Tools for Improved Quantum Metrology on Atomic
Platforms
byAthreya Shankar
B.Tech. (Hons), Indian Institute of Technology Madras, 2014
M.S., University of Colorado Boulder, 2017

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
2020

This thesis entitled:
Tools for Improved Quantum Metrology on Atomic Platforms
written by Athreya Shankar
has been approved for the Department of Physics

Prof. Murray J. Holland

Dr. John Bollinger

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Shankar, Athreya (Ph.D., Physics)
Tools for Improved Quantum Metrology on Atomic Platforms
Thesis directed by Prof. Murray J. Holland

The last three decades have witnessed the rapid rise of technologies based on quantum mechanical principles. In particular, quantum systems are expected to revolutionize the way we sense
and measure properties of our universe. The ongoing transition from fundamental studies of quantum systems to a quantum technology revolution is being powered by significant advancements inlaser cooling and trapping of atoms, and in the manipulation and readout of quantum degrees of
freedom. In this thesis, we contribute to this movement by presenting theoretical ideas of practicalrelevance, that will enable improved quantum metrology using atom-based platforms.
We advance capabilities in three areas of relevance to quantum metrology. First, under thetheme of sub-Doppler cooling of large quantums systems, we describe the numerical modeling ofa successful experiment for near ground-state cooling of trapped ion crystals with more than 100
ions in a Penning trap. Second, we propose a new readout technique using atom-cavity interactions
to continuously and precisely track the relative phase of a spin superposition. Third, we propose
a scheme to engineer squeezing on a platform where controllable atom-atom interactions havebeen hard to achieve, namely atomic Bragg interferometers. In this way, we introduce usefulentanglement into a sensing platform traditionally relying on single-atom physics.

Dedication

To Amma, Nanna, Myka, Rajamma aththa, Jothi avva and Venkatesh thatha.

v

Acknowledgements

A wave of gratitude sweeps over me as I express my heartfelt thanks to the people whohave helped me reach where I am today. Thank you, Murray, for giving me the freedom to explore
physics as a graduate student. He has always allowed me to venture freely in any direction I wanted
to, with the quiet assurance that a safety net was always in place in case things didn’t work out.
As an aspiring professor myself, I hope to be as supportive of my students’ ideas as Murray has
been of mine.
My mindset towards physics, research, scientific collaboration and mentoring has been shapedby some terrific scientists I have interacted with during grad school. Thank you, Jinx, for teachingme that the physics should dictate the math, and not the other way round. I am grateful to John
Bollinger for always making me feel equal in any discussion, even though he has been in the field
longer than I have been alive. Your humility has inspired me to stay grounded no matter what I
may achieve. Thanks to James Thompson for all the enthusiastic physics discussions that would
go on forever like a cricket Test match. I am sure that I will listen to my future students with thesame level of patience and interest that you have displayed during our conversations. Thanks to
Marilu Chiofalo and Nicola Poli for the fantastic hospitality in Pisa and Florence during my shorttrip to Italy. They have been very patient in collaborating with me and have always allowed me towork in my natural style. Thanks to Ana Maria Rey for several engaging discussions. I am very
grateful to Dr. K. S. Balaji and Prof. Lakshmi Bala for kindling my early interest in physics.
I have always struggled with my thoughts and emotions. If not for my mother, I would never
have been able to emerge successful from these struggles at various stages in my life; thank you,vi
Amma, for making it your life’s mission to see me successful. Also, thanks to the amazing resources
offered by the Counseling and Psychiatric Services at CU Boulder for helping me systematically
overcome my mental health problems.
During high school, I still remember how my father used to wake up at 4.30 AM in themorning just to make tea for me, so that I could wake up at 5 AM and study for my exams; thankyou, Nanna, for being the sturdy pillar on which I have always leaned on. Thanks to my sister,
Mythreyi, for the long, random phone conversations that helped me to relax after work; Myka, youhave played an enormous role in keeping loneliness and homesickness at bay. I am forever indebted
to my great aunt Rajamma atha and my grandparents, Venkatesh thatha and Jothi avva, for taking
care of me during my formative years and thereby supporting my working parents.
I am grateful to have been surrounded by an enthusiastic and approachable group of fellow
students and postdocs. Thanks to John Bartolotta, Haonan Liu, Lilian Chih, Jarrod Reilly, Chris
Langlett and Simon Jaeger for being collaborative and welcoming group mates. Thanks to David
Tieri for early advice on life as a grad student. Baochen and Graham— I have enjoyed the countless
discussions we have had on various physics ideas. Thanks to Elena, Arghavan and Kevin, whoshared my excitement for modeling an experiment as it was being performed. I am thankful to
Leo Salvi for all the Skype meetings from which I learned so much physics. I have also enjoyeddiscussing physics with Matt Affolter, Chen Tang, Yiheng Lin, Robert Lewis-Swan and Chengyi
Luo.
Life away from home would have been hard if not for my friends who were as good as family
for me. Kannan, Balaji and Harsha, it is not a stretch to say that you guys have been more thanbrothers to me. I will always remember the amazing badminton and cricket games I played with
Harish, Mano and Nishant. Thanks to Prasanth and Akshay for being adjusting and extremely
caring flat mates. Thanks to Suchit, Abhimanyu, Kiran and Sanni for all the get-togethers. Thanks
to Ramesh, Karuna, Susheera, Jayson and all the AID Colorado members. Sungoh, Tyko andAshik— thanks for always being there to help and to have a long conversation. I am indebted to
Nithin Sivadas for some very valuable conversations at various stages of my PhD.

vii

Contents

Chapter
1 Context

1

2 Introduction

3

2.1

2.2

2.3

2.4

Laser cooling of trapped ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1.1

The atom-laser interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1.2

The Lamb-Dicke regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.3

Motional sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.4

Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.5

Laser cooling with two electronic states . . . . . . . . . . . . . . . . . . . . .

8

2.1.6

Engineering the absorption spectrum using quantum interference . . . . . . . 11

Quantum Metrology with pseudospin-1/2 systems . . . . . . . . . . . . . . . . . . . . 14
2.2.1

The Ramsey sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2

Limitations of the Ramsey sequence . . . . . . . . . . . . . . . . . . . . . . . 17

Measurement precision in the Ramsey sequence . . . . . . . . . . . . . . . . . . . . . 19
2.3.1

Coherent spin states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2

Squeezed spin states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3

Mechanisms to prepare squeezed spin states . . . . . . . . . . . . . . . . . . . 22

Momentum pseudospins and Bragg interferometers . . . . . . . . . . . . . . . . . . . 25
2.4.1

Coherent manipulation of momentum states . . . . . . . . . . . . . . . . . . . 26

2.4.2

Phase accumulation in a gravitational field . . . . . . . . . . . . . . . . . . . 28

viii
3 Near ground-state cooling large trapped ion crystals: Theoretical modeling

31

3.1

Motivation: Why ground-state cooling? . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2

EIT cooling in a Penning trap: Expectation and challenges . . . . . . . . . . . . . . 32

3.3

Summary of predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4

Modeling the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5

3.6

3.4.1

NIST Penning trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.2

Master equation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.3

Parameters from the NIST EIT cooling experiment . . . . . . . . . . . . . . . 41

A single revolving ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5.1

Time-varying Doppler shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5.2

Timescale for internal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 43

EIT cooling of multiple ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.1

Results from the Gaussian model . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6.2

Benchmarking the Gaussian Model: Single-ion results . . . . . . . . . . . . . 46

3.6.3

Results from the Sampling model . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7

Cooling over the full bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.8

Sensitivity to laser alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.9

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Near ground-state cooling large trapped ion crystals: Experimental results
4.1

4.2

54

The Optical Dipole Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1

Physical mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.2

Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Thermometry with the ODF
4.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Analytical formula for the bright fraction . . . . . . . . . . . . . . . . . . . . 58

4.3

Temperature measurements on the COM mode . . . . . . . . . . . . . . . . . . . . . 59

4.4

Measurements over the full bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . 62

ix
4.5

Physical model for mode frequency fluctuations . . . . . . . . . . . . . . . . . . . . . 65
4.5.1

Classical dynamics of ions in a Penning trap

4.5.2

The thermal snapshot picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5.3

Molecular dynamics simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5.4

Further extensions of this study . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Real-time phase tracking: Formalism. . . . . . . . . . . . . . . . . . 65

76

5.1

Setup and basic mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2

Derivation of the QND Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.1

Simple picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.2

Accounting for ω0 /∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.3

Note concerning drive laser frequencies . . . . . . . . . . . . . . . . . . . . . . 83

5.3

Phenomenological free-space scattering model . . . . . . . . . . . . . . . . . . . . . . 84

5.4

Stochastic master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.5

Gaussian state approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.1

Operator means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5.2

Operator covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5.3

Some comments on the equations of motion . . . . . . . . . . . . . . . . . . . 88

5.5.4

Numerical evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Real-time phase tracking: Numerical experiments
6.1

90

Estimating the phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.1

Note concerning the cooperativity . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2

Parameters for numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3

Real-time phase tracking: A demonstration . . . . . . . . . . . . . . . . . . . . . . . 92

6.4

Use as a Ramsey sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.5

Analytic expression for variance in the difference measurement . . . . . . . . . . . . 96
6.5.1

Physical explanation for optimum measurement window . . . . . . . . . . . . 97

x
6.5.2
6.6

Absence of noticeable Zeno effect . . . . . . . . . . . . . . . . . . . . . . . . . 98

Advantage of a feedback loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.6.1

Atom number fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Squeezing on momentum pseudospins: Formalism

101

7.1

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.2

Basic working principle

7.3

Atom-cavity interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.3.1

Adiabatic elimination of the excited state . . . . . . . . . . . . . . . . . . . . 106

7.3.2

Replacing mode 1 by a c-number . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.4

Momentum width using the |n, qi notation . . . . . . . . . . . . . . . . . . . . . . . . 108

7.5

Numerical solution: Semiclassical Langevin equations . . . . . . . . . . . . . . . . . . 109
7.5.1

7.6

7.7

Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Effective atom-atom interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.6.1

Elimination of the cavity field â2 . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.6.2

Collective angular momentum operators . . . . . . . . . . . . . . . . . . . . . 117

Numerical solution: Cumulant theory for one and two-atom operators . . . . . . . . 118

8 Squeezing on momentum pseudospins: Results

122

8.1

Spin squeezing: Figure of merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2

Considerations for choosing parameters

8.3

Parameters for the 1 S0 − 3 P1 transition in 88 Sr . . . . . . . . . . . . . . . . . . . . . 124

8.4

Limits set by superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.5

Squeezing faster and faster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.6

Effect of momentum width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.7Collective physics with a many-body energy gap . . . . . . . . . . . . . . . . . . . . 132

8.8

Final comments and possible extensions . . . . . . . . . . . . . . . . . . . . . . . . . 134. . . . . . . . . . . . . . . . . . . . . . . . . 123

xi
9 Conclusion

137

Bibliography

138

Appendix
A Supplementary Material for Chapters 3 and 4

148

A.1 Equations of motion for first and second order moments . . . . . . . . . . . . . . . . 148
A.1.1 Internal moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.1.2 External moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
A.1.3 Hybrid moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.2 Analytic formula for the bright fraction: Derivation outline . . . . . . . . . . . . . . 153
B Supplementary Material for Chapters 7 and 8

156

B.1 Implementing instantaneous state rotations . . . . . . . . . . . . . . . . . . . . . . . 156
B.1.1 Multi-center model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B.1.2 Two-center model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B.2 Rabi oscillation model for population leakage . . . . . . . . . . . . . . . . . . . . . . 157
B.3 Relative importance of free-space scattering . . . . . . . . . . . . . . . . . . . . . . . 159

xii

TablesTable
3.1

Summary of approximate characteristic frequencies for the three branches of modes
for two dimensional crystals in the NIST Penning trap. . . . . . . . . . . . . . . . . . 36

3.2

List of moments classified according to the nature of the operators involved. The
equations for the moments marked with a # can be derived trivially by exchanging
g1 ↔ g2 in the appropriate equations of motion for the other moments. . . . . . . . . 45

3.3

Implementational differences between the Gaussian and Sampling models. HereGaus(0, σ) is a Gaussian distributed random variable with zero mean and standard deviation σ, and n̄n are the initial thermal mode occupations. In the sampling
model, the quantity hAi simply denotes the value of the respective phase space variable in that trajectory, and is not the mean value of the operator A. Instead, themean value of A is given by the average of hAi over many trajectories with random
initial conditions drawn from the initial phase space distribution. We note that weonly sample the initial thermal distribution of the normal modes, and initialize thejelectronic DOF in the same way as in the GM, i.e. hσαβi = 1 when α = β = g1 .

. . 48

A.1 Definition of partial sums to simplify notation and speed up computation. The
symbols Xm and λµjm are defined after Eq. 3.13. . . . . . . . . . . . . . . . . . . . . . 148
A.2 Additional partial sums, over the ions rather than modes, to simplify notation andspeed up computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xiii

FiguresFigure2.1

Sketch of absorption spectrum and locations of the carrier, motion adding and motion
removing sidebands in (a) Doppler cooling and (b) Sideband cooling. . . . . . . . . . 10

2.2

Engineering the absorption spectrum using quantum interference. (a) A weak probe
laser is scanned across the |gi ↔ |ei transition as the state |ei is dressed by a strong
coupling laser driving the |ri ↔ |ei transition in a blue detuned regime (∆c > 0).
(b) The corresponding absorption spectrum displays a Fano profile close to ∆p ≈ ∆c . 12

2.3

The two interfering pathways that transfer the atom from |gi to |ei. Subsequent
spontaneous emission returns the atom to |gi. . . . . . . . . . . . . . . . . . . . . . . 13

2.4

Bloch sphere representation of the pure state of a single spin-1/2 system . . . . . . . 16

2.5

State of the spin at different points in the Ramsey sequence. . . . . . . . . . . . . . . 17

2.6

Coherent spin state (CSS). (a) Visualization of a CSS as a cone ending in an uncertainty circle. (b) Quantum fluctuations in the components orthogonal to the mean
spin direction lead to limited resolution of small rotations. . . . . . . . . . . . . . . . 20

2.7

Squeezed spin state (SSS). (a) Visualization of an SSS as a cone ending in an uncertainty ellipse with unequal major and minor axes. (b) The minor axis can beoriented along the component accumulating phase in order to improve the resolutionof small rotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

xiv
2.8

Two common mechanisms to generate squeezed spin states involve (a) Squishing theuncertainty ellipse, as happens during squeezing by quantum nondemolition measurements, and (b) Shearing the uncertainty ellipse, as happens during squeezing by
one-axis twisting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.9

Bragg transitions. (a) Two counterpropagating lasers address an upward moving
atom with frequencies arranged to satisfy Eq. (2.44). (b) The two lasers drive aresonant transition between two momentum states (taken to be |0~ki and |2~kihere) in the same electronic manifold. The process is reminiscent of a two-photon
Raman transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Schematic of a two-path Bragg interferometer. The two paths are respectively denoted by A and B. The momentum labels indicate the ‘spin up’ (|↑i ≡ |p + 2~ki)and ‘spin down’ (|↓i ≡ |pi) states at different points during the interferometer sequence. Gravitational acceleration causes the momenta of the two states to linearlyvary over time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1

Two EIT lasers address the ion, with one driving the |g1 i ↔ |ei transition and theother driving the |g2 i ↔ |ei transition in a blue-detuned regime (∆ > 0). . . . . . . . 33

xv
3.2

EIT absorption spectrum with laser parameters relevant to the NIST EIT coolingexperiment. Two strong dressing lasers, with equal Rabi frequencies and equal detunings from their respective transitions, couple two long-lived states to an excitedstate in a closed three-level system. The absorption from a weak probe is plotted as the probe detuning ∆P is scanned across the dressing detuning ∆D . The
motion-removing (blue impulses) and motion-adding (red impulses) sidebands canbe interpreted as weak probes that sample this spectrum. Inset: Close-up near(∆P − ∆D )/2π = 0, with a magnified y-axis, showing the zero at the transparency
point as well as the asymmetry in the spectrum on either side of this point. The blue
dashed line is a mirror image of the spectrum on the blue-detuned side, drawn on the
red-detuned side to highlight the asymmetric growth of the absorption away from thetransparency point. The dressing lasers have equal detuning ∆D /2π ≡ ∆0 /2π = 360
MHz, and equal Rabi frequency Ωopt (∆0 )/2π ≈ 33.9 MHz. The Rabi frequency of
the weak probe is ΩP = 0.05 Ωopt (∆0 ). The decay rates from the excited state to
the two long-lived states are Γ1 /2π = 6 MHz, Γ2 /2π = 12 MHz. (See the discussion
in Section 3.4 and Eq. (3.14) for a detailed explanation of the parameters.) . . . . . 34

3.3

Experimental setup to cool ions in a Penning trap using electromagnetically induced
transparency (EIT). Two EIT lasers address the ion crystal at angles ±θ with respect
to the x-axis. The curved arrow indicates the rotation direction in the x-y plane. . . 37

3.4

Cooling of the transverse motion over time, for a single ion revolving around the trap
center at different radii. The cooling is slower at larger radii because of time-varying
Doppler shifts modulating the detunings of the EIT lasers as seen by the ion. Inset:
The steady-state occupation also increases with distance from the trap center. Here,∆0 /2π = 180 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 24 MHz. . . . . . . . . . . . . 43

xvi
3.5

Cooling curve for an ion at the trap center that experiences no Doppler shifts. The
separation of the timescales for the electronic and motional degrees of freedom is
not large enough to validate the adiabatic elimination of the electronic degrees
of freedom. The cooling is therefore much slower than the result predicted fromsuch an elimination procedure [83]. Here, ∆0 /2π = 180 MHz, Ω1 /2π = Ω2 /2π =Ωopt (∆0 )/2π ≈ 24 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6

Cooling curves for the center-of-mass (COM) mode for crystals with different ion
numbers, calculated using the Gaussian model (GM), showing rapid near groundstate cooling within 100 µs. However, the cooling rates are almost identical for allof these crystals. Here, ∆0 /2π = 360 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 33.9
MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.7

Cooling curve for a single revolving ion, located at r = 0, 20, 40 and 60 µm fromthe trap center, computed using three numerical approaches: (i) Time evolutionof the full density matrix (full DM), (ii) the Gaussian model (GM) and (iii) the
Sampling model (SM) using 2096 trajectories. The cooling curves from the GM do
not agree with the full DM curves. However, sampling the initial noise systematically(SM) accounts for beyond-Gaussian properties of the phase-space distribution of thesystem degrees of freedom, and reproduces the full DM curves very well. Here,∆0 /2π = 360 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 33.9 MHz. . . . . . . . . . . . 47

3.8

Cooling curves for the center-of-mass (COM) mode for crystals with different ion
numbers, calculated using the Sampling model (SM) with 2048 trajectories. The SM
predicts that the cooling rate increases with ion number N . The cooling curves fromthe GM (Fig. 3.6) are also shown for comparison, where the N -dependency of thecooling rate does not manifest. Inset: Cooling rate of an N -ion crystal RN relative
to the single-ion rate R1 extracted from the SM (markers). A power-law fit (solidline) shows that the cooling rate scales as ∼ N 0.3 for the parameters used. Here,∆0 /2π = 360 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 33.9 MHz. . . . . . . . . . . . 49

xvii
3.9

(a) Cooling curves for all the drumhead modes of a 37-ion crystal with bandwidth
approximately 2π×185 kHz, computed using the SM, showing efficient cooling within
100 µs. Here, ∆0 /2π = 360 MHz, Ω1 = Ω2 = Ωopt (∆0 ) ≈ 33.9 MHz. (b) GM coolingcurves for all the drumhead modes of a 120-ion crystal with bandwidth approximately
2π × 376 kHz, showing near ground-state, steady-state occupations of all the modes
after few hundred microseconds of EIT cooling. Note that the y-axis is plotted inlogscale. Here, ∆0 /2π = 400 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 35.7 MHz.
The same rotating wall frequency, ωr /2π = 180 kHz was used in both cases. . . . . . 50

3.10 Steady-state occupation as a function of the misalignment angle δθ for a single ion
revolving around the trap center at different radii. The final occupation is notvery sensitive to small misalignments (δθ ≤ 1◦ ) of the EIT wavevectors. Here,∆0 /2π = 400 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 35.7 MHz. . . . . . . . . . . . 52
4.1

Schematic of the NIST EIT cooling experiment, showing the EIT lasers and ODF
lasers incident on the ions at angles of ±10◦ to the crystal plane. Figure credit:
Elena Jordan, NIST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2

Schematic of the thermometry sequence involving the ODF interaction. Bloch spherepictures indicate the spin state of an ion at different points of the sequence. . . . . . 58

4.3

ODF spectra showing the bright fraction as the ODF difference frequency is stepped
across the COM mode frequency ω1 . The black and blue data are respectively
measured after Doppler cooling and 200 µs of additional EIT cooling on a crystal
with 158 ± 10 ions. The measured n̄ values in each case are obtained by fitting thedata with the formula Eq. (4.8). Figure and data credit: Bollinger group, NIST. . . 61

xviii
4.4

Measured thermal occupation n̄ of the COM mode (black) as a function of thecooling time for a crystal with 190 ± 10 ions. The measured cooling time constantof approximately 28 µs corresponds to a rate that is faster than that theoretically
expected from separate cooling of each ion in the trap (blue). Inset: We recall thecooling transients resulting from the Sampling Model presented in Chapter 3, where
the cooling rate of the COM mode increases with ion number. Experimental datacredit: Bollinger group, NIST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5

ODF spectrum measured over the full bandwidth of drumhead modes after Doppler
cooling only (red) and after 300 µs of additional EIT cooling (blue), for a crystal with158 ± 10 ions. The significant reduction in bright fraction over the entire bandwidth
after EIT cooling is evidence of strong sub-Doppler cooling of all the drumheadmodes. Data credit: Bollinger group, NIST. . . . . . . . . . . . . . . . . . . . . . . . 63

4.6

Qualitative differences between theory and experimentally measured ODF spectra,
exemplified using a 79 ion crystal. (a) Theoretically expected spectrum at high andlow n̄ values are spiky and partially resolve the modes. (b) Experimentally measured
spectra (red and blue) display a smooth continuum. Theory incorporating ad hoc
mode frequency fluctuations smear the sharp features seen in (a) and lead to curves(green and orange) that resemble the measured spectra. Experimental data credit:
Bollinger group, NIST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7

Simulated annealing to find the minimum energy configuration of a 53 ion crystal.
(a) Ion positions before (blue) and after (orange) the simulated annealing procedure.
(b) Annealing temperature and in-plane potential energy versus annealing iteration.

