appropriate substitutions.

5.4

Variant of quantum trajectories
We are now in a position to choose a simulation method for determining the semiclassicalevolution of particles in the SWAP MOT. In this section, we develop a variant of the Monte Carloquantum trajectories method in the time domain [63, 135]. The traditional Monte Carlo wave
function method is outlined in Section 2.2.4. This choice is motivated by this method’s ability to86create a time record for the emission of spontaneous photons by individual particles, i.e., a quantum
jump. We will use this knowledge about the jump time to simulate the motional dynamics as
accurately and efficiently as possible within our semiclassical framework.
In the quantum trajectory method, the density matrix ρ̂(t) is obtained by averaging theassociated pure states of many “quantum trajectories,” each described by a state vector |ψi:N

1 X
ρ̂(t) ≈
|ψi (t)i hψi (t)| ,N(5.17)

i=1

where N  1 is the number of simulated trajectories. The state vectors are continuously evolved
under a non-Hermitian Hamiltonian defined asĤeff ≡ Ĥ −i~ X †Ĵi Ĵi2(5.18)iin addition to discrete quantum jumps that correspond to the various decay channels in the system[63, 135]. As we present below, our method retains these features and introduces fictitious quantum
jumps of |ψi into one of the elements of W (p), directly followed by an update of the coordinates x
and p based on the results of the projection. These additional operations allow us to calculate aneffective, coarse-grained, single phase space trajectory (x(t), p(t)) for each quantum trajectory.
In the case of distinguishable spontaneous emission processes, the jump operators Ĵi take theform Ĵi =√γ σ̂i− . In our model, the spontaneous photons due to the decay of the the excited states|±i to the ground state |0i are distinguishable in principle by their polarization, so our effective
semiclassical Hamiltonian isĤeff (t) ≡ Ĥs (t) −


i~γ
e h+|
e + |−i
e h−|
e .
|+i
2

(5.19)The time evolution protocol for each quantum trajectory is as follows. Note that k > 0(k < 0) in the first (second) half of the sweep.
(1) Initialize the classical phase space coordinate (x, p) by sampling from a chosen distribution.
Using this choice of p, prepare the quantum state in a state vector that lies in the basis
W (p). Record the quantum state as |ψr i, the time as tr , and the initial phase spacecoordinate as (xr , pr ).

87(2) Continuously evolve |ψ(t)i under Ĥeff (t) [Eq. (5.19)] with the external coordinate, (xr , pr ),held constant. This evolution can be interrupted at at a time tj when one of two possible
discrete quantum jump events occur:
Case A The middle or end of a sweep is reached.
(i) Simulate a fictitious quantum jump of |ψ(tj )i into one of the basis states inthe subspace W (pr ) using populations to determine the relative likelihood, i.e.,|ψ(t)i → |ψP i ∈ W (pr ).
(ii) Calculate the corresponding impulse ∆p and displacement ∆x according to∆p = hψP |p̂|ψP i − hψr |p̂|ψr i ∈ {0, ~k, −~k};Z
1
1 pavg
1
∆p
0
0
∆x =
p(t ) dt ≈
∆t =
pr +
(tj − tr )
m
m 2
m
2

(5.20)

so that the new phase space coordinate is(x, p) = (xr + ∆x, pr + ∆p).

(5.21)(iii) Relabel the basis subset W (pr ) with the updated momentum p.
(iv) Replace |ψr i , tr and (xr , pr ) with their updated values |ψP i, tj , and (x, p).
Case B A spontaneous photon is detected according to the quantum trajectory procedure [63, 135].
e , |−i}e the system occupied and hence decayed(i) Simulate which state |eei ∈ {|+ifrom using probabilities weighted by the excited state populations, i.e., project|ψ(tj )i → |ψP i = |eei.
(ii) Apply ∆p and ∆x according to Eqns. (5.20) and update (x, p) to correspond to
this choice.
(iii) Simulate the spontaneous emission process further by updating p to account forthe momentum recoil by selecting one of the three possibilities in the discretely
represented dipole radiation pattern. Assign |ψ(tj )i = |e0i.

88(iv) Relabel the basis subset W (pr ) with the updated momentum p.
(v) Replace |ψr i , tr and (xr , pr ) with their updated values |e0i, tj , and (x, p).
(3) Repeat as needed from step 2 until a desired final time is reached.
We choose to perform the operations described in Case A at the middle and end of each sweep
because the particle ideally undergoes one adiabatic transfer per half sweep by the design of the
SWAP procedure, and hence experiences the impulse of a single photon momentum.
As a framework for comparison and consistency, we have developed an analogous model for a
traditional 1D MOT, which we use in Section 5.6. In that case, the phase space variables are only
updated whenever a spontaneous event occurs, as this is typically the only mechanism whereby
the system can traverse momentum space. The resulting cloud size, temperature, and spontaneous
photon record are in close agreement with other models [126]. Moreover, the SWAP MOT algorithm
we present agrees well with a fully quantum σ + −σ − SWAP cooling model with no magnetic trapping(as opposed to the configuration previously developed in [13, 141] with linearly polarized lasers),
which can easily be implemented computationally due to its translational invariance. Further
investigation of the validity of this three-state model is presented in the following section.

5.5

Validity of three-state basis in SWAP MOT simulation
Here we justify the reduction of the subset to three basis states. Starting the particle in thestate |ψ0 i = |0, pi, we evolve it according to ĤI (t) [Eq. (5.8) with x̂ → x = const for simplicity]
over the first half of a sweep in the 7-state basis
W7 ≡ {|0, pi , |0, p ± 2~ki , |+, p ± ~ki , |−, p ± ~ki}.

(5.22)For our choice of |ψ0 i, these are the only states that can possibly become occupied in one sweep,
so it is equivalent to using the full basis set in [13]. At the middle of the sweep, the state is+|ψ1/2 i = c01/2 |0, pi + c−
1/2 |−, p − ~ki + c1/2 |+, p + ~ki ,(5.23)

89
where the ci1/2 ’s are the probability amplitudes of each state. Note that this is the same state wewould have calculated had we only tracked the 3-state basis
W3 ≡ {|0, pi , |+, p + ~ki , |−, p − ~ki}.

(5.24)

Next, we calculate the state at the end of the sweep in two ways. In one case (the “7-state
formalism”), we simply propagate |ψ1/2 i under ĤI (t) with b → −b and k → −k (i.e., switching
the magnetic field and laser polarizations according to our protocol) in W7 over the second halfof a sweep to yield the state |ψf7 i; this is the final state found by tracking the full 7-state basis
throughout the entire sweep. The other case (the “3-state formalism” used in Chapter 5) is more
complicated because we must project |ψ1/2 i into one of the states in W3 using probabilities weighted
by their populations, which means the final state of the particle will be a mixed state. Here, we
propagate three states0|ψ1/2
i = |0, pi ;−i = |−, p − ~ki ;|ψ1/2+
i = |+, p + ~ki|ψ1/2(5.25)

according to ĤI (t) with b → −b and k → −k in W7 over the second half of a sweep. Labeling theresults as |ψf0 i , |ψf− i, and |ψf+ i, the final state for the 3-state formalism is
+
+ 2
+
−
−
2
ρ̂3f = |c01/2 |2 |ψf0 i hψf0 | + |c−
1/2 | |ψf i hψf | + |c1/2 | |ψf i hψf | .

(5.26)The state ρ̂3f is what is calculated according to our simulation procedure, aside from the additional
complexity of updating x at the middle of the sweep.
We now compare the resulting final states ρ̂7f = |ψf7 i hψf7 | and ρ̂3f using two metrics. The firstis the fidelity of quantum states√σρ σ,(5.27)F (ρ̂3f , ρ̂7f ) = hψf7 |ρ̂3f |ψf7 i(5.28)F (σ, ρ) = Trq√which reduces tosince the final state in the 7-state formalism is pure. The other is a comparison of the final state
probability distributions Pij = hi|ρ̂jf |ii according to the chi-squared test. That is, we compare the

90
chi-squared value
2

χ =

7
X
(P 3 − P 7 )2
i

i=1

i

Pi7

(5.29)

to the number of degrees of freedom df = 6 to quantitatively compare the final probability distributions.
Fig. 5.4a displays F and χ2 for a region of classical phase space centered on the phase space
origin. We refer the reader to Fig. 5.5 and the cited equations therein to gain intuition about the
expected dynamics of particles in phase space. The fidelity F of the states ρ̂3f and ρ̂7f (Fig. 7.3)
generally increases with |δ m |, which implies that the 3-state formalism becomes more accurate indetermining both the population and coherences of the particle’s quantum state when the stimulated
absorption and emission processes are time-resolved. This is expected because the particle becomes
distributed over more quantum states when the stimulated processes overlap, which is not capturedby the 3-state formalism.
On the other hand, we find that the chi-squared values χ2 are much smaller than the numberof degrees of freedom df = 6 in the entire region of phase space (Fig. 5.4b). This suggests thatthe final population distributions Pi3 and Pi7 are nearly identical. For example, at the phase spacecoordinate (kx, p/~k) = (2750, 20), which has a relatively high χ2 value of 6 · 10−9 , we find that Pi3
and Pi7 are identical to four decimal places. These results imply that the 3-state formalism very
accurately captures the final population distribution of a particle over the states W7 over one sweep,
which is promising because these populations directly determine the momentum (and therefore theposition) distribution in the SWAP MOT. We expect that this result extends to more than one
sweep.

5.6

Phase space dynamics
Now that we have developed and motivated a detailed semiclassical model, we are in a position

to simulate and characterize the dynamics in a SWAP MOT. In this section, we first provide generalinsight for appropriate choices of the many experimental parameters. Then, as a relevant physical

91
40

40

0.8
20
0.7
0.6
0

0.5
0.4

- 20

0.3

6. × 10-9
momentum (units of hk)

momentum (units of hk )0.9
20

5. × 10-9
4. × 10-90

3. × 10-9-20

2. × 10-9

0.2

- 40

0.1

- 4000 - 2000

0

2000

4000

1. × 10-9-40-4000 -2000

position (units of 1 /k)(a) The fidelity F of the final states ρ̂7f and ρ̂3f ,which is bounded according to 0 ≤ F ≤ 1.

0

2000

4000

position (units of 1/k)(b) The chi-squared value χ2 between the final probability distributions Pij .

Figure 5.4: Phase space contour plots of various metrics that compare the final states of the 3-state
and 7-state formalisms. The relative simulation parameters are identical to those used in Fig. 5.5
except γ → 0.

example, we simulate the dynamics of the SWAP MOT procedure as applied to the molecule
yttrium monoxide (YO) using parameters for one of its narrow linewidth transitions. We use thesimulation results to further elucidate the dynamics in various regions of phase space, which areseparated by diagonal lines with slopes determined by the motional detuning (see Fig. 5.5):δ m = bx + kv = M  + 2βωr ⇒

dβ
M
=−
.
d
2ωr

(5.30)

Here, M ≡ b/k is the frequency that characterizes the magnetic field gradient, and  ≡ kx andβ ≡ p/~k are the position and momentum expressed in the appropriate dimensionless units corresponding to the optical transition. Finally, we compare the results of the trap loading process andphase space compression to those in a traditional MOT.

5.6.1

Choosing appropriate experimental parameters

The essential constraints for the experimental parameters arise from the requirement of
rapid adiabatic passage, which forms the foundation of the SWAP method. Landau and Zener
proved [194] that population is efficiently transferred between two stable quantum states as long92
as the laser frequency is linearly swept over the energy splitting of the states with the following
condition satisfied:
κ≡

Ω2
> 1,
α

(5.31)

where we define κ as the adiabaticity parameter and α ≡ ∆s /Ts is the slope of the frequency
ramp [194]. Experimental parameters that satisfy Eq. (5.31) are said to operate within the adiabatic
regime.
Additionally, it can be shown [185] that the transfer process takes a time τj = 2Ω/α in the
adiabatic regime. Because it is necessary for the system to remain in the excited state without
spontaneously decaying for at least a time τj , it must satisfy τj  1/γ. Combining this result with
Eq. (5.31), we find that the Rabi frequency must satisfyΩ  γ.

(5.32)

This is exactly what is meant by adiabatic rapid passage [2].
We now discuss the choice of the sweep range ∆s . In Section 5.2, we described how a particle
can experience a coherent force and a consequent impulse of 2~k per sweep if it begins in the internal
ground state |0i. However, it will still experience an impulse that leads to cooling and trapping,albeit smaller in magnitude, even if it begins with some excited state population. By symmetry,
this impulse will aid in cooling and trapping a particle only if its wave function predicts that it ismore likely to begin a sweep in |0i. One way to guarantee this condition is to enforce an imbalancesuch that the particle spends more time outside the two photon resonances, allowing spontaneousemission to preferentially optically pump the system to |0i for the next sweep. The most obvious
way to do this is to introduce a waiting period in between each sawtooth ramp. However, because
it is often desirable to cool and trap as quickly as possible, we only consider situations with nowaiting period. In this case, we achieve a time imbalance whenever4|δ m | ≤ ∆s ,since the resonances are spaced in frequency by 2|δ m |.

(5.33)

93
Next, we consider restrictions on the sweep period Ts . In the sequence discussed in Section 5.2,
the particle spends a time τe ≈ 2|δ m |/α in one of the excited states in between the two resonances.
(This can be seen by writing the detuning during a single sweep as δ(t) = αt and noting that thelasers resonate with the particle when δ(t) = ±δ m .) Therefore, in order for the particle to maintaina low probability of spontaneous emission in between resonances, τe must satisfy
τe ≈2|δ m |
< 1/γ.
α⇒4|δ m | <

2∆s
.
γTs

(5.34)The resetting to the internal ground state enforced by the time imbalance is only useful if theparticle does not have a significant chance of decaying between resonances during the next sweep.
So, applying Eq. (5.33) and Eq. (5.34), a reasonable restriction on the sweep period Ts is given byγTs < 2 ,(5.35)in order to obtain fast results. It has been shown experimentally that a cooling force twice as large
as the radiation pressure force was achieved in a 163 Dy system with the choice γTs ≈ 0.1 [146].
However, some experiments may require the much stricter condition of minimal spontaneous events.
In these cases, one should instead come close to saturating the bound in Eq. (5.35) so that there
is substantial time to reset to the ground state via spontaneous emission in the event of incorrecttime-ordering of absorption and emission.
We now derive a condition for the choice of magnetic field gradient. As will be explained in
Section 5.6.3.1, there is a specific region of phase space, which we call the “ideal region,” whereparticles generally undergo a change in momentum of 2~k per sweep via the coherent dynamics
discussed in Section 5.2. Here, it is a good approximation to expand the momentum to first orderin time, i.e.,β(τ ) ≈ β0 +dβ
τ ≈ β0 − 2sgn(δ0m )τ.
dτ(5.36)In Eq. (5.36), τ ≡ t/Ts labels the number of sweeps, and we have made the approximation that the
momentum transfer ∆β = 2 is uniformly distributed over a sweep instead of localized near photon
resonances. Note that the sign function arises because the sign of δ m determines which laser the

94system interacts with first, and therefore the direction of the impulse. This family of trajectories
has an instantaneous slope of magnitude1dβ1≤(τ ) =,(5.37)d
ωr Ts |β0 − 2sgn(δ0m )τ |
ωr Ts |β0 |RRwhich was derived by applying the integral relationship x = v dt, equivalent to  = 2ωr Ts β dτ .
Any trajectory with a slope whose magnitude is smaller than Eq. (5.30) has a substantial likelihood of moving out of the ideal region and into an undesirable “heating region,” described in
Section 5.6.3.3. Therefore, it is reasonable to bound the slope of our trajectories, as given in
Eq. (5.37), from below by the the slope of the diagonal lines that define different phase space
regions [Eq. (5.30)], which leads to the condition
M Ts ≤ 2/|β0 |.

(5.38)

Because the minimum achievable temperature in SWAP cooling is the recoil temperature [13], it issafe to assume that the momenta of most of the particles satisfy |β0 | > 1.
As mentioned earlier, it is necessary to choose a sweep period Ts that comes close to saturating
Eq. (5.35) when it is a priority to minimize spontaneous events. Combining Eqns. (5.35) and (5.38)
in this limit yields the restriction
M ≤ γ.

(5.39)

Together, Eqns. (5.32) and (5.39) motivate our omission of the coherent Zeeman force in Section 5.3.

5.6.2

Cooling and trapping of the molecule YO

Figure 5.5 displays simulated phase space trajectories for a cloud of Yttrium Monoxide (YO)particles being cooled and trapped in a SWAP MOT using its 5.9 kHz linewidth X 2 Σ+ → A0 2 ∆3/2
transition [60]. Motivated by recent experimental progress [60], the initial particle distribution wascharacterized by Gaussians with standard deviations σ0 = 9.1·103 and σβ0 = 11.5, and is displayed
as yellow target symbols “+” in Fig. 5.5. By comparing the initial and final particle distributions,the latter displayed as purple triangles, it is clear that phase space compression was achieved. We
quantify this result in Section 5.6.4.

95

80

(ii)

60

momentum β

40

(i)

20

(i)

0

−20

(ii)

−40
−60
−80

−2

−1

0

position ǫ

1

2×104

Figure 5.5: Phase space dynamics in a SWAP MOT. The variables β and  label a particle’s
momentum as a multiple of the number of photon momenta and number of inverse wavenumbers
from the center of the trap, respectively. The diagonal lines, which have slopes determined by
Eq. (5.30), separate phase space into regions that are labeled above the plot and characterized
by Eq. (5.42). Several sample simulated phase space trajectories are displayed as green lines.
The black arrows show the flow of time. Particles in the “ideal” region, labeled (i), are generallycooled and trapped, while those outside the “ideal” region, labeled (ii), are heated out of the
trap. Each of the particles’ initial momenta and positions (yellow targets) were sampled fromGaussian distributions with standard deviations σβ0 = 11.5 and σ0 = 9.1 · 103 , respectively. Values
are: ωr = 0.67γ, M = 0.05γ, Ω = 55γ, ∆s = 2000γ, Ts = 1.5/γ, and tmax = 375/γ. For thisset of parameters, the particles are slightly cooled and become significantly more trapped (final
coordinates are displayed as purple triangles).

The simulation parameters were chosen under the conditions derived in Section 5.6.1. Because
we wish to highlight the SWAP MOT’s ability to cool and trap with fewer scattered photons, thesweep period was chosen to satisfy γTs = 1.5, which is close to the bound in Eq. (5.35). The
magnetic field gradient was consequently chosen to be M = 0.05γ in order to yield significant
trapping with a capture range roughly corresponding to any particles with momenta smaller inmagnitude than 3σβ0 [see Eq. (5.38)]. These choices bounded the motional detuning of most96
particles to the interval|δ m | ≤ M σ0 + 2σβ0 ωr ≈ 470γ.

(5.40)

Therefore, the choice ∆s = 2000γ guaranteed that most particles underwent the dynamics describedin Section 5.2 [Eq. (5.33)] and did not frequently emit spontaneous photons in between the two
resonances [Eq. (5.34)]. Finally, to operate in the regime of adiabatic rapid passage, the Rabi
frequency was chosen to be Ω = 55γ [see Eqns. (5.31) and (5.32)].

5.6.3

Characteristic dynamics in different phase space regions

The diagonal lines in Fig. 5.5 divide phase space into regions with different characteristic
behavior, which we now discuss in the following sections.

5.6.3.1

Region of ideal coherent dynamics

Upon investigation of the coherent dynamics under Eq. (5.12) in the adiabatic limit, we candeduce that a particle initialized in the internal ground state |0i populates a single excited state
during a sweep only if |δ m | > |Ω|. (In our previous work, we referred to a similar condition as the“high-velocity regime” [13].) Otherwise, power broadening would bring both lasers into resonancewith the particle simultaneously, distributing population among all three states. Specifically, aparticle with δ m > |Ω| (δ m < −|Ω|) first absorbs a photon from the σ − (σ + ) laser. Then, after
the laser polarizations and the magnetic field are abruptly switched, the particle eventually emitsa photon into the other laser, which now has the correct polarization (see Fig. 5.3). Thus, theparticle experiences a negative (positive) impulse of magnitude 2~k after each sweep in the absenceof spontaneous emission.
When spontaneous emission is included, the particle must not spend too much in the excitedstate τe [see Eq. (5.34)] or else dissipation will occur. In order to retain mostly coherent dynamics,
we propose that τe must be smaller than the time log(2)/γ, as half of the particles will decay on
average before resonating with the other laser in this time. Combining these results, we expect

97
coherent dynamics to approximately describe any particles with motional detunings δ m that satisfy|Ω| < |δ m | < α log(2)/2γ.

(5.41)

We will call this the “ideal” region, as most particles that are initialized in this region are eventuallytrapped and cooled. As seen in Fig. 5.5, particles in the ideal region, which are labeled (i), follow
trajectories that generally exhibit an effective “attraction” to the lines δ m = ±|Ω|. It should be
noted that the black arrows display the direction of the flow of time during the simulations.

5.6.3.2

Dynamics near δ m = 0

In this regime, the Zeeman and Doppler shifts are comparable in magnitude but opposite insign. Thus, the resonances with each laser are no longer time resolved, resulting in population ofboth excited states and therefore frequent spontaneous emission events. Nevertheless, we observe an
average attraction of the trajectories to the line δ m = 0 in phase space. This attraction paired withballistic expansion causes the particle to eventually drift toward the phase space origin in a similarmanner to the dynamics within a traditional MOT. After many cycles, the system equilibrates
about the phase space origin, as discussed in Section 5.7.

5.6.3.3

Dynamics outside the ideal region

The condition |δ m | < ∆s /2 is necessary to ensure that the particle resonates with both lasers
at some point during the sweep. Obviously, particles outside of this regime will undergo purely
ballistic expansion, as seen by the trajectories labeled (ii) at later times in Fig. 5.5.
If a particle is in the region defined by α log(2)/2γ < |δ m | < ∆s /2, there is a significant chanceof spontaneous emission before resonating with the second laser in a sweep. If the spontaneousemission does occur, the particle will absorb a photon when it resonates with the second laser,
directing it even further from the ideal region. In this sense, the particle is heated and will eventuallyescape the trap, as evidenced by the trajectories labeled (ii) at earlier times in Fig. 5.5. The mostextreme case of heating occurs when |δ m | ≈ ∆s /2; here, the absorption-emission sequence becomes

98
centered at the extremes of the sawtooth wave instead of at the center, leading to coherent transferof the particle away from the phase space origin.
Assembling these considerations together, we arrive at the following complete picture of thephase space regions, labeled in Fig. 5.5:
|δ m | ≤ |Ω|α log(2)2γα log(2)∆s< |δ m | <2γ2∆s< |δ m |2|Ω| < |δ m | <(overlap, scattering)(ideal coherent dynamics)(heating)(ballistic expansion).

(5.42)

Figure 5.6: Semilog plots of the evolution of the phase space density metric G [see Eq. (5.44)]
for the YO system subject to the SWAP MOT and traditional MOT (Doppler). The used SWAP
MOT simulation parameters were identical to those in Fig. 5.5. The detunings of the lasers in
the traditional MOT simulation were set to δ = −Ω, and all other parameters were identical to
the SWAP MOT simulation. Quantities were averaged over all trajectories that become trapped
within the overlap region [see Eq. (5.42)] from Ntot = 2000 total trajectories. (a) G vs. time t
for all trapped particles. The SWAP MOT equilibrates more quickly than the traditional MOT.
Inset: The SWAP MOT captures a higher fraction f (t) of particles. (b) G vs. the average numberof spontaneous emissions per particle. Spontaneous emissions of non-trapped particles are notincluded. The SWAP MOT equilibrates with significantly fewer spontaneous emissions per trapped
particle.

In summary, particles that begin in the ideal region are transferred to the overlap region
with few scattering events. Once there, the particle frequently scatters photons and the system
eventually equilibrates. Any particles that begin outside the ideal region are most likely heated out

99of the trap.

5.6.4

Phase space compression

The ultimate signature of a laser cooling method’s ability to effectively trap and cool a systemis the ability to overcome the limits imposed by Liouville’s theorem for Hamiltonian evolution andexhibit compression in terms of occupied phase space volume. One useful metric that can be used
to quantify the particle density in phase space is the root-mean-square (rms) interparticle spacingin phase space, lrms . If we label the position of a particle in phase space with the dimensionlessvector r ≡ {, β}, then lrms is

v
uN
X
1u
lrms ≡ t|r i − r j |2 ,N

(5.43)

i>jwhere N is the number of trajectories considered out of the total number Ntot . For our purposes,
we only need to consider the trajectories that ultimately become trapped within the overlap region.
We define a metric G for the particle phase space density asG ≡ 1/lrms ,(5.44)

and we label the fraction of trapped particles with f = N/Ntot .
In order to gain intuition into the interpretation of G, let us briefly consider its form fora system with a phase space distribution F (r) that can be modeled as a product of Gaussian
distributions in  and β with standard deviations σ and σβ :
"
!#
N
1 2
β2
F (r) =.
exp −+2πσ σβ2 σ2 σβ2
The continuous form of Eq. (5.43) iss Z
1 1
lrms =
dr1 dr2 F (r 1 )F (r 2 )|r 1 − r 2 |2 ,N 2

(5.45)(5.46)

so using Eq. (5.45) in Eq. (5.46) results inq
G = 1/ σ2 + σβ2 .

(5.47)

100
Further, let us consider the case when the system is in the Bose-Einstein condensate (BEC) regime.
Here, the system would occupy a single volume element of phase space, which corresponds to∆x∆p/h = ∆∆β/2π ≈ 1. In the symmetric case ∆ = ∆β, we find that G is on the order ofunity. These results help with interpreting the resulting scale of the calculated values for G in the
SWAP MOT, where G will always be considerably below the quantum degenerate value.
Figures 5.6(a) and 5.6(b) show the evolution of G and f for the YO transition introduced in
Section 5.6.2 subject to both a traditional MOT and the SWAP MOT as a function of time alongwith the average number of spontaneous emissions per particle. Only particles that ultimately
settle within the overlap region [see Eq. (5.42)] are considered. The insets display the fraction f ofparticles that are currently within the overlap region. It is evident that both procedures compressed
phase space, but the SWAP MOT did so more than three times faster and with roughly a thirdof the spontaneous emissions required by the traditional MOT for this set of parameters. This
result shows that the SWAP MOT can apply higher dissipative forces and remove entropy with ahigher photon efficiency, i.e., it removes more energy and momentum per scattered photon than a
traditional MOT.

5.6.5

MOT loading

The previous subsection showcases the SWAP MOT’s ability to efficiently trap and cool. Here,
we demonstrate its ability to coherently translate high momentum states toward zero momentum,
which is desirable in, e.g., the loading of molecules from an oven or supersonic nozzle for which a
low number of scattered photons is a priority.
Commonly utilized particle loading devices, such as the Zeeman slower, rely on radiationpressure to reduce the particle speed and yield a force that is fundamentally limited by the linewidthof the transition [126]. Consequently, such a process requires roughly one spontaneous photon perabsorbed slowing photon. In contrast, the SWAP protocol is fundamentally limited by the rate at
which one can can stimulate rapid adiabatic passage. Hence, operating in the regime Ω  γ yields
significantly higher coherent forces and therefore smaller slowing distances with fewer scattering

101

SWAP

200

150

100

50

0

average momentum βavg

momentum β

Doppler

−10

200
150
Doppler

100

SWAP

50
0

80 160 240 320

scattered photons per particle

−8

−6

−4

position ǫ−20 ×104

Figure 5.7: Simulated phase space trajectories of particles being loaded into a SWAP MOT andtraditional MOT (Doppler). The trajectories generally flow toward the phase space origin, which islabeled by a purple circle. The particles subject to the SWAP MOT experience much higher forcescompared to the radiation pressure force intrinsic to the traditional MOT. Common parameters
are: ωr = 0.67γ, Ω = 60γ, β0 = 200, σ0 = 1000, σβ0 = 10. SWAP MOT parameters are: M =0.012γ, ∆s = 1600γ, Ts = 1/γ, tmax = 175/γ, 0 = −2 · 104 . Traditional MOT parameters are:M = 0.0025γ, δ = −60γ, tmax = 1000/γ, 0 = −105 . Inset: The average momentum number βavg
vs. the average number of spontaneous emissions per particle. The SWAP MOT is loaded withfewer spontaneous photons and in a shorter distance.

events.
Figure 5.7 shows the results of simulating the slowing of YO on the 5.9 kHz transition
using both a traditional MOT and a SWAP MOT. We chose the Rabi frequency Ω = 60γ forboth procedures to allow for high coherent forces in the SWAP MOT and rapid absorption in
the traditional MOT. The laser detunings in the traditional MOT were set to δ = −Ω so thatthe particles were close to resonance with the counterpropagating laser at the beginning of thesimulation. The SWAP magnetic field gradient M = 0.012γ was chosen so that the trajectories layon the border between the overlap and ideal regions for the entire slowing process, which minimized
the amount of time the particles spent in the excited states while keeping the resonances sufficiently

102
separated. It is evident that this procedure applied forces to the particles in the SWAP MOT thatwere roughly five times larger than the traditional MOT and consequently slowed them in roughly
one-fifth of the distance. As shown in the inset of Fig. 5.7, the SWAP procedure produced roughlya fifth of the spontaneous photons required by the traditional MOT in order to reach the region of
the phase space origin for this set of parameters. Therefore, the SWAP MOT may potentially bea strong candidate for molecular slowing, as it produces high coherent forces with few spontaneous
photons, which may thereby reduce the adverse effect of optical pumping into dark states.

5.7

Equilibrium temperatures and cloud sizes
The simulated phase space diagram shown in Fig. 5.5 and the resulting phase space densities

shown in Fig. 5.6 suggest that particles captured in a SWAP MOT eventually equilibrate to a final,
fixed phase space distribution. In this section, we describe the associated equilibrium momentum
and position distributions by analyzing their dependence on both laser intensity and the ratio ωr /γ,
where the latter quantity characterizes the separation between the external and internal timescales.
The equilibrium momentum and position probability distributions P (β) and P () from a YO
SWAP MOT simulation are displayed in Figs. 5.8(e) and 5.8(f). The parameters were identical tothose in Fig. 5.5, except Ω = 60γ here. Both distributions can approximately be described by anormalized Gaussian function with zero mean, as shown by the fits displayed as blue curves. In
this way, we can consider the measured width to the momentum distribution as approximately thestandard deviation of a thermal 1D Maxwellian distribution. From the equipartition theorem,
p2rms
1
2
= ~ωr βrms
= kB T,2m
2(5.48)

thereby defining an effective temperature T . We find this type of fit to be much more accurate
for the momentum distribution than is observed for the position distribution. Nevertheless, thepoints displayed as purple circles in Figs. 5.8(a-d) are extracted by fitting each resulting probability
distribution to a Gaussian with zero mean.
Figures 5.8(a) and 5.8(c) display the equilibrium root-mean-square momenta βrms and po-

103sitions rms for various values of Ω/γ using the YO transition introduced in Section 5.6.2. Theseplots provide information about the effects of the laser intensity I on the equilibrium phase space√
distribution, since Ω/γ ∝ I [126]. One reason these plots are of interest is because the lowest
temperature in other cooling methods is typically achieved in the limit of low saturation, i.e., Ω → 0(although in this limit the cooling may take a long time and the trapping effect is reduced), which
is not usually experimentally reasonable due to the demands of adiabaticity [Eq. (5.31)] and rapid
adiabatic passage [Eq. (5.32)]. The vertical, black dashed line separates the plots into diabatic(left) and adiabatic (right) regimes. We observe that the minimum temperature and cloud size isachieved slightly within the adiabatic region, where particles are efficiently transferred into resonant states with minimal negative effects from power broadening. A similar result for the minimum
temperature was previously discovered for SWAP cooling [13, 141].
Figures 5.8(b) and 5.8(d) display the equilibrium root-mean-square momenta βrms and positions rms for various values of ωr /γ when Ω = 60γ. This ratio is a fundamental property of thecooling transition, so this investigation transcends our focus on the specific YO transition used
elsewhere in this work and provides a better understanding of the SWAP protocol’s efficacy on
an arbitrary transition. As seen in Fig. 5.8(b), βrms decreases as ωr /γ increases, which provides
direct evidence for our claim that SWAP procedures applied to narrow linewidth transitions resultin lower temperatures T . On the other hand, we observe an increase in cloud size in Fig. 5.8(d).
In the deeply adiabatic regime, we observe that the root-mean-square values scale asrβrms ∝Ω;
ωrrms ∝ Ω.

(5.49)

Numerical fits according to the equationrβrms =Ω
Cωr(5.50)are displayed as green dot-dashed curves in Figs. 5.8(a) and 5.8(b). Figure 5.8(c) is numerically fit
to the equation
rms = CΩ.

(5.51)

104

Figure 5.8: (a-d): Equilibrium root mean square momenta βrms and positions rms as a functionof Rabi frequency Ω [(a) and (c)] and recoil frequency ωr [(b) and (d)], both in units of thelinewidth γ. The vertical, dotted black lines in subplots (a) and (c) display the Rabi frequency
for which the adiabaticity parameter κ = 1 [see Eq. (5.31)]. Numerical fits to plots (a-c), which
are displayed as green dot-dashed curves, are discussed in the text. Common parameters are:M = 0.05γ, ∆s = 2000γ, and Ts = 2/γ. The recoil frequency in plots (a), (c), (e) and (f) isωr = 0.67γ, and the Rabi frequency in plots (b), (d), (e) and (f) is Ω = 60γ. Each point (purple
circles) was averaged over data taken from 800 independent trajectories, each being sampled once
per sweep over 1,500 sweeps. (e-f): Equilibrium momentum and position probability distributions
P (β) and P (). Gaussian fits (blue curves), which were used to extract root-mean-square values,
are discussed further in the text. Histogram data was taken from 800 independent trajectories, eachbeing sampled once per sweep over 1,500 sweeps. In (f), the black curve is the effective potential
Veff () [Eq. (5.59)] that generates the non-Gaussian position distribution [Eq. (5.54), red curve].

The scaling constants C for each independent fit are displayed within each plot. Combining
Eq. (5.48) with the observed trend in Eq. (5.49), we find thatkB T ∝ ~Ω,(5.52)

105
which agrees with previous studies of SWAP cooling (see Ref. [13] and Chapter 3).
In a traditional MOT, the equilibrium temperature Ttrad scales with the linewidth γ [126]:
kB Ttrad ∝ ~γ.

(5.53)

providing this is larger than the recoil temperature. Comparing this to Eq. (5.52), we find that thetemperature reached in a SWAP MOT must be larger since Ω  γ. As was discussed in the original
SWAP cooling papers [13, 141], we again reach the conclusion that the utility of the SWAP MOT
is not its low temperatures, but its ability to apply larger forces and reach equilibrium faster andwith fewer scattered photons. This is especially applicable to systems with narrow line transitions,
i.e., small γ.

5.7.1

SWAP MOT effective potential

As shown by the red curve in Fig. 5.8(f), the position distribution is actually more accurately
described by the sum of two Gaussians of different widths:




w
x2
x2
(1 − w)
X(x) ≈ √exp − 2 + √exp − 2 ,
2σ1
2σ2
2πσ1
2πσ2

(5.54)

where 0 ≤ w ≤ 1 describes the relative weight of the two Gaussians, and σ1 and σ2 are the widths ofthe two distributions. Note that we have normalized the distribution for simplicity. This suggeststhat the effective potential Veff (x) describing both the conservative and dissipative effects of the
SWAP MOT has an exaggerated dip near x = 0, similar to that of a dimple trap [188]. Here wederive an effective potential Veff (x) from the resulting equilibrium position distribution, which we
include as a black curve in Fig. 5.8(f).
In a state of thermodynamic equilibrium, the classical phase space distribution ρ of a system
is given by [168]ρ(x, p) = Z −1 e−H(x,p)/kB T ,(5.55)

where Z is the partition function and H is the system Hamiltonian. If we consider a system witha Hamiltonian of the form
H(x, p) =p2+ Veff (x),2m

(5.56)

106
where Veff (x) is an effective potential, then the position distribution X(x) satisfies
X(x) ≡Z

ρ(x, p) dp ∝ e−Veff (x)/kB T .

(5.57)

Equivalently,
Veff (x) ∝ − log X(x).

(5.58)In the case of the SWAP MOT, we find that




w
(1 − w)x2
x2
Veff (x) ∝ − log √exp − 2 + √exp − 2
2σ1
2σ2
2πσ1
2πσ2


(5.59)

by using Eq (5.54). We plot this effective potential against the simulated position space distribution
arising in the SWAP MOT in Fig. 5.8(f).

5.8Conclusion
We have proposed a novel cooling, trapping, and slowing scheme for neutral particles, which

we have termed the SWAP MOT. We have outlined an experimental protocol and developed asimple and efficient semiclassical model and simulation that demonstrates its ability to generate
phase space compression. We have also specified a protocol for determining appropriate experimental parameters for application to an arbitrary transition. Our results demonstrate the SWAP
MOT’s ability to generate large coherent and dissipative forces with fewer scattered photons than
traditional MOTs. We provided a discussion of the resulting equilibrium 1D temperature and cloudsize scaling properties with Rabi and recoil frequencies and compared the results to those foundfrom our theoretical model of the traditional MOT.
Aside from eventual experimental implementation, there are several topics that could bestudied in future work. We have not explicitly simulated 3D dynamics, but we conjecture that
applying the SWAP MOT protocol to each dimension successively in time is a viable option [171].
Also, the incorporation of shortcuts to adiabaticity could speed up the protocol with a low energetic
cost [75]. We consider one such shortcut to adiabaticity approach in the next chapter. Moreover,there are a number of additions to the simulation that would be required to more accurately

107
capture the dynamics in the quantum regime (aside from developing a computationally efficientalgorithm to simulate the fully quantum model developed in this work), such as spontaneous photon
reabsorption, particle collisions, and system loss to states that are not resonant with the coolinglaser frequencies.
Furthermore, we identify that the stimulated emission process can be realized in our model
via any operation that reverses the sign of the magnetic dipole potential energyĤB (x̂) = −µ̂ · B̂(x̂)(5.60)

at the middle of the sweep, whereµ̂ = −

gJ µB ˆJ~

(5.61)

is the magnetic dipole moment of the particle and Jˆ is the total electronic angular momentum. We
incorporated the sign flip by magnetic field switching, but this is technically difficult in an experimental setting and it is fundamentally impossible to perfectly, diabatically switch the magnetic
field direction. Other potentially more experimentally viable options are applying π-pulses at the
middle of each sweep in order to transfer the population between the excited Zeeman sublevels,
thereby flipping the sign of Jˆ, or optically pumping into a nearby hyperfine manifold with a sping-factor of opposite sign. Alternatively, it is possible that a vertically-shifted, sinusoidal magnetic
field could also achieve similar dynamics.

Chapter 6

Speeding up particle slowing using shortcuts to adiabaticity

The cooling and trapping mechanisms discussed in Chapters 3-5 inherently rely on the dynamics to be adiabatic, and can therefore take a potentially unfeasible amount of time. In thischapter, we propose a method that relaxes the adiabatic condition by applying laser pulses that
achieve the same state transfer effects in a shorter amount of time. We focus on the simpler caseof particle slowing, i.e., reducing its kinetic energy but not necessarily its temperature, although
we anticipate that it can be applied within cooling and trapping protocols. This work resulted ina publication in Physical Review A (Ref. [14]) in October 2020.
More specifically, this method can potentially generate large forces without the associatedmomentum diffusion that results from the random directions of spontaneously scattered photons.
We consider time-resolved laser pulses with periodically modified detunings that address an ultranarrow electronic transition to reduce the particle momentum through repeated absorption andstimulated emission cycles. We implement a shortcut to adiabaticity approach that is based on
Lewis-Riesenfeld invariant theory. This affords our scheme the advantages of adiabatic transfer,where there can be an intrinsic insensitivity to the precise strength and detuning characteristicsof the applied field, with the advantages of rapid transfer that is necessary for obtaining a shortslowing distance. For typical parameters of a thermal oven source that generates a particle beamwith a central velocity on the order of meters per second, this could result in slowing the particles tonear stationary in less than a millimeter. We compare the slowing scheme to widely-implemented
slowing techniques that rely on radiation pressure forces and show the advantages that potentially109
arise when the excited state decay rate is small. Thus, this scheme is a particularly promising
candidate to slow narrow-linewidth systems that lack closed cycling transitions, such as occurs incertain molecules.

Figure 6.1: (a) Schematic of the normalized velocity distributions P (v) of the particle beam before(solid) and after (dashed) the slowing process. We characterize the initial distribution by a Gaussianfunction with a standard deviation σv and mean velocity v̄. (b) Particles exit an atom or moleculesource and are collimated into the slowing region. The spatial setup of the sequentially-pulsed
counterpropagating lasers, which have time-dependent frequencies ω1 (t) and ω2 (t), and a sample
particle with velocity v in the laboratory frame are displayed in the slowing region. The circularinset shows the two-level internal structure of each particle.

6.1

Introduction
For decades, laser cooling atoms and molecules to near absolute zero has been at the forefrontof research into the interactions of light and matter [41, 46, 123, 149]. Ultracold ensembles provide
testbeds for exploring fundamental physics [164], can create low-temperature superfluids such as
Bose-Einstein and fermionic condensates [9, 157], and can be used as platforms for quantum simulators [94]. Since atom and molecule sources typically produce particles with high velocities, it isusually necessary to first perform a precursor stage that removes a substantial fraction of the kinetic
energy associated with the random thermal motion, as shown in Figure 6.1(a). A specific example
is sodium atoms produced in an oven at thermal velocities of a few hundred meters per second
that need to be slowed to a few meters per second before being transferred into a magnetic and/or
optical trap [150, 160, 193]. An assortment of particle slowing methods have been developed for thisprecursor stage, including Stark and Zeeman decelerators [7, 81, 138, 148], centrifuge decelerators

110[40], electrostatic trapping methods [20], frequency-chirped laser slowing [181], white light slowing[49, 79], and angled slowing [112], among others. Once slowed, the particles may possess sufficiently
low kinetic energy that they can be efficiently loaded into a finite-depth electromagnetic trap [126],
such as a magneto-optical trap (MOT), and then be cooled using light [176, 183], cooled throughevaporation that redistributes energy through two-body collisions [177], or sympathetically cooledwith another species [64].
The ability to produce a large number of ultracold particles is made difficult by practicalshortcomings of slowing methods, such as a large slowing distance that requires significant physicalspace, or substantial spread in the final velocities of the particles. Furthermore, the main hindrance
to slowing particles that lack closed cycling transitions is that there may be leakage of population todark electronic states that are not coupled with the fields that perform the laser cooling and slowing,
resulting in the loss of the particle from the system. Even if this does not occur, spontaneousemission of many photons creates momentum diffusion due to the random emission direction, andthis results in heating and a finite limit on the achievable temperatures. These issues make methods
that increase slowing forces and minimize the number of scattering events through enhanced controlof tailored coherent dynamics, such as sawtooth-wave adiabatic passage (SWAP) cooling [12, 13,74, 137, 141, 171], the Allen-Eberly scheme [8], stimulated Raman adiabatic passage (STIRAP), theadiabatic passage force, and the bichromatic force [33, 125, 129, 172, 173, 187] enticing candidates
to consider for particle slowing. However, one concern is that in order to satisfy an intrinsic
adiabaticity condition, the time evolution should typically be slow and this could result in a longstopping time and associated large stopping distance.
In order to speed up adiabatic processes while still achieving substantial population transfer,
there has been a growing interest in so-called “shortcuts to adiabaticity” [75], i.e., processes thatare able to transform systems from an initial quantum state to the same final quantum state as an
ideal adiabatic process but in much less time. In this chapter, we present a fast, simple, and robust
particle slowing scheme that employs one such shortcut method: inverse engineering based on LewisRiesenfeld invariants (LRI) [37, 38, 39, 76, 103, 105]. The slowing scheme involves driving sped-up

111transitions from a stable ground state to an excited state and back using counter-propagating,pulsed lasers with intensity and detuning profiles prescribed by the LRI shortcut method, as shown
schematically in Figure 6.1(b). By applying this protocol many times, the particle can in principle
be subject to the impulse of many photon momenta without emitting spontaneous photons. In
Section 6.2, we further motivate the use of our slowing scheme. In Section 6.3, we outline the general
theory of LRI-based inverse engineering. In Section 6.4, we present our theoretical model and derive
the laser intensity and detuning profiles that result from the LRI shortcut method. In Section 6.5,
we study the resulting classical forces, slowing times, and slowing distances, demonstrating that
we are able to generate higher forces than what is achieved with radiation pressure as employed
in methods such as Zeeman slowing. In Section 6.6, we simulate particle slowing dynamics byapplying our method to various atomic and molecular species with narrow linewidth transitions.
In Section 6.5, we study our slowing scheme’s robustness to various systematic errors that one may
encounter when implementing the protocol in an experimental setting, and compare these results
to slowing with π-pulse transitions and SWAP slowing.

6.2

Motivation for the speed-up protocol
The concept of coherent transfer between quantum states is a ubiquitous component of mostquantum control techniques. The most basic way to achieve such a transfer between two electronic
quantum states is to apply a resonant, coherent light pulse for a time t = π/Ω, where Ω is the
Rabi frequency. While this Rabi-flopping “π-pulse” method completely transfers population, atleast in principle, it is not robust to small errors in Ω, the frequency ω of the light source, or
coupling to other states outside of the two-level manifold. A further detrimental effect arises when
applying this method to quantum particles with motion due to the associated Doppler shift kv of
the laser light experienced by the particle, where v is the velocity of the particle as measured in thelaboratory frame and k = 2π/λ is the wavenumber of the transition with wavelength λ. This shifts
the laser light away from resonance with the particle, which changes both the rate and maximum
amount of population transfer between the states. This issue can be remedied by tuning the laser

112
light back into resonance with the particle, but this is of limited utility when the goal is to transfer
population in many particles that possess a distribution of velocities.
A robust solution to these problems is to chirp the laser frequency through the resonance, as
was shown by Landau and Zener in their theory of adiabatic passage [194]. However, such a method
is limited in the sense that the laser chirping must be adiabatic, i.e., sufficiently slow compared tothe strength of the coupling Ω. More specifically, they showed that the population of state |2i, P2 ,after the laser is chirped from a detuning of minus infinity to positive infinity, isπ Ω2P2 = 1 − exp −
,
2 α

(6.1)

where α is the frequency chirping rate in rad/s2 . Thus, if the experiment is operated in the limitΩ2  α, then P2 ≈ 1, regardless of the particle’s initial velocity, a phenomenon known as adiabatic
passage. The requirement of infinite time can be relaxed if the laser frequency is adiabaticallychirped over a frequency range larger than 2Ω in the moving frame of every particle, as can be seen
in a dressed state picture. The velocity-independent feature of adiabatic passage motivated thedevelopment of SWAP cooling [12, 13, 74, 137, 141, 171], which utilizes periodic frequency chirps
to coherently transfer a particle between quantum states in such a way that its momentum canbe significantly reduced. A natural extension of SWAP cooling is to employ its ability to apply
large coherent forces in the process of particle slowing. However, there are several issues that arisein this venture. The obvious issue is the requirement of adiabaticity, which limits the timescale
for population transfer and can lead to undesirably large slowing distances and times. Anotherpotentially detrimental problem that arises from this slow evolution is the particle’s tendency torelax from the quantum state with higher energy to the quantum state with lower energy through theprocess of spontaneous emission, which would disturb the coherent slowing process and introduce
unwanted momentum diffusion. This motivates transferring as quickly as possible, and thereby
satisfying the requirements, i.e., Ω  Γ, for adiabatic rapid passage [27]. Furthermore, in theneighborhood of the resonance, there are significant oscillations in the populations that result
from the precession of the Bloch vector as it travels along the Bloch sphere [67, 185], which can

113complicate the amount of population transfer. A detailed investigation of these oscillations andother important timescales in Landau-Zener transition theory is presented in [185].
As we will show, our derivation of an alternative scheme for coherent transfer, by use of
LRI theory, ameliorates every issue we have discussed thus far. We drive a different path along
the Bloch sphere (see Section 6.4.4) in order to speed up the transfer time and remove the highfrequency population oscillations while utilizing the same amount of laser power. But perhaps
the most appealing feature of our scheme is the ability to exactly transfer population between
quantum states in a finite time, which in principle permits high-fidelity slowing of a particle with
arbitrarily high momentum to near rest.