4.8

69

Metropolis-Hastings algorithm to find thermal snapshots of the in-plane configuration. (a) Thermal snapshot (orange) with T⊥ = 10 mK compared to the minimum
energy configuration (blue). (b) Thermalization of the in-plane potential energy as
the MH algorithm repeatedly scans over all the ions in the crystal. . . . . . . . . . . 70

xix
4.9

Histograms of sorted drumhead modes for 2000 realizations of the in-plane configurations with (a) T⊥ = 0 mK, (b) T⊥ = 1 mK and (c) T⊥ = 10 mK. The bin size is
0.5 kHz. In the 0 mK case, some of the bins carry counts that are not a multiple of2000 possibly because of round-off errors affecting the binning algorithm or becauseof slight fluctuations in mode frequencies as the MH algorithm might have foundlower energy configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.10 ODF spectra computed by averaging over the bright fraction expected from thenormal modes of 48 thermal snapshots of the in-plane configuration. Here, (a) T⊥ = 0
mK, (b) T⊥ = 1 mK and (c) T⊥ = 10 mK. We have assumed n̄ = 6 for all the modes.
Also shown are the mode frequencies for the minimum energy configuration. . . . . . 71
4.11 Power spectra of the drumhead motion from molecular dynamics simulations. Here,Tz = 0.5 mK, and (a) T⊥ = 0 mK, (b) T⊥ = 1 mK and (c) T⊥ = 10 mK. . . . . . . . 73
4.12 ODF spectra computed from molecular dynamics simulations. Here, Tz = 0.5 mK,
and (a) T⊥ = 0 mK, (b) T⊥ = 1 mK and (c) T⊥ = 10 mK. . . . . . . . . . . . . . . . 74
5.1

Schematic for continuous, real-time phase tracking. Two lasers drive a collection of
atoms to interact with a cavity mode. The relative phase φ(t) can be continuously
tracked by homodyne detection of the field leaking out. . . . . . . . . . . . . . . . . 78

5.2(a) Cavity-assisted Raman transitions: The red (blue) pathway leads to the emissionof a cavity photon accompanied by a spin flip |↓i → |↑i (|↑i → |↓i). (b) Hierarchyof frequencies.

5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Classical Bloch vector picture: The red and blue pathways set up balanced, opposing
superradiance pathways that lead to a coherent cancellation of the intracavity field
when the Bloch vector (green) is along the y-axis (φ = 0). When the Bloch vector
has a small x-component (φ 6= 0), the intracavity field from the two pathways add
constructively, giving rise to non-zero output field. . . . . . . . . . . . . . . . . . . . 80

xx
5.4

Free-space scattering processes considered in our model: Rayleigh scattering (blue),
Raman scattering (red) and atom loss (black).

6.1. . . . . . . . . . . . . . . . . . . . . 84

Real-time continuous tracking of a time-varying phase. (a) A single experimental
run: A squeezed state is prepared during [−50T0 , 0], with the initial measured phase(m)

φ0(blue triangle) varying in each run. Subsequently, a phase modulation φ(a) (t) =15 mrad×sin(t/40T0 ) (black line) is applied e.g. using a time-varying magnetic field.
(m)

(m)

The blue, filled (red, hollow) markers are estimates φSSS (φCSS ) of the phase using themeasured photocurrent in windows of duration 8T0 that account for (do not account(m)for) the initial offset φ0 . The gray shaded region indicates the 1-σ SQL tolerance
for this applied signal. Representative Bloch spheres for t ≤ 0 indicate the state
before and after the state preparation stage. For t > 0, Bloch spheres indicate thedeflection of the spin as a result of the phase modulation (black dots on the spheres
indicate the zero phase reference), as well as the equivalent spin state used for the
(m)

(m)

(m)

respective estimates φCSS , φSSS . (b) Histogram of phase errors φSSS − φ(a) (blue) and(m)

φCSS − φ(a) (red) over 2048 runs in one particular measurement window [48T0 , 56T0 ].
(m)(c) Single-run precision gain of the estimates φSSS relative to the SQL at different
window centers t. Here, ∆φ2SSS is the variance of Gaussian fits to histograms such as
the blue histogram in (b). Decoherence results in decreased gain over time. . . . . . 93

xxi
6.2

(a) A sudden jump in the phase with amplitude φJ = 40 mrad at TJ = 50T0 is
tracked in the same run using moving windows of durations TW = 2T0 (red) andTW = 20T0 (blue), showing the faster response of the shorter window. (b) Protocol(m)to estimate φJ . (c) Histograms, over 2048 runs, of φJ

for TW = 2T0 (red) andTW = 20T0 (blue), demonstrating the greater precision of the longer window. For
TW = 2T0 , W2 was offset by a small time 0.2T0 to allow transients on timescales ofκ−1 to decay. (d) Gain in precision over a CSS in Ramsey mode as the duration ofW1 and W2 is varied, for fixed Cγsc and different values of C. Analytic results (lines)
calculated using Eqs. (6.3) and (6.11) are in excellent agreement with simulations(markers). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3

Effect of number fluctuations on real-time phase tracking. Single-run precision gain
computed using error histograms of 2048 experimental runs (compare to Fig. 6.1(c)).
Parameters are from Fig. 6.1, except that the number of atoms in each run is variable,with a mean of N̄ = 105 and ∆N/N̄ indicated by the percentages. Data points(markers) extracted from the numerical experiments are in very good agreement withsemi-analytic results (solid lines) obtained using the simple expression in Eq. (6.13)
that accounts for number fluctuations. The black dashed line plots the applied phasemodulation, for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.1

Experiment setup and working principle. (a) A cloud of atoms interacts with twocounterpropagating modes of a ring cavity. One mode (mode 1) is driven at frequency
ωl , while the counterpropagating mode (mode 2) is in vacuum, i.e. not pumped. The
scheme enables cavity-mediated interactions between every pair of atoms. (b) The
excitation or de-excitation of a single atom is off-resonant. However, the exchangeof excitation between two atoms is a resonant process. . . . . . . . . . . . . . . . . . 103

xxii
8.1

Interplay of squeezing and superradiance for different R = κ/2δn↓ values. (a) Evo2 for R = 0.025, 0.05, 0.1, 0.2. Inset: Maximum metrological gain (in dB)
lution of ξR

and time taken to achieve this gain. (b) Population in n↓ , n↑ for R = 0.2, with total
population in all centers adding up to N = 103 . (c) Population in n+1 for different
R values. In this panel, N = 103 , C = 1, |β|2 ≈ 5.4 × 105 . Solid (dashed) lines
represent MCM (TCM) results. Four centers, n↓ , n↑ , n+1 , n+2 , were tracked in the
MCM simulations, with negligible population in n+2 . . . . . . . . . . . . . . . . . . . 126
8.2

2 for |β|2 /104 = 2, 4, 8, 16. (b) PopSqueezing faster and faster. (a) Evolution of ξR

ulation in n↓ , n↑ for |β|2 /104 = 16, with total population in all centers adding up
to N = 103 . Solid (dashed) lines represent MCM (TCM) results. (c-d) Populationin, respectively, n−1 and n+1 centers, for various drive strengths. (e) Comparison ofsimulated n±1 populations to analytic result of Rabi oscillation model (see Text). In
this panel, N = 103 , C = 10 and R = 0.05. Six centers, n−2 , n−1 , n↓ , n↑ , n+1 , n+2 ,were tracked in the MCM simulations with very low populations in n±2 .
8.3. . . . . . 128

2 in the case of
Squeezing in the presence of momentum width. (a) Evolution of ξR|β|2 /104 = 4 for σ̃q = 0, 0.025, 0.05, 0.1. Solid (dashed) lines represent MCM (TCM)
results. (b) Maximum metrological gain as a function of drive strength for different
2 in the TCM for σ̃ = 0.1 and |β|2 /104 = 2 whenσ̃q values. (c) Evolution of ξRqNE = 0, 1, 2, 4 echo pulses are inserted. The NE = 1, 2, 4 cases evolve identically to
the NE = 0 case until the first echo is applied (at different times in the three cases).
The gray broken line shows the σ̃q = 0 case with no echoes. (d) Evolution of the
2 in the TCM when N = 2 echo pulses are inserted.
constituents, C and Vmin of ξR
E

Other details are the same as in Fig. 8.2.

. . . . . . . . . . . . . . . . . . . . . . . . 130

xxiii
8.4

Manifestation of a many-body energy gap. (a) Evolution of C⊥ for σ̃q = 0.05 fordifferent values of |β|2 /104 = 2, 4, 8. (b) TCM results using the same parameters asin (a), but with the gap Hamiltonian ĤG turned off. The gray broken line in eachcase shows the decay of C⊥ under free evolution. Solid (dashed) lines represent MCM(TCM) results. Other details are the same as in Fig. 8.2.

. . . . . . . . . . . . . . . 133

Chapter 1

Context

We are currently entering an era where quantum technologies are expected to revolutionize theway we communicate, compute, measure and understand our universe. Tremendous advancementsin the theoretical understanding and experimental control of physics in the quantum regime has led
to a ‘There And Back Again’ scenario: Early experiments in the 20th century typically worked witha large number of poorly controlled atoms or molecules, but greatly contributed to understanding
central concepts in quantum physics. The late 20th and early 21st centuries witnessed significant
advances in laser cooling and trapping [126], heralded by elegant theoretical schemes for the coolingof single atoms and unprecedented experiments with a single or a few atoms [29, 69, 109]. We are
now ready to reconsider working with large quantum systems by asking the following two very
exciting questions:(1) Can we leverage the lessons learned with small systems and scale up these techniques toexquisitely control and manipulate large quantum systems?
(2) If we had such exquisite control, can we engineer and observe intrinsically many-bodyfeatures predicted by quantum mechanics, such as entanglement, and exploit such features
for novel practical applications?
Indeed, modern theory and experiments have answered both of these questions in the affirmative. The last decade has witnessed rapid advancements in studies and applications of quantum
many-body systems. The demonstration of quantum computers claiming quantum supremacy over2
known classical algorithms [3], quantum simulations of exotic states of matter such as discrete time
crystals [135] and the operation of optical lattice clocks with record breaking precision [86] areimportant but nevertheless only representative examples of quantum many-body systems playing
a central role in quantum computing, simulation and metrology.
In particular, quantum metrology with atoms stands to benefit immensely with the development of large, well controlled quantum systems. Typically, metrology with atoms involves preparing
an initial quantum state with high accuracy, perturbing the state with a signal to be measured,and reading out the change in the state as a result of the applied signal. Working in the spiritof the questions raised above, we broadly identify three crucial themes in the context of quantum
metrology, where advancements will lead to new and improved practical applications:(1) Sub-Doppler laser cooling of large quantum systems,(2) Novel techniques to read out changes in the prepared state, and(3) Engineering entanglement to improve measurement precision in traditionally non-interacting
quantum systems.
In this thesis, we present our contributions to advancing state-of-the-art capabilities in eachof these three areas. We begin with an introduction in Chapter 2, where we establish the backdrop
against which our work has been carried out. Subsequently, under the first theme, we report on
the numerical modeling of an experiment to ground-state cool large two dimensional trapped ion
crystals in a Penning trap. This work is detailed in Chapters 3 and 4. Under the second theme,
in Chapters 5 and 6, we describe and numerically demonstrate a proposal for continuously andprecisely tracking the relative phase of a spin superposition using atom-cavity interactions. As
a paradigm for the third theme, we propose in Chapters 7 and 8 a scheme for entangling spinsencoded in the center-of-mass motion of atoms, for use in a Bragg interferometer. For each of theseprojects, we comment on possible future directions at the end of the relevant chapters. We conclude
with a brief summary in Chapter 9.

Chapter 2

Introduction

In the last chapter, we identified three central themes in quantum metrology that we will
be exploring in this thesis. In this chapter, we provide an overview of concepts that are essential
to understanding our work on these themes. Accordingly, we first review advancements in laser
cooling from a trapped ion perspective. To warm up to our second and third themes, we introduce
the notion of spin-1/2 systems and how such systems can be encoded in a variety of atomic degrees
of freedom. Next, we describe the Ramsey sequence, a protocol that is currently the workhorse forquantum metrology with atomic systems. Despite its widespread application, we point out somesituations where it is not effective and alternative protocols may be required. We show how theuse of uncorrelated atoms in this sequence leads to limits on measurement precision. We discuss
how precision beyond this limit can be achieved via the concept of spin squeezing. As examples,
we describe a couple of mechanisms by which such squeezed spin states can be generated. Finally,
we briefly describe the operation of an atomic Bragg interferometer, a platform where controllable
atom-atom interactions are hard to engineer and spin squeezing has not been experimentally realized
till date.

2.1

Laser cooling of trapped ions
To illustrate general features of laser cooling, we consider a toy model of a single trapped ion

with two internal (electronic) states |gi , |ei and with one external (motional) degree of freedom,say along the z direction. The harmonic trapping potential enables us to treat the ion’s motion as

4that of a quantum harmonic oscillator. The Hamiltonian for this isolated trapped ion is
ĤA =

~ωa
(|ei he| − |gi hg|) + ~ω0 b̂† b̂,
2

(2.1)

where ~ωa is the energy difference between the ground (|gi) and excited states (|ei) and ω0 is
the trapping frequency. The external degree of freedom is described using bosonic creation andannihilation operators b̂† , b̂ satisfying [b̂, b̂† ] = 1 from which a position operator ẑ for the ion can beconstructed.

2.1.1

The atom-laser interaction

The essential idea behind laser cooling is that the absorption and emission of photons ingeneral simultaneously affects both the electronic and motional state of an ion. We assume that a
traveling wave laser with frequency ωl tuned close to ωa and wavevector kl = kl ẑ drives the |gi ↔ |eitransition. The electric field associated with this laser is given by E(z, t) = E0 cos (kl z − ωl t).
Within the electric dipole approximation, the Hamiltonian for the interaction of the atom with thelaser field is given by the expression −d̂·E, where the atomic dipole operator d̂ = d |ei hg|+d∗ |gi he|
induces transitions from |gi to |ei and vice versa [41]. The transition dipole matrix element d is avector and captures details about the spatial profile of the wavefunctions associated with the twoelectronic states. A subtle point to note is that only the center-of-mass coordinate of the entire ion
is represented as a quantum harmonic oscillator. The valence electron that is making transitions
between the two electronic states should be pictured as tagging along closely with the ion, thereby
making the electric dipole approximation valid. In other words, the dipole formed by the valence
electron and the ionic shell can still be regarded as a point dipole, but the location of the point
dipole itself is captured by the center-of-mass coordinate ẑ of the ion.
Except at very strong laser intensities, energy conservation can be imposed by assuming thatthe |gi → |ei (|ei → |gi) transition is accompanied by the absorption (emission) of a laser photon.
Rigorously, this assumption amounts to making the rotating wave approximation (RWA) [41].
Within the electric dipole and rotating wave approximations, the Hamiltonian for the atom-laser5interaction is given by
ĤAF =~Ω∗~Ω|ei hg| ei(kl ẑ−ωl t) +|gi he| e−i(kl ẑ−ωl t) ,
2
2

(2.2)

where Ω = −d̂ · E0 /~ is called the Rabi frequency and captures the driving strength. Without lossof generality, we assume Ω to be real in the subsequent discussion as its complex phase can always
be absorbed into an initial phase offset of the electric field. One way to understand the signs of
the complex exponentials multiplying the atomic transition operators is to consider the case when
the laser field is quantized and is described by bosonic creation and annihilation operators â† , â.
The laser field is then governed by a Hamiltonian ĤL = ~ωl â† â. From ĤL , it is easy to see thatthe time evolution of the Heisenberg picture operators â(t), â† (t) is given by â(t) = âe−iωl t andâ† (t) = â† eiωl t . Matching signs with the exponentials in Eq. (2.2), we see that the raising operator|ei hg| is accompanied by photon annihilation or absorption while the lowering operator |gi he| isaccompanied by photon creation or emission. Since the laser field is assumed to carry a large
number of photons, the operators â, â† which should have appeared in Eq. (2.2) have been replaced
by a macroscopic c-number, that has been absorbed into the definition of the classical electric field
E0 .
Since the laser electric field is z dependent, the ion experiences a varying electric field as it
oscillates along the z direction. This piece of physics is captured by the e±ikl ẑ terms entering theatom-laser interaction in Eq. (2.2), where ẑ is the position operator for the ion. Clearly, from theform of Eq. (2.2), the laser affects both the electronic and motional state of the ion and therefore
we can envisage the possibility of cooling the ion using the laser.
Time dependent Hamiltonians such as Eq. (2.2) are harder to analyze, so we first transformto an interaction picture with free Hamiltonian H0 = ~ωl /2 (|ei he| − |gi hg|). In this picture, thetotal Hamiltonian is time independent and is given byĤI = −~∆~Ω (|ei he| − |gi hg|) + ~ω0 b̂† b̂ +|ei hg| eikl ẑ + |gi he| e−ikl ẑ ,22where ∆ = ωl − ωa is the laser-atom detuning.

(2.3)

6
2.1.2

The Lamb-Dicke regime

When the typical displacement of the ion, as it oscillates, is small compared to the wavelengthof the laser light, the exponentials e±ikl ẑ can be expanded in a Taylor series. Mathematically, thispcondition translates to kl hẑ 2 i  1, and is called the Lamb-Dicke criterion [126]. Furthermore,
expressing the position operator ẑ in terms of the creation and annihilation operators, we can write
kl ẑ = η b̂ + b̂† ,where η = kl

(2.4)

p
~/2M ω0 is called the Lamb-Dicke parameter. An important point is that a small

Lamb-Dicke parameter, η  1, is necessary but not sufficient to satisfy the more restrictive LambDicke criterion defined above [126]; from the definition of η, we see that it is the ratio of the RMS
zero-point displacement to the laser wavelength. However, the Lamb-Dicke criterion also requiresthat the number of vibrational quanta in the harmonic oscillator remains small.
The atom-laser interaction in Eq. (2.3) can now be expanded in powers of η as
 η2 
2~Ω††|ei hg| 1 + iη b̂ + b̂ −+ . . . + H.c.
b̂ + b̂
2
2
2.1.3

(2.5)

Motional sidebands

Eq. (2.5) implies that the atom can be excited from |ei hg| not just by the absorption of a laser
photon, but also by the simultaneous absorption and emission of vibrational quanta, or phonons[126, 36]. The O(η) term describes two processes where the excitation of an atom is accompaniedby the absorption of a photon and a phonon or by the absorption of a photon and emission of aphonon. Similarly, the O(η n ) term describes various processes involving n phonons and a single
laser photon. To understand what these terms entail from an energy perspective, let us consider
the O(η) term as an example. The energy required to excite the atom is ~ωa . The laser photon
supplies ~ωl , while the absorption (emission) of a phonon supplies (demands) ~ω0 , leading to a
net excitation energy of ~ (ωl + ω0 ) (~ (ωl − ω0 )). If the laser is appropriately detuned from theatomic transition frequency, one of these two processes can be tuned closer to resonance than theother. From a cooling perspective, we would prefer to make the phonon absorption process more

7
resonant, as this would ensure that every atomic excitation is more likely to remove, rather thanadd, one quantum of motional energy. From the atom’s perspective, the laser now appears tohave sidebands at frequencies ωl ± ω0 . Almost all laser cooling schemes rely on strongly enhancing
atomic excitation via the motion removing sideband at ωl + ω0 while simultaneously suppressing
excitations via the motion adding sideband at ωl − ω0 .
2.1.4

Spontaneous emission

A purely Hamiltonian process cannot irreversibly remove energy from the system, as thereverse process (the Hermitian conjugate term) is also energetically equally favorable. Instead,such irreversible removal of energy occurs through spontaneous emission from the excited state |eithat resets the atom to the ground state |gi, thereby completing one absorption-emission cycle. As
an example for a single cooling cycle, an atom in an n-phonon state |g, ni is likely to be excited tothe state |e, n − 1i, from where spontaneous emission brings the state to |g, n − 1i. In this way, laser
cooling can be pictured as a sequence of absorption-emission cycles, with each cycle more likely toremove a phonon rather than add one. For this picture to be accurate, spontaneous emission itself
should not change the vibrational quantum number. To see the effect of spontaneous emission on
the motion, we consider the Lindblad dissipator associated with this process, given byDρ = Γ |gi he|Z 1
duNµ (u)e−1−iksc ẑuρeiksc ẑu11|ei hg| − |ei he| ρ − ρ |ei he| .
2
2

(2.6)

Here, the density matrix ρ describes the joint electronic and motional state of the ion, Γ is the
spontaneous decay rate of |ei → |gi, ksc is the wavevector associated with the spontaneously emitted
photon, u = cos θsc , with θsc the angle between ksc and the z axis, and N (u) is the normalized
dipole radiation pattern associated with the emission. Assuming ksc ≈ kl , we can expand thecomplex exponentials in Eq. (2.6) to O(η 2 ) to write Dρ ≈ D0 ρ + K2 ρ, where
11
D0 ρ = Γ |gi he| ρ |ei hg| − |ei he| ρ − ρ |ei he| ,
2
2

(2.7)

8
is the familiar Lindblad dissipator for spontaneous emission that neglects effects on the motion,
andK2 ρ =
η 2 Γhu2 i|gi he| 2X̂ρX̂ − X̂ 2 ρ − ρX̂ 2 |ei hg| ,
2

(2.8)

is a term that accounts for changes in motional state at O(η 2 ). The O(η) term vanishes because
N (u) is an even function. Therefore, at O(η 0 ), spontaneous emission predominantly does notchange the motional state, thereby providing support to the cooling cycle picture described above.
Yet another interpretation of η helps understand the associated physics [126]. Upon spontaneously emitting a photon, the kinetic energy of the atom is reduced by ∼ ~ωr , where ωr = ~kl2 /2M(ksc ≈ kl ) is the recoil frequency. The energy required to change the harmonic oscillator state by
one quantum is ~ω0 . The ratio of the two frequencies is
~kl2
ωr
=
= η2.
ω0
2M ω0

(2.9)

Therefore, η  1 is also equivalent to assuming that the recoil energy of the scattered photon is
too small to cause significant changes in the motional state.