6.3

Lewis-Riesenfeld Invariant-based inverse engineering
In this section, we present a brief overview of the LRI shortcut method. This approach allows

us to derive specific laser intensity and detuning profiles that can be used to generate fast dynamics
with the same results as adiabatic processes.
ˆ
A dynamical invariant I(t)is a Hermitian operator with a time-independent expectation
value, i.e.,ˆ = hΨ(t)| I(t)
ˆ |Ψ(t)i = const,
hIi

(6.2)

where |Ψ(t)i is the state vector evolved by the Hamiltonian Ĥ(t). It can be shown that the states|ψn (t)i, defined by a gauge transformation|ψn (t)i = eiαn (t) |φn (t)i

(6.3)

ˆ
of the eigenbasis |φn (t)i of I(t),are each a solution to the time-dependent Schrödinger equation.
In Eq. (6.3), the “Lewis-Riesenfeld phases” αn (t) are defined as1αn (t) =~Z tt0

hφn (t0 )| i~∂− Ĥ(t0 ) |φn (t0 )i dt0 ,
∂t0

(6.4)

where t0 is some initial reference time and ~ is the reduced Planck constant. It follows that a

114
general solution |Ψi to the Schrödinger equation can be decomposed as|Ψ(t)i =Xncn |ψn (t)i =Xncn eiαn (t) |φn (t)i ,(6.5)

where cn are time-independent amplitudes [105]. From Eq. (6.5), the unitary time-evolution operator in the invariant basis isÛ (t) =Xn

eiαn (t) |φn (t)i hφn (t0 )| ,(6.6)

which can be used to solve for the Hamiltonian:Ĥ(t) = i~

∂ Û (t) †Û (t).
∂t

(6.7)

ˆ are
These results indicate numerous benefits for this method. In particular, if Ĥ(t) and I(t)designed to commute at t0 and some final time tf , i.e.,hi hiˆ 0 ) = Ĥ(tf ), I(tˆ f ) = 0,
Ĥ(t0 ), I(t(6.8)

then the final state |Ψ(tf )i will maintain the initial populations for each eigenstate [75], and wetherefore recover the results of an adiabatic process without the requirement of slow time-evolution.
Also, by combining Eqns. (6.6) and (6.7), we can explicitly inverse-engineer the Hamiltonian in theinvariant basis:Ĥ(t) = − ~Xnα̇n |φn (t)i hφn (t)| + i~Xn|∂t φn (t)i hφn (t)| .

(6.9)

Eqns. (6.7)-(6.9) define what is meant by invariant-based inverse engineering. Equating the Hamiltonian in the invariant basis to the original Hamiltonian creates a map between the physical parameters and the auxiliary parameters that define the invariant operator. In Section 6.4.3, We will use
this procedure to derive the laser intensity and detuning profiles for each single-photon transitionin our slowing scheme.

6.4

Model
We consider the experimental setup depicted in Fig. 6.1(b). As shown in the circular inset,

we consider each particle to possess two internal electronic states labeled |gi and |ei with an energy

115

Figure 6.2: Left: frequency diagram of an isolated subset of states in the lab frame. The laser
frequencies ωi (p̄, t) [see Eqns. (6.10)] are dynamically updated according to the solution derived fromthe Lewis-Riesenfeld invariant shortcut method to promote quick, coherent transfer from (a) |g, p̄i to|e, p̄ − ~ki, followed by (b) |e, p̄ − ~ki to |g, p̄ − 2~ki. Right: experimental parameters and particle
dynamics over a |p̄i → |p̄ − 2~ki sequence of period 2T . (c) Square-pulse Rabi frequency profilesΩ1 (t) (red) and Ω2 (t) (blue) with amplitude Ω [see Eq. (6.37)]. (d) Lewis-Riesenfeld detuning
profile δ(t) [see Eq. (6.38)] with cutoff frequency ±δcut [see Eq. (6.51)] for each laser. (e) Ideal
excited state fraction Pe dynamics. (f) Ideal average momentum hp̂i dynamics. Parameters are:T = 0.032/ωr , Ω = 100ωr , δcut = 250ωr , Γ = 0, p̄ = 100~k, and β = 0.85 π/2.

separation of ~ωa , where ωa is a frequency assumed to be in the optical domain. The excitedstate |ei can decay to the ground state |gi at a rate given by the natural linewidth Γ. We track
motion along one dimension, for which the particle has position and momentum operators ẑ andp̂. A thermal beam of particles exits an atom or molecule source (e.g., an oven) at a high averagevelocity v̄ and potentially large velocity spread σv . The goal is to quickly remove kinetic energy
from the particles such that an appreciable fraction of the final distribution is centered on zero
velocity [see Fig. 6.1(a)]. In order to achieve this goal, the particle beam sequentially interacts
with pulsed slowing lasers aligned parallel to the beam axis upon entering the slowing region. The
essential dynamics are displayed in Fig. 6.2 and are as follows: The counterpropagating laser (laser
1) is switched on, and the particle absorbs a photon and transits to |ei. Then, laser 1 is switched off
as the copropagating laser (laser 2) is switched on, and the particle emits a photon by stimulated
emission, transitioning back to |gi. By conservation of momentum, the particle has experienced

116
an impulse of 2~k by the end of the sequence. By repeating many times, we can remove many
photon momenta from the particle without the emission of spontaneous photons, contingent thatthe operation occurs quickly compared to any decoherence processes.
We set the Rabi frequency of each laser Ωi (t) (with i = 1, 2) to follow a square pulse temporalprofile. The peak value Ω [see Fig. 6.2(c)] will be derived from the LRI shortcut method. This choiceof waveform relaxes the requirement of exactly aligning the centers of periodic Rabi frequency andlaser detuning profiles in time as is needed in other protocols [129, 167, 173], and thus makes theprocess more robust. We parameterize the instantaneous frequencies ω1 (t) and ω2 (t) of each laser
field in the form
ω1 (p̄, t) = ωa − δ(t) − kv̄ + ωr

(6.10)

ω2 (p̄, t) = ωa + δ(t) + kv̄ − 3ωr ,so that, after accounting for the mean particle velocity v̄ = p̄/m, each laser is resonant with thetransitions displayed in Fig. 6.2(a) and Fig. 6.2(b) when the detuning δ(t) is zero. We will derive
an explicit form for δ(t) from the LRI shortcut method. Here, ωr ≡ ~k 2 /2m is the recoil frequencyof the transition, i.e. ~ωr is the kinetic energy obtained by an atom at rest from the absorption of asingle photon, and m is the particle mass. Also, we have made the typical approximation that thewavenumber ki (t) of each laser is equal to the constant k at all times, even though the frequency
is varying. It should be pointed out that although we have chosen specific signs for δ(t) in eachlaser frequency ωi (p̄, t) [see Eq. (6.10)], these are not the only conventions that can be employed toresult in slowing.

6.4.1

System dynamics

We refer the reader to Sections 2.1 and 2.2 for general details on modeling light-matterinteractions. When applied to a system with momentum p̄ = mv̄, the Hamiltonian takes thegeneral formĤ(p̄, t) = Ĥself + Ĥint (p̄, t),(6.11)

117
whereĤself =

p̂2
~ωa z
+
σ̂
2m
2

(6.12)

is the particle’s free evolution Hamiltonian. The quantity σ̂ z ≡ |ei he| − |gi hg| is the usual Paulispin matrix. Under the dipole and rotating wave approximations, the particle-field interaction
Hamiltonian Ĥint (p̄, t) in the Schrödinger picture is given byĤint (p̄, t) =i
~ −hσ̂ Ω1 (t)ei[kẑ+η1 (p̄,t)] + Ω2 (t)e−i[kẑ−η2 (p̄,t)] + h.c.
2

(6.13)

where σ̂ − ≡ |gi he| is the lowering operator. The quantity ηi (p̄, t) is the accumulated phase froman initial time t0 for each laser field [see Eq. (6.10)]:ηi (p̄, t) ≡Z t

ωi (p̄, t0 )dt0 .

(6.14)

t0

In order to capture the relevant physics in the first half of the sequence, we omit the second
term in Eq. (6.13) (since laser 2 is off) and transform into the interaction picture defined by the
HamiltonianĤ0 (t) =~p̂2+ [ωa − δ(t)] σ̂ z .
2m 2(6.15)The resulting interaction picture Hamiltonian Ĥ1 (p̄, t) is
Ĥ1 (p̄, t) =~δ(t) z ~Ω1 (t)  −σ̂ exp (i [kẑ + k(v̂ − v̄)t + ωr t]) + h.c. ,
σ̂ +
2
2

(6.16)

where v̂ = p̂/m is the velocity operator. The interaction picture Hamiltonian for the second halfĤ2 (p̄, t) is found by the substitutions 1 → 2, δ(t) → −δ(t), k → −k, and ωr → −3ωr . After thecomplete evolution with Ĥ1 (p̄, t) followed by Ĥ2 (p̄, t), the momentum p̄ is updated to p̄ − 2~k forthe laser detunings [i.e., see Eq. (6.10)] under the assumption that the particle momentum has beenchanged accordingly, and the process is repeated.
We incorporate the effects of incoherent dynamics due to spontaneous emission and its associated recoil by evolving the density matrix operator ρ̂ of the system under the quantum masterequation,
i
dρ̂
1 h
=
Ĥ, ρ̂ + L̂ρ̂,dt
i~(6.17)

118
where the Lindblad superoperator L̂ is
L̂ρ̂ = Γ

1 ikẑ − + −ikẑ 1 −ikẑ − + ikẑ 3 − + 1  + −σ̂ σ̂ , ρ̂e σ̂ ρ̂σ̂ e+ eσ̂ ρ̂σ̂ e + σ̂ ρ̂σ̂ −
5
5
5
2


.

(6.18)

Here, Γ is the optical transition linewidth (we reserve the symbol γ for an auxiliary variable), and,for convenience, we have approximated the continuous dipole radiation pattern as allowing only
the three discrete recoil possibilities that correspond to whole intervals of photon quanta. For
states |ei and |gi coupled by a circularly-polarized standing wave, the correct coarse discretizationof the radiation pattern yields possible impulses of ∆p of −~k, 0, and ~k along the slowing axis
with associated probabilities of 51 , 53 , and 15 , respectively. We refer the reader to Ref. [135] andSection 2.2.3 for more details on the dicretization procedure.

6.4.2

Eigensystem in a momentum subspace

Our next task is to find appropriate forms for the Rabi frequencies Ωi (t) and detuning δ(t)by following the shortcut protocol. In order to do this, we will perform analytical calculations fortransfer 1 in a small, isolated subset W (p̄) of the full composite Hilbert space:
W (p̄) = {|g, p̄i , |e, p̄ − ~ki} = {|Gi , |Ei}.

(6.19)W 0 (p̄) = {|e, p̄ − ~ki , |g, p̄ − 2~ki}(6.20)The subset

is used to calculate the corresponding quantities for transfer 2 [see Fig. 6.2(b)], but we do not
explicitly show the derivation here as it follows in a straightforward manner. We expect theresulting shortcuts to be useful for application to the entire Hilbert space because the dynamics
that result from the interaction with each traveling wave are relatively simple, i.e., there are no
multiphoton resonances that would cause population transfer to other quantum states [134]. In
Section 6.6, we apply the resulting Rabi frequencies and laser frequencies from this calculation to
the entire system in a numerical simulation.

119
The interaction Hamiltonian Ĥ1 (t) written in the subspace W (p̄) is given by
(W )

Ĥ1

(t) =

~δ(t) z~Ω1 (t) x
σ̂W +
σ̂W
2
2

(6.21)z ≡ |Ei hE| − |Gi hG| and σ̂ x ≡ |Gi hE| + |Ei hG|. As we will show in subsection 6.4.3, we
where σ̂WW
(W )cast the invariant eigenvectors in a similar form as the eigenvectors of Ĥ1(W )shortcut method. The eigenvalues of Ĥ1(t) areE± (t) = ±where Ω̃1 (t) ≡(t) to apply the LRI~Ω̃1 (t),2(6.22)p
δ(t)2 + Ω1 (t)2 is the generalized Rabi frequency. The corresponding instantaneous

eigenvectors areχ|Ei + sin

χ

|Gi ,
2
2
χ
χ
|−i = sin|Ei − cos
|Gi ,22|+i = cos

(6.23)

with the mixing angle defined by cos χ ≡ δ(t)/Ω̃1 (t).
In general, if a system begins in an eigenstate of the Hamiltonian and the time evolution obeys
the conditions of the adiabatic approximation, the state of the particle will simply adiabaticallyfollow, only picking up an inconsequential global phase. Therefore, if one chooses experimental
parameters such that the particle is initialized in one of the dressed states and the dressed state
adiabatically evolves from |Gi to |Ei (or vice versa), then we can change the particle momentum
by ~k. However, these adiabatic states must evolve slowly to satisfy the adiabatic condition thatarises from Landau-Zener theory. We shall now speed up this adiabatic passage by constructing adriving Hamiltonian based on the LRI shortcut method.

6.4.3

Deriving detuning profiles from the auxiliary equations

We are now in a position to design an invariant operator in order to end up with the desired
final populations using the formalism established in Section 6.3. Its construction and associatedeigenvalue equation can be a difficult process, and various methods have been introduced to overcome this [56, 69, 95, 105]. However, since we have the form of the Hamiltonian and its eigenvectors,

120
ˆ and |φn (t)i in the same functional forms. Thus, in a similar manwe need only to parameterize I(t)ner to [38], we use the inverse engineering approach to parameterize δ(t) and Ω(t) as to begin andend the sweep with the desired populations. (Note that this derivation holds for time-dependentΩ(t) unless noted otherwise.)ˆ in parallel to Eq. (6.23):
We parameterize the eigenvectors of the invariant I(t)
γ 

γ 
eiβ |Ei + sin
|Gi ,
2
2
γ 
γ 
|Ei − cose−iβ |Gi ,|φ− (t)i = sin
2
2|φ+ (t)i = cos(6.24)

where γ = γ(t) and β = const are auxiliary angles. We have introduced the unitary phase eiβparameterized by β as an additional degree of freedom to explore with the shortcut protocol [seeSection 6.4.4]. From Eqns. (6.24), it follows that the invariant can be written in the basis of the
Hamiltonian as


ˆ =
I(t)

 cos γ
2
sin γ e−iβ~Ω0

sin γ eiβ− cos γ
(6.25)

ˆwith eigenvalues λ± = ~Ω0 /2, where Ω0 is an arbitrary constant frequency in order to keep I(t)with units of energy. Substituting Eqns. (6.21) and (6.24) into Eq. (6.4), we calculate the LewisRiesenfeld phases for transfer 1 as
α± (t) = ∓

1
2

Z t

(δ cos γ + Ω1 sin γ cos β) dt0 .

(6.26)0

Substituting these phases into Eq. (6.9), we find that the Hamiltonian is parameterized by


iβ
Be 
~ A
(W )
Ĥ1 (t) = (6.27)
,2
B ∗ e−iβ −A
where
A = δ(t) cos2 γ + Ω1 (t) cos γ sin γ cos β,(6.28)

B = δ(t) cos γ sin γ + Ω1 (t) sin2 γ cos β − iγ̇.

(6.29)

121
Equating the two forms of the Hamiltonian [Eqns. (6.21) and (6.27)], we arrive at the auxiliary
equationsγ̇ = Ω1 (t) sin β,(6.30)

δ = Ω1 (t) cot γ cos β.

(6.31)

Next, we determine the boundary conditions on the auxiliary variables γ and β required forstate transfer from state |Gi to |Ei. As seen from Eq. (6.24), γ must satisfyγ(t0 ) = πn;γ(tf ) = γ(t0 ) + π(2m + 1),(6.32)

where n and m are integers. For simplicity and without loss of generality, we choose n = m = 0
so that γ evolves from 0 to π and causes the state to align with |φ− (t)i at t = t0 , tf . Since the(W )commutator between Ĥ1
h

(W )

Ĥ1

ˆ is
(t) and I(t)i ~2 Ω0 h†zˆ(t), I(t)=− iΩ sin γ sin β σ̂W+ (δ sin γeiβ − Ω cos γ)σ̂W2i+ (Ω cos γ − δ sin γe−iβ )σ̂W ,(6.33)

where σ̂W ≡ |Gi hE|, we should also imposeΩ(t∗ ) sin γ(t∗ ) sin β = 0,

(6.34)

β = qπ(6.35)(W )for t∗ = t0 and t∗ = tf and integer q to align the eigenbases of Ĥ1

ˆ at the beginning(t) and I(t)and end of the shortcut process [see Eq. (6.8)]. The condition (6.34) is automatically satisfied by(6.32), whereas the condition (6.35) is not necessary to enforce in the case of complete state transfer
because it only affects non-physical, global phases.
We have now parameterized the Hamiltonian in terms of the auxiliary angles and foundboundary conditions for these angles for the interaction with laser 1. The corresponding quantities
for the subsequent interaction with laser 2 are derived in a similar manner.
Together Eqns. (6.30)-(6.32) directly determine the experimental parameters δ(t) and Ω1 (t).
In the general case of Ω1 (t) 6= const, we would be faced with the task of choosing, from an infinite set

122
of Hamiltonians, a particular form of γ(t) and β(t) that satisfies Eqns. (6.30)-(6.32) [38]. However,
we will now simplify the discussion by restricting the solution to the form Ω1 (t) = Ω = const,
whereby the auxiliary equations (6.30) and (6.31) can be solved analytically, which we now show.
Let us label the time required to perform the shortcut as T . Defining t0 = 0 (and therefore tf = T ),the auxiliary equation for γ using γ(0) = 0 yields
γ(t) = (Ω sin β)t.

(6.36)The other boundary condition γ(T ) = π [see Eq. (6.32)] requires thatπ = ΩT sin β.

(6.37)

Combining Eqns. (6.36) and (6.37) and substituting the result into Eq. (6.31), we find that
δ(t) =

π cot β
cot
T



πt
T


.

(6.38)

We provide a plot of δ(t) for a particular choice of β and T in Fig. 6.2(d).
We have now found a particular Hamiltonian that will drive the transition |Gi → |Ei withdynamics that need not be adiabatic. The analogous solution can be used for laser 2 to drive thetransition |e, p̄ − ~ki → |g, p̄ − 2~ki using precisely the same theoretical formalism. The method
may be optimized further with respect to different cost functions (e.g., see Ref. [175]).

6.4.4

Bloch sphere trajectories

Different choices of the auxiliary angle β [see Eq. (6.24)] can result in very distinct dynamics
as the particle is transferred from |Gi to |Ei, which can be seen by investigating the associated
Bloch sphere trajectories. The resulting trajectories under the interaction Hamiltonian Ĥ1 (t) [seeEq. (6.21)] are purely longitudinal and are parameterized by the azimuthal angle φ(β) = π + β.
For a more appropriate comparison, we instead work in the time-independent interaction picture
defined by the self-energy of the particle [see Eq. (6.12)], which results in the interaction picture
HamiltonianĤI (t) =~Ω  iθ(t)e|Ei hG| + h.c.
2

(6.39)

123

Figure 6.3: Bloch sphere trajectories (thick curves) between the initial state |Gi and final state |Eiparameterized by the auxiliary angle β [see Eq. (6.24)] in the free-energy interaction picture [seeEq. (6.39)]. We set the Rabi frequency Ω to be equal for all trajectories, and the slowing periods
1 π
1π
T are given by Eq. (6.37). The cutoff detunings for β = 40
2 and β = 2 2 are δcut = 318Ω andδcut = 225Ω, respectively [see Eq. (6.51)].

in the subspace W (p) [see Eq. (6.19)], whereθ(t) ≡Z t

δ(t0 ) dt0(6.40)

t0

is the accumulated phase of the detuning δ(t) derived from the LRI shortcut method [see Eq. (6.38)].
Bloch sphere trajectories for three choices of β are displayed in Figure 6.3. For the sakeof comparison, we chose to fix the Rabi frequency Ω across all trajectories so that the slowing
periods T are completely determined by β [see Eq. (6.37)]. A choice of auxiliary angle satisfying
β = (r + 12 )π for integer r minimizes the slowing period T ; this is simply resonant Rabi-flopping(δ(t) = 0). For r = 0, i.e., a resonant π-pulse, the system follows the purely longitudinal Bloch
sphere trajectory (yellow, dotted curve). This choice of β can lead to experimental challenges, suchas inefficient transfer due to sensitivity to errors in Ω (see Section 6.5). On the other hand, a choiceof auxiliary angle satisfying β ≈ sπ for integer s causes T to tend to infinity, as in an adiabatic
1 π
process. We approximate the choice s = 0 by using the very small auxiliary angle β = 40
2 , whichresults in a trajectory that precesses significantly (blue, solid curve). This choice also comes withcomplications, such as a high chance of spontaneous emission and a large slowing distance. We

124take all of these considerations into account and choose from the range 0  β < π/2 in our slowing
simulations in Section 6.6, which results in an intermediate trajectory (red, dash-dotted curve).

6.5

Slowing dynamics in classical phase space
The usefulness of the LRI shortcut method is ultimately determined by whether or not it

can achieve a practical device that is competitive with the current state of the art. With this inmind, we calculate the resulting classical force, slowing time, slowing distance, scattering rate, andnumber of scattered photons of this method and compare the results to commonly-implemented
processes that rely on radiation pressure (RP), such as a Zeeman or Stark decelerator. These
quantities are derived below and collected in Table 6.1.
We use Eq. (6.37) in order to calculate the classical force FLRI exerted on the particle during
each sweep:
FLRI =~kΩ~k sin β=.
Tπ(6.41)

Γ~k,2(6.42)

The classical force from RP is given by
FRP = Γ~kρee ≤where ρee is the excited state fraction, and the inequality is saturated at infinite laser power [126].
Next, we consider the time required to transfer the particle to zero momentum. We assume
the particle begins the slowing process in the state |g, p̄0 i. It receives an impulse of ~k against its
motion after every sweep for a total time
∆tLRI = T p̄0 /~k =πζ0,Ω sin β(6.43)

where ζ0 ≡ p̄0 /~k. Assuming that the force FRP remains constant, the slowing time for RP is∆tRP =

ζ0,Γρeesince approximately one scattered photon is required per slowing photon.

(6.44)

Γρeeωr ζ02λ2πρee Γζ0
Γρee

Γ~kρeeRadiation pressureΓ2

ωr ζ02λ2Ω sin βπζ0Ω sin βΩ~k sin βπ

Lewis-Riesenfeld

scattering rate Rs

slowing time ∆tforce F

slowing method

slowing distance ∆x

ζ0

Γ π
ζ0
Ω 2 sin β

number of scattered photons N

Table 6.1: Comparison of slowing dynamics between the shortcut slowing scheme based on Lewis-Riesenfeld invariants and radiationpressure slowing. If Ω sin β  πΓρee , the shortcut slowing scheme results in larger forces, shorter slowing times, shorter slowing distances,and fewer scattered photons than RP.

125

126
We now calculate the distance a particle in the final momentum wave packet travels in physicalspace throughout the slowing process in order to reach zero momentum. We approximate p(t) to
be linear:p(t) ≈ p̄0 − ~kt/T(6.45)

so we may write the slowing distance as∆xLRI =pavg
ωr T ζ02
∆t =
λ,
m
2π

(6.46)

where pavg = p̄0 /2 is the time-averaged momentum and λ is the wavelength of the transition.
Because laser power is typically the limiting experimental factor, we use Eq. (6.37) to rewrite
Eq. (6.46) as∆xLRI =

ωr ζ02λ.
2Ω sin β(6.47)

Using Eq. (6.42), we can obtain a similar expression for the RP slowing distance:∆xRP =

ωr ζ02λ.
2πρee Γ(6.48)

Comparing the LRI and RP results, we find that our scheme is able to exert higher forces, andtherefore slow particles in less time and in a shorter distance, with fewer scattered photons whenΩ sin β  πΓρee . For choices of sin β and ρee on the order of unity, this amounts to the limit Ω  Γ.
In the LRI analysis, we have not incorporated any effects due to dissipation. Since the slowing
mechanism is purely coherent in nature, it would be ideal if there were no spontaneous emission
events. This can be achieved in several ways: the use of ultra-narrow linewidth transitions, applying
the entire slowing protocol in a time shorter than the lifetime of the excited state (which typicallyrequires extremely high laser power but also results in very small slowing distances), or applying
the shortcut protocol on a Raman transition between internal ground states. The last method isbeyond the scope of this work, but may extend the application of this protocol to other atomic andmolecular species. Nevertheless, the scattering rate Rs of each slowing technique is approximately
s
RLRI
≈

Γ
;
2

s
RRP
≈ Γρee ≤Γ,2(6.49)

127
since the particles in the LRI scheme are in the excited state for roughly half of the time, suggestingthat the scattering rates are on the same order. However, the expected number of scattered photons,
N = Rs ∆t, for each technique is
NLRI ≈Γ π
ζ0 ;Ω 2 sin β

NRP ≈ ζ0 ,(6.50)

which means that the LRI scheme generates fewer scattered photons when Ω sin β  Γ.

6.6Slowing Examples
By repeatedly driving the transition |g, p̄i → |e, p̄ − ~ki followed by |e, p̄ − ~ki → |g, p̄ − 2~ki

according to our shortcut solution, we now show that we are able to slow a significant number ofparticles when the full composite Hilbert space is taken into account. After each pair of transitions,
we update the momentum p̄ to p̄ − 2~k in the laser frequencies [see Eq. (6.10)], as we assume thatthe initial population in |g, p̄i has moved to |g, p̄ − 2~ki. Note that this procedure results in theslowing of a pulse, as opposed to a steady-state ensemble, of particles. We assume that the jump inlaser frequency at the end of each ramp is perfectly diabatic. Since there is very little populationtransfer when δ(t)  Ω, we do not track the detuning profile’s asymptotic behavior, as this cancause numerical errors. We thus take δ(t) to be equal to a constant ±δcut in these regions, definedbyδ(t < t0 + tcut ) = −δ(t > tf − tcut ) = δcut

(6.51)

for some cutoff time tcut , as shown in Fig. 6.2(d). We evolve the master equation [Eq. (6.17)] using
the method of quantum Monte Carlo wave functions [135]. This involves unraveling the masterequation into trajectories based on stochastic simulation of spontaneously emitted photons.
Fig. 6.4 presents the momentum distribution of an ensemble of particles before (blue) andafter (orange) application of our slowing scheme over 100 sweeps in the case of purely coherent(left column) and dissipative (right column) dynamics. In the dissipative case, we chose Γ = ωr
as an order-of-magnitude estimate for a narrow-linewidth transition. We chose to initialize themomentum distribution as either a momentum eigenstate at 100~k (top row) or a Gaussian profile

128
with an average momentum hp̂i = 100~k and a standard deviation of σp,0 = 10~k (bottom row),which typically corresponds to an initial average particle speed on the order of 1 m/s. Whileactual physical systems may have much higher initial particle beam speeds, this choice sufficiently
demonstrates the slowing effects of our protocol. We have chosen to use the average momentumof the Gaussian as the initial momentum used in the laser frequencies (p̄0 = hp̂i) because it resultsin the largest fraction of slowed particles. For experimental accessibility and robustness, we choseβ = 0.85 π/2 and Ω = 100Γ, which results in a shortcut time T = 0.032/Γ [see Eq. (6.37)] typicallyon the order of microseconds, much shorter than the associated timescale of radiation pressure
slowing on the transition. This choice of Ω also sets the expected number of scattered photons N
to be on the order of unity [see Eq. (6.50)].
As seen in Fig. 6.4, a substantial fraction of the distribution is slowed to near zero momentum.
We first discuss the results in the case of purely coherent dynamics (left column). If the system
begins in the eigenstate |g, 100~ki [Fig. 6.4(a)], about 99.3% of the population ends in the zero
momentum eigenstate. (We attribute the lack of 100% transfer to our use of a cutoff frequency
δcut = 244ωr .) If the system begins in a Gaussian state [Fig. 6.4(c)], about 35% of the populationends with momentum |p| ≤ 10~k, which corresponds to about half of the population within one
standard deviation of the average momentum in the initial distribution.
In the case of dissipative dynamics, 29% of the population ends with momentum |p| ≤ ~k if
the system is initialized in the eigenstate |g, 100~ki [Fig. 6.4(b)], and 13% of the population ends
with momentum |p| ≤ 10~k if the system is initialized in the Gaussian state [Fig. 6.4(d)]. In bothcases, there was an average of 1.6 scattered photons per particle, which agrees with the predictedscattering rate given in Table 6.1. These results demonstrate that our protocol can potentiallyslow a significant fraction of particles to zero momentum in a very short distance and with few
spontaneous emission events. An appealing feature of our protocol is its ability to apply highslowing forces with a virtually negligible reliance on spontaneous emission.
As a more realistic demonstration, we present several atomic and molecular species that arereasonable candidates for our slowing protocol in Table 6.2. We simulated the slowing of eachX 2 Σ+ → A02 ∆3/2

YO

BaH

174 YbX 2 Σ + → A2 Π

1S → 3P
0
1

1S → 3P
0
1

1S → 3P
1
0

40 Ca

88 Sr

Transition

Species

1061

556

690

689

657

Wavelength (nm)

1200

180

5.9

7.5

0.4

Linewidth (kHz)

1.3

3.7

4.0

4.8

11.5

Recoil frequency (kHz)

1.3 × 10−4

1.4 × 10−4

2.3 × 10−6

3.0 × 10−6

1.8 × 10−7

Saturation intensity (W/cm2 )Table 6.2: Fundamental properties for various atomic and molecular candidates for shortcut slowing on a narrow transition.

129

Rabi frequency (MHz)

0.2

1.0

0.9

3.4

23.6

Species

40 Ca

88 Sr

YO

174 Yb

BaH

0.54

0.82

1.1

1.3

3.0

Initial speed at 200~k (m/s)

4

30

119

106

494

Slowing time (µs)

1

12

66

70

746

Slowing distance (µm)

1.6 × 10−5

6.8 × 10−6

2.1 × 10−1

1.5 × 10−1

5.6 × 10−1

Capture fraction

Table 6.3: Slowing results for various atomic and molecular candidates using a 0.1 W/cm2 laser intensity and starting from a momentum of
200~k. The Rabi frequency, slowing time, and slowing distance follow from Eqns. (6.52), (6.43), and (6.47) respectively, with β = 0.85 π/2.
The simulated capture fraction, defined as the fraction of particles with momentum |p| ≤ 3~k after the slowing process, was calculated
over 1,000 trajectories for Ca, Sr, and YO, and 10,000 trajectories for Yb and BaH.

130

131

Figure 6.4: Initial (blue) and final (orange) momentum distributions P (p) of a system subject to theslowing protocol. The system is initialized in the internal ground state |gi and with a momentum
distribution of either the 100~k eigenstate (top row) or a Gaussian state with average momentum
hp̂i = 100~k and width σp = 10~k (bottom row). The excited state linewidth is Γ = 0 in (a)and (c), and Γ = ωr in (b) and (d). Further details are discussed in text. Other parameters
are: Ω = 100ωr , T = 0.032/ωr , ttot = 1.6/ωr , and δcut = 244ωr , β = 0.85 π/2, and p̄0 = 100~k.
Subplots (b) and (d) are averaged over 1,000 trajectories.

species with an initial momentum of 200~k using a 0.1 W/cm2 laser intensity, and provide theexpected slowing times ∆t, slowing distances ∆x, and capture fractions C in Table 6.3. The
saturation intensity Isat , laser intensity I, and Rabi frequency Ω can be calculated according to
[126]
 2
Ω
I
2
=
;
Γ
Isat

Isat ≡πhcΓ
,3λ3(6.52)

where h is Planck’s constant and c is the speed of light. The calculated slowing times and distances
become smaller as the transition dipole matrix element increases, as can be seen for the species
with larger linewidths. However, the required temporal control of the Rabi frequency and detuning

132
profiles in these cases may be an experimental challenge. One solution is to simply reduce the laser
power, but this can quickly move the parameters away from the regime Ω  Γ, which increases the
chance of spontaneous emission and hence reduces the capture fraction C.
Due to our method’s sensitivity to spontaneous emission, a rough estimate for the capture
fraction C is given by the fraction of particles that do not emit a single spontaneous photon
throughout the slowing process. Using Eq. (6.50), this is approximatelyΓ πC ≈ exp (−NLRI ) = exp −ζ0 .
Ω 2 sin β(6.53)

This formula is on the same order as the simulated capture fractions given in Table 6.3 for Ca,
Sr, and YO, but it underestimates the results for Yb and BaH, suggesting that particles can still
be slowed after emitting spontaneous photons. Eq. (6.53) is not a fundamental limit, as it can be
improved with a more sophisticated implementation of the shortcut process, such as introducing
occasional waiting periods so that the particles return to the internal ground state (as is done inother methods [129]), allowing the laser pulses to overlap in time (see Fig. 6.6), or repeating theslowing protocol over a range of momentum states. For example, we were able to increase thesimulated 174 Yb capture fraction by over an order of magnitude to C = 1.7 × 10−4 by introducing
a laser pulse overlap fraction f = 0.2 as discussed in Section 6.7.

6.7

Robustness
We now study the robustness of the shortcut slowing scheme to various systematic errorsthat may arise in an experimental setting. Specifically, we modify the Rabi frequency amplitude,
then separately consider the result of slowing a particle with a momentum p that is not equal tothe momentum accounted for in the laser frequencies p̄ (Fig. 6.5). We also consider the scheme’srobustness to the temporal overlap f of the square pulses (Fig. 6.6), which may minimize scattering
events since it potentially reduces the amount of time the particle remains in the internal excitedstate. For simplicity, we focus on a single |gi → |ei → |gi process and calculate the resulting
impulse ∆p experienced by the particle. Moreover, we employ a phase relation between the laser

133

Figure 6.5: Robustness comparison of π-pulse slowing (purple, plus), SWAP slowing (green, cross),and the shortcut scheme (blue, circle with cross) over a p̄ → p̄ − 2~k transfer process. Ideally,
the resulting impulse ∆p satisfies ∆p/2~k = −1. The evolution is purely coherent (Γ = 0), themomentum of the particle (which is addressed by the laser frequencies) is p̄ = 2~k, and the Rabi
frequency is Ω = 10ωr for all processes. (a) Impulse ∆p experienced by the particle, in units of
the ideal impulse magnitude 2~k, as a function of the error in the Rabi frequency amplitude  [seeEq. (6.55)]. The shortcut scheme is the most robust protocol when  > 0. (b) ∆p/2~k as a functionof the relative momentum of the particle δp with respect to the momentum p̄ = 2~k used in thelaser frequencies [see Eqns. (6.10) and (6.56)]. For this set of parameters, the shortcut and π-pulse
scheme are generally more robust than SWAP slowing. π-pulse parameters are: T = 0.314/ωr andβ = π/2. SWAP slowing parameters are: T = 1/ωr and ∆ = 50ωr . Shortcut parameters are:T = 0.44/ωr , δcut = 230ωr , and β = 0.5 π/2.

pulses such that the Rabi frequency of each pulse is purely real. The correct detuning profiles
are employed throughout this section. We set the Rabi frequency to Ω = 10ωr across all slowing
methods, but necessarily allow for different slowing times.
Using the auxiliary variable β = 0.5 π/2, we compare the robustness of our scheme to both
π-pulse and SWAP slowing. As discussed in Section 6.4.4, The π-pulse solution is a special caseof our slowing method with the choice β = π/2, which results in a fixed detuning [see Eq. (6.38)].
SWAP slowing is simulated by using a sawtooth-wave detuning profile with full range ∆ and period
T for each single-photon transfer. Unlike SWAP cooling, the lasers are sequentially pulsed as inFig. 6.2(a) and 6.2(b), and the laser detunings are centered on the momentum p̄.

134
While there is not a unique way to choose the SWAP slowing parameters, we used thefollowing method. Landau and Zener showed [194] that the population of the second (initiallyunoccupied) state P2 if the laser is linearly chirped from a detuning of minus infinity to positive
infinity, is given by Eq. (6.1). Therefore, to obtain a transfer probability of at least 95%, we chose to
set Ω2 /α = 2. Next, we increased the chance of population transfer by setting the shortcut period
to be several times larger than the approximate time τj required to transfer population between
quantum states in the adiabatic regime [185]:
τj =

2Ω
0.4
=
α
ωr

⇒

T =

1
> τj .
ωr(6.54)

These choices constrained the sweep range to be ∆ = 50ωr . Note that the resulting SWAP slowing
period T = 1/ωr is significantly longer than the π-pulse (T ≈ 0.3/ωr ) and shortcut slowing (T ≈0.4/ωr ) periods [see Eq. (6.37)].
Fig. 6.5(a) presents the resulting particle impulse ∆p as a function of the error in the Rabi
frequency amplitude, which is characterized by the small parameter :Ω → Ω(1 + ).

(6.55)

While the shortcut scheme applies a similar impulse ∆p compared to the π-pulse method for  < 0,
it is the most robust method for  > 0. The SWAP slowing result, while being the most robust
method for  < 0, can change significantly with small changes to the parameters. This instability is
as a result of high-frequency population oscillations [185]. Moreover, when dissipation is included,
the much longer SWAP slowing period increases the chance of spontaneous emission, which candisrupt the slowing process. These results demonstrate the utility of the shortcut scheme, as it
is robust to small errors  and takes much less time than SWAP slowing. Note that intensity
modulators typically have errors less than approximately 5%.
Fig. 6.5(b) presents the effects of applying each slowing method to a particle with a momentum p which is not equal to the momentum p̄ accounted for in the laser frequencies [see Eq. (6.10)].
Such an error occurs when slowing a cloud of particles with a distribution of momenta. We param-

135
eterize this difference in momentum with the variable δp as:
p = p̄ + δp.

(6.56)In this case, we find that the shortcut scheme is most robust for small δp, but SWAP slowing
becomes more robust for |δp| > 3~k. This change in the trend for SWAP slowing near the particularvalues δp = ±3~k is due to population oscillations, and the results can again change significantlywith small changes in the parameters. In the limit of adiabatic dynamics and time-resolved transfers(which necessarily takes a long time), SWAP slowing is generally most robust to this error since
the rate of change of the detuning profile in SWAP slowing is constant, thereby removing the needto align the center of the laser detuning profile with the Doppler shift of the particle. It should be
noted that it is not necessarily a problem that the shortcut scheme is less robust to large deviationsof δp from zero because we scan through values of p/~k in integer steps by design. What is more
important is that the shortcut scheme is more robust when |δp| < ~k than SWAP slowing, as
particles will not necessarily have integer momentum.

Figure 6.6: Impulse ∆p experienced by the particle, in units of the ideal impulse magnitude 2~k,
as a function of the laser pulse overlap fraction f . The impulse is calculated under both coherent(Γ = 0) and dissipative (Γ = ωr ) dynamics. Other parameters are: T = 0.44/ωr , δcut = 230ωr , Ω =10ωr , p̄ = 2~k, and β = 0.5 π/2. All points are averaged over 1,000 trajectories.

Fig. 6.6 presents the effects of allowing the laser pulses to overlap in time. We define the136overlap fraction of the square pulses f such that f = 0 when the pulses are completely time-resolved
but occur sequentially with no delay, and f = 1 when the pulses occur at the same time for theentire pulse duration. We find, in the case of purely coherent dynamics (Γ = 0), that a pulse
overlap fraction as large as f = 0.2 does not change the impulse experienced by the particle. In thecase of dissipative dynamics, we find that there is an optimal f that maximizes the impulse ∆p.
This optimal f , which depends on system specifics, must be small enough to allow transfer to |ei
before the transfer back to |gi, but large enough to minimize the time the particle spends in the
excited state, thereby minimizing the chance of spontaneous emission. It should be noted that apulse overlap introduces the possibility of multiphoton, or Doppleron, resonances [134] which may
interfere with the single-photon slowing dynamics.
When slowing a particle with a large initial momentum p  ~k to rest, we emphasize thatthe slowing efficiency is significantly affected by even a small deviation from the ideal impulse∆p = −2~k since the error compounds exponentially with the number of transfer processes, andthe particle is generally not transferred back to |gi for the next pulse sequence. It may be possible tofurther enhance the robustness of the slowing protocol to the rapidly changing detuning profile at the
beginning and end of the transfers by using a Rabi frequency profile that satisfies Ω(t0 ) = Ω(tf ) → 0,such as a Gaussian or a sinusoidal function [129, 167, 173]. However, in order to satisfy the auxiliary
equations and boundary conditions [see Eqns. (6.30)-(6.32)], the peak Rabi frequency may need tobe larger than what we consider here.

6.8

Conclusion and Outlook
In this work, we proposed a purely coherent particle slowing scheme. By utilizing the methodof inverse-engineering based on Lewis-Riesenfeld invariant theory, we demonstrated that these slowing designs are able to achieve effective adiabatic dynamics but with a short slowing time. We
illustrated how to design Rabi frequency and detuning profiles that achieve the desired dynamics
and demonstrated that our slowing blueprint is a promising alternative to conventional slowing
schemes that rely on radiation pressure for narrow linewidth systems or systems that lack a closed

137
cycling transition. We theoretically examined the effective classical forces exerted on the particles
during the slowing processes and demonstrated that, when the Rabi frequency is large comparedto the excited state decay rate, our scheme is able to exert significantly higher forces than radiation pressure while maintaining a very low number of scattered photons. We also showed that our
proposed slowing scheme is robust to various systematic errors.
A possible practical implementation of our scheme is to directly apply it to a particle beam
exiting a supersonic nozzle or buffer gas cell. Our scheme may also be utilized as a second slowing
stage after particles exiting an effusive oven have been slowed in a precursor stage to the order of
10 m/s and the initial spread in velocity has been greatly reduced. Additionally, it may be possible
to implement our protocol as a steady-state slowing procedure by compensating for the changing
particle velocity with a magnetic field gradient instead of the time-dependent laser frequencies, ina similar approach to a Stark or Zeeman decelerator.
A natural next step would be to optimize our shortcut solutions with respect to different criteria. Time-optimization of different adiabatic shortcuts has been studied previously for two-levelsystems [161, 162], STIRAP [136], and frictionless cooling in harmonic traps [175]. Another optimization criterion comes from the fact that adiabatic shortcuts cannot be implemented without anenergetic cost, an intrinsic relationship that has been rigorously studied for Berry’s transitionlessquantum driving algorithm [28], various other shortcuts [1], and in its applications to quantum
computing [52]. Thus, minimizing the energetic cost for a fixed sweep period is, in effect, finding the most efficient shortcut. Additionally, quantum optical control based protocols could be
implemented to derive further solutions [72].
Another appealing scheme is a similar shortcut protocol applied to a system with an internal
state structure comprised of two stable ground states coupled to an excited state, such as found in
a J = 1 → J 0 = 1 transition, where J is the total electronic angular momentum quantum number.
Speeding up a Raman transition between the ground states could remove twice as much momentum
per transition. This model, while generally requiring more laser power, has the additional benefitof an engineered excited state linewidth, which can be made arbitrarily small if the lasers are

138
sufficiently detuned from the excited state. Moreover, applying anti-symmetric detuning sweeps
could allow for control of the slowed velocity range, potentially resulting in a slowed distribution
with a low temperature.
There are similarities between our results and others [111, 129, 167, 173] in the sense that
two-level inversion is achieved in the diabatic limit. A more thorough investigation could further
our understanding of the connection between these solutions and adiabatic shortcuts.
Furthermore, it is impossible to implement the LRI scheme to the entire momentum Hilbert
space by hand. An intriguing possible solution to this problem is to employ advanced optimization
techniques, such as quantum optimal control and reinforcement learning, to find a solution thatmaximizes the slowing scheme’s capture range or minimizes the energetic cost of implementing theshortcut for a given sweep period.

Chapter 7

Entropy transfer from a quantum particle to a classical coherent light field

In the previous chapters, we studied various laser cooling (Chapters 3 and 4), trapping(Chapter 5), and slowing (Chapter 6) techniques. In the cases involving cooling, we saw that theensemble of quantum particles eventually equilibrated to some temperature and cloud size. As
discussed in Section 2.2.2, this was a consequence of the dissipative spontaneous emission process,which irreversibly carried away entropy from the ensemble by its interaction with the environment
and allowed the system to reach a steady-state configuration. We can also ask the question as
to whether the degrees of freedom of the laser light could also remove entropy from the particle
ensemble. However, this would require a quantized model of the laser light (see Section 2.26) anda relaxation of the assumption that the laser light is unperturbed by the particle-laser interaction.
We adopt this perspective and investigate its consequences in this chapter.
In this chapter, we develop and theoretically analyze a simple Gedankenexperiment which
involves the interaction of a coherent state with a quantum particle in an optical cavity. We quantify
the resulting alteration of the light field by measuring the fidelity of its initial and equilibrium states.
Using Bayesian inference, we demonstrate the information transfer through photon measurements.
In addition, we employ the concepts of quantum entropy and mutual information to quantify theentropy transfer from the particle to the light field. In the weak coupling limit, we validate theusually assumed negligible alteration of the light field and entropy transfer. In the strong coupling
limit, however, we observe that the information of the initial particle state can be fully encoded in
the light field even for large photon numbers. Nevertheless, we show that spontaneous emission is a140
sufficient mechanism for removing the entropy initially stored in the particle. Our analysis provides
a deeper understanding of the entropy exchange between quantum matter and classical light.

7.1

Introduction
When studying a quantum system, it is generally desirable for it to be prepared in a low-

entropy configuration. Such systems can populate a relatively small number of quantum states,
which makes their dynamics more controllable, predictable, and comprehensible [22, 195]. If thesystem of interest is a gas consisting of atoms or molecules, its entropy is typically reduced byutilizing laser cooling and optical pumping techniques. In these processes, the particles absorb
light from an applied laser field and incoherently scatter the light into free space [126]. Whilethese methods can irreversibly remove entropy from the gas, the second law of thermodynamics
requires that the total entropy of the universe does not decrease, and therefore that the entropyof the gas must have been absorbed by some other system. In this context, the most common
explanation is that the entropy of the gas is absorbed by the vacuum modes of the quantizedelectromagnetic field [190]. This process is often cast in the framework of open quantum systems
whereby the quantized electromagnetic field is treated as an external reservoir, allowing for an
irreversible reduction of the gas’s entropy through the process of spontaneous emission [24, 65, 155].
It is typically explained that the reservoir can absorb a substantial amount of entropy due
to the large number of possible emission configurations [93, 126]. Moreover, it is often stated thatthe coherent light field is not perturbed, and hence does not absorb entropy, because the quantum
counterpart to the laser field, the coherent state, is unaffected by the absorption of photons by thegas. However, there are some studies that predict entropy removal from the gas via interaction
with the laser field [35, 50, 100, 124, 128]. In an effort to further understand the underlying physics
of laser cooling and optical pumping, we propose a simple Gedankenexperiment that probes thechange of a light field that coherently interacts with a particle containing nonzero initial entropy.
Our results conclude that the entropy initially possessed by the particle can be imprinted on theapplied light field, but the same information is also encoded in the spontaneous photons emitted

141
by the particle.

�Figure 7.1: (a) A schematic of the experimental setup. A particle (circle) is placed in a lossless(κ = 0) optical cavity that contains a coherent light field. The particle can undergo spontaneousemission into free space at rate γ. (b) Energy diagram of the particle’s internal state structure. It
is coupled to the cavity with strength g on the bright transition (|ei ↔ |bi) and has a linewidth γ
on the dark transition (|ei → |di).