2.1.5

Laser cooling with two electronic states

The toy model we developed using just two electronic states forms the basis for the two mostpopular laser cooling techniques, Doppler cooling and sideband cooling, which are closely related.
The absorption spectrum of the two level system is a useful tool for understanding the coolingmechanism [36]. This spectrum is obtained by addressing the |gi ↔ |ei transition with a laser
and measuring the scattered intensity as the laser frequency is varied across the atomic resonance.
Essentially, this measurement is equivalent to probing the excited state population ρee . For a laser
detuning ∆, decay rate Γ and Rabi frequency Ω  Γ, the steady state excited population takes theform of a Lorentzian given byΩ2 /4
ρee ≈ Γ2.
2+∆
4

(2.10)

From Eq. (2.5), the action of the first motion removing sideband on the state |g, ni isi~Ωηi~Ωη √|ei hg| b̂ |g, ni =n |e, n − 1i ,2
2

(2.11)

9√suggesting that the Rabi frequency associated with this sideband drive is Ω−
n = iΩη n. Similarly,√
for the motion adding sideband, the associated Rabi frequency is Ω+
n = iΩη n + 1. Combinedwith Eq. (2.10), the rates for cooling and heating cycles are respectively given by Rn− = nη 2 A− andRn+ = (n + 1)η 2 A+ , whereA − = Γ Γ24

Ω2 /4
+ (∆ + ω0 )2, A + = Γ Γ24

Ω2 /4
+ (∆ − ω0 )2,(2.12)

where the modified detunings now account for the respective sideband frequencies. These rates arelinear in n, therefore we can immediately write down a rate equation for the average occupation n̄
of the harmonic oscillator asdn̄ = −Rn̄− + Rn̄+dt
= −W n̄ + η 2 A+ ,(2.13)

where W = η 2 (A− − A+ ) is the cooling rate provided that A− > A+ . The steady-state occupationn̄SS is given byn̄SS =

A+
A− − A+.

(2.14)In addition to these cooling and heating cycles, there is a third diffusive cycle that first
involves absorbing just a laser photon (the ‘carrier’ transition, O(η 0 ) term in Eq. (2.5)) and leaving
the motional state unaffected, and then changing the motional state via the spontaneously emitted
photon. As the K2 term, Eq. (2.8) shows, this process is also O(η 2 ). However, such a process has
equal probability for heating and cooling, and therefore leaves the net cooling rate W unaffected.
However, n̄SS depends additionally on A+ which is modified because of this cycle. Therefore, n̄SS
increases by a small numerical factor of O(1).

2.1.5.1

Doppler cooling

When Γ  ω0 , the Lorentzian absorption spectrum with width ∼ Γ is so broad that several
motional sidebands can fit within the broad peak. The sidebands are said to be unresolved andcooling in this regime is traditionally called Doppler cooling. The maximum ratio of A− /A+ , and10

Figure 2.1: Sketch of absorption spectrum and locations of the carrier, motion adding and motion
removing sidebands in (a) Doppler cooling and (b) Sideband cooling.

correspondingly, the lowest n̄SS is obtained for the optimum value ∆ = −Γ/2 [36]. Intuitively, thisoptimum value can be understood by noting that for a Lorentzian function f (x) = 1/(1 + x2 ), themaximum fractional change between two points x ± δx, characterized by 2δxf 0 (x)/f (x), occurs at
the the half maximum point corresponding to x = ±1 and takes the simple value 2δx. Identifying
x = 2∆/Γ and δx = ±2ω0 /Γ, this result can directly be applied to the absorption spectrum,Eq. (2.10), in the regime where Γ  ω0 . The minimum value of n̄SS is then simply given by
n̄min
SS ∼

Γ
 1.
2ω0

(2.15)

We avoid writing a strict equality since a rigorous derivation must also account for the previously
discussed diffusion cycles. Therefore, in Doppler cooling, the mean occupation, or equivalently, thetemperature kB T = n̄~ω0 ∼ ~Γ, is limited by the lifetime of the excited state.
2.1.5.2

Sideband coolingIn the opposite limit when Γ  ω0 , the absorption spectrum is so narrow that with the
choice of ∆ = −ω0 , the motion removing sideband is exactly on resonance, while the carrier andmotion adding sideband are strongly suppressed [36]. The sidebands are now said to be resolved
and cooling in this regime is called resolved sideband cooling. At this value of ∆, clearly A− ∼ Ω2 /Γ11
whereas A+ ∼ A− Γ2 /ω02  A− . Therefore, the minimum occupation is given byn̄SS ≈

A+
Γ2
∼
 1.
A−
ω02

(2.16)

Therefore, sideband cooling is an attractive choice for cooling the ion to near ground state occupancies. We note that a popular technique to implement sideband cooling involves creating an artificalnarrow line transition using stimulated Raman transitions on a three-level system. However, thegeneral features from the two-level model still apply in that case [126].

2.1.6

Engineering the absorption spectrum using quantum interference

An attractive question to ask is the following. Is it possible to somehow engineer the absorption spectrum so that the asymmetry between the motion removing and motion adding sidebands isgreatly enhanced? Furthermore, can the diffusion arising from the carrier transition be completely
suppressed? It turns out that these features can be engineered by coupling the state |ei to a third
long-lived state |ri as shown in Fig. 2.2(a) [84]. A strong coupling laser with Rabi frequency Ωccouples the |ri ↔ |ei transition and is blue detuned so that ∆c > 0. To obtain the absorption
spectrum, a weak probe laser with Rabi frequency Ωp  Ωc couples the |gi ↔ |ei transition andthe excitation probability is studied as its detuning ∆p is swept across ∆c . The excited state is assumed to decay with rates Γg and Γr respectively to the two long-lived states, and its total lifetimeis Γ = Γg + Γr . An analytic expression for the steady state value of ρee is tedious to derive, but has
been done, for example in Refs. [57, 73]. For Ωp  Ωc , the expression can be compactly written asρee ≈4δ 2 Ω2p Γ
1,
Γg 4δ 2 Γ2 + [Ω2c − 4δ(∆c + δ)]2

(2.17)

where δ = ∆p − ∆c is the two-photon detuning from Raman resonance. Figure 2.2(b) shows theabsorption spectrum as a function of ∆p for a specific choice of parameters. The spectrum featuresa broad resonance, around ∆p ≈ 0 when Ωc  ∆c , and a narrow resonance for ∆p slightly greaterthan ∆c . Remarkably, the absorption completely vanishes at Raman resonance, when ∆p = ∆c .
The vanishing of the absorption probability from a probe laser when the excited state is coupled
to an auxiliary state by a second laser is called electromagnetically induced transparency (EIT).

12

Figure 2.2: Engineering the absorption spectrum using quantum interference. (a) A weak probe
laser is scanned across the |gi ↔ |ei transition as the state |ei is dressed by a strong coupling
laser driving the |ri ↔ |ei transition in a blue detuned regime (∆c > 0). (b) The corresponding
absorption spectrum displays a Fano profile close to ∆p ≈ ∆c .

In the bare basis, the transparency point can be understood as arising from destructive
interference between two pathways [73]. In the first pathway, a probe photon is absorbed from|gi and immediately scattered into free space via |ei, returning the atom to |gi. In the second, aprobe photon absorption from |gi is followed by the emission and absorption of a coupling laser
photon, before free space scattering from |ei returns the atom to |gi. These pathways are depicted
in Fig. 2.3.
The two resonances can be understood by considering the dressed states |ψ± i of the systemcomprising the atom and the coupling laser [84]. With the zero energy reference chosen to be theexcited state |ei, the energies of the two dressed states are given byp∆c ∓ ∆2c + Ω2c
E± = ~.
2(2.18)

With ~∆c the bare energy of the state |ri, |ψ+ i (|ψ− i) can be identified with the broad (narrow)resonance in the absorption spectrum shown in Fig. 2.2(b). Further confirmation can be obtained
from the form of |ψ± i, which for Ωc  ∆c simplify to
1
1|ψ+ i = p(Ωc |ri − 2∆c |ei) , |ψ− i = p(2∆c |ri + Ωc |ei) .
4∆2c + Ω2c
4∆2c + Ω2c(2.19)

From the excited state coefficient, the lifetime of these states can be approximately identified as
Γ+ ≈ Γ and Γ− ≈ ΓΩ2c /4∆2c  Γ+ , corresponding respectively to the broad and narrow peaks inthe spectrum.

13

Figure 2.3: The two interfering pathways that transfer the atom from |gi to |ei. Subsequent
spontaneous emission returns the atom to |gi.

2.1.6.1

EIT cooling

From the perspective of cooling, an important feature of the spectrum is that the energy
difference between the transparency point and the sharp narrow peak can be tuned via the Rabi
frequency Ωc of the coupling laser [84]. Consider the case when the probe laser is fixed at the
detuning ∆p = ∆c . Then, by arranging∆c +p∆2c + Ω2c
− ∆ c = ω0 ,2

(2.20)

the motion removing sideband at ∆p + ω0 can be brought on resonance with the narrow peak and istherefore strongly enhanced. Simultaneously, the very low absorption probability at ∆p −ω0 strongly
suppresses the motion adding sideband. Naturally, this cooling mechanism is called EIT cooling.
Notably, diffusion from the carrier transition is nonexistent as the absorption vanishes exactly at∆p = ∆c . This latter property ensures that for the same cooling rates, EIT cooling achieves
lower n̄SS compared with resolved sideband cooling. Under the optimum condition Eq. (2.20), theminimum value of n̄SS = (Γ/4∆c )2 is attained, indicating that the lowest occupations are obtained
when the atom is dressed with a far detuned laser [84]. In contrast to Doppler and sideband cooling,
EIT cooling works only for blue detuned probe and coupling lasers. When the coupling laser is red
detuned, the absorption spectrum is a mirror image of the one shown in Fig. 2.2(b) (see Ref. [73]),
and with ∆c = ∆p , the motion adding sideband is instead enhanced, leading to heating.
As in the case of Doppler and sideband cooling, analytic expressions for the A± rates and n̄SS

14have also been derived for EIT cooling. The rates can be motivated from the form of Eq. (2.17),
although a formal derivation is tedious and involves projection operator techniques [84, 83]. An
important feature of the corresponding rate equation is that the Lamb-Dicke parameter η dictating
the cooling rate corresponds to the projection of an effective wavevector along the motional axis.
Specifically,rη = |kp − kc | cos φ

~
,
2M ω0

(2.21)

where kp , kc are respectively the wavevectors of the probe and coupling lasers, and φ is the angle ofthe difference wavevector along the motional axis. Furthermore, the results have been generalized
to the case when Ωp ≈ Ωc , an attractive parameter regime for rapid cooling since the cooling rateq
scales with the product Ωc Ωp for a fixed value of Ω = Ω2c + Ω2p [83]. However, the physics is not
as intuitive as in the Ωp  Ωc limit presented here.

2.2

Quantum Metrology with pseudospin-1/2 systems
A two-level system such as the toy model we introduced in Section 2.1 can be conveniently

mapped onto a spin-1/2 system by defining the following operatorsσ̂ x = (|ei hg| + |gi he|) , σ̂ y = −i (|ei hg| − |gi he|) , σ̂ z = (|ei he| − |gi hg|) .

(2.22)

These operators are precisely the Pauli matrices that are ubiquituous in the description of spin1/2 systems. Specifically, they satisfy the commutation relations [σ̂ j /2, σ̂ k /2] = ijkl σ̂ l , implyingthat the vector σ̂/2 = (σ̂ x /2)x̂ + (σ̂ y /2)ŷ + (σ̂ z /2)ẑ is a legitimate angular momentum operator.
2
Furthermore, since σ̂ j = 1, the quantity σ̂ 2 /4 ≡ S(S + 1) = 3/4, revealing that the total spin
S = 1/2. We can also define ladder operators σ̂ ± = σ̂ x /2 ± iσ̂ y /2 that correspond to exciting andde-exciting the atom. To connect with spin terminology, we will henceforth generically refer to thetwo states as |↑i and |↓i in this chapter.
Atoms and atom-like systems provide a variety of degrees of freedom in which spin-1/2
systems can be encoded. Such ‘pseudospin-1/2’ systems find applications throughout quantum
computing, simulation and metrology. For example, optical atomic clocks based on 87 Sr atoms rely

15
on encoding a pseudospin-1/2 system in an ultranarrow linewidth optical transition with |↓i ≡ 1 S0
and |↑i ≡ 3 P0 [86]. As a second instance, quantum computing and simulations with trapped ionsare often enabled by the ability to encode spins in atomic hyperfine levels [46, 10]. For example, in
Chapter 4, we will encounter a motion sensing technique that relies on encoding a spin-1/2 systemin the 2s2 S1/2 (mJ = −1/2) and 2s2 S1/2 (mJ = +1/2) hyperfine states of 9 Be+ ions trapped in ahigh magnetic field [105, 42, 58]. A very interesting spin encoding is utilized in the case of atomic
Bragg interferometers, where a discrete spin-1/2 system is mapped on to a continuous variable,
namely the center-of-mass momentum of the atom. Important applications of such interferometers
include their use as gravimeters and gravity gradient sensors [1, 31, 27]. We will provide a brief
introduction to Bragg interferometers in Section 2.4 as our work in Chapters 7 and 8 revolves
around such momentum pseudospin based systems.

2.2.1

The Ramsey sequence

The Ramsey sequence is by far the most important and most common technique for metrology
with spin-1/2 systems. The sequence is best illustrated by visualizing pure states of a spin-1/2
system as points on the surface of a Bloch sphere. For the purpose of this discussion, a pure state
can be parameterized in spherical coordinates as|ψ (θ, φ)i = cosθθ|↑i + eiφ sin |↓i ,22(2.23)

where θ and φ represent polar and azimuthal angles respectively (see Fig. 2.4). In this representation, the south (north) pole corresponds to |↓i (|↑i), while points on the equator correspond toequal superpositions of the form|ψ (θ = π/2, φ)i =|↓i + eiφ |↑i√.
2(2.24)

Measurements with spin-1/2 systems rely on mapping the signal of interest on to the azimuthal phase φ. For example, let us suppose that the |↓i and |↑i are nominally separated in16

Figure 2.4: Bloch sphere representation of the pure state of a single spin-1/2 systemenergy by E0 = ~ω0 . Then, the time evolution of a initial state with θ = π/2, φ = 0 is given by|ψ(T )i =|↓i + eiω0 T |↑i√.
2(2.25)

Therefore, the spin transition frequency is mapped on to a total accumulated phase of φ(T ) = ω0 T .
Extending this example further, let us suppose that the energy difference of the spin states increases
linearly in the presence of a weak magnetic field. In a frame rotating at ω0 , the accumulated phaseφ(T ) = 0 in the absence of a magnetic field and φ(t) ∝ B × T in the presence of an applied field.
That is, the magnetic field strength is mapped on to the accumulated phase.
There still remains the question of how this accumulated phase can be measured. In the
Ramsey sequence, the accumulated phase is converted into a difference in population between thetwo states |↑i and |↓i. Mathematically, the angle φ is mapped on to θ. The sequence is illustrated
in Fig. 2.5. The spin is initially prepared in |↓i. A π/2 rotation about the −y axis prepares thespin in an equal superposition of the two spin states with φ = 0. A signal field µ(t) interacts withthe spin for a total time T , leading to a net accumulated phaseZ T
φ(T ) = Gdtµ(t),(2.26)

0where G is a proportionality constant. Finally, a second π/2 rotation about the −y axis takes thestate out of the equatorial plane. If φ(T ) = 0, the spin ends up in the north pole after the second
rotation and the final polar angle θf = 0. If φ(T ) 6= 0, θf > 0 after the second rotation. The

17

Figure 2.5: State of the spin at different points in the Ramsey sequence.

probability to find the spin in the state |↑i at the end of the sequence isP↑ = cos2

1 + cos θf
θf
1 + cos φ(T )
=
=
.
2
2
2

(2.27)

If φ(T ) is linear in the accumulation time T , P↑ oscillates between 0 and 1 as T increases, leading
to the so called Ramsey fringes.
With just a single spin, one run of the Ramsey sequence will simply collapse the spin to |↓i or|↑i. In order to measure the underlying probabilities, experiments typically employ a large numberof spins in each run of the Ramsey sequence, and also repeat the sequence several times for eachaccumulation time T . The random collapse of each spin means that the measurement precision islimited by quantum noise. We will further discuss this notion of measurement precision, as well as
how to improve it using entanglement, in Section 2.3.

2.2.2

Limitations of the Ramsey sequence

A big advantage of the Ramsey sequence is that, in the end, the experimenter has to only
measure state populations. Such measurements are facilitated by well established techniques, forexample, using fluorescence measurements on cycling transitions [126]. However, let us now analyze
some features of the traditional Ramsey sequence, which makes it unsuitable for use in certain
situations. Here, we assume that N  1 spins are participating in each sequence.
In Section 2.2.1, we emphasized that the Ramsey sequence is designed to measure the net
phase φ(T ) accumulated up to some time T . In a single run of the sequence, the population

18
measurement collapses the state of individual spins, and the next run must begin with re-initializing
all spins in the |↓i state. In order to study the time variation of a signal field µ(t), the sequence
has to be repeated at least once for several different accumulation times T so that the time series ofφ(T ) versus T can be constructed. The derivative of φ(T ) then yields information about µ(t). This
procedure seems straightforward, but it intrinsically assumes that the signal field is reproducible.
If µ(t) is, for example, derived from a rare phenomenon, then we would like to have a protocol
by which the entire time series of µ(t) can be measured in a single experimental run without
re-initializing the spins at all.
A second problem with the Ramsey sequence concerns the variation of P↑ with φ(T ). From
Eq. (2.27), the mapping of φ(T ) on to P↑ is one to one only for φ(T ) ∈ [0, π]. Phase excursions
beyond such an unambiguous interval cannot be uniquely measured within the traditional Ramsey
paradigm.
Recent work has attempted to increase the length of the unambiguous interval beyond π by amodification of the Ramsey sequence [112, 63]. First, a phase φ(T ), within the unambiguous interval, is allowed to accumulate over time T . Next, an intermittent rotation converts the phase into a
population difference. The subsequent population measurement, however, is coherence preserving
in nature. This means that a measurement of the total population in |↑i is performed without
gaining knowledge about the state of each individual spin. After measurement, the state is rotatedback on to the equatorial plane. The inferred phase is then fed back as an azimuthal rotation that
returns the phase of the spins to zero. In this process, the time history of applied phase corrections
can be tracked, and the net accumulated phase can be uniquely measured even when it is welloutside the unambiguous interval. We note that this procedure also partially overcomes the firstproblem discussed above as several rotation-measurement-feedback sequences are possible before
decoherence effects force re-initialization of the spins and restarting of the experiment.
While this workaround is elegant, it naturally begs the following question. Can we design ascheme where the phase can be continuously monitored even as it accumulates? That is, can wemove beyond the Ramsey paradigm of converting the accumulated phase φ(T ) into a population19
difference, and attempt to directly measure the instantaneous phase φ(t)? In Chapters 5 and 6,
we present such a scheme for continuous real-time tracking of the instantaneous phase φ(t). Furthermore, it can potentially be combined with a feedback loop to extend the unambiguous interval
for phase measurements while avoiding intermittent rotation-measurement-feedback sequences thatinterfere with the phase accumulation.

2.3

Measurement precision in the Ramsey sequence
Upon measurement, a single spin in a superposition state will collapse to either |↑i or |↓iwith fixed probabilities. Because of this inherent randomness, measurements of the accumulated
phase are always noisy and it is meaningful to define a measurement precision ∆φ.
To do so, we first introduce the concept of a collective spin associated with N spin-1/2 systems. The vector components of this collective spin are described by collective angular momentumPjoperators Jˆj = N
r=1 σ̂r /2, j = x, y, z. These operators also satisfy the angular momentum commutation relations [Jˆj , Jˆk ] = ijkl Jˆl . Analogous to the spin-1/2 case, raising and lowering operators
can be defined as Jˆ± = Jˆx ± iJˆy .
2.3.1

Coherent spin states

Let us suppose that all of the N spins are initialized in the state |ψ(θ = π/2, φ = 0)i (Eq. (2.23),
that is, they are directed along the x axis. This state is an example of a coherent spin state (CSS)|Ψ(θ, φ)i, where the collective spin is simply built as a direct product of N identical, uncorrelated
spins. Classically, the collective spin can be visualized as the expectation value of the angularmomentum vector, hĴi = hJˆx ix̂ + hJˆy iŷ + hJˆz iẑ = (N/2)x̂. Although the mean values of Jˆy , Jˆz are
zero, the quantum fluctuations in these quantities do not vanish. For the present state, we have
that





h Jˆy

2


N
X
X
1
N
i =  h(σ̂ry )2 i +hσ̂ry σ̂sy i = .
4
4
r=1

r6=s

(2.28)

20

Figure 2.6: Coherent spin state (CSS). (a) Visualization of a CSS as a cone ending in an uncertainty
circle. (b) Quantum fluctuations in the components orthogonal to the mean spin direction lead tolimited resolution of small rotations.

 2
In calculating the above variance, we have used the properties that σ̂rj = 1 and in this particular 2state, hσ̂ry σ̂sy i = hσ̂ry ihσ̂sy i = 0. Similarly, we can compute h Jˆz i = N/4.
Therefore, quantum mechanically, a CSS can be visualized as a cone as shown in Fig. 2.6,
with the radius of the circular base representing the isotropic standard deviations in the directions
perpendicular to the mean spin direction (MSD) [62]. Rigorously, such states can be represented
on a collective Bloch sphere using quasiprobability distributions [76]. However, the notion of amean spin and an uncertainty circle (or more generally, an uncertainty ellipse) is sufficient for ourpresent discussion, especially since we generally consider N  1 spins, in which case the curvatureof the spherical phase space can be neglected.
The measurement precision of such a CSS can be determined by considering a rotation by asmall angle φ about the z axis. Classically,J y = J x tan φ ≈ J x φ.

(2.29)

By the propagation of errors, we have that
∆φ =

∆J y
.
∂J y /∂φ(2.30)

Using the quantum mechanical value of (∆J y )2 = N/4 and ∂J y /∂φ = J x = N/2, we get
1∆φSQL = √ .
N(2.31)

21
The measurement precision ∆φSQL associated with a coherent spin state is called the standard
quantum limit (SQL) in the metrology community [76]. Going through the full sequence of state
rotations involved in the Ramsey technique still leads to the very same measurement precision for
an initial CSS pointing along the ±z axes.
2.3.2

Squeezed spin states

The Heisenberg uncertainty relation dictates that for a collective spin (assuming MSD along
the x axis),


∆Jˆy

2 

∆Jˆz

2

≥

|hJˆx i|2.
4(2.32)

States which satisfy the equality are called minimum uncertainty states. We restrict the subsequentdiscussion to such minimum uncertainty states. The CSS |Ψ(θ = π/2, φ = 0)i is a trivial example
of such a state.
The concept of squeezed spin states (SSS) arises with the realization that the equality in
Eq. (2.32) only imposes a constraint on the product of the variances in the orthogonal components.
This means that the variance in one of these components can be significantly reduced at the expenseof increasing the variance in the other component. In other words, as shown in Fig. 2.7, the baseof the cone can be ‘squeezed’ into an ellipse with vastly different lengths for the major and minor
axes such that the area of the ellipse is still equal to that of the original circle [62].
Depending on the context, various types of spin squeezing parameters have been introduced
to quantify the degree of squeezing. From a metrology perspective, the relevant parameter is theWineland squeezing parameter ξR [76]. This parameter has a practical definition as the improvement in phase sensitivity of the prepared squeezed state, compared to a coherent spin state withthe same number of spins. Mathematically,
2
ξR
=

(∆φ)2
.
(∆φSQL )2(2.33)

Values of ξR < 1 =⇒ ∆φ < ∆φSQL , signifying an improvement in measurement precision beyond
the standard quantum limit.

22

Figure 2.7: Squeezed spin state (SSS). (a) Visualization of an SSS as a cone ending in an uncertainty
ellipse with unequal major and minor axes. (b) The minor axis can be oriented along the component
accumulating phase in order to improve the resolution of small rotations.

For a collective spin state, the Wineland parameter can be written in terms of the angularmomentum components as
2
ξR
=

min

n̂⊥n̂MSD
N × Var Ĵ · n̂|hĴi|2,(2.34)

where n̂MSD is the mean spin direction. The factor of N is simply the inverse of the baseline valueof (∆φSQL )2 . The remaining fraction corresponds to the minimum possible value of (∆φ)2 given
the present state.1

2.3.3

Mechanisms to prepare squeezed spin states

Taking the y component as an example, Eq. (2.28) implies that in order to reduce the varianceP
below the N/4 level, spin-spin correlations have to be engineered so that r6=s hσ̂ry σ̂sy i < 0. While
many methods to engineer spin-spin interactions have been proposed and demonstrated, a broadclass of spin squeezing techniques rely on the availability of a common bosonic channel, such as acavity mode [11] or a shared vibrational mode [12], which couples to all the constituent spin-1/2
particles.
1
The formula assumes that the direction corresponding to the minimum variance can always be rotated to alignwith the spin component that accumulates phase. For example, consider a SSS with MSD along the x axis and with

2

2
2
∆Jˆz = N/20, ∆Jˆy = 5N/4 and hJˆx i ≈ N/2, so that ξR= 1/5. In the Ramsey sequence, phase accumulates

in the x − y plane. Therefore, the SSS must be first rotated by 90◦ about the MSD so that the minimum variancecomponent aligns with the y axis and enhances the measurement precision.

23

Figure 2.8: Two common mechanisms to generate squeezed spin states involve (a) Squishing theuncertainty ellipse, as happens during squeezing by quantum nondemolition measurements, and(b) Shearing the uncertainty ellipse, as happens during squeezing by one-axis twisting.

Two typical squeezing mechanisms enabled by such a channel are collective quantum nondemolition (QND) measurements [67, 51, 11, 26] and twisting protocols that work using a non-linear
interaction [12, 71, 107].

2.3.3.1

Squeezing by quantum non-demolition measurements

Squeezing by QND measurements arises naturally in our proposal for real time phase tracking
and we therefore extensively discuss this mechanism in Chapters 5 and 6. Briefly, such schemes
involve coupling a Hermitian operator of the common channel with a spin component by engineeringa QND Hamiltonian of the form
ĤQND = ~g X̂ Jˆz .

(2.35)

 √Here, X̂ could be, for example, the amplitude quadrature, given by â + â† / 2. Measurementof the conjugate phase quadrature can then enable estimation of the collective spin component Jˆzbelow the N/4 level. In such schemes, the squeezing dynamics proceeds by continuously ‘squishing’
the measured component of the uncertainty ellipse while elongating the orthogonal component. For
simple QND schemes, the variance in the squeezed component falls as 1/T with the measurement
time T .

24
2.3.3.2

Squeezing by one-axis twisting

The one axis twisting (OAT) Hamiltonian is a common non-linear mechanism to generate
spin squeezing [62]. Here, we will briefly describe the OAT mechanism in its simplest form. The
Hamiltonian is of the form 2
ĤOAT = ~χ Jˆz .

(2.36)Let us consider an initial CSS along the x axis. The squeezing mechanism can be visualized byconsidering the dynamics of different points (jy , jz ) within the uncertainty ellipse. A Hamiltonianof the form ΩJˆz leads to a rotation of the CSS around the z axis. Over a short time interval ∆t,
the transformation is (jy , jz ) → (jy + N Ω∆t/2, jz ). Analogously, HOAT can be visualized as a nonlinear rotation where the rotation rate of a point depends on its jz coordinate. The transformationin the OAT case is given by (jy , jz ) → (jy + N χjz ∆t, jz ). As a result, the uncertainty ellipse is‘sheared’ or twisted over time, leading to reduced variance in one component at the expense of theorthogonal one. The minor axis of the ellipse rotates over time and gets closer to the z axis.
Simple calculations can reveal the rate of squeezing in this mechanism. For sufficiently shorttimes, the state is accurately described by a mean spin, assumed along the x axis and of approximate
length N/2, and a covariance matrix that captures the quantum fluctuations in the y − z plane.
The diagonal components of this covariance matrix evolve asd ˆy 2d ˆz 2h(J ) i = 2N χhJˆy Jˆz is ,h(J ) i = 0.
(2.37)dtdtHere, hÂB̂is = hÂB̂i + hB̂ Âi /2 is the symmetric combination of the second order moments.
Clearly, all moments involving only Jˆz commute with ĤOAT . The off-diagonal element hJˆy Jˆz isevolves asd ˆy ˆzhJ J is = N χh(Jˆz )2 i.
dt(2.38)In deriving these equations, we have substituted Jˆx → N/2 wherever it appears on the right handside. The solution to these equations are given byh(Jˆz )2 i(t) =N
N 2 χtN
N 3 χ2 t 2, hJˆy Jˆz is (t) =, h(Jˆy )2 i(t) =
+
.
4
4
4
4

(2.39)

25
The squared lengths of the major and minor axes of the corresponding uncertainty ellipse are givenby the eigenvalues of the covariance matrix


y
2
y
z
ˆ
ˆ
ˆ
 h(J ) i hJ J is 
Σ=
.
y
z
z
2
ˆ
ˆ
ˆ
hJ J is h(J ) i(2.40)The eigenvalues are given by"1/2 #N4
N 2 χ2 t2 N 2 χ2 t2
λ± =1+
1+ 2 2 2
.
±
4
2
2
N χ t

(2.41)For t  1/N χ, N χt  1 and the expression under the square root can be Taylor expanded tosecond order. The two eigenvalues are then respectively given by
λ+ ≈ N 2 χ2 t2 × N/4, λ− ≈N/4.
N 2 χ2 t2(2.42)

Therefore, the minor axis of the uncertainty ellipse shrinks as 1/t2 with interaction time t. Furthermore, we can identify N χ as a characteristic rate for the dynamics. The assumption that the
dynamics in the y − z plane can be decoupled from the x axis fails when the major axis becomescomparable to the mean spin length. Therefore, this analysis is valid for times such thatλ+
 1 =⇒ N χ2 t2  1.
N 2 /4(2.43)

In fact, the dynamics of an initial CSS under the action of ĤOAT can be computed exactly. Expressions for the minimum attainable variance and spin squeezing parameters that account for thecurvature of the collective Bloch sphere can be found in Refs. [76, 62].

2.4

Momentum pseudospins and Bragg interferometers
In Chapters 7 and 8, we will engineer the OAT interaction between pseudospins encodedin the center-of-mass momentum of the atomic wavepackets. We will then study the efficiency
of squeezing accounting for the complications arising from the setup and the nature of the spins
themselves. Here, we introduce the notion of momentum pseudospins and explain how they form
the basis of Bragg interferometers.

26
Atomic Bragg interferometers are so named because the basic working principle can be visualized as the diffraction of atomic matter waves from periodic potentials of light [27]. Compared to,say, X-ray diffraction from a crystal, where electromagnetic waves are diffracted from periodic arrays of atoms, here the roles of atoms and light are interchanged. Atomic Bragg interferometers arewidely used for applications such as tests of fundamental physics [90, 64, 4, 134] and precision measurements of gravitational acceleration [1, 31]. These systems are attractive because of the uniqueencoding of the spin-1/2 system in two momentum states associated with the center-of-mass motionof the atomic wavepacket. They operate by splitting the wavepacket into two momentum states—that propagate along different spatial paths accumulating a relative phase—and finally recombining
them to obtain interference fringes. Throughout the interferometer operation, the atom is confinedto the same metastable electronic state, typically the ground state. Although an atom’s momentum
is a continuous variable, a pseudospin-1/2 system with two discrete states can be mapped on to
the external motion in a Bragg interferometer. This mapping requires an initial atomic momentum
distribution that is a sharp peak about a central value, so that the distribution serves as one ofthe pseudospin states. Subsequently, pulses of light resonantly couple this distribution to a second
narrow peak that is shifted by an even multiple of the well-defined photon momentum. This shifted
distribution serves as the other pseudospin state.

2.4.1

Coherent manipulation of momentum states

To understand how the momentum of the atom can be manipulated, let us consider a toy
model of a two-level atom in free space. The initial state of the atom is assumed to be |g, pi, wherethe variable p denotes the momentum of the atom along the z axis in the lab frame. By addressing
the atom with two suitable counterpropagating lasers as shown in Fig. 2.9(a), the atom can becoherently transferred to the state |g, p + 2~ki. These lasers drive the |gi → |ei transition in a far
detuned regime so that the state |ei is negligibly occupied. However, as Fig. 2.9(b) illustrates, thetwo-photon process involving the absorption of a photon from the upward propagating laser andemission into the other laser is arranged to be a resonant process: Momentum conservation dictates

27
that the atom has gained a momentum of 2~k after such a two-photon process. Therefore, energy
conservation requires that the two-photon process must also supply the associated excess energy
for the atom to occupy the new momentum state. Therefore, the frequencies of the lasers, ω1 andω2 must be arranged such that~ω1 − ~ω2 =(p + 2~k)2
p2
−
.
2M
2M

(2.44)

Assuming that the Rabi frequency associated with either laser driving the |gi ↔ |ei transition isΩ, and that the single photon detuning is ∆, the effective two-photon Rabi rate for the |g, pi ↔
|g, p + 2~ki transition is Ωeff ∼ Ω2 /∆. The recoil frequency ωr = ~k 2 /2M characterizes the energy
increase of a stationary atom as a result of absorbing or emitting a single photon. ProvidedΩeff  8ωr , the |g, pi ↔ |g, p − 2~ki and |g, p + 2~ki ↔ |g, p + 4~ki transitions are far off resonanceand the |g, pi, |g, p + 2~ki manifold can be considered a closed two-level system. A two-photon
Bragg transition between these two states is then analogous to two-photon Raman transitions
mediated by a far off resonant excited state.

Figure 2.9: Bragg transitions. (a) Two counterpropagating lasers address an upward moving atomwith frequencies arranged to satisfy Eq. (2.44). (b) The two lasers drive a resonant transition
between two momentum states (taken to be |0~ki and |2~ki here) in the same electronic manifold.
The process is reminiscent of a two-photon Raman transition.

Although the above example illustrates the basic concept, significant theoretical and experimental research has been devoted to the design of momentum transfer pulses [85, 118]. Furthermore,experiments are now using large momentum transfer pulses that use a succession of higher-order28
multi-photon Bragg pulses [21, 93, 65]. Such pulses enable the spin-1/2 system to be encoded instates with a large momentum difference, greatly increasing the sensitivity to external fields suchas gravity.

2.4.2

Phase accumulation in a gravitational field

We conclude this chapter with an illustration of how Bragg interferometers typically operate
and how they are capable of sensing gravitation fields (see Fig. 2.10). The sequence of state
manipulations is exactly as in a two-path Mach Zehnder interferometer. Here, the beam splitterand mirror operations are achieved using Bragg pulses that rotate the state of each atom by 90◦ or180◦ in a Bloch sphere formed by the two states |↓i ≡ |pi and |↑i ≡ |p + 2~ki.
First, the atoms are prepared in a narrow momentum distribution about |pi. We ignore
the momentum spread in this simple discussion. Two counterpropagating lasers resonantly drivea Bragg transition and place all the atoms in an equal superposition of |pi and |p + 2~ki. In
the presence of a gravitational field, the momentum of these states linearly decreases with time.
Therefore, the energy difference between the two states at a time t is given byE↑ (t) − E↓ (t)p − M gt
= 4ωr 1 +,~~k

(2.45)

where g is the gravitational acceleration and M is the atomic mass. Knowing the approximate valueof g, the difference frequency of the two lasers is chirped at a rate α that roughly compensates forthis changing transition frequency, thereby ensuring that the lasers always drive near resonant
Bragg transitions [78]. The laser frequency difference therefore varies in time asp 
ω1 (t) − ω2 (t) = 4ωr 1 +
− αt.
~k

(2.46)

The atoms are then allowed to freely evolve for time T . During this time, the two momentum states
propagate along different spatial paths that we will denote as A and B as shown in Fig. 2.10.
From Eq. (2.45) and Eq. (2.46), the net phase accumulated relative to the two-photon drive
is given by(1)
φBA = −




Z T
Mg
1
Mg
4ωr
− α t dt = −4ωr− α T 2.
~k2~k
0(2.47)

29

Figure 2.10: Schematic of a two-path Bragg interferometer. The two paths are respectively denotedby A and B. The momentum labels indicate the ‘spin up’ (|↑i ≡ |p + 2~ki) and ‘spin down’(|↓i ≡ |pi) states at different points during the interferometer sequence. Gravitational acceleration
causes the momenta of the two states to linearly vary over time.

At time T , a Bragg pulse is applied so that the momenta of the two paths are interchanged. The
atoms are then allowed to freely evolve again for time T . The relative phase accumulated between
the two paths in this arm is given by(2)(2)
φBA = −φAB =Z 2T 
Mg
3
Mg
4ωr− α t dt =4ωr
− α T 2.
~k
2~k
T

(2.48)The sign of the accumulated phase in the second arm is opposite to that in the first because themomenta of the two paths have been flipped. Therefore, the net phase accumulated up to time 2T
is given by
(1)

(2)

Φ = φBA + φBA = (2kg − α) T 2 .

(2.49)The net phase accumulated depends on the precise value of g and scales with the square of theinterferometer arm time T . Finally, a third Bragg pulse converts the accumulated phase into a
population difference. In practice, the net phase accumulated also depends on the phases of thethree Bragg pulses, leading to the formula [78]
Φ = (2kg − α) T 2 + φ1 − 2φ2 + φ3 .

(2.50)In a Bloch sphere picture, the phases φj , j = 1, 2, 3 set the axis of rotation in the x − y plane for30
the three Bragg pulses. Ramsey-type fringes of the population in one of the momentum states are
typically obtained by varying the phase of the final Bragg pulse while keeping other parameters
fixed.
Finally, we note that the entire sequence can be viewed as a Ramsey sequence with an extra
spin echo pulse halfway through the phase accumulation time. The spin echo pulse ensures thatdephasing arising from a momentum width is reversed during the second half of the sequence. In
real space, such a cancellation ensures that the two atomic wavepackets spatially overlap at theend of the sequence, thereby closing the interferometer.
In Chapters 7 and 8, we will discuss how to engineer squeezing on momentum pseudospins
for use in such Bragg interferometers.

Chapter 3

Near ground-state cooling large trapped ion crystals: Theoretical modeling

3.1

Motivation: Why ground-state cooling?
Trapped ions have rapidly evolved to become a leading platform for quantum computing,quantum simulation and metrology [46, 10]. Specifically, ions stored in Penning traps have been
demonstrated to be ideal for analog quantum simulation as well as quantum-enhanced sensing, inpart because large ion crystals are routinely formed and controlled in this device [8, 77, 5]. For
example, planar crystals of tens to hundreds of 9 Be+ ions have been used to simulate spin-spin
as well as spin-boson models including the Ising [17, 12], transverse-field Ising, as well as Dicke
models [101, 24]. Quantum information studies on the growth of entanglement [39] as well as
investigations on preparing ground states of exotic Hamiltonians [101, 24] have shown that exciting
many-body physics can be studied with this versatile quantum simulator. In addition, ions in
Penning traps serve as excellent motion sensors capable of resolving, in a single experimental trial,
motional amplitudes smaller than the zero-point fluctuations of the normal modes dictating the
motion transverse to the crystal plane [42], thus enabling the detection of extremely weak forcesand electric fields.
For implementing these protocols with the NIST Penning trap, the spin is encoded in two
hyperfine ground states of 9 Be+ [8]. Spin-spin interactions are mediated by the motional drumheadmodes, transverse to the crystal plane, that arise from the interplay of the trap potential and theinter-ion Coulomb repulsion, with the spin-motion coupling generated using suitable drive lasers[17]. As a result, excess thermal energy in these normal modes adversely affects the science protocol

32
being investigated, higlighting the need for sub-Doppler, near ground-state cooling. For example,
the fidelity of preparing the ground state of the Dicke model is significantly reduced by the thermal
occupation of n̄ ≈ 6 of the center-of-mass (COM) mode, which is close to the Doppler cooling limit[101, 24]. Estimates show that the fidelity significantly improves if the COM mode is cooled down
to n̄ ≈ 0. From a metrology standpoint, near ground-state cooling should also greatly improve themotion sensing capability of this platform.