7.2

Motivation
We first consider the level of complexity necessary to demonstrate the state alteration andentropy transfer processes with the goal of removing complications or details of any particularcooling or pumping scheme that might obscure the core physics of interest. While other candidates
may exist, we model the initial state of the laser field as a coherent state, as this is the quantum
state that shares the most similarities with a classical coherent laser field [6]. The coherent stateis prepared in a lossless optical cavity, after which any external driving fields are turned off. The
cavity is an efficient way to incorporate quantization, and we anticipate that many of our findings
also apply to free-space coherent light fields. Because the essential physics of interest is the same,
we focus only on the effects of optical pumping on the evolution of the particles’ internal states.
To this end, we model the gas as a single, motionless particle that exists at an anti-node of the
optical cavity. With these considerations in mind, we construct the minimal experimental setup
depicted in Fig 7.1(a). The particle possesses two ground states and a single excited state, as
shown in Fig 7.1(b). The |bi ↔ |ei (bright) transition is resonant with the optical cavity, which
encapsulates the coherent interaction between the cavity field and the particle, while the |ei → |di(dark) transition is mediated only by the spontaneous emission process, which models the incoherent
interaction between the background radiation field and the particle. The particle always transitions

142
to the dark state |di upon emission of a spontaneous photon so that the cavity isn’t trivially depletedof photons.
By studying this model, we aim to quantify the alteration of the cavity field due to its
interaction with the particle and to determine if entropy is transferred from the particle subspace
to the cavity subspace through their resulting correlations. To achieve these goals, we employ both
statistical and information theoretic techniques. We first calculate the final cavity field state andquantify its distinguishability from the initial cavity state through their quantum fidelity. Then,
we demonstrate that information about the particle becomes encoded in the cavity field by using
Bayesian inference. Lastly, we quantify the amount of entropy transferred from the particle to thecavity field by calculating the quantum mutual information shared between the initial particle stateand final cavity state.

7.3

Model
We refer the reader to Sections 2.1 and 2.2 for details on modeling light-matter interactions.

The system of interest [Fig. 7.1(a)] consists of a three-level particle [Fig. 7.1(b)] coupled to a lightfield in a lossless (κ = 0) optical single-mode cavity. Let us denote the Hilbert space of eachsubspace as A and L, respectively. In the time-independent interaction picture, the system is
evolved according to the quantum master equation
i
dρ̂AL
1 h
=
ĤAL , ρ̂AL + L̂ρ̂AL .
dt
i~

(7.1)

Here, the coherent cavity-particle interaction is described by the Jaynes-Cummings Hamiltonian
ĤAL =

~g |bi he| â† + |ei hb| â ,
2

(7.2)

where g is the coupling strength, and â is the annihilation operator for the cavity field. As isachieved in the moving frame of a particle in Doppler cooling, we have set the cavity to be resonantwith the particle’s bright state transition. The environment surrounding the system, which wemodel as an infinite bandwidth bosonic bath that is coupled to the particle’s dark transition, has

143
been traced out under the Born-Markov approximation [109, 127]. Its effects are incorporated
through the Lindblad superoperator

o
1n †
†
,
L̂ρ̂ = γ L̂ρ̂L̂ −L̂ L̂,̂ρ
2i.e., the only jump operator is Jˆ =√γ L̂ =√(7.3)

γ |di he|. This term describes the spontaneous emissionof photons into free space by the particle that occurs at a rate γ. We consider both analytical andnumerical solutions to Equation (7.1) in the sections that follow. We use both MATLAB and the
QuantumOptics package in the Julia programming language for numerical calculations [101, 120].
To model the entropy initially possessed by the particle, it is prepared in the mixed stateρ̂A (0) = x |bi hb| + (1 − x) |di hd| ,(7.4)

where 0 ≤ x ≤ 1 is the probability that the particle begins in the bright state |bi. As alreadymentioned, the cavity field is initialized in a coherent state2|αi = e−|α| /2∞X
αn√ |ni ,
n!
n=0

(7.5)

where |ni is the n-photon Fock state, so that its initial density matrix readsρ̂L (0) = |αi hα| .

7.4

(7.6)

Calculating the final cavity field state
In order to derive an analytic form of the cavity field state in the limit t → ∞, we separate

the density matrix into Jordan blocks:ρ̂b = hb|ρ̂|bi |bi hb| + hb|ρ̂|ei |bi he| + he|ρ̂|bi |ei hb| + he|ρ̂|ei |ei he| ,(7.7)

ρ̂d = hd|ρ̂|di |di hd| ,in which we momentarily drop the AL subscript for notational simplicity. This simplifies theformalism because the initial particle-cavity stateρ̂(0) = [x |bi hb| + (1 − x) |di hd|] ⊗ |αi hα|

(7.8)

144
explicitly assumes there are no coherences between the three internal states. Taking a time derivative and using the master equation Eq. (7.1) in the form

∂ ρ̂
1 
†
=
Ĥeff ρ̂ − ρ̂Ĥeff+ L̂jump ρ̂,
∂t
i~

(7.9)

we findρ̂˙ d = γ he|ρ̂|ei |di hd| ,i
1 h†ρ̂˙ b =Ĥeff ρ̂b − ρ̂b Ĥeff.
i~(7.10)

Here, we used hi| L̂jump |ji = 0 for i, j ∈ {b, e} and definedĤeff = Ĥ −i~γ|ei he| ,2

L̂jump ρ̂ = γ he|ρ̂|ei |di hd| .

(7.11)

We first diagonalize the non-Hermitian matrix Ĥeff in the subspace {|e, n − 1i , |b, ni}√~  −iγ g n (7.12)
Ĥn = 

√
2
g n
0
(n)

(n)

for n > 0. The eigenvalues are λ± = −i~Λ± , withγi(n)
Λ± ≡ ±
42

r
ng 2 −

γ2,4(7.13)(n)and the kernel of the matrix Ĥn − λ± În is spanned by
√
−g 2n



(n)
~  −iΛ+
V̂n =

√
cn
g n
2

(n)

−iΛ+


(7.14)






P
(n) 2
with normalization factor c2n = −i~Λ++ ~2 g 2 n/4. We now define V̂ = n V̂n ⊗ În so that thebright manifold matrix can be written as ρ̂b = V̂ ρ̂V V̂ † .
We can use the method of Laplace transforms, L[Â](s) =R∞0dte−st Â, to solve for the trans-

formed density matrix:˙sL[ρ̂] = L[ρ̂](s)+ ρ̂(0),(7.15)

which we then can solve for steady-state:˙ρ̂(t → ∞) = lim L[ρ̂](s)+ ρ̂(0).
s→0(7.16)

145
Projecting with hd| and |di and using the master equation Eq. (7.1), hd|ρ̂b |di = 0, and he|ρ̂d |ei = 0,
we findρ̂d (t → ∞) = γ lim he|L[ρ̂b ](s)|ei + hd|ρ̂d (0)|di .

(7.17)s→0

To solve for the first term on the right-hand side of Eq. (7.17), we findhi
X
L[ρ̂V ](s) = L V̂ −1 ρ̂b (V̂ † )−1 (s) =R̂n0 ,n ⊗ Iˆn0 ,n ,(7.18)

n,n0 ≥1

where Iˆn0 ,n is a matrix with 1 at position n0 , n and zeros everywhere else, andR̂n0 ,n =~2 
cn0 c∗n √(n)∗
ig n0 Λ+√g 2 nn0



(n0 )

(n0 )

(n)∗

4(s+Λ+ +Λ+ )√ (n0 )
ig nΛ+

−

(n0 )

(n)∗

2(s+Λ− +Λ+(n)∗2(s+Λ+ +Λ−))(n0 ) (n)
Λ+ Λ+
(n0 )
(n)∗
(s+Λ− +Λ− ) 0 hn |αi hα|ni .
(7.19)

We next calculateL[ρ̂b ](s) =Xn,n0 ≥1

V̂n0 R̂n0 ,n V̂n† ⊗ Iˆn0 ,n ,(7.20)and project with he| and |ei. Since the excited state in our basis in Eq. (7.12) is |e, n − 1i, Eq. (7.17)
becomesρ̂d (∞) = |di hd| ⊗ (1 − x) |αi hα| + x|α2 |Xl,l0 ≥0

Kl,l0 hl0 |αi hα|li |l0 i hl| ,(7.21)with−1
l + l0 (l − l0 )2+,
Kl,l0 ≡ 1 +28m(7.22)

in which
1
m=
2

 2
γ
g

(7.23)

is the particle-cavity critical photon number. The quantity m characterizes the nonlinearity of theparticle-cavity interaction in the sense that the particle will undergo many (few) Rabi oscillations
before emitting a spontaneous photon in the strong (weak)-coupling regime m  1 (m  1) [96].
Projecting hb| and |bi on Eq. (7.16) and noting that the steady-state bright state population
results from the cavity being in the vacuum state |0i, we have2ρ̂b (∞) = xe−|α| |bi hb| ⊗ |0i h0| .

(7.24)

146
Tracing over the atomic states, we find that the final cavity field state isρ̂L (∞) = (1 − x) |αi hα| + xρ̂c ,(7.25)

in which2ρ̂c = e−|α| |0i h0| + |α|2Xl,l0 ≥0

Kl,l0 hl0 |αi hα|li |l0 i hl|(7.26)

characterizes the effects of the interaction.

7.5

Q-function for the final cavity field state
To gain intuition for the differences between the initial and final cavity states, we calculate

their Husimi Q-functions, which are defined as
Q(β) ≡

hβ|ρ̂|βi.
π(7.27)

Here, the coherent states |βi form a basis for the 2-dimensional optical phase space (Re(β), Im(β)).
The initial cavity state has Q-function
Q0 (β) =hβ|ρ̂L (0)|βi| hα|βi |21
2== e−|α−β| ,πππ(7.28)

while the Q-function for the final cavity state is
Q∞ (β) =hβ|ρ̂L (∞)|βi.
π(7.29)The latter has a more complicated form since it is generally no longer a coherent state. Fig. 7.2
presents numerical plots of Q∞ (β) for various m. To focus on the effects of the interaction, wechoose x = 1. We also choose α = 10 so that Q0 (β) is localized and centered on the coordinate(Re(β), Im(β)) = (10, 0) with a spread on the order of unity.
Fig. 7.2(a) displays Q∞ (β) for the intermediate-coupling case m = 1. We find that Q∞ (β)
does not differ significantly from Q0 (β), which indicates that the cavity field nearly remains in a
coherent state. In the case of stronger coupling [m = 10−2 , Fig. 7.2(b)], however, Q∞ (β) remains
localized near |β| = |α|, but is spread over a larger phase range ∆φ. In the infinite-coupling limit

147
m → 0 [Fig. 7.2(c)], we find that Q∞ (β) has a uniform phase distribution, and therefore differs
substantially from Q0 (β).
Let us now interpret these results. The function Q∞ (β) always remains localized near thecircle defined by |β| = |α| = 10 because the average number of photons in the cavityn̄ = hâ† âi(7.30)

doesn’t significantly change. This is a consequence of the particle’s internal state structure [seeFig. 7.1(b)], which prevents a reduction of n̄ by more than one, and our choice of a large initialintracavity photon number: n̄0 ≡ n̄(0) = 100 = |α|2 . On the other hand, we observe diffusion of
the phase φ as m decreases because the particle undergoes more Rabi oscillations with each Fockstate before emitting a spontaneous photon, thereby scrambling the cavity field’s phase coherence.
The coherences vanish completely in the limit m → 0 (as can be seen from the m-dependence ofKl,l0 [Eq. (7.22)]), resulting in the uniform phase distribution shown in Fig. 7.2(c). This potentiallyextreme change in the phase distribution is largely responsible for the distinguishability of theinitial and final cavity states, which we quantify in terms of their fidelity in the next section.

Figure 7.2: The Q-function Q∞ (β) for the final cavity state ρ̂L (∞) [see Eq. (7.29)] for various m.
√
We set x = 1 and α = n̄0 = 10. Although the amplitude of the state remains localized near√|β| = n̄0 = 10, the spread in phase φ increases as m decreases.

148

7.6

Calculating the fidelity between the initial and final cavity field states
We now quantify the alteration of the cavity field by calculating the Uhlmann-Jozsa fidelity
 q

p p 2
F (ρ̂, σ̂) = Trρ̂σ̂ ρ̂(7.31)

between its initial and final states, which is a generalization of the transition probability for purestates [89, 182]. The fidelity satisfies 0 ≤ F (ρ̂, σ̂) ≤ 1 for any two density matrices ρ̂ and σ̂, withF (ρ̂, σ̂) = 1 if and only if ρ̂ = σ̂, and F (ρ̂, σ̂) = 0 if ρ̂ and σ̂ have support on orthogonal subspaces.
In particular, if we find that F (ρ̂, σ̂) 6= 1, then the cavity field is no longer in the coherent state|αi.
Because the cavity field is initially in a pure state, its fidelity with the final cavity stateρ̂L (∞) is simply
F ≡ F [ρ̂L (0), ρ̂L (∞)] = hα|ρ̂L (∞)|αi .

(7.32)We point out that F = πQ∞ (α), i.e., the fidelity is (up to a factor of π) the Q-function of the final
cavity state evaluated at β = α. We can therefore qualitatively predict features of the fidelity F
through the plots in Fig. 7.2. For example, we expect F to decrease as m decreases due to theassociated diffusion of the Q-function.
As seen from Eq. (7.25),
F = 1 − x [1 − f (n̄0 , m)] ,in whichf (n̄0 , m) ≡ e−2n̄0 1 + n̄0Xl,l0 ≥00n̄l+lKl,l0 0 0 
l!l !

(7.33)(7.34)

is the fidelity conditioned on the particle beginning in the bright state. We discuss this result in
Section 7.7.

7.6.1

Fidelity in the thermodynamic limit

We now calculate an analytic expression for F in the thermodynamic limit n̄0 → ∞, which isof interest if there are many photons in the cavity (as in laser cooling and optical pumping). Using149
the Laplace transform of Kl,l0 [see Eq. (7.22)] and the Fourier transform of the resulting Gaussian
term to reduce its order, we find that the sum in Eq. (7.34) can be rewritten as
Z ∞
Z ∞ r
0
X
n̄l+l
2m −s 2n̄0 e−s/2 cos u −2mu2 /s
0
Kl,l0 0 =
du
ds
e e
e
.
l!l
!
πs
−∞
0
0

(7.35)

l,l ≥0In the limit n̄0  1, the integrand in Eq. (7.35) is localized around s = 0 and u = 2πp, where
p is an integer. This can be seen from the dominating term exp 2n̄0 e−s/2 cos u . Expanding theargument of this term to lowest order in s and u, we find thate2n̄0 e−s/2 cos u≈ e2n̄0 e−sn̄0X2

e−n̄0 (u−2πp) .

p∈Z(n̄0  1)

(7.36)√
From this result, we see that the relevant domains are 0 < s < 1/n̄0 and |u − 2πp| < 1/ n̄0 .
Let us now consider the size of the integrand in Eq. (7.35) for fixed s. The Gaussian term2e−2mu /s dampens the integrand for increasing u. Along with the term in Eq. (7.36), the integrandis maximized when u = 0, and the next largest maxima occur when u = ±2π. Therefore, we
2

can neglect all the components other than p = 0 in Eq. (7.36) if the Gaussian term e−2mu /s issufficiently small for all p 6= 0. This is true when2

1  e−8π m/s

=⇒

m>

s
.
8π 2

(7.37)

Because the largest relevant value is s = 1/n̄0 , this condition is strictest whenm > mmin ≡1.
8π 2 n̄0(7.38)

Because we are interested in the thermodynamic limit n̄0 → ∞, we can safely approximate
Eq. (7.36) with only the p = 0 component, and our result will be valid for all m. Using theseresults, the sum in Eq. (7.35) becomes
Z ∞
Z ∞ r
0
X
2m −n̄0 s −(n̄0 + 2m )u2
n̄l+l
2n̄0
0
s
e
e
Kl,l0 0 ≈ e
du
ds
l!l !
πs
−∞
0
0
l,l ≥0=e2n̄0 √n̄0√

2πme2m erfc( 2m).

(m > mmin ; n̄0  1)

(7.39)

Substituting the result into Eq. (7.33) and applying the thermodynamic limit n̄0 → ∞, wefind thathi√√lim F = 1 − x 1 − 2πme2m erfc( 2m) .

n̄0 →∞(7.40)

150

7.7

Analyzing the fidelity result
We now analyze the fidelity F between the initial and final cavity states after its interaction

with the particle, which was calculated in the previous section. But first, we address a common
argument that claims that the cavity state should be unchanged (F = 1). It is often argued that a
coherent state is unaffected by the interaction because it is an eigenstate of the annihilation operatorâ by definition: â |αi = α |αi, and therefore is unperturbed by photon absorption. However, what
this argument fails to consider is the stimulated emission of a photon back into the cavity, which
can occur if the particle-cavity coupling rate is sufficiently large. With this additional process in
mind, we emphasize that a change of the cavity state is expected since it implies the action of thecreation operator â† [see Eq. (7.2)] on the coherent state, which yields nontrivial dynamics [5].
Nevertheless, we have calculated the fidelity F between the initial and final cavity states [seeEq. (7.33)], and provide a numerical contour plot for the case x = 1 (so that F = f ) in Fig. 7.3(a).
We find that f  1, and therefore that the cavity state is significantly altered, when n̄0 ≥ 1 andm < 1. This region of parameter space corresponds to a cavity containing at least one photon (onaverage) and a strong cavity-particle coupling, respectively.
Because a cooling laser field contains many photons, we focus on the thermodynamic limitn̄0 → ∞ for the remainder of our investigation, although we include some interesting results forfinite n̄0 towards the end of the chapter in Section 7.11. Fig. 7.3(b) displays the corresponding
fidelity [Eq. (7.40)], again for the choice x = 1, and a numerical result for n̄0 = 10, which agree
very well when m is sufficiently large, i.e., it satisfies Eq. (7.38).
We now consider the strong (m  1) and weak (m  1) coupling limits. From Eqns. (7.33)
and (7.40),
F ≈



√


1 − x(1 − 2πm),

m1


x


1 −
,
4m

m1

.

(7.41)

Therefore, in the infinite-coupling limit m → 0, the fidelity becomes F = 1 − x. This shows thatthe cavity field can be altered by the interaction provided that the particle has a chance of starting

10!1

100

Fidelity F

10!1(b) 100

Fidelity F

70
n

101

10!1

151

100

(a) 10210!2n
7 0 = 10n
70 ! 1
mmin

10!2
10!4 10!3 10!2 10!1
m

100

101

Figure 7.3: Fidelity F between the initial and final cavity states ρ̂L (0) and ρ̂L (∞) [Eq. (7.31)] when
the particle begins in the bright state [x = 1, see Eqns. (7.33) and (7.34)] as a function of the initialintracavity photon number n̄0 and the critical photon number m. (a) Contour plot of F . Whenm < 1 and n̄0 > 1, we find that F  1, signaling a significant change in the cavity state. (b)Numerical (solid, n̄0 = 10) and analytical [dashed, Eq. (7.40)] results for F as a function of m. Form < mmin [dot-dashed, Eq. (7.38)], the results diverge.

in the bright state (0  x ≤ 1). However, in the zero-coupling limit m → ∞, we find that F = 1,
and the field is not altered.
It is also interesting to consider how small the critical photon number must be for the initialand final cavity states to be substantially distinguishable, e.g., for their fidelity to be F = 21 . For
simplicity and to achieve the greatest effect, we define such a critical photon number m1/2 in thecase when the particle is prepared in the bright state. From Eq. (7.40), we find that this occurs
when
m = m1/2 ≈ 0.0936.

(7.42)

One can therefore observe a substantial alteration of the cavity state in an experimental setting bystudying a system satisfying m ≤ m1/2 , or more generally, g ≥ 2.3γ  κ.
These results are direct evidence that the cavity state can be altered by interaction with aparticle, even in the limit of infinitely many photons, if the particle-cavity coupling is sufficiently

152
large. We interpret the deviation from F = 1 as result of the development of correlations between
the particle and cavity states through the coherent interaction ĤAL [see Eq. (7.2)].

7.8

Bayesian analysis
Now that we have demonstrated an alteration of the cavity state after interaction with theparticle, we consider if any information about the particle is imprinted on the cavity field. As
discussed in Section 7.5, the change of the cavity state manifests primarily in its phase. Thissuggests that simply measuring the field intensityhâ† â(∞)i = n̄0 − x(1 − e−n̄0 )(7.43)

would not provide an effective way to extract this information, as it does not access the phase of
the state. Moreover, the potentially broad photon number distribution could introduce photon shotnoise. To create a measurement with which we can distinguish differences in both amplitude andphase, we construct a noise-reduced method that can in principle measure the information encodedin the cavity after a single experimental iteration. We first reduce the number of cavity photons
by applying the inverse of the displacement operator defined by the initial coherent state:η̂L ≡ D̂(−α)ρ̂L (∞)D̂† (−α),(7.44)

which can be done, e.g., by removing a cavity mirror at t → ∞ and feeding the cavity state and acoherent state |αi into separate ports of a beam splitter [144]. (We emphasize that any information
about the particle contained in the cavity state is unaltered because this operation is unitary.)
Consider a situation wherein the particle is prepared in some diagonal mixed state [Eq. (7.4)],
and we are tasked with determining the initial particle state (“start in |bi” or “start in |di”) fora given experimental run by performing measurements exclusively on the displaced cavity stateη̂L . For simplicity, we calculate the results for the maximally mixed state (x = 12 ), but our
results can be generalized to any x. By symmetry, the most successful approach without performing any measurements would be to sample from a flat probability distribution (the “prior”):

153
P (start in |bi) = P (start in |di) =1
2,

for which the probability that we would be correct is

Pprior (correct) = 12 . What we show here is that a more accurate probability distribution canbe constructed by incorporating the results from a single measurement of the displaced cavity photon number distribution hn|η̂L |ni, thereby proving that information about the particle is present
in the cavity field.
For our purposes, it is sufficient to reduce the outcome space to a binary scenario: either wedetect (i) zero photons (n = 0, “no click”) or (ii) one or more photons (n ≥ 1, “click(s)”). This
simplification is appropriate because we gain no additional information about the particle’s initialstate by distinguishing between nonzero numbers of clicks. With this approach, we only need to
calculate the vacuum state population h0|η̂L |0i, which is conveniently equivalent to the fidelity F[see Eq. (7.33)]. Notice that one can experimentally probe F by measuring this population. The
probability distribution in the event of “no clicks” is then




1,
P (no click|start in |ii) =f (n̄0 , m),i=d,(7.45)

i=b

where f (n̄0 , m) is the conditional fidelity. The “click(s)” conditional probability distribution is complementary to Eq. (7.45). With these results, we can use Bayesian inference to construct a posterior
probability distribution for the initial particle state conditioned on the cavity measurement:
P (start in |ii |C) ∝ P (C|start in |ii)P (start in |ii);(7.46)i ∈ {b, d}; C ∈ {click(s), no click}.
Here, P (start in |ii) is the (initially flat) prior distribution for the initial particle state [83]. The
posterior probability distribution is
P (start in |di | click(s)) = 0;
P (start in |bi | click(s)) = 1;
1;
1 + f (n̄0 , m)f (n̄0 , m)
P (start in |bi | no click) =.
1 + f (n̄0 , m)

P (start in |di | no click) =(7.47)

154
It is only left to demonstrate that the posterior probability distribution predicts the initial
particle state more accurately than the prior probability distribution, i.e., that we would correctlypredict the initial particle state with a probability satisfying Ppost (correct) > Pprior (correct) = 12 ,by sampling from Eqns. (7.47). Using the posterior probability distribution, the probability that
we correctly predict the initial particle state is
Ppost (correct) =1.
1 + f (n̄0 , m)

(7.48)

Since 0 < f (n̄0 , m) < 1, we find that Ppost (correct) ≥ 21 , and therefore conclude that information
about the particle is present in the cavity field. We emphasize that we have increased our chanceof predicting the initial particle state with a single measurement of the cavity state.
We now focus on the thermodynamic limit n̄0 → ∞, as often occurs in laser cooling andoptical pumping. In this limit, the conditional fidelity f is given by Eq. (7.40). Using this form off in Eq. (7.48), we find that


√


1 − 2πm,
Ppost (correct) ≈



1
1


1+
,

2
8m

m1

.