3.2

EIT cooling in a Penning trap: Expectation and challenges
Electromagnetically induced transparency (EIT) promises a path for cooling the entire band-

width of drumhead modes close to their ground states. In contrast to sideband cooling where
the modes are cooled one-by-one by sweeping the two-photon detuning across the bandwidth ofmodes, EIT cooling can potentially cool the full bandwidth of modes in a single experimental application with no time-varying parameters, allowing for simpler implementation and faster cooling.
The naive expectation comes from the well understood physics of EIT cooling of a single trappedion [83], which we now recall briefly. The ion is assumed to have a closed three-level electronic
manifold consisting of two long-lived states, such as the hyperfine ground states of 9 Be+ , and an
excited state (see Fig. 3.1). Two strong dressing lasers couple the long-lived states to the excitedstate and are equally blue detuned from their respective transitions. EIT cooling can be understood by considering the absorption of a fictitious weak probe coupling one of the long-lived statesto the excited state. As shown in Fig. 3.2, the steady-state absorption spectrum has a uniqueprofile as the probe detuning ∆P is swept, with the absorption exactly vanishing when the probe
detuning equals the dressing detuning ∆D . A sharp peak immediately follows this transparency
point and the separation between this peak and the transparency point can be tuned using the
dressing laser powers. For a trapped ion, the motion-adding and motion-removing sidebands of the
dressing lasers serve as weak probes that sample this absorption spectrum. Tuning the separation
between the sharp peak and the transparency point to match the motional frequency causes the
motion-removing sideband to be strongly enhanced and the motion-adding sideband to be strongly

33

Figure 3.1: Two EIT lasers address the ion, with one driving the |g1 i ↔ |ei transition and theother driving the |g2 i ↔ |ei transition in a blue-detuned regime (∆ > 0).

suppressed, leading to highly efficient cooling. In the Penning trap, by tuning this separation tocoincide with the frequency of the COM mode, which is the highest frequency drumhead mode,
all the drumhead modes have highly asymmetric motion-removing and motion-adding sidebands,
which should produce efficient cooling over the full bandwidth of these modes. In this chapter,
we theoretically investigate this idea under realistic experimental conditions employed in the NIST
Penning trap.
The unique challenges confronted in implementing EIT cooling in a Penning trap further
motivate our theoretical study of the prospects for its success. First, ions stored in a Penning
trap are constantly revolving around the trap center and therefore, in general, experience timevarying Doppler shifts on the applied dressing lasers. Second, experimental constraints as well as acompromise between the speed of cooling and the final temperature dictate that the timescales forthe electronic and motional degrees of freedom may not be sufficiently well separated to adiabaticallyeliminate the electronic degrees of freedom, as was done in the initial analysis of trapped-ion EIT
cooling [83, 84]. Third, although EIT cooling has been used to cool all the radial modes of a linear
chain of up to 18 ions [68], the dynamics of simultaneously EIT cooling several tens to hundredsof normal modes that can interact via the applied lasers is not well understood and could be very
different from the single-ion case.

34

Figure 3.2: EIT absorption spectrum with laser parameters relevant to the NIST EIT coolingexperiment. Two strong dressing lasers, with equal Rabi frequencies and equal detunings fromtheir respective transitions, couple two long-lived states to an excited state in a closed threelevel system. The absorption from a weak probe is plotted as the probe detuning ∆P is scanned
across the dressing detuning ∆D . The motion-removing (blue impulses) and motion-adding (redimpulses) sidebands can be interpreted as weak probes that sample this spectrum. Inset: Close-up
near (∆P − ∆D )/2π = 0, with a magnified y-axis, showing the zero at the transparency point aswell as the asymmetry in the spectrum on either side of this point. The blue dashed line is a mirror
image of the spectrum on the blue-detuned side, drawn on the red-detuned side to highlight theasymmetric growth of the absorption away from the transparency point. The dressing lasers haveequal detuning ∆D /2π ≡ ∆0 /2π = 360 MHz, and equal Rabi frequency Ωopt (∆0 )/2π ≈ 33.9 MHz.
The Rabi frequency of the weak probe is ΩP = 0.05 Ωopt (∆0 ). The decay rates from the excitedstate to the two long-lived states are Γ1 /2π = 6 MHz, Γ2 /2π = 12 MHz. (See the discussion in
Section 3.4 and Eq. (3.14) for a detailed explanation of the parameters.)

3.3

Summary of predictions
At the outset, we summarize the major predictions from our study. Under typical experi-mental conditions, EIT cooling leads to near ground-state occupancies for all the drumhead modes,
spread over a bandwidth of hundreds of kilohertz, of large ion crystals in a Penning trap. The cooling of the COM mode has a time constant of few tens of µs. Further, under suitable experimental
conditions, the cooling rate of the COM mode increases with the number of ions in the crystal.
This latter result leads us to predict that the measured cooling of multi-ion crystals will be fasterthan the rate expected if the ions cooled independently. These predictions have been verified by
the successful demonstration of EIT cooling with more than 100 ions [58], where significant subDoppler cooling, strongly suggestive of near-ground state cooling, has been observed over the full35
bandwidth of drumhead modes. Quantitative measurements on the COM mode reveal occupationsof n̄ ≈ 0.3 ± 0.2, and a measured cooling constant τ ≈ 28 µs. The measured cooling rate is fasterthan the expected single-ion rate under the same experimental conditions.
The rest of this chapter is devoted to the modeling procedures and the numerical results. In
Section 3.4, we describe the NIST Penning trap and proceed to set up the master equation for the
EIT cooling of multiple ions. In Section 3.5, we use a toy model of a single revolving ion to illustrate
the degrading effects of the time-varying Doppler shifts as well as demonstrate the invalidity of theadiabatic elimination procedure in our system. In Section 3.6, we first build a Gaussian model for
approximately studying the cooling dynamics for multiple ions, and demonstrate the near-groundstate cooling of the COM mode for crystals with up to N = 37 ions. By comparing the coolingtransients from our Gaussian model to full density matrix calculations for a single ion, we show
the build-up of beyond Gaussian correlations between the system degrees of freedom, which we areable to reproduce by systematically extending our approximate model beyond the Gaussian regime.
Our improved model predicts a surprising enhancement in the cooling rate of the COM mode withincreasing number of ions that is not captured by the Gaussian model. In Section 3.7, we show
how EIT cooling works efficiently over the full bandwidth of drumhead modes of crystals with as
many as 120 ions, resulting in near ground state occupancies for all these modes. In Section 3.8, we
briefly demonstrate the expected robustness of EIT cooling to small misalignments of the dressing
lasers. We conclude with a brief summary in Section 3.9, where we also discuss possible future
extensions of our work.

3.4

Modeling the experiment

3.4.1

NIST Penning trap

The NIST Penning trap is used to routinely produce, control and manipulate planar ion
crystals of tens to hundreds of 9 Be+ ions [8]. A static electric quadrupole field is used to achievetransverse confinement, while the addition of a strong transverse magnetic field ensures radial36
confinement. Potentials applied to electrodes arranged symmetrically around the z-axis generate
the required electric fields and a superconducting magnet produces a strong magnetic field of 4.46 T.
The E × B drift of the ions arising from the combination of the magnetic field and the azimuthally
symmetric electric fields causes the ions to revolve around the trap center. The frequency of thecrystal rotation can be precisely controlled and stabilized by applying a weak rotating potential on
the electrodes. Typically, this ‘rotating wall’ potential is used to stabilize the rotation frequency ofthe crystal to ωr /2π = 180 kHz. When the radial confinement is weak compared to the transverse
confinement, the ions form a 2D planar crystal with an approximate triangular lattice (see Fig. 3.3
for an illustration). The strength of the transverse harmonic confinement is characterized by atrapping frequency that is also the frequency of the transverse center-of-mass (COM) mode, ωCOM ,of the ion crystal. A planar ion crystal with N ions has 3N normal modes of motion, 2N ofwhich are in-plane modes superposed on the crystal rotation, and N of which are drumhead modes
transverse to the crystal plane. The COM mode is the highest-frequency drumhead mode. For theexperiments we model here, ωCOM /2π ≈ 1.57 − 1.59 MHz, tunable using trap parameters. The
2N in-plane modes consist of N high frequency cyclotron modes and N low frequency magnetronmodes. Typical frequencies characterizing the three branches are summarized in Table 3.1. Further
discussion of these modes is postponed to Chapter 4.
BranchDisplacement

Typical frequency

Drumhead
Cyclotron
Magnetronz axisx − y planex − y plane1 − 1.59 MHz
7.2 − 7.3 MHz
0 − 170 kHz

Table 3.1: Summary of approximate characteristic frequencies for the three branches of modes fortwo dimensional crystals in the NIST Penning trap.

3.4.2

Master equation model

We consider singly-charged positive ions with a closed three-level electronic structure loaded
in a Penning trap (see Fig. 3.1). The two hyperfine ground states of each ion are labeled |g1 i and37

Figure 3.3: Experimental setup to cool ions in a Penning trap using electromagnetically induced
transparency (EIT). Two EIT lasers address the ion crystal at angles ±θ with respect to the x-axis.
The curved arrow indicates the rotation direction in the x-y plane.

|g2 i. Two EIT lasers are respectively blue-detuned from the |g1 i ↔ |ei and |g2 i ↔ |ei transitions,where |ei is an excited state separated from the two ground states by optical frequencies. The
two EIT lasers are incident on the planar ion crystal at angles ±θ with respect to the plane of thecrystal, which we take to be the x-y plane. The {|ei , |g1 i , |g2 i} manifold is a closed system, withdecay rates of Γ1 and Γ2 for the |ei → |g1 i and |ei → |g2 i pathways respectively. We will adopt
the shorthand notation σαβ to denote the internal state operator |αi hβ|.
In the Schrödinger picture, the Hamiltonian for the interaction of the EIT lasers with a single
ion is H S = H0S + H1S (t), where
H0S = −Xµωe − ωgµ σgµ gµ

(3.1)


ei(kµ ·r(t)−ωµ t) σegµ + H.c. ,

(3.2)

and
H1S (t) =X Ωµ 
µ2

where the index µ = 1, 2 accounts for the two ground states and the two EIT lasers coupling themto |ei. Here, ωe − ωgµ are the |gµ i ↔ |ei transition frequencies, Ωµ are the Rabi frequencies for thelaser-ion interaction, and ωµ and kµ are the angular frequencies and propagation vectors for the two
EIT lasers. Eventually, we will describe the motion along the z-direction using a set of quantizednormal modes, and add the self-energy terms associated with these quantum harmonic oscillators
to the interaction Hamiltonian. Throughout this Chapter, we have set ~ = 1, unless we explicitly
specify otherwise. For brevity, we have also dropped hats on the operators in this Chapter.
Since any given ion in the Penning trap is undergoing a coherent rotation in the x-y plane,

38
its position r(t) is




Z t
Z t
0
0
0
0
dt vy (t ) ŷ + z(t)ẑ,dt vx (t ) x̂ + y(0) +r(t) = x(0) +

(3.3)

0

0

Here we are neglecting any thermal motion in the plane of the ion crystal and assume vx (t0 )and vy (t0 ) arise from the coherent circular motion caused by the E × B drift, which is the dominant
in-plane motion. We assume that the laser beams are propagating in the x-z plane, so that k1 =k1,x x̂ + k1,z ẑ and k2 = k2,x x̂ + k2,z ẑ.
We then transform to an interaction picture with a time-dependent ‘free evolution’ Hamiltonian HfS (t) given by
HfS (t) =−Xµ{ωµ − kµ,x vx (t)} σgµ gµ .

(3.4)

The interaction picture transformation is very similar to the usual case with a time-independent freeevolution Hamiltonian because HfS (t), HfS (t0 ) commute at all times t, t0 . The interaction HamiltoRt

0

S

0

Rt

0

S

0

nian in this frame, given by the transformation H I (t) = ei 0 dt Hf (t ) (H0S +H1S (t)−HfS (t))e−i 0 dt Hf (t ) ,is
H I (t) =Xµ∆µ (t)σgµ gµ +oX Ωµ neikµ,x x(0) eikµ,z z(t) σegµ + H.c. ,
2
µ

(3.5)

where ∆µ (t) = ∆0µ − kµ,x vx (t) are the effective detunings of the EIT lasers as seen by the ion,with ∆0µ = ωµ − (ωe − ωgµ ). For an ion with initial position {x(0), y(0)}, the x-component of thevelocity is vx (t) = ωr (y(0) cos ωr t − x(0) sin ωr t), where ωr is the angular frequency of the rotatingwall potential.
To perform EIT cooling, we tune the EIT lasers to satisfy the two-photon resonance on theblue-detuned side [83], i.e. ∆01 = ∆02 ≡ ∆0 > 0. Further, the lasers are aligned such that theirdifference wavevector lies along the z-axis. This implies k2,x = k1,x and k2,z ≈ −k1,z . However, we
will develop the theory without these two simplifications, and only apply these conditions numerically. For N ions in the Penning trap, the interaction Hamiltonian generalizes straightforwardly asa sum over all ions.

39
The equilibrium crystal configuration results from the balance of the trap potential and theinter-ion Coulomb repulsion. The transverse motion of the N ions about their equilibrium positions
are not independent, instead being described by a set of N collective normal modes with frequencies
ωn , n = 1, 2, . . . , N and amplitudes Mjn at each ion j. The frequencies ωn and the column vectors ofthe matrix M are respectively obtained as the eigenvalues and eigenvectors of the potential energy
matrix associated with the coupled transverse harmonic motion of the ions [124]. The transverse
displacement of any ion j can be expressed in terms of the N quantized drumhead modes of theion crystal aszj (t) =

N
X
n=1

r



~
Mjn bn e−iωn t + b†n eiωn t ,
2M ωn

(3.6)

where b†n , bn are the creation and annihilation operators for the normal mode n.
The time-dependent exponentials in Eq. (3.6) can be recast as self-energy terms, leading to
the total interaction Hamiltonian
H I (t) =Xnωn b†n bn +Xj,µ∆µ,j (t)σgjµ gµ +

o
X1n
j
+
H.c.
,
Ωµ,j eikµ,z zj σeg
µ
2

(3.7)

j,µ

where the displacement operators zj are now simply zj =P pn~/2M ωn Mjn

bn + b†n.

Here, the instantaneous detunings experienced by each ion is different, depending on the xcomponent of the ion’s velocity at that time point. In writing Eq. (3.7), the complex phase factors
associated with the initial positions of the ions have been absorbed into the (now complex) Rabi
frequencies. Further, for large ion crystals, the spatial profile of the EIT lasers over the extent
of the crystal may be important, and therefore Ω1(2),j ≡ Ω01(2) (xj (t), yj (t)) eik1(2),x xj (0) , i.e. theamplitude of the Rabi frequency is in general a function of the instantaneous in-plane position of
the rotating ion.
Spontaneous emission from the excited level |ei to the two ground states is accounted forusing dissipation terms written in Lindblad form, i.e. for any jump operator O, the dissipation
term takes the form D[O]ρ = OρO† − 21 O† Oρ − 21 ρO† O, where ρ is the density matrix of the systemat hand. Since we are interested in the effect on the motion along the z-direction, the dissipation
terms must account for the recoil momentum along the z-axis due to spontaneous emission [28].

40
Therefore, the Lindblad term for ion j, including recoil associated with spontaneous decay of |ei
to |gµ i, is
Dµ,j ρ = Γµ

Z 1
−1

1 j
1 j
j
duNµ (u)σgjµ e e−iksc zj u ρeiksc zj u σeg− σeeρ − ρσee
µ
2

2


,

(3.8)

where Γµ is the spontaneous decay rate of |ei → |gµ i, ksc is the wavevector associated with the
spontaneously emitted photon, u = cos θsc , with θsc the angle between ksc and the z-axis, andNµ (u) is the normalized dipole radiation pattern associated with the transition. Finally, the masterequation for EIT cooling of N ions in a Penning trap is given byρ̇ = −i[H I (t), ρ] +X

Dµ,j ρ,

(3.9)

j,µ

where H I (t) and Dµ,j ρ are as in Eq. (3.7) and Eq. (3.8) respectively.

3.4.2.1

Lamb-Dicke regime

When the condition h(kµ,z zj )2 i1/2  1 is satisfied for every ion, the motion is in the LambDicke regime [126] and the master equation can be expanded in a series expansion in kµ,z zj [83]. In
our setup, the angles ±θ are such that the z-components kµ,z are (a) not too large to cause beyond
Lamb-Dicke regime dynamics, and (b) not too small that the cooling is weak. For any wavevector,
it is useful to recast its coupling to the z-motion in terms of the Lamb-Dicke parameters associatedwith the drumhead modes as
kz zj =

NX
i=1

where ηnkz = kz

qηnkz Mjn (bn + b†n ),(3.10)~
2M ωn is the Lamb-Dicke parameter [126] for mode n, associated with a wavevector

whose z-component is kz . Expanding the master equation Eq. (3.9) up to second-order in the LambDicke parameters results inρ̇ = L0 ρ + L1 ρ + L2 ρ.

(3.11)

41
Here,
X

L0 ρ

= −i[H0 (t), ρ] +L1 ρ= −i[H1 , ρ],L2 ρ= −i[H2 , ρ] + K2 ρ,µ,jΓµ D[σgjµ e ]ρ,(3.12)

with
H0 (t) =Xωn b†n bn +

n

H1 =

H2 = −

2X Ωµ,j
j,µ

2

j
σeg
+ H.c.,µj
Xn σeg+ H.c.,µX λµjn λµjk Ωµ,j
j,n,k,µ∆µ,j (t)σgjµ gµ +j,µX iλµjn Ωµ,j
j,n,µ

X

4

j
Xn Xk σeg
+ H.c.,
µ

X Γµ
sc
j
j
K2 ρ =
hu2 iegµ λsc
jn λjk K2 ρσgµ e (2Xn ρXk − Xn Xk ρ − ρXn Xk ) σegµ ,2(3.13)

j,n,k,µkkscwhere Xn = bn + b†n and λµjn = ηnµ,z Mjn , λsc
jn = ηn Mjn are dimensionless electronic-motional

coupling strengths. The quantity hu2 iegµ is the variance of u = cos θsc taken with respect tothe dipole radiation pattern Negµ (u) associated with the |ei → |gµ i decay. The master equation,
Eq. (3.11), is the starting point for our analysis of EIT cooling of ions in the Penning trap.

3.4.3

Parameters from the NIST EIT cooling experiment

In the NIST EIT cooling experiment with 9 Be+ ions, the transverse magnetic field of B =4.46 T leads to a splitting of 124 GHz between the 2s2 S1/2 (mJ = −1/2) and 2s2 S1/2 (mJ = +1/2)
levels [8], labeled as |g1 i and |g2 i respectively. The |g1 i ↔ |ei transition frequency is ωg1 e /2π ≈957 THz. The two EIT lasers, with σ + and π polarizations are oriented at ±10◦ with respect tothe x-axis and respectively couple the |g1 i and |g2 i levels to the excited level 2p2 P3/2 (mJ = +1/2),
labeled as |ei. They are blue detuned with equal detuning ∆0 from their respective transitions by
hundreds of megahertz. These lasers generate sufficient power to give Rabi frequencies of tens ofmegahertz. The beam diameters (≈ 1 mm) of the EIT lasers are large compared to the diameters42of the ion crystals (≤ 300 µm) so that we can assume constant laser intensities over the spatial
extent of the crystal. The decay rates out of |ei are Γ1 /2π ≈ 6 MHz and Γ2 /2π ≈ 12 MHz, with
hu2 ieg1 = 2/5 and hu2 ieg2 = 1/5.
For all the results in this chapter, we operate at the expected optimum EIT cooling condition
for the COM mode of a stationary ion [83] given by Ω21 + Ω22 = 4ωCOM (ωCOM + ∆0 ), and assume
equal Rabi frequencies, i.e. Ω1 = Ω2 = Ωopt , so that
Ωopt (∆0 ) =p
2ωCOM (ωCOM + ∆0 ),(3.14)

where we use the value ωCOM /2π = 1.59 MHz. Further, we assume the rotation frequency of thecrystal is ωr /2π = 180 kHz.

3.5

A single revolving ion

3.5.1

Time-varying Doppler shifts

A toy model of a single ion revolving around the trap center in the x-y plane can shed lighton the impact of the in-plane motion on the cooling of the transverse motion. We recall that thecircular in-plane motion of the ion causes a sinusoidally modulated Doppler shift, with the precise
form of the modulation detailed in the paragraph immediately following Eq. (3.5). For a single ion,
we are able to perform full density matrix computations using the master equation, Eq. (3.11).
We set the detuning ∆0 /2π ≈ 180 MHz, and operate with equal Rabi frequencies Ωopt (∆0 )given by Eq. (3.14). We assume that a preceding Doppler cooling stage initializes the transverse
motion of the ion to a thermal state with n̄ = 5. Typically, EIT cooling is applied after initializing
the ion(s) in |g1 i by optical pumping. Figure 3.4 shows the decrease in the thermal occupationn̄ with time as the EIT lasers address a single revolving ion, for different distances of the ionfrom the trap center. For a detuning ∼ 180 MHz, the ion experiences effective red detunings forparts of its trajectory for a radius r & 50 µm. Consequently, the ion undergoes heating in theseregions, leading to slower cooling and higher final occupancies at larger radii (see inset of Fig. 3.4).

43

Figure 3.4: Cooling of the transverse motion over time, for a single ion revolving around the trap
center at different radii. The cooling is slower at larger radii because of time-varying Doppler shifts
modulating the detunings of the EIT lasers as seen by the ion. Inset: The steady-state occupation
also increases with distance from the trap center. Here, ∆0 /2π = 180 MHz, Ω1 /2π = Ω2 /2π =Ωopt (∆0 )/2π ≈ 24 MHz.

Therefore, sufficiently large detunings have to be used, so that ions at the outer boundary of large
crystals still experience an effective blue detuning of the EIT lasers.

3.5.2

Timescale for internal dynamics

An ion located at the trap center experiences no Doppler shifts, and hence, we could argue
that analytical expressions derived elsewhere [83] for EIT cooling of a single ion might be valid
in such a situation. With the EIT wavevectors making an angle of ±10◦ with the x-axis, theLamb-Dicke parameters are η k1,z ≈ −η k2,z ≈ 0.066. Combined with the typical Rabi frequencies
used in the experiment, in the range of tens of megahertz, the wide separation of electronic andmotion timescales demanded by an adiabatic elimination procedure is not satisfied in our system[83]. Figure 3.5 shows the disagreement between the cooling curves obtained with (black dashedline) and without (red solid line) adiabatic elimination of the electronic degrees of freedom (DOF)
for an ion at the trap center. The insufficient separation of timescales can also be seen qualitatively
by simultaneously examining the transient dynamics of the population, in say, |g1 i, on a common
time axis, as shown in Fig. 3.5.

44n̄, no eliminationn̄, with elimination|g1 i population

4

1
0.8
0.6

2

0.4

1

0.2

n̄

3

0

0

5

10
time (µs)

15|g1 i population

5

0
20

Figure 3.5: Cooling curve for an ion at the trap center that experiences no Doppler shifts. The
separation of the timescales for the electronic and motional degrees of freedom is not large enoughto validate the adiabatic elimination of the electronic degrees of freedom. The cooling is therefore
much slower than the result predicted from such an elimination procedure [83]. Here, ∆0 /2π = 180
MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 24 MHz.

3.6

EIT cooling of multiple ions
The NIST Penning trap routinely stores and manipulates tens to hundreds of ions. Since thedensity matrix, now consisting of the electronic degrees of freedom of all the ions and their drumheadmodes, scales exponentially in the ion number, full density matrix solutions are impossible, and weare forced to resort to approximate techniques.
From the master equation, Eq. (3.11), we write down the equations of motion for the meansof all the system operators (first order moments) and the products of operator pairs (second ordermoments). In general, these equations will couple to higher order moments, for example, means of
products of triplets of operators, and so on. We truncate the hierarchy at second order by neglecting
all cumulants higher than means and covariances, and close the set of equations by approximating
higher order moments using sums of products of first and second order moments [80, 132]. As an
example, for a product of three operators this would imply
hABCi ≈ hABihCi + hACihBi + hBCihAi − 2hAihBihCi.

(3.15)

Here h. . .i denotes the mean value of an operator or product of operators. We will refer toTypeMoments

Internal

hσgj1 g1 i, hσgj1 g2 i, hσgj1 e i, hσgj2 g2 i# , hσgj2 e i#External

hbn i, hbn bk i, hb†n bk i

Hybridhbn σgj1 g1 i, hbn σgj1 g2 i, hbn σgj1 e i hbn σgj2 g1 i# ,jj#
hbn σgj2 g2 i# , hbn σgj2 e i# , hbn σeg
1 i, hbn σeg2 i45Table 3.2: List of moments classified according to the nature of the operators involved. The
equations for the moments marked with a # can be derived trivially by exchanging g1 ↔ g2 in theappropriate equations of motion for the other moments.

the equations obtained for the first and second order moments using this truncation scheme as
the Gaussian model (GM), since the scheme neglects third and higher order cumulants. We note,however, that we factorize second order moments involving the electronic degrees of freedom ofjk i as hσ j ihσ k i for j 6= k. We classify the remaining moments
σγδdifferent ions of the type hσαβγδαβ

into three categories: internal, external and hybrid moments. These are listed in Table 3.2. In
Appendix A.1, we write down the equations of motion (EOM) for the moments not marked witha # in Table 3.2, from which the remaining EOM can be obtained by appropriately exchanging
g1 ↔ g2 .
3.6.1

Results from the Gaussian model

We set the detuning ∆0 /2π = 360 MHz, which ensures that the cooling rate for a single
revolving ion does not change appreciably over the spatial extent of the small crystals we consider
here. Figure 3.6 shows the cooling of the COM mode for crystals with N = 1, 2, 19 and 37 ions.
For N = 1, 2 we simply take the ion(s) to be revolving at a distance of 20 µm from the trap
center, and diametrically opposite each other in the N = 2 case. In the case of multi-ion crystals(N > 2), the equilibrium configuration of the crystal and the mode frequencies and eigenvectors
are solved for following the procedure in Ref. [124]. We assume that a preceding Doppler coolingstage initializes the COM mode to a thermal state with n̄ = 5. We choose the initial n̄ of theremaining drumhead modes assuming that they are initially in thermal equilibrium with the COM

46

6

N=1
N=2
N = 19N = 37

5

n̄

4
3
2
1
0

0

20

40
60
time (µs)

80100Figure 3.6: Cooling curves for the center-of-mass (COM) mode for crystals with different ion
numbers, calculated using the Gaussian model (GM), showing rapid near ground-state coolingwithin 100 µs. However, the cooling rates are almost identical for all of these crystals. Here,∆0 /2π = 360 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 33.9 MHz.

mode. Qualitatively, the COM mode rapidly cools to near ground-state occupations within 100
microseconds. However, the cooling curves for the different crystals are nearly identical, showingthat the net cooling rate of the COM mode, within the Gaussian framework, is approximately
independent of the number of ions. For the multi-ion crystals, we observe an initial heating effectthat is also observed experimentally [58]. We attribute the initial heating to transient electronictransitions until the electronic state of the ions reaches close to the steady state, at which point
the cooling begins.

3.6.2

Benchmarking the Gaussian Model: Single-ion results

Since we are able to perform full density matrix (DM) computations in the single-ion case, we
proceed to compare the cooling transients from the GM with the full DM results. Figure 3.7 shows
the cooling curves from the GM (black, dashed) and the full DM calculation (red, solid) for a single
revolving ion at r = 0, 20, 40 and 60 µm from the trap center, with ∆0 /2π = 360 MHz. While bothmodels qualitatively indicate that the cooling rate is roughly the same at these different radii, thecooling rate obtained from the GM is quantitatively very different from the full DM result. This
indicates that keeping track of only means and covariances is not sufficient to accurately capture

47
the cooling curve.
The GM assumes that in the combined phase space of all the system degrees of freedom(DOF), the joint (quasi)-probability distribution of these DOF remains Gaussian at all times. In
reality, while the initial distribution is Gaussian, evolution under the subsequent dynamics generally
distorts the distribution so that it is no longer Gaussian at later times. A systematic way to capture
this effect is to construct moments by averaging the evolution of the corresponding phase spacevariables (or products of variables) over a large number of trajectories, where the initial conditions ofthese variables in each trajectory are chosen randomly from their initial distribution (see Table 3.3).
Such a sampling and averaging procedure captures the build-up of non-trivial third and higher order
cumulants that are neglected in the GM. We note that this approach is in the same spirit as the
Truncated Wigner Approximation (TWA) used in calculating the dynamics of spin-spin and spinboson models [94, 9, 106, 92]. Moreover, we track separate phase space variables corresponding
to system operators as well as operator pairs, and evolve these variables using the same equationsof motion as in the GM (Appendix A.1), but for many trajectories. We thereby perform beyond
6full DM
GM
SM

5

n̄

4
3
2
1
0

0

20

40
60
time (µs)

80100Figure 3.7: Cooling curve for a single revolving ion, located at r = 0, 20, 40 and 60 µm from the trap
center, computed using three numerical approaches: (i) Time evolution of the full density matrix(full DM), (ii) the Gaussian model (GM) and (iii) the Sampling model (SM) using 2096 trajectories.
The cooling curves from the GM do not agree with the full DM curves. However, sampling the initialnoise systematically (SM) accounts for beyond-Gaussian properties of the phase-space distributionof the system degrees of freedom, and reproduces the full DM curves very well. Here, ∆0 /2π = 360
MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 33.9 MHz.

48
mean-field calculations [92] that capture those contributions to the covariances between system
operators which develop as a result of the subsequent diffusive-dissipative dynamics. We refer tothis method as the Sampling model (SM). Table 3.3 summarizes the implementational differences
between the Gaussian and the beyond-Gaussian Sampling model.
Model

Trajectories

Initial condition

Gaussian (GM)1

hbn i(0) = 0,hb†n bn i(0) = n̄n (0)ManypRe{hbn i(0)} = Gaus(0, pn̄n /2),
Im{hbn i(0)} = Gaus(0, n̄n /2),hb†n bn (0)i = |hbn i(0)|2

Sampling (SM)

Table 3.3: Implementational differences between the Gaussian and Sampling models. HereGaus(0, σ) is a Gaussian distributed random variable with zero mean and standard deviation σ,
and n̄n are the initial thermal mode occupations. In the sampling model, the quantity hAi simply
denotes the value of the respective phase space variable in that trajectory, and is not the meanvalue of the operator A. Instead, the mean value of A is given by the average of hAi over many
trajectories with random initial conditions drawn from the initial phase space distribution. We
note that we only sample the initial thermal distribution of the normal modes, and initialize thejelectronic DOF in the same way as in the GM, i.e. hσαβi = 1 when α = β = g1 .

In Fig. 3.7 we show the cooling curves from the SM along with the full DM as well as GM
results for the single revolving ion. Since the SM involves averaging over multiple trajectories withrandomly drawn initial conditions, the cooling curves are shown as 1-σ confidence intervals insteadof a line plot. The SM cooling curves agree very well with the full DM result, indicating thatbeyond-Gaussian correlations develop during the cooling process that lower the cooling rate.

3.6.3

Results from the Sampling model

The SM predicts that the cooling rate of the COM mode increases with the number of ions N
in the crystal. In Fig. 3.8, we plot the cooling curve for the COM mode for crystals with N = 1, 2, 19
and 37 ions. The cooling is faster in the N = 2 case than in the single ion case, and even faster inthe 19 and 37 ion crystals. The inset shows the ratio of the cooling rate RN of an N -ion crystal

49
to the rate R1 for a single ion. With the parameters used, and for these small crystals, the coolingrate scales as ∼ N 0.3 , highlighting that EIT cooling of multiple ions cannot be explained triviallyas the net cooling resulting from the individual ions. Rather, the N -dependency of the cooling rate
indicates the intrinsic many-body nature of this problem. We note, however, that for fixed laserparameters, the adverse effects of the time-varying Doppler shifts is expected to slow down thecooling enhancement with increasing N as ions are added further from the trap center. Moreover,with different laser parameters, the scaling with N could also vary. Finally, the rapid nature of
EIT cooling is evident from the time constant τ ≈ 21 µs for the cooling curve of the 37 ion crystal.

3.7

Cooling over the full bandwidth
The full bandwidth of drumhead modes are typically cooled to near ground-state occupancies

in a single experimental application of EIT cooling with a fixed set of parameters. We recall thatthe cooling parameters are set according to the expected optimum condition, Eq. (3.14), for cooling
the COM mode, which is the highest frequency drumhead mode with ωCOM /2π = 1.59 MHz. In
6
RN /R1

5

n̄

4
3

3
2.5
2
1.5
1

RN /R1 ∼ N 0.3

0

10

2

30

40

GM
N =1
N =2
N = 19N = 37

1
0

20
N

0

20

40
60
time (µs)

80100Figure 3.8: Cooling curves for the center-of-mass (COM) mode for crystals with different ion
numbers, calculated using the Sampling model (SM) with 2048 trajectories. The SM predicts
that the cooling rate increases with ion number N . The cooling curves from the GM (Fig. 3.6)
are also shown for comparison, where the N -dependency of the cooling rate does not manifest.
Inset: Cooling rate of an N -ion crystal RN relative to the single-ion rate R1 extracted from theSM (markers). A power-law fit (solid line) shows that the cooling rate scales as ∼ N 0.3 for theparameters used. Here, ∆0 /2π = 360 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 33.9 MHz.

50

Figure 3.9: (a) Cooling curves for all the drumhead modes of a 37-ion crystal with bandwidth
approximately 2π ×185 kHz, computed using the SM, showing efficient cooling within 100 µs. Here,∆0 /2π = 360 MHz, Ω1 = Ω2 = Ωopt (∆0 ) ≈ 33.9 MHz. (b) GM cooling curves for all the drumheadmodes of a 120-ion crystal with bandwidth approximately 2π ×376 kHz, showing near ground-state,steady-state occupations of all the modes after few hundred microseconds of EIT cooling. Note thatthe y-axis is plotted in logscale. Here, ∆0 /2π = 400 MHz, Ω1 /2π = Ω2 /2π = Ωopt (∆0 )/2π ≈ 35.7
MHz. The same rotating wall frequency, ωr /2π = 180 kHz was used in both cases.

Fig. 3.9(a), we show the cooling transients for all the tranverse modes of a 37-ion crystal, calculatedusing the SM. The bandwidth of the drumhead modes is approximately 2π×185 kHz. All the modes
are observed to reach near ground-state occupations within 100 microseconds.
The computational complexity of a single trajectory in the SM scales as N 3 with the numberof ions N , thereby making trajectory computations for large crystals (& 60) untractable. However,the GM and SM will result in the same steady-state results since they only differ in the initialconditions, and eventually dissipation leads to the system of equations losing memory of its initialconditions. Therefore, the GM can be used to study the final temperatures that result from EIT
cooling of large ion crystals, as shown in Fig. 3.9(b) for a 120-ion crystal. From initial occupationsin the range n̄ ≈ 5 − 7, all the modes are cooled down to n̄ < 0.1, showing the efficient coolingover the full bandwidth of drumhead modes, which in this case is approximately 2π × 376 kHz.
We note that the experimentally observed occupations are expected to be somewhat higher than
the steady-state values attained here because of the approximate model used in the simulations.
Although the GM cooling transients are not completely reliable, they nevertheless indicate that a

51
few hundred microseconds of EIT cooling is sufficient to achieve these very low occupations.

3.8

Sensitivity to laser alignment
In modeling the EIT cooling in Sections 3.5, 3.6 and 3.7, we have assumed that the lasers

are perfectly aligned, i.e. k2,x = k1,x , so that their difference wavevector is along the z-axis. In
practice, a small misalignment of the EIT lasers could result in a component of the difference
wavevector along the in-plane x-axis, because k2,x 6= k1,x . As a result, the detunings of thesedressing lasers are now modulated unequally by Doppler shifts arising from the large amplitude
in-plane rotation of the ion crystal, so that the instantaneous detunings of the two lasers as seenby the ion, ∆µ (t) = ∆0µ − kµ,x vx (t) with µ = 1, 2, are no longer identical.
We study the effect of such a misalignment by considering a single ion revolving at different
distances r from the trap center. We introduce a small misalignment δθ that modifies the perfectly(m)aligned k2 → k2

such that,
k2,x

(m)

= k2,x cos δθ + k2,z sin δθ(m)= k2,z cos δθ − k2,x sin δθ,k2,z(3.16)

where the subscript (m) denotes the misaligned k2 vector.
Figure 3.10 shows the final steady-state occupation n̄ of a single ion revolving around thetrap center at different radii r, as the misalignment angle δθ is varied. We set the detuning∆0 /2π = 400 MHz, which ensures that for r ≤ 110 µm, the revolving ion experiences effective
blue detuning throughout its trajectory. The steady-state n̄ begins to significantly increase only
when |δθ| & 1◦ , at which point the plot still indicates n̄  1. As r varies from 25 µm to 75 µm,
the sensitivity to δθ also increases with r. For r = 100 µm, the ion experiences only small blue
detunings for parts of its trajectory, resulting in inefficient cooling in these regions even with perfectalignment. As a result, the final temperature is not very sensitive to small misalignments such that|δθ| . 0.5◦ . However, the final temperature grows sharply as |δθ| increases beyond this value.
In the NIST EIT cooling experiment, the misalignment between the two EIT wavevectors

52

Figure 3.10: Steady-state occupation as a function of the misalignment angle δθ for a single ion
revolving around the trap center at different radii. The final occupation is not very sensitive tosmall misalignments (δθ ≤ 1◦ ) of the EIT wavevectors. Here, ∆0 /2π = 400 MHz, Ω1 /2π = Ω2 /2π =Ωopt (∆0 )/2π ≈ 35.7 MHz.

was estimated to be < 0.2◦ [58], ensuring that the cooling is negligibly affected by the in-plane
crystal rotation. We note, however, that our analysis of the laser misalignment does not consider
the potential adverse effect of the EIT lasers on the in-plane modes, which in turn could degrade
the cooling of the drumhead modes. Such an analysis might result in a more stringent restriction
on the tolerable range of δθ.

3.9

Conclusion
Our numerical study shows that EIT cooling is a robust technique for cooling all the drumheadmodes of two-dimensional ion crystals in Penning traps to near ground-state occupancies. EIT
cooling relies on quantum interference effects for its operation, and prior to our work, the chances
for its success, dependent on delicate cancellations of absorption amplitudes, in a challenging setting
such as a Penning trap were very uncertain. Multiple factors could have potentially led to the failureof EIT cooling, namely, Doppler shifts, insufficient separation of electronic and motional timescales,
as well as simultaneous cooling of multiple ions. Our predictions for the success of EIT cooling havebeen validated by the successul experimental demonstration of EIT cooling of crystals with more

53
than 100 ions in the NIST Penning trap [58]. Quantitative measurements of the cooling rate andfinal occupation of the COM mode, as well as qualitative features observed over the full bandwidthof modes are consistent with the expectations from our numerical study. These theoretical andexperimental results highlight the robustness of EIT cooling and make it an attractive scheme tocool large chains or arrays of trapped ions in other settings [89, 96].
In the future, an important aspect to investigate is the effect of higher-order anharmonic
terms in the trap potential, and more importantly, in the Coulomb interaction. These anharmonic
terms not only result in additional coupling of the different drumhead modes to each other, but also
couple these modes to the thermal motion associated with the in-plane modes. An understandingof these anharmonic couplings, at least for small ion crystals, will provide great insight into whether
EIT cooling of the drumhead modes can also indirectly cool the in-plane modes, and also conversely,
how the temperature of the in-plane modes could limit the achievable drumhead mode temperatures
as well as cooling rates. Finally, while we have numerically observed and also found experimental
support for the enhancement in the cooling rate with the number of ions, an intuitive explanation
for this surprising feature will greatly illuminate the role played by many-body physics in the coolingdynamics.

Chapter 4

Near ground-state cooling large trapped ion crystals: Experimental results

In the last chapter, we presented numerical results that demonstrated the effectiveness of EIT
cooling when implemented on large ion crystals in Penning traps. Here, we will present results fromthe EIT cooling experiment performed with the NIST Penning trap. The focus in our presentation
will be on how the data validates the features predicted by our theoretical modeling. Details about
the experimental implementation can be found in Ref. [58].
The thermometry of the drumhead modes in the NIST Penning trap is performed in a rather
unique manner, using the so-called optical dipole force (ODF). In order to understand the data, abasic introduction to the ODF and its application in thermometry is required. We will therefore
begin this chapter with a description of the thermometry technique. Next, we will present quantitative measurements on the center-of-mass mode of large crystals that reveal mean occupations
close to zero, and cooling rates that are faster than single-ion rates. We will then present dataacquired on the full bandwidth of drumhead modes that provides qualitative evidence for significant sub-Doppler cooling of all these modes. We will discuss how the smearing of sharp features in
the ODF spectrum inhibit a quantitative analysis over the full bandwidth of modes. As a starting
point for future studies, we will present a model based on in-plane thermal motion that attempts
to explain the smearing out of the spectrum.

55

Figure 4.1: Schematic of the NIST EIT cooling experiment, showing the EIT lasers and ODF lasers
incident on the ions at angles of ±10◦ to the crystal plane. Figure credit: Elena Jordan, NIST.

4.1

The Optical Dipole ForceOne way to interpret the optical dipole force (ODF) is as a ‘spin’-dependent force, where thetwo states composing the pseudospin correspond to 2s2 S1/2 (mJ = −1/2) and 2s2 S1/2 (mJ = +1/2).
Therefore, we will switch to labeling these states by |↓i and |↑i respectively, instead of the |g1 i,|g2 i notation used in the previous chapter.
4.1.1

Physical mechanism

The optical dipole force (ODF) is generated by two lasers, termed the ODF lasers, that are
incident on the crystal at angles of ±10◦ with respect to the crystal plane, as illustrated in Fig. 4.1.
These lasers couple the spin states to optically excited states in the 2p2 P3/2 manifold and lead to
AC Stark shifts on the spin states.
First, let us consider the case when only one of the ODF lasers is on. For any particular
detuning, there exists a laser polarization such that the two spin states experience the exact same
AC Stark shift [105]. Next, consider the case when both lasers simultaneously address the crystal.
From the geometry, it is clear the difference wavevector δk of the two lasers lies along the z axis,
similar to the pair of EIT lasers. In addition, the difference frequency µr of the ODF lasers canbe tuned using an acousto-optic modulator. The net intensity of the two lasers now includes aninterference term which has the form ∼ sin (δkz + µr t). For a specific choice of detuning and at

56
fixed z, t, this interference term results in additional AC Stark shifts for the spin states that haveequal magnitude |δAC |, but are opposite in sign [105]. For a single ion, the Hamiltonian for thisinteraction can be written as
ĤODF = ~|δAC | sin (δkẑ + µr t) σ̂z .
In the Lamb-Dicke regime (δk

(4.1)

p
hẑ 2 i  1), this interaction simplifies to
ĤODF ≈ F0 cos (µr t) ẑσ̂z ,(4.2)

where F0 = ~|δAC |δk has dimensions of force. The difference frequency µr can be tuned on resonancewith the various frequency components of the drumhead motion, and is therefore typically variedin the range ∼ 900 kHz to 1.6 MHz. As a result, the z-independent term that appears in the Taylor
expansion is ’rapidly oscillating’ at frequency µr and therefore has been dropped while going from
Eq. (4.1) to Eq. (4.2).
For a crystal with N ions, the ODF Hamiltonian straightforwardly generalizes to
ĤODF = F0 cos (µr t)NXẑj σ̂jz .