(7.49)

m1

In the infinite-coupling limit m → 0, Ppost (correct) = 1. Therefore, a single measurement of thecavity state can in principle predict the initial particle state with 100% accuracy, regardless of thehigh-noise photon statistics. This is possible because in this limit the equilibrium cavity states
resulting from the particle starting in |bi and |di have orthogonal support, as evidenced by theirvanishing fidelity {see Eq. (7.33) and Ref. [108]}. In the zero-coupling limit m → ∞, however,Ppost (correct) = 12 = Pprior (correct), so the cavity measurement does not increase our chance ofpredicting the initial particle state.

7.9

Calculating entropies and mutual information
In the next two sections, we switch gears and investigate the particle-field interaction throughthe lens of information theory, which provides us with a way to quantify the entropy transfer process. This perspective is appropriate because because we posit that the amount of transferred

155
entropy is characterized by the mutual information [139] shared between the initial particle state
and the equilibrium cavity state. To connect the formalism of this section with the Bayesian inference approach of the previous section, we first calculate the amount of mutual information shared
between the initial particle state and final cavity field as determined by the classical conditional
probability distribution we derived through the photon number measurements [Eqns. (7.47)]. Then,
we calculate their quantum mutual information from a density matrix approach, which incorporates
all particle-cavity correlations. We reserve the analysis of these results for Section 7.10.
We briefly reminder the reader of some essential concepts from information theory. In general,mutual information is defined asI(Y : Z) = S(Y ) + S(Z) − S(Y Z) = S(Y ) − S(Y |Z),(7.50)

in which Y and Z are probability distributions, {S(Y ), S(Z)} are their information entropies,
S(Y Z) is their joint entropy, and S(Y |Z) is the conditional entropy of Y given Z. The entropy S
is given by the Shannon entropy
S(Ycl ) = −X

P (y) ln P (y)(7.51)yfor a classical probability distribution Ycl with events y and probabilities P (y), whereas it is givenby the von Neumann entropyS(ρ̂Y ) ≡ −Tr [ρ̂Y ln ρ̂Y ] .

(7.52)

for a quantum probability distribution, which is described by a density matrix ρ̂Y . Intuitively,
I(Y : Z) is equal to zero in the absence of any correlations between the two distributions, and ismaximized if the higher-entropy distribution contains all of the information of the lower-entropy
distribution:
0 ≤ I(Ycl : Zcl ) ≤min[S(Ycl ), S(Zcl )],(7.53)

0 ≤ I(ρ̂Y : ρ̂Z ) ≤ 2 min[S(ρ̂Y ), S(ρ̂Z )].
Notice that the upper bound of quantum mutual information is twice as large as its classicalcounterpart [106].

156
7.9.1

Mutual information from cavity measurements

Here we calculate the (classical) mutual information [Eq. (7.50)] between the initial particle
state and equilibrium cavity state as determined by the cavity measurements introduced in Section 7.8. Using the click probabilities from Eq. (7.33), the notation of Eq. (7.46), and generalizing
the conditional probability distribution in Eqns. (7.47) to any x, the conditional entropy of theinitial particle state given the cavity measurements is
S[A(0)|L(∞)] = −XC,i

P (C)P (start in |ii |C) ln[P (start in |ii |C)]

(7.54)




fx
1−x
= −xf ln
− (1 − x) ln.
1 − x(1 − f )1 − x(1 − f )From the classical mixture in Eq. (7.4), the Shannon entropy of the initial state isS[A(0)] ≡ S0 = −x ln x − (1 − x) ln(1 − x).

(7.55)

Together, Eqns. (7.54) and (7.55) can be used to calculate the mutual information [Eq. (7.50)]
between the initial particle state and equilibrium cavity state as determined by the cavity measurements:
I[A(0) : L(∞)] = S[A(0)] − S[A(0)|L(∞)].
7.9.2(7.56)

Quantum mutual information

Let us now explain how to calculate the quantum mutual information between the initial
particle state and final cavity state. As seen from Eq. (7.50), this calculation requires knowing
the joint distribution of these two states. Because these states are defined at different times, we
calculate their joint distribution by incorporating an non-interacting auxiliary Hilbert space R
which purifies ρ̂A (0) and hence contains the entropy of ρ̂A (0) at all times. We purify the initial
particle state through its Schmidt decomposition. That is, we view the particle ensemble as the
reduced density matrix of a pure state, ρ̂A (0) = TrR |ui hu|AR , where|uiAR =

Xp
βi |ψi iA ⊗ |φi iR .
i(7.57)

157
Here, |ψn iA is an eigenvector of ρ̂A (0) with eigenvalue βn and |φn iR ∈ R for an auxiliary Hilbert
space R. It can be shown that the reduced density matrices ρ̂A (0) and ρ̂R have the same eigenvalues[139], which motivates the interpretation of ρ̂R = TrA |ui hu|AR as an identically prepared ensembleto ρ̂A (0), but with particles that are not interacting with the field. The total system’s density
matrix becomes ρ̂ARL , and the master equation [Eq. (7.1)] is edited by incorporating the identity
operator of R:
Jˆ → Jˆ ⊗ ÎR .

ĤAL → ĤAL ⊗ ÎR ;(7.58)

This updated master equation is then used to evolve the initial pure stateρ̂ARL (0) = |ui hu|AR ⊗ |αi hα|L .

(7.59)

We can then calculate the entropy of any subset of the ARL composite Hilbert space by performing
the appropriate trace operations.
Using this formalism, the quantum mutual information
I(ρ̂R : ρ̂L ) = S(ρ̂R ) + S(ρ̂L ) − S(ρ̂RL ),(7.60)

can then be calculated as follows. Because we prepare the initial particle state asρ̂A (0) = x |bi hb| + (1 − x) |di hd| ,(7.61)

then it follows that an appropriate Schmidt decomposition is|uiAR =√

x |b, bi +√1 − x |d, di .

(7.62)

The corresponding initial (pure) state of the entire ARL (particle + auxiliary particle + cavity)
system is thenρ̂ARL (0) = |ui hu|AR ⊗ |αi hα|nop= x |b, bi hb, b| + (1 − x) |d, di hd, d| + x(1 − x) [|b, bi hd, d| + h.c.] ⊗ |αi hα| .

(7.63)

158
We now evolve each term in Eq. (7.63) to its corresponding equilibrium state by using Eqns. (7.21)
and (7.24):
ρ̂ARL (∞) =x |bi hb|R ⊗ e−n̄0 |bi hb|A ⊗ |0i h0|L + n̄0 |di hd|A ⊗+ (1 − x) |d, di hd, d| ⊗ |αi hα| +Xl,l0 ≥0

Kl,l0 hl0 |αi hα|li |l0 i hl|L  (7.64)px(1 − x)e−n̄0 /2 (|b, bi hd, d| ⊗ |0i hα|L + h.c.) .

From this result, the reduced density matrices in the RL, R, and L subspaces areρ̂RL (∞) = x |bi hb| ⊗ ρ̂c + (1 − x) |di hd| ⊗ |αi hα| ;(7.65)

ρ̂R (∞) = x |bi hb| + (1 − x) |di hd| ;ρ̂L (∞) = xρ̂c + (1 − x) |αi hα| ,where ρ̂c is given in Eq. (7.26). Notice that ρ̂RL (∞) is separable, and is therefore not an entangled state. The quantum mutual information I(ρ̂R : ρ̂L ) [Eq. (7.60)] can be calculated (at least
numerically) directly from this result. We can simplify the result, however, since some of the von
Neumann entropies can be expressed in terms of the quantity S(ρ̂c ):S[ρ̂RL (∞)] = S0 + xS(ρ̂c );(7.66)

S[ρ̂R (∞)] = S0 ,in which S0 ≡ S[ρ̂A (0)]. We do not investigate the behavior of S(ρ̂c ), but we emphasize that ρ̂c isgenerally not pure, and hence has nonzero entropy. The von Neumann entropy of the L subspace
S[ρ̂L (∞)] is harder to calculate analytically because ρ̂c and |αi hα| do not always have orthogonal
support. We can, however, use the identity [139]
!
X
X
X
X
pi S(ρ̂i ) ≤ S
pi ρ̂i ≤pi S(ρ̂i ) −pi ln pii

i

i

(7.67)ito place an upper and lower bound on the cavity field entropy. From Eqns. (7.65) and (7.67),
xS(ρ̂c ) ≤ S[ρ̂L (∞)] ≤ S0 + xS(ρ̂c ).
The upper bound is saturated only if ρ̂c and |αi hα| have orthogonal support.

(7.68)

159

7.10

Analyzing the entropies and mutual information
We now analyze the results of the entropic perspective introduced in the previous section.

Because the cavity state is inherently quantum, the photon click measurements (which correspondto the diagonal elements of the cavity density matrix) may not reveal all of the information contained in the light field. Consequently, Eq. (7.56) underestimates the amount of mutual information
shared between the particle and cavity, so we will mostly use the results from the quantum calculation. However, Eq. (7.56) is useful for approximating the quantum mutual information in thethermodynamic limit n̄0 → ∞, for which numerical calculations are intractable due to the infinite
dimension of the cavity Hilbert space.
It is useful to first consider the upper and lower bounds of the quantum mutual information.
Because ρ̂RL is separable [see Eq. (7.65)], the quantum mutual information satisfies Eq. (7.53)
without the factor of 2 [139]:
0 ≤ I[ρ̂R (∞) : ρ̂L (∞)] ≤ min{S0 , S[ρ̂L (∞)]} ≤ S0 ,(7.69)

in which S0 ≡ S[ρ̂A (0)]. In other words, somewhere between none [I(ρ̂R : ρ̂L ) = 0] or all [I(ρ̂R :ρ̂L ) = S0 ] of the entropy initially contained in the particle becomes encoded in the cavity field. An
important consequence of the classicality of Eq. (7.69) is that the spontaneous photons escaping
the system contain all of the information that becomes imprinted on the final cavity state, which
we demonstrate in Section 7.10.1.
Fig. 7.4(a) presents various quantum entropic quantities as a function of time t for the choices√√
x = m = 21 and α = n̄0 = 5. These results were calculated by numerically evolving the quantum
master equation [Eq. (7.1)] with the aforementioned adjustments [Eqns. (7.58) and (7.59)], and thencalculating the von Neumann entropies [Eq. (7.52)] and quantum mutual information [Eq. (7.60)] of
the various subspaces. For convenience, we have scaled all quantities by the initial particle entropy
S0 . As expected, the particle entropy S[ρ̂A (t)] decreases (as occurs in laser cooling and opticalpumping), the cavity state entropy S[ρ̂L (t)] increases, and the entire ARL space entropy increases
as it evolves under the non-unitary dynamics. Importantly, the quantum mutual information

1.5
1

S(^
;ARL )
S(^
;A )
S(^
;L )
I(^
;R : ;^L )

0.5
0

0
5
10
15
time t (units of 1=.)

;R : ;^L ) (units of S0 )I(^

Entropies (units of S0 )

160

(b)

(a)

1

0.8
0.6

n
70 = 1n
7 0 = 10n
7 0 = 100n
70 ! 1

0.4
0.2
0
10!6

10!3
100
m

Figure 7.4: Entropic quantities scaled by the initial particle entropy S0 ≡ S[ρ̂A (0)]. (a) von Neumann entropies S[ρ̂M (t)] [see Eq. (7.52)] of various subspaces M and quantum mutual information
between the field and auxiliary particle I(ρ̂R : ρ̂L ) [see Eq. (7.60)] as a function of time t. Parameters: x = 0.5, m = 0.5, and n̄0 = 5. (b) Equilibrium I(ρ̂R : ρ̂L ) as a function of m for various n̄0with x = 0.5. The n̄0 → ∞ curve is given by Eq. (7.56). The quantum mutual information reaches
its maximum value I(ρ̂R : ρ̂L ) = S0 [see Eq. (7.69)] in the thermodynamic, infinite-coupling limit:(n̄0 , m) → (∞, 0).

I(ρ̂R : ρ̂L ) increases and equilibrates to a nonzero value, which indicates the imprinting of theparticle’s initial entropy onto the cavity field.
We present numerical results for the equilibrium value of I(ρ̂R : ρ̂L ), scaled by S0 , as a functionof m for various n̄0 in Fig. 7.4(b) (non-solid curves). These are calculated from the equilibrium
density matrices [Eq. (7.65)]. We find that I(ρ̂R : ρ̂L ) can be as large as S0 /2, and hence that asignificant amount of information about the initial particle state can be imprinted on the cavity
field, for n̄0 ≥ 1, provided that m is sufficiently small. As n̄0 increases, we find numerically that theequilibrium quantum mutual information [Eq. (7.60) with the density matrices given by Eq. (7.65)]
and mutual information as determined by the cavity measurements [Eq. (7.56)] converge. Therefore,
we use Eq. (7.56) to analytically calculate the equilibrium value of I(ρ̂R : ρ̂L ) in the thermodynamic

161
limit n̄0 → ∞ (solid curve). In the strong and weak-coupling limits, this simplifies to
"


2 #

1
−
x
1


S0 − (x, m), m  min 1,
2πxI(ρ̂R : ρ̂L ) ≈
,


x
ln
x

−
,
m1
4m
in which




√
√
x
−1
(x, m) = − 2πm ln 2πm + ln
1−x(7.70)(7.71)

is a small positive number. The additional constraint on m in the strong-coupling limit is aconsequence of the highly nonlinear behavior of I(ρ̂R : ρ̂L ) for small m. In the zero-coupling limit
m → ∞, we find that I(ρ̂R : ρ̂L ) = 0, so the cavity field does not contain the information entropyof the initial particle state, as expected. However, in the infinite-coupling limit m → 0, we findthat I(ρ̂R : ρ̂L ) = S0 , which means that the cavity field contains complete information about theinitial particle state. This agrees with our Bayesian inference result in Section 7.8.
Let us now prove analytically that the cavity contains complete information about the particle
only in the thermodynamic, high-coupling limit, i.e., that I(ρ̂R : ρ̂L ) = S0 if n̄0 → ∞ and m → 0.
From Eq. (7.68), this occurs only if the density matrices ρ̂c and |αi hα| have orthogonal support [139].
This condition is equivalent toF (ρ̂c , |αi hα|) = 0,
where F is the fidelity [see Eq. (7.31)]. Using Eq. (7.25), this can be rewritten as


l+l0
X
n̄
e−2n̄0 1 + n̄0
Kl,l0 0 0  = 0.
l!l !
0

(7.72)(7.73)

l,l ≥0Because the term in parentheses is positive, the only way to satisfy Eq. (7.73) is to consider thethermodynamic limit n̄0 → ∞, for which the exponential tends to zero. In this limit, we can thenuse Eq. (7.39) to simplify Eq. (7.73) to√√2πme2m erfc( 2m) = 0,(7.74)

which is true only when m → 0. Therefore, we come to the conclusion that I(ρ̂R : ρ̂L ) = S0 , andtherefore that the cavity state contains all of the information about the initial particle state, onlyin the high-coupling (m → 0), thermodynamic (n̄0 → ∞) limit.

162
7.10.1

Entropy of the spontaneous photon

We previously claimed that the spontaneous photon emitted by the particle contains all of
the information that becomes encoded in the cavity state. We prove this claim here.
Consider the quantum state describing the particle A, cavity field L, and environment P(which we modeled as an infinite bandwidth bosonic bath). We remind the reader that in Section 2.1.5 we discussed how the environment is responsible for the emission of spontaneous photons
by the particle. In our Gedankenexperiment, these three subspaces constitute the entire universe,
i.e., they form a closed system and undergo unitary evolution. If we purify the particle state (asdone previously) with the auxiliary Hilbert space R, then the density matrix of the entire system
at any time t isρ̂ARLP (t) = |ΨARLP (t)i hΨARLP (t)| ,(7.75)

which is obviously pure. It follows that the entanglement entropy between the ARL and P subsystems satisfies
S[ρ̂ARL (t)] = S[ρ̂P (t)].

(7.76)

Suppose we propagate the entire ARLP system to equilibrium. For n̄0  1, the equilibrium
particle state ρ̂A (∞) is approximately pure (or completely pure for n̄0 → ∞) because the particle
almost certainly ends in the dark state |di:ρ̂A (∞) = xe−n̄0 |bi hb| + (1 − xe−n̄0 ) |di hd| ≈ |di hd| .

(7.77)

It immediately follows thatρ̂ARL (∞) ≈ |di hd| ⊗ ρ̂RL (∞).

(7.78)

S[ρ̂ARL (∞)] ≈ S[ρ̂RL (∞)] = S0 + xS(ρ̂c ),(7.79)The entanglement entropy is thenin which we used Eq. (7.66). Combining this result with Eqns. (7.68) and (7.76),
S[ρ̂L (∞)] ≤ S0 + xS(ρ̂c ) = S[ρ̂RL (∞)] ≈ S[ρ̂ARL (∞)] = S[ρ̂P (∞)],(7.80)

163orS[ρ̂L (∞)] ≤ S[ρ̂P (∞)].

(7.81)

Eq. (7.81) states that the environment S[ρ̂P (∞)] always absorbs at least as much information
entropy as the equilibrium cavity state. Therefore, spontaneous emission is always a sufficientmechanism for the removal of entropy from the system.

7.11

Additional calculations
In this section, we provide some additional calculations that may be of interest in future work.

First, we calculate the complete photon number distribution of the displaced cavity state hn|η̂L |ni
in the large (but not necessarily infinite) n̄0 limit for arbitrary m, and then in the infinite-coupling
limit m → 0 with arbitrary n̄0 . We then calculate the fidelity F in the infinite-coupling limit forarbitrary n̄0 , which may be particularly interesting for studies of cavity QED interactions between
low-photon states and single particles.
Two useful equations in the infinite-coupling limit are the simplification of Eq. (7.22) toKl,l0 = δl,l0

1
,
1+l

(m = 0)where δl,l0 is the Kronecker delta, and the simplification of Eq. (7.25) toX n̄n+10ρ̂L (∞) = (1 − x) |αi hα| + xe−n̄0 |0i h0| +|ni hn| .
(n + 1)!

(7.82)(m = 0)(7.83)

n≥0

7.11.1

Displaced final cavity field photon distribution

Here we calculate the photon number distribution of the final cavity field state after actionof the displacement operator D̂(−α):η̂L ≡ D̂(−α)ρ̂L (∞)D̂(α).

(7.84)

 −2n̄0 nen̄0+ n̄0 Tn ,hn|η̂L |ni = (1 − x)δn,0 + xn!

(7.85)As seen from Eq. (7.25),

164
in whichXTn ≡l,l0 ≥0

Kl,l0 hl0 |αi hα|li hn|D̂(−α)|l0 i hl|D̂(α)|ni .

(7.86)

Using the ladder operationsâ† |li =√l + 1 |l + 1i ;â |li =√l |l − 1i(7.87)and displacement operator identitiesD̂† (α)âD̂(α) = â + α;D̂(α)âD̂† (α) = â − α;D̂(−α) = D̂† (α),(7.88)

we find thatne−2n̄0 n̄n0 X
n!

Tn =

p,q=0

  
0
X
n n
n̄l+l
(−1)q+p
Kl+p,l0 +q 0 0
q
p
l!l !
0

(7.89)

l,l ≥0Using the Laplace transform of Kj+p,j 0 +q [see Eq. (7.22)] and the Fourier transform of the resulting
Gaussian term to reduce its order, we find thate−2n̄0 n̄n0
Tn =
n!

Z ∞

du

Z ∞

−∞

r
ds

0

n
2m −s −s/2 cos u −2mu2 /s
1 + e−s − 2e−s/2 cos u e2n̄0 e
e
.
e
πs

(7.90)In the following two subsections, we compute hn|η̂L |ni under approximations which dependon the values of m and n̄0 .

7.11.1.1n̄0  1, m > mmin

If we are interested in the regime n̄0  1, we can use the same argument as in Section 7.6.1
to simplify Tn . Using Eq. (7.36) with only the l = 0 term in Eq. (7.90), we find that we canapproximate Tn asn̄n
Tn ≈ 0
n!

Z ∞

du

−∞

Z ∞
0

r
ds

2m −n̄0 s 2n −(n̄0 + 2m )u2
s
e
u e
.
πs

(m > mmin ; n̄0  1, m, n)(7.91)

Integrating over u and using Tricomi’s confluent hypergeometric function [154], which has integral
representation
1
U (a, b, z) =
Γ(a)

Z ∞
0

dt e−zt ta−1 (1 + t)b−a−1 ,(Re a > 0)

(7.92)

165
we can write the approximate form of Tn as
2(2n)!m
Tn ≈ nU
4 n!n̄03
n + 1, , 2m .
2(n̄0  1, n; m > mmin )(7.93)

Substituting this result into Eq. (7.86) and taking the thermodynamic limit n̄0 → ∞, we find that2(2n)!mlim hn|η̂L |ni = (1 − x)δn,0 + x
Un̄0 →∞
4n n!


3n + 1, , 2m .
2(7.94)

Invoking Stirling’s approximation, this can be further simplified for n ≥ 1 to√lim hn|η̂L |ni ≈ x 2 2e−n nn m U

n̄0 →∞

7.11.1.2


3
n + 1, , 2m .
2(n ≥ 1)(7.95)

m → 0 and finite n̄0

When m → 0, we can no longer neglect the l 6= 0 terms in Eq. (7.36). In this case, we instead
use Eq. (7.83) and find that
X n̄k+120hn|η̂L |ni = (1−x)δn,0 +xe−n̄0  hn|D̂(−α)|0i +hk|D̂(α)|ni  .
(k + 1)!


2(m = 0) (7.96)

k≥0

Using the identityrhk|D̂(α)|ni =n! k−n −|α|2 /2 (k−n)α
eLn(|α|2 )
k!

(7.97)

when k ≥ n [26, 55], Eq. (7.96) simplifies to
(
n−1
i2
n
X 1 h (n−k)
n̄n+1−2n̄0 n̄00+
Lk
(n̄0 )
hn|η̂L |ni = (1 − x)δn,0 + xe
n!
n!
1+k
k=0
 k
2 )
n! X 1n̄0 (k−n)+ n−1
L(n̄0 ),
1 + k k! nn̄0(m = 0)(7.98)k≥n(q)where Lp (r) are the generalized Laguerre polynomials. Using the identity(−x)n (n−k)(−x)k (k−n)
Lk
(x) =
Ln
(x)
n!
k!

(7.99)

which is valid for all integer n and k, we can simplify Eq. (7.98) tohn|η̂L |ni = (1 − x)δn,0 + xn̄n0 e−2n̄0[1 + n̄0 Sn (n̄0 )] ,n!

(m = 0)(7.100)166
in which
Sn (n̄0 ) =

X
k≥0

i2
1 h (n−k)e2n̄0 (2n)! 1
1√.
Lk(n̄0 ) ≈ n+1
n
1+kn! 4 2 πn̄0n̄0(7.101)The last equality, which is valid when n̄0  n, is demonstrated in Section 7.11.1.3. Using thisresult in Eq. (7.100), we arrive at the displaced final cavity photon distribution:



2n 1
x
√
hn|η̂L |ni ≈ (1 − x)δn,0 +.
nn 4 2 πn̄0(m = 0; n̄0  n)(7.102)For the Fock state populations 1  n  n̄0 , we can expand the n-dependent terms Equation (7.102):hn|η̂L |ni ≈7.11.1.3x√.
2π nn̄0(m = 0; 1  n  n̄0 )(7.103)

Evaluating the Laguerre polynomial sum Sn (n̄0 )To derive the asymptotic limit of the sum in Eq. (7.101), we first introduce the following
identity [30]:
∞
X

(n−k)

Lk

(n̄0 )tk = (1 + t)n e−n̄0 t .