(4.3)

j=1

In Eq. (4.3), the position operator for each ion is time-dependent, and can be expanded in termsof the N drumhead modes asẑj (t) =

X
n

4.1.2

r



~
Mjn b̂n e−iωn t + b̂†n eiωn t .
2M ωn

(4.4)

Interpretation

The ODF Hamiltonian, Eq. (4.2), can be interpreted in two equivalent ways. First, thegrouping [F0 cos (µr t) σ̂z ] ẑ leads to its interpretation as a force acting on the ion, whose sign
is dependent on the electronic spin state. This interpretation gives rise to the ‘optical dipole
force’ or ‘spin-dependent force’ terminology. Alternatively, the grouping [F0 cos (µr t) ẑ] σ̂z gives
an interpretation of a spatially dependent differential AC Stark shift between the two spin states.
That is, as the ion oscillates about the crystal plane z = 0, an equal superposition state of |↓i

57
and |↑i, initialized say along the x-axis of the Bloch sphere, will precess on the equatorial plane
and accumulate a relative phase that sensitively depends on the nature of the ion’s out-of-planemotion. The contribution of specific drumhead modes can be selectively enhanced and probed bytuning the difference frequency µr close to these mode frequencies. We note that when µr is tuned
exactly on resonance with a mode, the resulting AC Stark shift is effectively mixed down to DC
and results in spin precession at a constant rate. The AC Stark shift interpretation is particularly
useful in understanding how the ODF can be used for thermometry.

4.2

Thermometry with the ODF
In order to measure the temperature of a mode, the ODF interaction is applied as part ofa Ramsey-style sequence as shown in Fig. 4.2 [105, 58]. First, all spins are initialized in |↑i. A
π/2 rotation about the y axis results in an equal superpositon of |↓i and |↑i along the x axis.
The ODF interaction is then applied for an arm time τ , during which the spins precess on theequatorial plane because of the motion of the ions. Next, an echo pulse of duration tπ is applied
about the x axis. This pulse is intended to cancel unwanted phase accumulations, say because ofmagnetic field inhomogeneities. The echo is followed by a second arm of duration τ during which
the ODF interaction is applied again. Finally, a second π/2 rotation about the y axis converts theaccumulated phase into a population difference. If no phase was accumulated, all the spins end upin |↓i. Otherwise, there is a non-zero probability to detect the spins in |↑i and this population canbe detected via fluorescence. For this reason, |↑i is referred to as the ‘bright’ state and the fractionof ions in this state is called the ‘bright fraction’. In principle, an ODF spectrum for a mode canbe generated by plotting the measured bright fraction as a function of the difference frequency µr ,as it is stepped across the mode frequency. A single measurement merely collapses each spin to |↓ior |↑i. In order to precisely measure the bright fraction, the cooling and ODF sequence is repeateda large number of times (∼ 100) at each µr value so that sufficient statistics accumulate to reveal
the true bright state probability.
We assume that the drumhead modes can be described by thermal states both after Doppler

58

Figure 4.2: Schematic of the thermometry sequence involving the ODF interaction. Bloch spherepictures indicate the spin state of an ion at different points of the sequence.

cooling as well as after EIT cooling. One interpretation of thermal motion is that in each trial of theexperiment, the ions are left with varying motional energies after the cooling. As a result, in eachtrial of the ODF sequence, any particular spin will accumulate varying amounts of relative phasebetween the two spin states. When averaged over a large number of cooling and measurement
sequences, the net effect is that the spin appears to have ‘dephased’. Therefore, measuring thebright fraction is often termed as a measurement of the ‘motion-induced spin dephasing’ [104].

4.2.1

Analytical formula for the bright fraction

The Hamiltonian, Eq. (4.3), is special because the propagator Û (t) associated with it can becomputed exactly [123, 34]. This property makes the ODF Hamiltonian one of the best studied
many-body spin-boson models, with analytical formulae calculable for all observables of interest. In
this section, we present an analytical formula for the bright fraction at the end of the thermometry
sequence, which can be used to fit an experimentally measured ODF spectrum in order to measuremode temperatures. This formula is a straightforward adaptation of analytical expressions thathave been rigorously derived elsewhere by other researchers [123, 34, 102], so we briefly sketch the59derivation in Appendix A.2.
The bright fraction at the end of the thermometry sequence is given by!
Njj
1 X 1 − e−2Γτ Css
Csm
P (|↑i) =
,
N
2

(4.5)

j=1

j
where the spin-motion contribution Csm
is an exponential involving a summation over all thedrumhead modes, given by
(
j
Csm
= exp

−2

N
X

)
2

|αnj | (2n̄n + 1)

(4.6)

n=1

j
and the spin-spin contribution Css
is a product over the ions, given by
j
Css
=

Y

cos (4Jjk ) .

(4.7)

k6=j

In writing the formula for the bright fraction, we have introduced new symbols αnj and Jjk for which
exact analytical expressions are presented in Appendix A.2. We use these expressions to numericallycompute the bright fraction in order to compare with and fit the experimentally measured ODF
spectrum. Importantly, these coupling constants depend on (a) the ODF difference frequency µr ,which appears in the expressions for these quantities in the form δn = µr − ωn and (b) the mode
occupations n̄n . These dependencies enable the use of the ODF spectrum in thermometry. In
addition, they also depend on the ODF amplitude F0 , arm time τ , echo time tπ , and also explicitly
on the drumhead mode frequencies ωn and displacements Mjn . The factor e−2Γτ in Eq. (4.5)
accounts for decoherence arising from free-space scattering induced by the ODF lasers.

4.3

Temperature measurements on the COM mode
For values of µr close to the COM mode frequency, denoted as ω1 here, contributions fromother modes can be neglected. The identical coupling of the COM mode to all ions results in simple
expressions Csm = exp −2|α|2 (2n̄ + 1) and Css = [cos (4J)]N −1 that are independent of j. For
simplicity, here we denote n̄ ≡ n̄1 . Once again, we provide expressions for α and J in Appendix
A.2. The bright fraction is simply given by
P (|↑i) =1
1 − e−2Γτ Css Csm .
2

(4.8)

60
As µr is stepped across the COM mode, Eq. (4.8) predicts qualitatively different lineshapes
for the spectrum of P (|↑i) versus µr in regimes of high and low n̄ values, when other parameters areheld fixed. A crucial insight into this feature is obtained by noting that the term Css is independentof n̄ while Csm decreases exponentially with n̄. Furthermore, the product of these two terms isthen subtracted from a constant value of 1. At low n̄, Csm does not suppress the features resulting
from Css and leads to a complex lineshape. At high n̄, Csm is very small and strongly suppresses
features resulting from Css , leading to a simpler lineshape. Moreover, the larger suppression of
Css results in greater values of P (|↑i) as n̄ increases. In fact, P (|↑i) → 1/2 as n̄ → ∞, except fordiscrete values of µr where |α|2 = 0. Therefore, the spectrum after Doppler cooling can be expected
to be higher as well as simpler in structure, whereas the spectrum after EIT cooling is expected
to be lower and relatively more complex. In general, with similar parameters used for the ODF
sequences after Doppler and EIT cooling, the above argument leads us to expect a lower brightfraction across the entire bandwidth of drumhead modes after EIT cooling.
Such features are indeed observed in the experimentally measured ODF spectrum for µr close
to the COM mode, an example of which is shown in Fig. 4.3 for a crystal of 158 ± 10 ions. The data
set in black is measured immediately after Doppler cooling, whereas the blue data set is measured
after an additional 200 µs of EIT cooling. The spectrum after Doppler cooling is higher and two
peaked, whereas the spectrum after EIT cooling is lower but features four peak-like structures.
The same formula, Eq. (4.8), is used to fit the spectrum in both cases, resulting in measured n̄values of 5.6 ± 1.1 in the Doppler cooling case and 0.28 ± 0.18 in the EIT cooling case. These
measurements quantitatively demonstrate near ground-state cooling of the COM mode within justa few hundred microseconds of EIT cooling. We note that, although we considered the case of an
equal ODF amplitude F0 in motivating the expected lineshapes in the two cases, the spectrum after
EIT cooling is typically taken with a somewhat higher F0 value.
By measuring n̄ after varying durations of EIT cooling, a cooling rate can be measured for the
COM mode. Figure 4.4 shows a measurement of n̄ versus cooling time for a crystal with 190 ± 10
ions. An exponential fit to the data reveals a cooling time constant of τcool = 27.6 ± 1.7 µs for this61

Figure 4.3: ODF spectra showing the bright fraction as the ODF difference frequency is stepped
across the COM mode frequency ω1 . The black and blue data are respectively measured after
Doppler cooling and 200 µs of additional EIT cooling on a crystal with 158±10 ions. The measuredn̄ values in each case are obtained by fitting the data with the formula Eq. (4.8). Figure and datacredit: Bollinger group, NIST.

Figure 4.4: Measured thermal occupation n̄ of the COM mode (black) as a function of the coolingtime for a crystal with 190 ± 10 ions. The measured cooling time constant of approximately 28 µs
corresponds to a rate that is faster than that theoretically expected from separate cooling of eachion in the trap (blue). Inset: We recall the cooling transients resulting from the Sampling Modelpresented in Chapter 3, where the cooling rate of the COM mode increases with ion number.
Experimental data credit: Bollinger group, NIST.

particular set of EIT cooling parameters [58]. The rapid cooling is consistent with results from ourtheory, where the observed time constant is typically a few tens of microseconds.

62
An exciting observation is that the measured cooling rate is faster than the average rate
theoretically expected if each ion was independently cooled in the trap (Fig. 4.4, blue curve inmain panel). For this comparison, we compute single-ion cooling transients for ions revolving atdifferent radii from the trap center, and then average by weighting each transient by the numberof ions approximately situated at that radius. The stepwise behavior is merely an artifact causedby choosing the same initial phase of revolution for each transient. The faster rate of cooling in
the multi-ion crystal is consistent with the numerical observation that the cooling rate of the COM
mode increases with ion number, a result that we recall in the inset of Fig. 4.4. In the future,better understanding of this speed-up phenomenon can be obtained by performing cooling ratemeasurements on small ion crystals (N < 60), where direct theory-experiment comparisons arefeasible.

4.4

Measurements over the full bandwidth
Our modeling predicts that with a fixed set of experimental parameters, EIT cooling should be

able to cool down the entire bandwidth of drumhead modes close to their ground states within a few
hundred microseconds. To verify this experimentally, the cooling and ODF sequence is repeatedly
applied, keeping the cooling parameters fixed while scanning the ODF difference frequency overthe full bandwidth of the drumhead modes. Figure 4.5 shows the ODF spectrum obtained using
a crystal with 158 ± 10 ions, after Doppler cooling only (red) and after 300 µs of additional EIT
cooling (blue). A steep drop in the bright fraction is observed across the entire bandwidth of modes,
providing evidence for strong sub-Doppler cooling of all the drumhead modes.
A natural next step is to use this spectrum and the formula, Eq. (4.5), to quantitatively
estimate the mean occupations for various drumhead modes. However, the measured spectrum
is not easily amenable to such a quantitative analysis. The qualitative differences between thetheoretically expected and measured spectra already manifest in smaller ion crystals, where acomparative study is easier to carry out. Figure 4.6(a) shows the theoretically expected ODF
spectra with a 79 ion crystal, for the cases when all the modes have n̄ = 6.0 and n̄ = 0.26

63

Figure 4.5: ODF spectrum measured over the full bandwidth of drumhead modes after Doppler
cooling only (red) and after 300 µs of additional EIT cooling (blue), for a crystal with 158 ± 10
ions. The significant reduction in bright fraction over the entire bandwidth after EIT cooling isevidence of strong sub-Doppler cooling of all the drumhead modes. Data credit: Bollinger group,
NIST.

(green and orange curves respectively). The ODF parameters correspond to the experimental data
presented in Fig. 4.6(b). The theory curves confirm two of our intuitive expectations. One, thebright fraction at low n̄ is consistently lower than at high n̄ over the full bandwidth.1

Two, theODF spectrum displays sharp peaks that are expected when µr is tuned close to a mode or a clusterof modes.
In comparison, Fig. 4.6(b) shows the spectra measured using a crystal with 79 ± 5 ions, after
Doppler cooling only (red) and after additional EIT cooling (blue). In the latter spectrum, themeasured n̄ for the COM mode is 0.26 ± 0.38, motivating our choice of n̄ for the corresponding
theory curve. While the spectrum after EIT cooling is indeed lower than that after Doppler cooling
only, the sharp features predicted by the theory have smeared out into a smooth continuum. A
quantitative analysis of this spectrum is not possible without first understanding the reason behind
this qualitative difference between theory and experiment.
The smearing of the ODF spectrum suggests fluctuations in the drumhead mode frequencies. To investigate this possibility, we carry out a numerical simulation that is inspired by theexperimental protocol used to obtain the spectrum. We assume that the mode frequencies remain
1This feature manifests even though the effective amplitude F0 for the n̄ = 0.26 case is somewhat larger than then̄ = 6.0 case.

64

Figure 4.6: Qualitative differences between theory and experimentally measured ODF spectra,
exemplified using a 79 ion crystal. (a) Theoretically expected spectrum at high and low n̄ values
are spiky and partially resolve the modes. (b) Experimentally measured spectra (red and blue)
display a smooth continuum. Theory incorporating ad hoc mode frequency fluctuations smear
the sharp features seen in (a) and lead to curves (green and orange) that resemble the measured
spectra. Experimental data credit: Bollinger group, NIST.

constant within the duration of a single ODF sequence, but fluctuate from one trial to the next.
To simulate this, we first numerically obtain the drumhead mode frequencies ωn and amplitudes
Mjn for the equilibrium crystal configuration [124], which are used to generate the theory curves
in Fig. 4.6(a). Next, for each value of µr , we add independent Gaussian distributed fluctuations toeach mode frequency and compute the bright fraction. We repeat this several tens of times at everyµr value and average over all the trials. We assume that the magnitude of fluctuations increases
with decreasing mode frequency. Repeating this process for µr spanning the full bandwidth leads tothe theory curves shown in Fig. 4.6(b) (green and orange). These curves were generated assuming
frequency fluctuations that linearly increase from 1 kHz for the second-highest frequency mode to
80 kHz for the lowest frequency mode. Clearly, these curves suggest that frequency fluctuations
strongly affect the measured spectrum.
The similar profiles of the theory (orange) and data (blue) measured after EIT cooling enable us to make qualitative assessments of the drumhead mode temperatures. The measured brightfraction after EIT cooling is below the corresponding theory curve, strongly suggesting near groundstate cooling over the entire bandwidth of drumhead modes. In the case of only Doppler cooling,
there is qualitative agreement at higher frequencies, but the measured bright fraction at low frequencies is consistently higher than theory predictions with reasonable n̄ values. We have observed
these general trends over a range of data sets measured after both types of cooling, and with varying

65
ion numbers.
For improving the analysis of these spectra, the ad hoc frequency fluctuations assumed so
far in this study have to be replaced by a physically motivated model. To explain the EIT data,
the effect of frequency fluctuations on the spin-spin interaction signal, which is dominant at low
temperatures, is required. In the Doppler cooling case, the large temperatures should allow fora numerical model that neglects quantum effects [105], making a first analysis far easier in thiscase. Some initial steps towards developing a physical model of mode frequency fluctuations andmodeling the spectrum measured after Doppler cooling are described in Section 4.5.

4.5

Physical model for mode frequency fluctuations
The Coulomb interaction is intrinsically anharmonic, and therefore introduces coupling be-

tween the in-plane and out-of-plane motion even when the trapping potential is purely harmonic.
We speculate that this coupling is the primary reason for the observed drumhead mode frequency
fluctuations. To understand the motivation for this hypothesis, a brief introduction to the classicaldynamics of ions in a Penning trap is first required.

4.5.1

Classical dynamics of ions in a Penning trap

Following Ref. [124], we write down the Lagrangian for a collection of N ions in a Penning
trap. The Lagrangian for this system is
L=

N 
X
1
j=1

2


mj ṙj · ṙj − eφj + eAj · ṙj .

(4.9)

Here, Aj = (B × rj )/2 is the vector potential associated with the magnetic field B = Bz ez in thesymmetric gauge. Therefore, Aj = −Bz (yex − xey )/2. The potential φj experienced by each ion is
1 2
ke e X 1
2
φj (t) = V0 zj − ρj + VW ρ2j cos [2(θj + Ωt] +
.
2
2
rkj

(4.10)

k6=j

The first term represents the quadrupole electric potential, the second term is the rotating wall
potential while the third term describes the inter-ion Coulomb potential. Here, we have used the66
shorthand ke = 1/(4π0 ) with 0 the permittivity of free space. In writing Eq. (4.10), we have usedq
the radial coordinate ρj = x2j + yj2 and the angular coordinate θj = tan−1 (yj /xj ).
4.5.1.1

Transformation to the rotating frame

In a frame rotating at angular frequency Ω, the Lagrangian becomes time independent.
R
The coordinates in the rotating frame xRj (t), yj (t) are related to the coordinates in the lab framexj (t), yj (t) as [124]
  
 
R
xj   cos Ωt sin Ωt  xj 
 =
 .
yj
− sin Ωt cos ΩtyjR(4.11)

Direct substitution of this relation into Eq. (4.9) and Eq. (4.10) gives the Lagrangian in the rotating
frame as
L =

N 
X
1
j=1

2



eBeff,j [Ω] R R
RR
mj ṙR(ẋj yj − ẏjR xR
j · ṙj − eφj −
j )
2

,

(4.12)

where Beff,j [Ω] = Bz − 2mj Ω/e andR 2
eφR
j = eV0 (zj ) +
ke e2 X 1
1
2
R 2
R 2
.
eBz Ω − mj Ω2 − eV0 (ρR
)
+
eV
[(x
)
−
(y
)
]
+
W
j
j
j
R
2
2
rkj

(4.13)

k6=j

Henceforth, we drop the R subscript, and work in the rotating frame, unless explicitly mentioned
otherwise.

4.5.1.2

Dimensionless units

We now make the problem dimensionless by introducing certain characteristic scales. The
axial trapping strength can be characterized in terms of the trapping frequency ωz for a single 9 Be+
ion with mass mBe , i.e. eV0 = mBe ωz2 /2. We define the scale factors

m0 = mBe , l0 =2ke e2mBe ωz2

1/3, t0 = 1/ωz ,(4.14)

for mass, length and time respectively. In these new units, frequencies are specified relative to ωz .
A characteristic energy scale that can be constructed out of these base units is1
2
E0 = m0 l02 t−2
0 ≡ eV0 l0 .
2(4.15)

67
We normalize the Lagrangian, Eq. (4.12), by this characteristic energy to obtain L̃ = L/E0 , given
by

N 
X
∼
0
˙
˙
˙
˙
L̃ =
m̃j r̃j · r̃j − eφj − ω̃c,j [Ω](x̃j ỹj − ỹj x̃j ) ,(4.16)

j=1

0 =where the ˜· indicates normalization with respect to the respective characteristic scales. Here, ω̃c,jω̃c − 2m̃j Ω̃ is an effective normalized cyclotron frequency with ωc = eBz /mBe , and ω̃c = ωc /ωz ,Ω̃ = Ω/ωz and m̃j = mj /mBe . The effective potential energy term is∼

eφj = z̃j2 +
 1X 112
ω̃c Ω̃ − m̃j Ω̃ −
ρ̃2j + CW x̃2j − ỹj2 +
.
2
2
r̃kj

(4.17)

k6=j

Here CW = VW /V0 is the dimensionless rotating wall strength. Note how simple the Coulombinteraction becomes in these scaled variables. We note that the Lagrangian, Eq. (4.16), genericallyallows for the presence of ions with different masses and can therefore be used to study the effectof impurity ions [79].

4.5.1.3

Euler-Lagrange equations of motion

Defining the mechanical momentum Π̃j = m̃j r̃˙j , we obtain the equations of motion along
the three directions as a set of first order coupled differential equations. The positions are updated
trivially as
1r̃˙j =
Π̃jm̃j(4.18)

by construction. The momentum components evolve as


0 [Ω]
ω̃c,j
1 X x̃j − x̃k˙ x = − ω̃ Ω̃ − m̃ Ω̃2 − 1 + C
x̃
+
Πyj +
,
Π̃
j
c
j
W
j
3
2
m̃j
2
r̃kj
k6=j


0
X
ω̃c,j [Ω] x 1ỹj − ỹk˙ y = − ω̃ Ω̃ − m̃ Ω̃2 − 1 − C
Π̃
Πj +
,
c
j
W ỹj −
j
3
2
m̃j
2
r̃kj
k6=j

Π̇zj = −z̃j +1 X z̃j − z̃k

2

k6=j

3
r̃kj

.

Here, the · now indicates the time derivative with respect to the scaled time t̃.