(n−k)(n̄0 )e(k+1)x = ex (1 + ex )n e−n̄0 e .

(7.104)k=0

The choice t = ex leads tog(x) ≡∞
X

Lk

x(7.105)k=0

We next calculate the antiderivative G(x) of g(x) with the ansatz
G(x) =nXx

cl (1 + ex )l e−n̄0 e .

(7.106)l=0

By enforcing G0 (x) = g(x), we obtainn

1 n! X n̄l0xG(x) = −(1 + ex )l e−n̄0 e .
nn̄0 n̄0l!

(7.107)l=0

From the series representation of g(x) in Eq. (7.105) we findG(x) =∞(n−k)
X
L(n̄0 )e(k+1)x
k

k=0

k+1

n

−

1 n! X n̄l0
n̄0 n̄n0l!
l=0(7.108)167
as a series for G. From Eqns. (7.105) and (7.108), we can write down the following Fourier series
∞
(n−k)
X
L
(n̄0 )eikx
k

k=0
∞
X

k+1
(n−k)

Lk

n

i
e−ix n! X n̄l0 h
ix l −n̄0 eix
=
1
−
(1
+
e
)
e
,
n̄0 n̄n0l!

(7.109)l=0ix(n̄0 )eikx =(1 + eix )n e−n̄0 e .

(7.110)k=0

By the orthogonality of complex exponentials, we can write∞Xi2
1 h (n−k)Lk(n̄0 )
k+1
k=0
Z 2π
n
i
−ix n! X1n̄l0 h−ix eix l −n̄0 eix
dx(1 + e−ix )n e−n̄0 e
=
1
−
(1
+
e
)
e
.
2π 0
n̄0 n̄n0l!

Sn (n̄0 ) =(7.111)l=0

In the large n̄0  n limit we only need to consider highest order term l = n. This is appropriate
because we increase n̄0 without increasing n. With this we can approximate
Z 2π

1
Sn (n̄0 ) ≈ −2πn̄0

dx e−ix (1 + e−ix )n (1 + eix )n e−2n̄0 cos(x) .

(7.112)0

We now use a saddle point method. In the large n̄0 limit the measure will concentrate aroundx = π. In that case we can use2e−2n̄0 cos(x) ≈ e2n̄0 −n̄0 (x−π) .

(7.113)

Consequently we obtaine2n̄0
Sn (n̄0 ) ≈
2πn̄0

Z ∞

e2n̄0

Z ∞

−∞

2

du e−iu (1 − e−iu )n (1 − eiu )n e−n̄0 u
2

du u2n e−n̄0 u

≈

2πn̄0

≈

e2n̄0 (2n)! 1
1√n+1 n! 4n 2 πn̄ ,n̄00(7.114)−∞

which we use in Eq. (7.101).

7.11.2

Fidelity in the infinite coupling limit m → 0

If we are instead interested in the case m → 0 with finite n̄0 , we can use Eq. (7.82) to simplify
Eq. (7.33) toX 1  n̄n 2 0
 .
F = 1 − x 1 − e−2n̄0 1 + n̄0
1 + n n!
n≥0(m = 0)(7.115)

168
Using the modified Bessel function of the first kind,
I1 (2y) = y

X
n≥0

1
1+n

 n 2y,n!

(7.116)

Eq. (7.115) becomesF = 1 − x 1 − e−2n̄0 [1 + I1 (2n̄0 )] .

(m = 0)(7.117)

We now consider two cases for n̄0 : (i) the thermodynamic limit n̄0 → ∞, and (ii) the quantum
limit n̄0 → 0, for which we can use the asymptotic expansions
e2n̄0
I1 (2n̄0 ) ≈ √2 πn̄0(n̄0  1);I1 (2n̄0 ) ≈ n̄0(n̄0  1)

respectively. Therefore, to leading order in n̄0 , we conclude that the fidelity is given by1 − xn̄0 ,n̄0 → 0F ≈(m = 0)11 − x 1 − √,n̄0 → ∞.

(7.118)(7.119)2 πn̄0We also consider the limit m → ∞, for whichn + n0 −1
lim Kn,n0 = 1 +.
m→∞
2(7.120)In this case the double sum in Eq. (7.33) simplifies and


−2n̄0

lim F = 1 − x 1 − em→∞

1 − e2n̄0 (1 − 2n̄0 )
1+
2n̄0


.

(7.121)

To leading order, the fidelity is thenlim F =

m→∞

7.12





1 − xn̄ ,n̄0 → 0
1 − 2n̄x0 ,n̄0 → ∞.

0(7.122)Conclusion
We have demonstrated through a simple Gedankenexperiment under what conditions theentropy of a quantum system can be imprinted on a classical coherent light field, which we model

169
as a coherent state in a lossless cavity. We have quantified the cavity state’s alteration due tothe particle-cavity interaction through the measurement of fidelity and proved that the cavity fieldcontains information about the initial state of the particle by the method of Bayesian inference
and using quantum information theoretic techniques. Our results demand reconsideration of theunderlying physics of laser cooling and optical pumping in the strong particle-light coupling regime,
as it suggests that the assumption of an unperturbed light field is not necessarily accurate.
The entropy transfer from the particle to the light field could be realized in an experimental
setting by studying a system that satisfies κ  γ, g. Although we have mainly focused on thehigh-n̄0 limit, we believe that there is also interesting physics in the low-n̄0 regime. Our model
could also be generalized to further understand some cavity-based quantum memories [4, 88, 163]by, e.g., incorporating cavity pumping and loss (κ 6= 0) or initializing the cavity field in a different
quantum state. We also anticipate that this system can be used as a platform to generate andstudy novel photon-subtracted states [6].
There are many ways to extend this study for the purpose of modeling the laser coolingprocess more accurately. Most notably, the incorporation of cavity pumping and loss could permit
a nontrivial equilibrium solution even when the particle can relax back to the bright state, which
would bring the particle’s internal state structure closer to that of typical two-level models. In
this case, one could even remove the background radiation field altogether to further investigate
the entropy dynamics of the laser-particle system [121]. Of course, particle motion could also be
incorporated, potentially allowing for another Hilbert space to absorb or emit entropy.
One could also study this system in the framework of phase space compression. However, therich correlations generated by quantum mechanical processes preclude a clear phase space approachfor quantum systems [17, 18, 143]. This would require, for example, a deeper understanding of theconnection between Wigner trajectories and Liouville’s theorem [165]. On a related note, one could
consider the use of quantum discord as a measurement of quantum correlation [66] as opposed to
von Neumann entropy.

Chapter 8

Conclusion

In this thesis, we investigated three novel techniques for manipulating the motion of atoms
and molecules with laser light and electromagnetic traps, as well as the transfer of entropy from aquantum system to a classically-behaved yet inherently quantum light field. To contextualize andformalize our approach, we provided the essential background material. We began with building up
the physical model for the interaction of quantum particles with light as determined from quantum
theory. This required constructing the Hamiltonian for an isolated atom or molecule, demonstrating
the emergence of its internal state quantum structure, and then coupling it to an electromagnetic
field. We outlined the classical and quantum descriptions of the light field, including the background radiation field which is responsible for spontaneous emission. A discussion of the theoryof open quantum systems naturally followed, in which we recast the Schrödinger equation into aquantum master equation which efficiently models the system of interest and its interaction withthe environment. Next, we presented an algorithm for simulating the master equation called theMonte Carlo wave function method, which we extensively used for approximating the dynamics instudies involving particle motion. To further contextualize the proposed laser cooling and trapping
process, we also provided a brief description of the commonly-implemented Doppler cooling andconventional magneto-optical trap methods.
Our first contribution was the theoretical description of SWAP cooling, a technique which has
the capability to cool atoms and molecules with narrow optical transitions. We explained its basic
mechanism, outlined its operating regime, and demonstrated its ability to generate large slowing

171
forces without the need of spontaneous emission. Through numerical simulations, we demonstrated
its capacity to cool and characterized its minimum achievable temperature. To benchmark its
effectiveness, we compared it to Doppler cooling on the same transition and showed that it requires
fewer spontaneous photons to reach thermodynamic equilibrium. We then connected the theory
to an experimental realization of SWAP cooling on a narrow linewidth transition in 88 Sr. Our
numerical simulations agreed with the experimental data, which corroborated our explanationof SWAP cooling’s underlying physics. Next, we combined the SWAP cooling protocol with aquadrupole magnetic trap to realize a novel magneto-optical trap, which we named the “SWAP
MOT.” We explained the necessary alterations to the cooling protocol and developed an efficient
semiclassical model. We described its effects on particle motion from a phase space trajectory
perspective and proved its potential to exhibit phase space compression with a higher efficiency thana conventional MOT. Our numerical simulations showed that the minimum achievable temperatures
agreed with the SWAP cooling result and that the particles can become tightly confined in space.
We then relaxed the condition of adiabatic evolution by formulating a speedup protocol based on
Lewis-Riesenfeld invariants. We applied it in the context of laser slowing and characterized theparameter regime in which it displays superiority over other common slowing methods. Our final
investigation involved tracking the transfer of information entropy from a quantum particle to aclassically-behaved coherent light field in a lossless optical cavity. We quantified the alteration of
the light field, corroborated many of the typical assumptions regarding the interaction of quantum
matter with laser light, and demonstrated that that in some extreme cases, information about thequantum particle can become encoded in the light field.
We are optimistic that our proposed techniques will prove useful in advancing the scientificcommunity’s ability to control the motion of atoms and molecules. In particular, we hope thatthese procedures will increase the catalog of molecules that can be captured in an optical trap
for use in basic science and quantum computing experiments. Also, we anticipate that our study
on entropy transfer from quantum particles will motivate further investigation into the underlying
physics of (and possibly be used to propose new methods of) optical pumping and laser cooling.

Bibliography[1] Obinna Abah, Ricardo Puebla, Anthony Kiely, Gabriele De Chiara, Mauro Paternostro, andSteve Campbell. Energetic cost of quantum control protocols. New Journal of Physics,
21(10):103048, oct 2019.
[2] A. Abragam. The principles of nuclear magnetism. Clarendon Press, 1961.
[3] C.S. Adams and E. Riis. Laser cooling and trapping of neutral atoms. Progress in Quantum
Electronics, 21(1):1 – 79, 1997.
[4] Mikael Afzelius and Christoph Simon. Impedance-matched cavity quantum memory. Phys.
Rev. A, 82:022310, Aug 2010.
[5] G. S. Agarwal and K. Tara. Nonclassical properties of states generated by the excitations ona coherent state. Phys. Rev. A, 43:492–497, Jan 1991.
[6] G.S. Agarwal. Quantum Optics. Quantum Optics. Cambridge University Press, 2013.
[7] Nitzan Akerman, Michael Karpov, Liron David, Etay Lavert-Ofir, Julia Narevicius, and Edvardas Narevicius. Simultaneous deceleration of atoms and molecules in a supersonic beam.
New Journal of Physics, 17(6):065015, jun 2015.
[8] Leslie Allen and Joseph H. Eberly. Optical Resonance and Two-level Atoms. Dover, 1987.
[9] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of bose-einstein condensation in a dilute atomic vapor. Science, 269(5221):198–201,
1995.
[10] M. Arnal, V. Brunaud, G. Chatelain, C. Cabrera-Gutiérrez, E. Michon, P. Cheiney, J. Billy,and D. Guéry-Odelin. Evidence for cooling in an optical lattice by amplitude modulation.
Phys. Rev. A, 100:013416, Jul 2019.
[11] S. Balasubramanian, Shuyang Han, B. T. Yoshimura, and J. K. Freericks. Bang-bang shortcut
to adiabaticity in trapped-ion quantum simulators. Phys. Rev. A, 97:022313, Feb 2018.
[12] John P. Bartolotta and Murray J. Holland. Sawtooth-wave adiabatic passage in a magnetooptical trap. Phys. Rev. A, 101:053434, May 2020.
[13] John P. Bartolotta, Matthew A. Norcia, Julia R. K. Cline, James K. Thompson, and Murray J. Holland. Laser cooling by sawtooth-wave adiabatic passage. Phys. Rev. A, 98:023404,
Aug 2018.

173[14] John P. Bartolotta, Jarrod T. Reilly, and Murray J. Holland. Speeding up particle slowing
using shortcuts to adiabaticity. Phys. Rev. A, 102:043107, Oct 2020.
[15] Mark G. Bason, Matthieu Viteau, Nicola Malossi, Paul Huillery, Ennio Arimondo, Donatella
Ciampini, Rosario Fazio, Vittorio Giovannetti, Riccardo Mannella, and Oliver Morsch. Highfidelity quantum driving. Nature Physics, 8(2):147–152, Feb 2012.
[16] Peter G. Bergmann and Joel L. Lebowitz. New approach to nonequilibrium processes. Phys.
Rev., 99:578–587, Jul 1955.
[17] Alex E. Bernardini and Orfeu Bertolami. Phase-space continuity equations for quantum
decoherence, purity, von neumann and renyi entropies. Journal of Physics: Conference Series,
1275:012032, sep 2019.
[18] Alex E Bernardini and Caio Fernando e Silva. Anharmonic effects on phase-space quantum
profiles: an exact approach. Journal of Physics: Conference Series, 1612:012002, aug 2020.
[19] M V Berry. Transitionless quantum driving.
Theoretical, 42(36):365303, 2009.

Journal of Physics A: Mathematical and[20] Hendrick L. Bethlem, Giel Berden, Floris M. H. Crompvoets, Rienk T. Jongma, André
J. A. van Roij, and Gerard Meijer. Electrostatic trapping of ammonia molecules. Nature,406(6795):491–494, August 2000.
[21] Hendrick L. Bethlem, André J. A. van Roij, Rienk T. Jongma, and Gerard Meijer. Alternate
gradient focusing and deceleration of a molecular beam. Phys. Rev. Lett., 88:133003, Mar
2002.
[22] Immanuel Bloch, Jean Dalibard, and Sylvain Nascimbène. Quantum simulations with ultracold quantum gases. Nature Physics, 8(4):267–276, 2012.
[23] E. Boukobza and D. J. Tannor. Entropy exchange and entanglement in the jaynes-cummings
model. Phys. Rev. A, 71:063821, Jun 2005.
[24] Samuel L. Braunstein and Arun K. Pati. Quantum information cannot be completely hidden in correlations: Implications for the black-hole information paradox. Phys. Rev. Lett.,
98:080502, Feb 2007.
[25] Heinz-Peter Breuer and Francesco Petruccione. The theory of open quantum systems. Oxford
Univ. Press, 2010.
[26] K. E. Cahill and R. J. Glauber. Ordered expansions in boson amplitude operators. Phys.
Rev., 177:1857–1881, Jan 1969.
[27] J C Camparo and R P Frueholz. A dressed atom interpretation of adiabatic rapid passage.
Journal of Physics B: Atomic and Molecular Physics, 17(20):4169–4178, oct 1984.
[28] Steve Campbell and Sebastian Deffner. Trade-off between speed and cost in shortcuts toadiabaticity. Phys. Rev. Lett., 118:100601, Mar 2017.
[29] T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, and G. E.
Santoro. Optimal control at the quantum speed limit. Phys. Rev. Lett., 103:240501, Dec
2009.

174
[30] L. Carlitz. On the product of two laguerre polynomials. Journal of the London MathematicalSociety, s1-36(1):399–402, 1961.
[31] Lincoln D Carr, David DeMille, Roman V Krems, and Jun Ye. Cold and ultracold molecules:
science, technology and applications. New Journal of Physics, 11(5):055049, 2009.
[32] William B. Case. Wigner functions and weyl transforms for pedestrians. American Journalof Physics, 76(10):937–946, 2008.
[33] M. T. Cashen and H. Metcalf. Bichromatic force on helium. Phys. Rev. A, 63:025406, Jan
2001.
[34] Jasper Chan, TP Mayer Alegre, Amir H Safavi-Naeini, Jeff T Hill, Alex Krause, SimonGröblacher, Markus Aspelmeyer, and Oskar Painter. Laser cooling of a nanomechanicaloscillator into its quantum ground state. Nature, 478(7367):89–92, 2011.
[35] Thierry Chanelière, Daniel Comparat, and Hans Lignier. Phase-space-density limitation inlaser cooling without spontaneous emission. Phys. Rev. A, 98:063432, Dec 2018.
[36] Xi Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga. Shortcut to adiabatic
passage in two- and three-level atoms. Phys. Rev. Lett., 105:123003, Sep 2010.
[37] Xi Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin, and J. G. Muga. Fast
optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity. Phys. Rev.
Lett., 104:063002, Feb 2010.
[38] Xi Chen, E. Torrontegui, and J. G. Muga. Lewis-riesenfeld invariants and transitionlessquantum driving. Phys. Rev. A, 83:062116, Jun 2011.
[39] Ye-Hong Chen, Yan Xia, Qing-Qin Chen, and Jie Song. Shortcuts to adiabatic passage formultiparticles in distant cavities: applications to fast and noise-resistant quantum populationtransfer, entangled states’ preparation and transition. Laser Physics Letters, 11(11):115201,
sep 2014.
[40] S. Chervenkov, X. Wu, J. Bayerl, A. Rohlfes, T. Gantner, M. Zeppenfeld, and G. Rempe.
Continuous centrifuge decelerator for polar molecules. Phys. Rev. Lett., 112:013001, Jan
2014.
[41] Steven Chu. Nobel lecture: The manipulation of neutral particles. Rev. Mod. Phys., 70:685–
706, Jul 1998.
[42] Steven Chu, JE Bjorkholm, A Ashkin, and Alex Cable. Experimental observation of opticallytrapped atoms. Physical Review Letters, 57(3):314, 1986.
[43] Marvin L. Cohen. Essay: Fifty years of condensed matter physics.
101:250001, Dec 2008.

Phys. Rev. Lett.,[44] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg. Lagrangian and
Hamiltonian Approach to Electrodynamics, The Standard Lagrangian and the Coulomb
Gauge, chapter 2, pages 79–168. John Wiley & Sons, Ltd, 1997.

175[45] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg.
Quantum
Electrodynamics in the Coulomb Gauge, chapter 3, pages 169–252. John Wiley & Sons,
Ltd, 1997.
[46] Claude N. Cohen-Tannoudji. Nobel lecture: Manipulating atoms with photons. Rev. Mod.
Phys., 70:707–719, Jul 1998.
[47] J Cohn, A Safavi-Naini, R J Lewis-Swan, J G Bohnet, M Gärttner, K A Gilmore, J E Jordan,A M Rey, J J Bollinger, and J K Freericks. Bang-bang shortcut to adiabaticity in the dicke
model as realized in a penning trap experiment. New Journal of Physics, 20(5):055013, May
2018.
[48] Alejandra L. Collopy, Shiqian Ding, Yewei Wu, Ian A. Finneran, Loı̈c Anderegg, Benjamin L.
Augenbraun, John M. Doyle, and Jun Ye. 3d magneto-optical trap of yttrium monoxide.
Phys. Rev. Lett., 121:213201, Nov 2018.
[49] Alejandra L Collopy, Matthew T Hummon, Mark Yeo, Bo Yan, and Jun Ye. Prospects for anarrow line mot in yo. New Journal of Physics, 17(5):055008, 2015.
[50] Christopher Corder, Brian Arnold, Xiang Hua, and Harold Metcalf. Laser cooling without
spontaneous emission using the bichromatic force. J. Opt. Soc. Am. B, 32(5):B75–B83, May
2015.
[51] Christopher Corder, Brian Arnold, and Harold Metcalf. Laser cooling without spontaneousemission. Physical Review Letters, 114(4):043002, 2015.
[52] Ivan B. Coulamy, Alan C. Santos, Itay Hen, and Marcelo S. Sarandy. Energetic Cost ofSuperadiabatic Quantum Computation. Frontiers in ICT, 3, 2016.
[53] J Dalibard and C Cohen-Tannoudji. Atomic motion in laser light: connection between semiclassical and quantum descriptions. Journal of Physics B: Atomic and Molecular Physics,
18(8):1661–1683, apr 1985.
[54] J. Dalibard and C. Cohen-Tannoudji. Dressed-atom approach to atomic motion in laser light:
the dipole force revisited. J. Opt. Soc. Am. B, 2(11):1707–1720, Nov 1985.
[55] F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buek. Properties of displaced number
states. Phys. Rev. A, 41:2645–2652, Mar 1990.
[56] M. A. de Ponte, P. M. Cônsoli, and M. H. Y. Moussa. Method for the construction of
the lewis-riesenfeld time-dependent invariants and their eigenvalue equations. Phys. Rev. A,
98:032102, Sep 2018.
[57] Sebastian Deffner. Shortcuts to adiabaticity: suppression of pair production in driven diracdynamics. New Journal of Physics, 18(1):012001, Dec 2015.
[58] Luigi Delle Site. Formulation of liouville’s theorem for grand ensemble molecular simulations.
Phys. Rev. E, 93:022130, Feb 2016.
[59] F Diedrich, JC Bergquist, Wayne M Itano, and DJ Wineland. Laser cooling to the zero-point
energy of motion. Physical Review Letters, 62(4):403, 1989.

176[60] Shiqian Ding and Jun Ye. Private communication, 2019.
[61] Yan-Xiong Du, Xian-Xian Yue, Zhen-Tao Liang, Jia-Zhen Li, Hui Yan, and Shi-Liang Zhu.
Geometric atom interferometry with shortcuts to adiabaticity. Phys. Rev. A, 95:043608, Apr
2017.
[62] P. M. Duarte, R. A. Hart, J. M. Hitchcock, T. A. Corcovilos, T.-L. Yang, A. Reed, and R. G.
Hulet. All-optical production of a lithium quantum gas using narrow-line laser cooling. Phys.
Rev. A, 84:061406, Dec 2011.
[63] R. Dum, P. Zoller, and H. Ritsch. Monte carlo simulation of the atomic master equation for
spontaneous emission. Phys. Rev. A, 45:4879–4887, Apr 1992.
[64] Ethan R. Elliott, Markus C. Krutzik, Jason R. Williams, Robert J. Thompson, and David C.
Aveline. NASA’s Cold Atom Lab (CAL): system development and ground test status. npj
Microgravity, 4(1):16, August 2018.
[65] Massimiliano Esposito, Katja Lindenberg, and Christian Van den Broeck. Entropy production
as correlation between system and reservoir. New Journal of Physics, 12(1):013013, jan 2010.
[66] Alessandro Ferraro and Matteo G. A. Paris. Nonclassicality criteria from phase-space representations and information-theoretical constraints are maximally inequivalent. Phys. Rev.
Lett., 108:260403, Jun 2012.
[67] Richard P. Feynman, Frank L. Vernon, and Robert W. Hellwarth. Geometrical representation of the schrödinger equation for solving maser problems. Journal of Applied Physics,
28(1):49–52, 1957.
[68] R. Frisch. Experimenteller Nachweis des Einsteinschen Strahlungsrückstoßes. Zeitschrift fur
Physik, 86:42–48, January 1933.
[69] Xiao-Chun Gao, Jing-Bo Xu, and Tie-Zheng Qian. Geometric phase and the generalized
invariant formulation. Phys. Rev. A, 44:7016–7021, Dec 1991.
[70] Crispin W. Gardiner and Peter Zoller. Quantum noise: a handbook of Markovian andnon-Markovian quantum stochastic methods with applications to quantum optics. Springer,
2010.
[71] Christopher Gerry and Peter Knight. Introductory Quantum Optics. Cambridge University
Press, 2004.
[72] Steffen J. Glaser, Ugo Boscain, Tommaso Calarco, Christiane P. Koch, Walter Köckenberger,
Ronnie Kosloff, Ilya Kuprov, Burkhard Luy, Sophie Schirmer, Thomas Schulte-Herbrüggen,
Dominique Sugny, and Frank K. Wilhelm. Training Schrödinger’s cat: quantum optimal
control. Strategic report on current status, visions and goals for research in Europe. European
Physical Journal D, 69(12):279, December 2015.
[73] Kurt Gottfried and Tung-Mow Yan. Quantum mechanics: fundamentals. Springer Science
Business Media, 2005.
[74] G P Greve, B Wu, and J K Thompson. Laser cooling with adiabatic transfer on a ramantransition. New Journal of Physics, 21(7):073045, jul 2019.

177[75] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martı́nez-Garaot, and J. G.
Muga. Shortcuts to adiabaticity: Concepts, methods, and applications. Rev. Mod. Phys.,
91:045001, Oct 2019.
[76] Utkan Güngördü, Yidun Wan, Mohammad Ali Fasihi, and Mikio Nakahara. Dynamical
invariants for quantum control of four-level systems. Phys. Rev. A, 86:062312, Dec 2012.
[77] T. W. Hansch and A. L. Schawlow.
Communications, 13:68, January 1975.

Cooling of gases by laser radiation.

Optics[78] D. J. Heinzen and D. J. Wineland. Quantum-limited cooling and detection of radio-frequency
oscillations by laser-cooled ions. Phys. Rev. A, 42:2977–2994, Sep 1990.
[79] Boerge Hemmerling, Eunmi Chae, Aakash Ravi, Loic Anderegg, Garrett K Drayna,
Nicholas R Hutzler, Alejandra L Collopy, Jun Ye, Wolfgang Ketterle, and John M Doyle.
Laser slowing of CaF molecules to near the capture velocity of a molecular MOT. Journal of
Physics B: Atomic, Molecular and Optical Physics, 49(17):174001, aug 2016.
[80] L Henderson and V Vedral. Classical, quantum and total correlations. Journal of Physics A:
Mathematical and General, 34(35):6899–6905, aug 2001.
[81] S. D. Hogan, A. W. Wiederkehr, H. Schmutz, and F. Merkt. Magnetic trapping of hydrogen
after multistage zeeman deceleration. Phys. Rev. Lett., 101:143001, Sep 2008.
[82] Alexander S. Holevo.
Quantum Systems, Channels, Information:
Introduction. De Gruyter, 2 edition, 2019.

A Mathematical[83] M. J. Holland and K. Burnett. Interferometric detection of optical phase shifts at the heisenberg limit. Phys. Rev. Lett., 71:1355–1358, Aug 1993.
[84] Peter Horak, Gerald Hechenblaikner, Klaus M. Gheri, Herwig Stecher, and Helmut Ritsch.
Cavity-induced atom cooling in the strong coupling regime. Phys. Rev. Lett., 79:4974–4977,
Dec 1997.
[85] Matthew T Hummon, Mark Yeo, Benjamin K Stuhl, Alejandra L Collopy, Yong Xia, and
Jun Ye. 2d magneto-optical trapping of diatomic molecules. Physical Review Letters,
110(14):143001, 2013.
[86] Tetsuya Ido, Yoshitomo Isoya, and Hidetoshi Katori. Optical-dipole trapping of sr atoms at
a high phase-space density. Phys. Rev. A, 61:061403, May 2000.
[87] A. M. Jayich, A. C. Vutha, M. T. Hummon, J. V. Porto, and W. C. Campbell. Continuous alloptical deceleration and single-photon cooling of molecular beams. Phys. Rev. A, 89:023425,
Feb 2014.
[88] P Jobez, I Usmani, N Timoney, C Laplane, N Gisin, and M Afzelius. Cavity-enhanced storage
in an optical spin-wave memory. New Journal of Physics, 16(8):083005, aug 2014.
[89] Richard Jozsa. Fidelity for mixed quantum states. Journal of Modern Optics, 41(12):2315–
2323, 1994.

178[90] Alfred Kastler. Quelques suggestions concernant la production optique et la détection optiqued'une inégalité de population des niveaux de quantifigation spatiale des atomes. applicationà l'expérience de stern et gerlach et à la résonance magnétique. Journal de Physique et leRadium, 11(6):255–265, 1950.
[91] Hidetoshi Katori, Tetsuya Ido, Yoshitomo Isoya, and Makoto Kuwata-Gonokami. Magnetooptical trapping and cooling of strontium atoms down to the photon recoil temperature. Phys.
Rev. Lett., 82:1116–1119, Feb 1999.
[92] Hidetoshi Katori, Tetsuya Ido, and Makoto Kuwata-Gonokami. Optimal design of dipole potentials for efficient loading of sr atoms. Journal of the Physical Society of Japan, 68(8):2479–
2482, 1999.
[93] Wolfgang Ketterle. Nobel lecture: When atoms behave as waves: Bose-einstein condensationand the atom laser. Rev. Mod. Phys., 74:1131–1151, Nov 2002.
[94] K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin,
L.-M. Duan, and C. Monroe. Quantum simulation of frustrated ising spins with trapped
ions. Nature, 465(7298):590–593, Jun 2010.
[95] S.P Kim, A.E Santana, and F.C Khanna. Generalized invariants and quantum evolution ofopen fermionic systems. Physics Letters A, 272(1):46 – 52, 2000.
[96] H. J. Kimble. Strong interactions of single atoms and photons in cavity QED. Physica Scripta,
T76(1):127, 1998.
[97] Vladimir V. Kiselev. Collective effects in condensed matter physics. De Gruyter, 2018.
[98] Catherine Klauss. A little less spontaneous, 2018.
[99] J Kobayashi, K Aikawa, K Oasa, and S Inouye. Prospects for narrow-line cooling of krb
molecules in the rovibrational ground state. Physical Review A, 89(2):021401, 2014.
[100] E. Korsunsky. Laser cooling during velocity-selective adiabatic population transfer. Phys.
Rev. A, 54:R1773–R1776, Sep 1996.
[101] Sebastian Krämer, David Plankensteiner, Laurin Ostermann, and Helmut Ritsch. Quantumoptics.jl: A julia framework for simulating open quantum systems. Computer Physics
Communications, 227:109 – 116, 2018.
[102] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien. Quantum
computers. Nature, 464(7285):45–53, 2010.
[103] Y.-Z. Lai, J.-Q. Liang, H. J. W. Müller-Kirsten, and J.-G. Zhou. Time-dependent quantum
systems and the invariant hermitian operator. Phys. Rev. A, 53:3691–3693, May 1996.
[104] Joel L. Lebowitz and Abner Shimony. Statistical mechanics of open systems. Phys. Rev.,
128:1945–1958, Nov 1962.
[105] H. R. Lewis and W. B. Riesenfeld. An exact quantum theory of the time-dependent harmonic
oscillator and of a charged particle in a time-dependent electromagnetic field. Journal of
Mathematical Physics, 10(8):1458–1473, 1969.

179[106] Nan Li and Shunlong Luo. Classical and quantum correlative capacities of quantum systems.
Phys. Rev. A, 84:042124, Oct 2011.
[107] Yi-Chao Li and Xi Chen. Shortcut to adiabatic population transfer in quantum three-levelsystems: Effective two-level problems and feasible counterdiabatic driving. Phys. Rev. A,
94:063411, Dec 2016.
[108] Yeong-Cherng Liang, Yu-Hao Yeh, Paulo E M F Mendonça, Run Yan Teh, Margaret D Reid,and Peter D Drummond. Quantum fidelity measures for mixed states. Reports on Progress
in Physics, 82(7):076001, jun 2019.
[109] G. Lindblad. On the generators of quantum dynamical semigroups. Communications inMathematical Physics, 48(2):119–130, 1976.
[110] Thomas H. Loftus, Tetsuya Ido, Andrew D. Ludlow, Martin M. Boyd, and Jun Ye. Narrow
line cooling: Finite photon recoil dynamics. Phys. Rev. Lett., 93:073003, Aug 2004.
[111] T. Lu, X. Miao, and H. Metcalf. Nonadiabatic transitions in finite-time adiabatic rapid
passage. Phys. Rev. A, 75:063422, Jun 2007.
[112] William Lunden, Li Du, Michael Cantara, Pierre Barral, Alan O. Jamison, and Wolfgang
Ketterle. Enhancing the capture velocity of a Dy magneto-optical trap with two-stage slowing.
arXiv e-prints, page arXiv:1908.10433, Aug 2019.
[113] Daniel Manzano. A short introduction to the lindblad master equation. AIP Advances,10(2):025106, 2020.
[114] Peter J. Martin, Bruce G. Oldaker, Andrew H. Miklich, and David E. Pritchard. Bragg
scattering of atoms from a standing light wave. Phys. Rev. Lett., 60:515–518, Feb 1988.
[115] S. Martı́nez-Garaot, A. Ruschhaupt, J. Gillet, Th. Busch, and J. G. Muga. Fast quasiadiabatic
dynamics. Phys. Rev. A, 92:043406, Oct 2015.
[116] S. Martı́nez-Garaot, E. Torrontegui, Xi Chen, and J. G. Muga. Shortcuts to adiabaticity inthree-level systems using lie transforms. Phys. Rev. A, 89:053408, May 2014.
[117] Shumpei Masuda and Katsuhiro Nakamura. Fast-forward problem in quantum mechanics.
Phys. Rev. A, 78:062108, Dec 2008.
[118] Shumpei Masuda and Katsuhiro Nakamura. Fast-forward of adiabatic dynamics in quantum
mechanics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Sciences, 466(2116):1135–1154, 2010.
[119] Shumpei Masuda and Stuart A. Rice. Fast-forward assisted stirap. The Journal of Physical
Chemistry A, 119(14):3479–3487, 2015. PMID: 25775133.
[120] The Mathworks, Inc., Natick, Massachusetts.
2017.

MATLAB version 9.9.0.1467703 (R2020b),[121] P. Maunz, T. Puppe, I. Schuster, N. Syassen, P. W. H. Pinkse, and G. Rempe. Cavity coolingof a single atom. Nature, 428(6978):50–52, Mar 2004.

180[122] Gerard McCaul, Alexander Pechen, and Denys I. Bondar. Entropy nonconservation andboundary conditions for hamiltonian dynamical systems. Phys. Rev. E, 99:062121, Jun 2019.
[123] H. J. Metcalf and P. van der Straten. Laser cooling and trapping of atoms. J. Opt. Soc. Am.
B, 20(5):887–908, May 2003.
[124] Harold Metcalf. Entropy exchange in laser cooling. Phys. Rev. A, 77:061401, Jun 2008.
[125] Harold Metcalf. Colloquium: Strong optical forces on atoms in multifrequency light. Rev.
Mod. Phys., 89:041001, Oct 2017.
[126] Harold J. Metcalf and Peter van der. Straten. Laser cooling and trapping. Springer, 2002.
[127] Pierre Meystre and Murray Sargent. Elements of Quantum Optics. Springer, 2007.
[128] Jian-Guo Miao, Chun-Wang Wu, Wei Wu, and Ping-Xing Chen. Entropy exchange andthermodynamic properties of the single ion cooling process. Entropy, 21(7):650, Jul 2019.
[129] X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf. Strong optical forces from adiabatic rapid
passage. Phys. Rev. A, 75:011402, Jan 2007.
[130] P. W. Milonni. Why spontaneous emission? American Journal of Physics, 52(4):340–343,
1984.
[131] Peter W. Milonni, Jay R. Ackerhalt, and Wallace Arden Smith. Interpretation of radiative
corrections in spontaneous emission. Phys. Rev. Lett., 31:958–960, Oct 1973.
[132] Peter W. Milonni and Wallace Arden Smith. Radiation reaction and vacuum fluctuations inspontaneous emission. Phys. Rev. A, 11:814–824, Mar 1975.
[133] Z. K. Minev, S. O. Mundhada, S. Shankar, P. Reinhold, R. Gutiérrez-Jáuregui, R. J.
Schoelkopf, M. Mirrahimi, H. J. Carmichael, and M. H. Devoret. To catch and reverse aquantum jump mid-flight. Nature, 570(7760):200–204, 2019.
[134] V.G. Minogin and O.T. Serimaa. Resonant light pressure forces in a strong standing laser
wave. Optics Communications, 30(3):373 – 379, 1979.
[135] Klaus Mølmer, Yvan Castin, and Jean Dalibard. Monte carlo wave-function method inquantum optics. J. Opt. Soc. Am. B, 10(3):524–538, Mar 1993.
[136] Henrik Lund Mortensen, Jens Jakob W. H. Sørensen, Klaus Mølmer, and Jacob Friis Sherson.
Fast state transfer in a λ-system: a shortcut-to-adiabaticity approach to robust and resource
optimized control. New Journal of Physics, 20(2):025009, feb 2018.
[137] Juan A. Muniz, Matthew A. Norcia, Julia R. K. Cline, and James K. Thompson. A Robust Narrow-Line Magneto-Optical Trap using Adiabatic Transfer. arXiv e-prints, page
arXiv:1806.00838, Jun 2018.
[138] Edvardas Narevicius, Adam Libson, Christian G. Parthey, Isaac Chavez, Julia Narevicius, Uzi
Even, and Mark G. Raizen. Stopping supersonic beams with a series of pulsed electromagnetic
coils: An atomic coilgun. Phys. Rev. Lett., 100:093003, Mar 2008.

181[139] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information:
10th Anniversary Edition. Cambridge University Press, USA, 10th edition, 2011.
[140] Matthew A. Norcia. New tools for precision measurement and quantum science with narrow
linewidth optical transitions. PhD thesis, University of Colorado, Boulder, 8 2018.
[141] Matthew A Norcia, Julia R K Cline, John P Bartolotta, Murray J Holland, and James K
Thompson. Narrow-line laser cooling by adiabatic transfer. New Journal of Physics,
20(2):023021, 2018.
[142] B. Nölle, H. Nölle, J. Schmand, and H. J. Andrä. Atomic-beam deflection by double-π-pulse
laser technique. EPL (Europhysics Letters), 33(4):261, 1996.
[143] Maxime Oliva, Dimitris Kakofengitis, and Ole Steuernagel. Anharmonic quantum mechanicalsystems do not feature phase space trajectories. Physica A: Statistical Mechanics and its
Applications, 502:201 – 210, 2018.
[144] Matteo G.A. Paris. Displacement operator by beam splitter. Physics Letters A, 217(2):78 –80, 1996.
[145] O Penrose. Foundations of statistical mechanics. Reports on Progress in Physics, 42(12):1937–
2006, dec 1979.
[146] N. Petersen, F. Mühlbauer, L. Bougas, A. Sharma, D. Budker, and P. Windpassinger.
Sawtooth-wave adiabatic-passage slowing of dysprosium. Phys. Rev. A, 99:063414, Jun 2019.
[147] Francesco Petiziol, Benjamin Dive, Florian Mintert, and Sandro Wimberger. Fast adiabatic
evolution by oscillating initial hamiltonians. Phys. Rev. A, 98:043436, Oct 2018.
[148] M Petzold, P Kaebert, P Gersema, M Siercke, and S Ospelkaus. A zeeman slower for diatomic
molecules. New Journal of Physics, 20(4):042001, apr 2018.
[149] William D. Phillips. Nobel lecture: Laser cooling and trapping of neutral atoms. Rev. Mod.
Phys., 70:721–741, Jul 1998.
[150] William D. Phillips and Harold Metcalf. Laser deceleration of an atomic beam. Phys. Rev.
Lett., 48:596–599, Mar 1982.
[151] M. B. Plenio and P. L. Knight. The quantum-jump approach to dissipative dynamics inquantum optics. Rev. Mod. Phys., 70:101–144, Jan 1998.
[152] Gabriel N Price, S Travis Bannerman, Kirsten Viering, Edvardas Narevicius, and Mark G
Raizen. Single-photon atomic cooling. Physical Review Letters, 100(9):093004, 2008.
[153] David E. Pritchard and Phillip L. Gould. Experimental possibilities for observation of unidirectional momentum transfer to atoms from standing-wave light. J. Opt. Soc. Am. B,
2(11):1799–1804, Nov 1985.
[154] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev. Integrals and Series, Volume 3: More
Special Functions, volume 3, chapter 7, page 439. Gordon and Breach, 1990.
[155] Krzysztof Ptaszyński and Massimiliano Esposito. Entropy production in open systems: The
predominant role of intraenvironment correlations. Phys. Rev. Lett., 123:200603, Nov 2019.

182[156] E. L. Raab, M. Prentiss, Alex Cable, Steven Chu, and D. E. Pritchard. Trapping of neutral
sodium atoms with radiation pressure. Phys. Rev. Lett., 59:2631–2634, Dec 1987.
[157] C. A. Regal, M. Greiner, and D. S. Jin. Observation of resonance condensation of fermionic
atom pairs. Phys. Rev. Lett., 92:040403, Jan 2004.
[158] Florentin Reiter and Anders S. Sørensen. Effective operator formalism for open quantum
systems. Phys. Rev. A, 85:032111, Mar 2012.
[159] Simon M. Rochester, Konrad Szymański, Mark Raizen, Szymon Pustelny, Marcis Auzinsh,
and Dmitry Budker. Efficient polarization of high-angular-momentum systems. Phys. Rev.
A, 94:043416, Oct 2016.
[160] K. J. Ross and B. Sonntag. High temperature metal atom beam sources. Review of Scientific
Instruments, 66(9):4409–4433, 1995.
[161] A Ruschhaupt, Xi Chen, D Alonso, and J G Muga. Optimally robust shortcuts to population
inversion in two-level quantum systems. New Journal of Physics, 14(9):093040, sep 2012.
[162] A. Ruschhaupt and J.G. Muga. Shortcuts to adiabaticity in two-level systems: control andoptimization. Journal of Modern Optics, 61(10):828–832, 2014.
[163] M Sabooni, S Tornibue Kometa, A Thuresson, S Kröll, and L Rippe. Cavity-enhanced storage—preparing for high-efficiency quantum memories. New Journal of Physics, 15(3):035025,
mar 2013.
[164] M. S. Safronova, D. Budker, D. DeMille, Derek F. Jackson Kimball, A. Derevianko, andCharles W. Clark. Search for new physics with atoms and molecules. Rev. Mod. Phys.,
90:025008, Jun 2018.
[165] R. Sala, S. Brouard, and J. G. Muga. Wigner trajectories and liouville’s theorem. The Journalof Chemical Physics, 99(4):2708–2714, 1993.
[166] M. S. Sarandy, E. I. Duzzioni, and R. M. Serra. Quantum computation in continuous timeusing dynamic invariants. Physics Letters A, 375(38):3343–3347, Sep 2011.
[167] Denise Sawicki and J.H. Eberly. Perfect following in the diabatic limit. Opt. Express,
4(7):217–222, Mar 1999.
[168] F. Schwabl. Statistical Mechanics. Springer Berlin Heidelberg, 2006.
[169] Dries Sels. Stochastic gradient ascent outperforms gamers in the quantum moves game. Phys.
Rev. A, 97:040302, Apr 2018.
[170] E. S. Shuman, J. F. Barry, and D. Demille. Laser cooling of a diatomic molecule. Nature,467(7317):820–823, 2010.
[171] S. Snigirev, A. J. Park, A. Heinz, I. Bloch, and S. Blatt. Fast and dense magneto-opticaltraps for strontium. Phys. Rev. A, 99:063421, Jun 2019.
[172] J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon. Short-distanceatomic beam deceleration with a stimulated light force. Phys. Rev. Lett., 78:1420–1423, Feb
1997.

183
[173] Daniel Stack, John Elgin, Petr M. Anisimov, and Harold Metcalf. Numerical studies of opticalforces from adiabatic rapid passage. Phys. Rev. A, 84:013420, Jul 2011.
[174] Daniel A. Steck. Quantum and atom optics, 2007.
[175] Dionisis Stefanatos, Justin Ruths, and Jr-Shin Li. Frictionless atom cooling in harmonic
traps: A time-optimal approach. Phys. Rev. A, 82:063422, Dec 2010.
[176] Simon Stellmer, Benjamin Pasquiou, Rudolf Grimm, and Florian Schreck. Laser cooling toquantum degeneracy. Phys. Rev. Lett., 110:263003, Jun 2013.
[177] Benjamin K. Stuhl, Matthew T. Hummon, Mark Yeo, Goulven Quéméner, John L. Bohn,
and Jun Ye. Evaporative cooling of the dipolar hydroxyl radical. Nature, 492(7429):396–400,
December 2012.
[178] Kazutaka Takahashi. Hamiltonian engineering for adiabatic quantum computation: Lessons
from shortcuts to adiabaticity. Journal of the Physical Society of Japan, 88(6):061002, 2019.
[179] E. Torrontegui, S. Martı́nez-Garaot, and J. G. Muga. Hamiltonian engineering via invariants
and dynamical algebra. Phys. Rev. A, 89:043408, Apr 2014.
[180] E. Torrontegui, S. Martı́nez-Garaot, A. Ruschhaupt, and J. G. Muga. Shortcuts to adiabaticity: Fast-forward approach. Phys. Rev. A, 86:013601, Jul 2012.
[181] S Truppe, H J Williams, N J Fitch, M Hambach, T E Wall, E A Hinds, B E Sauer, and
M R Tarbutt. An intense, cold, velocity-controlled molecular beam by frequency-chirped
laser slowing. New Journal of Physics, 19(2):022001, Feb 2017.
[182] A. Uhlmann. The “transition probability” in the state space of a ∗-algebra. Reports on
Mathematical Physics, 9(2):273 – 279, 1976.
[183] Alban Urvoy, Zachary Vendeiro, Joshua Ramette, Albert Adiyatullin, and Vladan Vuletić. Direct laser cooling to bose-einstein condensation in a dipole trap. Phys. Rev. Lett., 122:203202,
May 2019.
[184] Lorenza Viola and Seth Lloyd. Dynamical suppression of decoherence in two-state quantum
systems. Phys. Rev. A, 58:2733–2744, Oct 1998.
[185] N. V. Vitanov. Transition times in the Landau-Zener model. Physical Review A, 59:988–994,
February 1999.
[186] Kurt R Vogel, Timothy P Dinneen, Alan Gallagher, and John L Hall. Narrow-line
doppler cooling of strontium to the recoil limit. IEEE Transactions on Instrumentation andMeasurement, 48(2):618–621, 1999.
[187] V.S. Voitsekhovich, M.V. Danileiko, A.M Negriiko, V.I Romanenko, and L.P. Yatsenko. Stimulated light pressure on atoms in counterpropagating amplitude-modulated waves. JETP,
72:219–227, Feb 1991.
[188] Tino Weber, Jens Herbig, Michael Mark, Hanns-Christoph Nägerl, and Rudolf Grimm. Boseeinstein condensation of cesium. Science (New York, N.Y.), 299(5604):232—235, January
2003.

184
[189] V. Weisskopf and E. Wigner. Berechnung der natürlichen linienbreite auf grund der diracschen
lichttheorie. Zeitschrift für Physik, 63(1-2):54–73, 1930.
[190] Viktor Weisskopf.
Probleme der neueren quantentheorie des elektrons.
Naturwissenschaften, 23(37):631–637, 1935.

Die[191] M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf. Measurement of the bichromatic
optical force on rb atoms. Phys. Rev. A, 60:R1763–R1766, Sep 1999.
[192] D. Wineland and H. Dehmelt. Proposed 1014 ∆ν < ν laser fluorescence spectroscopy on TI+
mono-ion oscillator III. Bull. Am. Phys. Soc., 20:637, 1975.
[193] Tao Yang, Kanhaiya Pandey, Mysore Srinivas Pramod, Frederic Leroux, Chang Chi Kwong,
Elnur Hajiyev, Zhong Yi Chia, Bess Fang, and David Wilkowski. A high flux source of cold
strontium atoms. The European Physical Journal D, 69(10), 2015.
[194] C. Zener. Non-Adiabatic Crossing of Energy Levels. Proceedings of the Royal Society of
London Series A, 137:696–702, September 1932.
[195] Xibo Zhang and Jun Ye. Precision measurement and frequency metrology with ultracold
atoms. National Science Review, 3(2):189–200, 03 2016.
[196] V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R.
Tarbutt, and B. E. Sauer. Laser cooling and slowing of caf molecules. Phys. Rev. A, 89:053416,
May 2014.
[197] Yuanjian Zheng, Steve Campbell, Gabriele De Chiara, and Dario Poletti. Cost of counterdiabatic driving and work output. Phys. Rev. A, 94:042132, Oct 2016.