(4.19)

68
4.5.1.4

Nature of normal modes for planar crystals

We can roughly motivate the typical frequency ranges of the three branches of modes, summarized in Table 3.1, using characteristic frequencies constructed out of the trap parameters. First,
we recall that the drumhead center-of-mass mode frequency is set by the axial trapping frequency
and is typically ω1 = ωz ≈ 2π × 1.57 − 1.59 MHz in this work. In the crystal plane, the highfrequency cyclotron modes have frequencies close to the cyclotron frequency ωc = eB/mBe . For
the NIST Penning trap, ωc ≈ 2π × 7.6 MHz. In addition, large amplitude in-plane motion occurs
on a slower time scale and is governed by the magnetron modes. These modes have frequenciespbelow the magnetron frequency of ωm = ωc − ωc2 − 2ωz2 /2. Typically, ωm ≈ 2π × 170 kHz forthe parameters of the NIST Penning trap. In the absence of a rotating wall potential, the lowest
frequency magnetron mode, called the rocking mode, acquires zero frequency and can be thoughtof as a Goldstone mode arising from the spontaneous breaking of rotational symmetry. In thepresence of a weak wall potential, this rocking mode acquires a small non-zero frequency.

4.5.2

The thermal snapshot picture

We are now ready to discuss an intuitive model for drumhead mode frequency fluctuations.
During a single time period associated with a typical magnetron frequency (∼ 10 − 100 kHz),every drumhead mode (& 1 MHz) completes several oscillations. In an extreme, even unrealistic,
simplification, we can imagine the ions to be frozen in plane at different points along their magnetron
motion trajectories, while they execute rapid motion out-of-plane. Further, we are assuming thatthe cyclotron motion has very small amplitude and can be neglected for the sake of this argument.
For each frozen configuration, we can follow the procedure of Ref. [124] and perform a normal mode
analysis in the out-of-plane direction and sort the drumhead modes in descending order. We canthen repeat this exercise a large number of times and histogram the sorted modes into frequency
bins to study the spread in the mode frequencies.
In order to execute this program, we extensively use the Metropolis-Hastings (MH) algo-

69

Figure 4.7: Simulated annealing to find the minimum energy configuration of a 53 ion crystal. (a)Ion positions before (blue) and after (orange) the simulated annealing procedure. (b) Annealing
temperature and in-plane potential energy versus annealing iteration.

rithm 2 . In fact, the MH algorithm is the building block of the global minimization technique ofsimulated annealing. Therefore, we first realize a minimum energy configuration using the simulated annealing technique and the potential energy for the x − y coordinates as the cost function.
Figure 4.7(a) shows a minimum energy crystal (orange) obtained using simulated annealing, starting from a random configuration (blue). The parameters used for this study are N = 53 ions,ωz = 2π × 1.575 MHz and Ω = 2π × 187 kHz, deviating from the conventional 180 kHz rotation
frequency. The wall potential is VW = 5.3 V. Figure 4.7(b) shows how the annealing temperature
is decreased with each successive iteration and also plots the decreasing in-plane potential energy
as the simulated annealing proceeds.
Next, we introduce a temperature T⊥ that characterizes the excess thermal energy in thein-plane configuration. We use the MH algorithm with the in-plane potential energy as the costfunction to generate a sequence of snapshots corresponding to the temperature T⊥ . Figure 4.8(a)captures the deviation of ion positions in a typical thermal configuration (orange) with T⊥ = 10 mK,
from the corresponding positions in the minimum energy configuration (blue). The MH algorithm
proceeds by attempting to perturb the position of each ion in succession. Going through each ion
2

See Ref. [91] for an introduction to this technique, which is referred to as the Metropolis Monte Carlo algorithm.

70

Figure 4.8: Metropolis-Hastings algorithm to find thermal snapshots of the in-plane configuration.
(a) Thermal snapshot (orange) with T⊥ = 10 mK compared to the minimum energy configuration(blue). (b) Thermalization of the in-plane potential energy as the MH algorithm repeatedly scansover all the ions in the crystal.

in the crystal completes one scan. Figure 4.8(b) plots the in-plane potential energy versus scannumber and confirms that typical configurations for the set temperature are reached within thefirst 104 scans. Compared to the minimum energy configuration, typical configurations at a set
temperature T⊥ correspond to an average potential energy increase of approximately kB T⊥ per ion,
accounting for the two spatial degrees of freedom.
For each value of T⊥ , we then perform a normal mode analysis on 2000 snapshots. Successive
snapshots are chosen 40 scans apart to ensure that they are sufficiently uncorrelated. Figure 4.9
shows the histograms of drumhead frequencies for T⊥ = 0, 1 and 10 mK. As T⊥ increases, thespectrum transitions from being spiky and fully resolved to a smooth and continuous distribution
where only the first few modes are resolvable.
We can also directly study the effect of T⊥ on the ODF spectrum. For this purpose, we
realize 48 different in-plane configurations at each value of T⊥ , feed in the normal mode spectrum
corresponding to each realization into the analytic formula for the ODF spectrum (Eq. (4.5)), andaverage the spectrum over all the realizations. The results are shown in Fig. 4.10. For comparison,
we also show the location of the normal modes for the minimum energy configuration. Clearly,

71

Figure 4.9: Histograms of sorted drumhead modes for 2000 realizations of the in-plane configurations with (a) T⊥ = 0 mK, (b) T⊥ = 1 mK and (c) T⊥ = 10 mK. The bin size is 0.5 kHz. In the 0mK case, some of the bins carry counts that are not a multiple of 2000 possibly because of round-off
errors affecting the binning algorithm or because of slight fluctuations in mode frequencies as theMH algorithm might have found lower energy configurations.

Figure 4.10: ODF spectra computed by averaging over the bright fraction expected from the normal
modes of 48 thermal snapshots of the in-plane configuration. Here, (a) T⊥ = 0 mK, (b) T⊥ = 1
mK and (c) T⊥ = 10 mK. We have assumed n̄ = 6 for all the modes. Also shown are the mode
frequencies for the minimum energy configuration.

the ODF spectrum smoothens out with increasing value of T⊥ , further supporting the case thatin-plane position fluctuations might explain the experimentally measured drumhead spectra.

4.5.3

Molecular dynamics simulations

After establishing an intuitive argument for why thermal in-plane fluctuations may lead to
drumhead mode fluctuations, we proceed to verify this idea by brute force molecular dynamics
simulations.
The initial conditions for the position and momentum degrees of freedom of the ions are

72
set using an adaptation of the MH algorithm. First, the MH algorithm is applied using the total Hamiltonian of the system as the cost function. The Hamiltonian can be obtained from the
Lagrangian, Eq. (4.16), using a Legendre transformation. When written in terms of the positions
and the mechanical momenta, instead of the canonical momenta, the Hamiltonian is separable andtherefore the positions and momenta can be sampled separately [33]. Note, however, that as in
the case of the in-plane potential energy, the sampling of the ion positions cannot be performed
independently for each ion because of the Coulomb interaction.
We would like to study the effect of changing the in-plane temperature T⊥ as the out-of-planetemperature Tz is held fixed. Therefore, we first apply the MH algorithm for the full set of 6N
degrees of freedom using T⊥ as the temperature. Then, we scale the position and momentum flucp
tuations of each ion in the out-of-plane direction by a factor of Tz /T⊥ in order to approximately
set an unequal temperature in the z direction. When T⊥ = 0, we first follow the above procedure fora value T⊥ 6= 0 and then additionally reset the fluctuations in the in-plane positions and momenta
to zero.
To study the dynamics, we use a 4th order Runge-Kutta integrator to evolve the EulerLagrange equations of motion (Eqs. (4.18 and (4.19). First, we consider the power spectrum,
obtained by taking the Fourier transform of the drumhead motion [119]. In Fig. 4.11, we compare
the power spectrum in the three cases when T⊥ = 0, 1 and 10 mK. In all cases, the out-of-planetemperature is held fixed at Tz = 0.5 mK. As T⊥ increases, the power spectrum broadens intoa smooth continuum where the modes beyond the highest frequency ones are no longer resolvable. This observation is consistent with the intuitive ‘thermal snapshot’ picture presented in
Section 4.5.2.

4.5.3.1

ODF spectrum from molecular dynamics simulations

We can also extract the expected ODF spectrum from the molecular dynamics simulation.
Since the motion is treated classically, this analysis ignores the effect of the zero-point motion andthe phonon-mediated spin-spin interaction on the bright fraction (see Eq. (4.5)).

73

Figure 4.11: Power spectra of the drumhead motion from molecular dynamics simulations. Here,Tz = 0.5 mK, and (a) T⊥ = 0 mK, (b) T⊥ = 1 mK and (c) T⊥ = 10 mK.

The ODF interaction Hamiltonian for ion j isj
= F0 cos(µR t)zj σ̂jz .
HODF(4.20)First, we investigate the effect of HODF on the motion. In addition to the trap dynamics, theequation for the momentum pzj picks up an additional termd zp = −F0 cos(µR t)hσ̂jz i.
dt j(4.21)

However, hσ̂jz i ≈ 0 since all the spins precess on the equator for the ODF sequence. Therefore,
the effect of the ODF interaction on the ion motion can be neglected when the motion is treatedclassically, as in the molecular dynamics simulations.
Next, we investigate the effect on the spin. For the quantity hσ̂j+ i, we getd +1hσ̂j i = 2iF0 cos(µR t)zj hσ̂j+ i.
dt~(4.22)

From this, we see that the time evolution of hσ̂j+ i is given byhσ̂j+ i(t) = hσ̂j+ i(0) exp



1
2iF0
~

Z t

0

0

0



dt zj (t ) cos(µR t ) ,(4.23)

0with the initial value hσ̂j+ i(0) = 1/2. Knowing that the bright fraction after the ODF sequence isrelated to hσ̂jx i before the final state rotation, we immediately obtain
Pj (↑) =1
1 − e−2Γτ cos Aj ,2

(4.24)

74

Figure 4.12: ODF spectra computed from molecular dynamics simulations. Here, Tz = 0.5 mK,
and (a) T⊥ = 0 mK, (b) T⊥ = 1 mK and (c) T⊥ = 10 mK.

where
2F0
Aj =
~

Z t

dt0 zj (t0 ) cos(µR t0 ).

(4.25)

0

We have included the effect of decoherence by introducing the factor of e−2Γτ as in Eq. (4.5). To
simulate the actual ODF sequence, the force F0 is multiplied by a factor g(t), which takes the value
1 for t < τ , 0 during the echo time and −1 for t > τ + tπ . The sequence ends at t = 2τ + tπ .
In Fig. 4.12, we plot the ODF spectrum for the three cases with T⊥ = 0, 1 mK and 10 mK.
Each spectrum was generated by averaging the bright fraction over 48 realizations of the initialconditions. The ODF spectrum at T⊥ = 10 mK is reminiscent of experimentally observed spectraand strongly suggests that thermal energy in the in-plane modes is the primary reason for drumheadmode frequency fluctuations.
In the case of a simple harmonic oscillator, thermal initial conditions can be realized bysimply providing velocity kicks corresponding to twice the temperature and setting the position coordinate to zero. The oscillator dynamics will eventually mix the velocity fluctuations into position
to give thermal snapshots of the harmonic oscillator. Because of the magnetic field, the in-plane
cyclotron and magnetron modes are not such simple harmonic oscillator modes. Therefore, we findthat initializing the ions by in-plane velocity kicks is qualitatively very different to additionallyinitializing the positions via the Metropolis-Hastings algorithm. We find that, even at T⊥ = 10mK, the velocity-kick initialization typically leads to a spiky power spectrum and ODF spectrum,in contrast to the MH based initialization. Based on videos of the in-plane dynamics, we speculate

75that the velocity-kick initialization predominantly excites the high-frequency, small-amplitude cyclotron modes, whereas the MH based initialization also excites the low-frequency, large amplitude
magnetron modes. The large amplitudes of the latter modes are responsible for significant changes
in the instantaneous in-plane configuration, leading to drumhead mode frequency fluctuations.

4.5.4

Further extensions of this study

In order to improve the qualitative and quantitative agreement with measured ODF spectra,
several extensions are possible. It is now known that when the EIT measurements were taken, thetrap was actually operating with a small anharmonic potential. The addition of such a C4 term may
further smear out the ODF spectrum. Further, the role of impurities has to be investigated. This
will require data sets where the number and type of impurities are measured. While here the initialconditions have been assumed to correspond to some temperature, a more realistic simulation could
first apply Doppler cooling as in the experiment and use the initial conditions resulting from thisprocess [119]. Studying the ODF spectra obtained after EIT cooling is more complicated because
quantum effects of the ion motion must be properly accounted for at such low temperatures.

Chapter 5

Real-time phase tracking: Formalism

Many applications in quantum metrology rely on the dynamics of pseudospin-1/2 systems
with two long-lived quantum states, |↑i and |↓i. After preparing an equal superposition of these
two states, a physical interaction is studied by investigating its effect on the relative phase φ(t), √with the state of each spin evolving in time as |ψ(t)i = |↓i + eiφ(t) |↑i / 2. In this chapter, we
will describe a novel scheme that enables continuous tracking of this relative phase. Our scheme
continuously and directly measures the real-time phase φ(t) unlike the widely used Ramsey sequence[95, 86, 49, 60, 6, 100, 108, 125, 97, 48, 43, 39], which indirectly measures the net accumulated
phase φ(T ) during an interrogation time T . The typically destructive readout in a Ramsey sequencerequires multiple state resets, rotations and repetitions of the sequence to infer the phase at different
times from a population difference. In contrast, a single run of our protocol yields a continuous
time series of phase measurements. Therefore, our scheme enables real-time tracking of time-varying
signals that are not reproducible.
As an added benefit, our scheme yields continuous phase estimates with precision well beyond√
the standard quantum limit (SQL) of ∆φSQL = 1/ N radians that limits readout precision with N
unentangled spins. In comparison to several proposals and experiments [62, 71, 107, 67, 11, 26, 52]
that have demonstrated squeezed states with precision beyond the SQL, our scheme enjoys theadvantage that the squeezing is produced, the phase accumulated, and the readout performed, allin the same spin quadrature.
Recent experiments have demonstrated phase tracking of a spin using quantum non-demolition

77(QND) measurements via a Faraday rotation angle [25]. In contrast, our proposal is based on interfering Raman transitions in a cavity and enables an intuitive interpretation of phase tracking interms of elementary atom-cavity interactions that nearly balance one another. Our scheme directly
reveals a phasor precessing in the equatorial plane of a Bloch sphere, in the spirit of the “hand on
a clock” analogy at the core of quantum metrology.
Such interfering cavity-assisted Raman transitions have been considered previously for deterministic squeezing schemes [115] and quantum simulations of the Dicke model [32, 7, 136]. The
frequency arrangement of our drive lasers is also related to two-tone drive schemes for back-action
evading measurements of mechanical oscillators [16, 23, 15] and for measuring the state of individual
superconducting qubits [45, 35].
We represent the collective angular momentum of N atomic spins by a classical Bloch vectorof length N/2 with components Jx , Jy , Jz . With all spins initially in the same equal superposition
state, the Bloch vector lies in the equatorial plane along a direction that we define as the y-axis.
As the phase evolves, the Bloch vector acquires a small x-component, Jx = N2 sin φ(t) ≈ N2 φ(t), forsmall deflections, and we propose a straightforward extension to large deflections in the conclusion.
We arrange atom-cavity interactions wherein a cavity field quadrature is sourced by Jx . Continuous
homodyne detection of this quadrature amounts to real-time, continuous, QND measurement ofφ(t).
We proceed by first introducing the setup under consideration. We then provide an intuitive
picture based on classical Bloch vectors that outlines the essential mechanism by which the phasecan be tracked. After rigorously deriving the quantum non-demolition Hamiltonian, we introduce
the stochastic master equation governing the dynamics of the atom-cavity system under continuous measurement. We employ a Gaussian state approximation to derive dynamical equations forobservables from this master equation.

78

Figure 5.1: Schematic for continuous, real-time phase tracking. Two lasers drive a collection of
atoms to interact with a cavity mode. The relative phase φ(t) can be continuously tracked byhomodyne detection of the field leaking out.

5.1

Setup and basic mechanism
We consider N atoms trapped at the antinodes of a cavity with resonance frequency ωc anddecay rate κ, as shown in Fig. 5.1. The states |↓i and |↑i have an energy separation ~ω0  ~κand form a pseudospin-1/2 system described by the Pauli spin operators σ̂i , i = x, y, z, withraising (lowering) operators σ̂+ (σ̂− ). The N atoms form a collective spin with total angularPjmomentum components Jˆx , Jˆy , Jˆz , with Jˆi = N
j=1 σ̂i /2. We assume the dipole-allowed transitions|↓i ↔ |ei and |↑i ↔ |ei with frequencies ω↓e and ω↑e to be respectively driven using lasers withfrequencies ω1 and ω2 in a far-detuned regime with detuning ∆  ω0 , κ, allowing for the adiabatic
elimination of |ei [56]. The two drive lasers differ by a frequency 2ω0 (Fig. 5.2(b)) and do not bythemselves drive |↓i ↔ |↑i Raman transitions; however, they are symmetrically detuned by ω0 fromωc and participate in cavity-assisted Raman transitions as illustrated in Fig. 5.2(b). When the Rabi
frequencies of the two drive lasers are balanced, i.e. Ω1 = Ω2 = Ω0 , the atom-cavity Hamiltonian,
to leading order in 1/∆, is simply the sum of a Jaynes-Cummings and an anti-Jaynes-Cummings
interaction and is given by
ĤQND =~ΩQND ˆX̂ Jx .
2

(5.1)√
Here X̂ = (â + â† )/ 2 is the amplitude quadrature, with â, â† the annihilation and creation√
operators for the cavity mode, and Ŷ = (â − â† )/ 2i is the conjugate phase quadrature such

79

Figure 5.2: (a) Cavity-assisted Raman transitions: The red (blue) pathway leads to the emissionof a cavity photon accompanied by a spin flip |↓i → |↑i (|↑i → |↓i). (b) Hierarchy of frequencies.

that [X̂, Ŷ ] = i. The atom-cavity interaction strength is ΩQND =√2Ω0 g0 /∆ with g0 the single

atom-cavity vacuum Rabi frequency. For the interested reader, a detailed derivation of ĤQND
is presented in Section 5.2. If the two drive lasers have initial phases ψ1 and ψ2 , the cavity
quadrature â† ei(ψ1 +ψ2 )/2 + H.c. is coupled to the spin component Jˆ+ ei(ψ1 −ψ2 )/2 + H.c. , whereJˆ+ = Jˆx + iJˆy . Here we assume ψ1 = ψ2 = 0 without loss of generality.
Classically, the intracavity fields established by the two balanced drives exactly cancel when
Jx = 0 (Fig. 5.3). However, even with hJˆx i = 0, hJˆx2 i =6 0, i.e. quantum fluctuations source the Y
quadrature of the cavity field. In the regime κ2  N Ω2QND , Ŷ is slaved to Jˆx as
Ŷ (t) ≈ −ΩQND ˆJx (t) + F̂(t),
κ

(5.2)

where the noise operator F̂(t) arises from coupling of the cavity mode to external modes throughthe lossy mirror (Fig. 5.1) [133, 82]. The field leaking out is to be monitored via balanced homodyne
detection using a local oscillator at frequency (ω1 + ω2 )/2 with phase tuned to detect the output
field quadrature that is sourced by the intracavity Y quadrature. The photocurrent thus recorded
is a measurement of the Y quadrature which, from Eq. (5.2), amounts to measuring Jx .
Measurement back-action in the Jz quadrature arises because of the indistinguishability of thetwo pathways that give rise to the intracavity field (Fig. 5.1(b)): The field leaking out is consistent
with equal probability amplitudes for tipping the Bloch vector above or below the equator and80

Figure 5.3: Classical Bloch vector picture: The red and blue pathways set up balanced, opposing
superradiance pathways that lead to a coherent cancellation of the intracavity field when the Blochvector (green) is along the y-axis (φ = 0). When the Bloch vector has a small x-component (φ 6= 0),
the intracavity field from the two pathways add constructively, giving rise to non-zero output field.

therefore increases the spread in Jz without affecting its mean value.

5.2

Derivation of the QND HamiltonianHere, we systematically derive ĤQND starting from basic atom-cavity interactions and byusing effective Hamiltonian theory. This section can be skipped if the reader is not interested in
the details of this derivation.
The basic Hamiltonian for the interaction of the atoms with the drive lasers and cavity modeis (~ = 1 in this Section)Ĥ = ωc â† â − ω↓eXj|↓ij h↓| − ω↑eXj|↑ij h↑|X  Ω1Ω2−iω1 t−iω2 t+|eij h↓| e+|eij h↑| e+ H.c.
22j
X  g1g2+
â |eij h↓| + â |eij h↑| + H.c. .
2
2

(5.3)

j

The drive laser frequencies are arranged such that ω1 = ωc +ω0 and ω2 = ωc −ω0 . We assume
that the splitting ω↓e − ω↑e between the spin states, nominally ω0 , can be slightly modified, e.g. by
a weak external magnetic field that we wish to sense, i.e. ω↓e − ω↑e = ω0 + 2δ, where δ (−δ) is theshift of the |↑i (|↓i) state.
The detunings of the drive lasers from the atomic transitions are given by ∆1 = ω1 − ω↓e =81−∆ + ω0 /2 − δ and ∆2 = ω2 − ω↑e = −∆ − ω0 /2 + δ. Similarly, the detunings of the cavity mode
from the atomic transitions are ∆c1 = ωc −ω↓e = −∆−ω0 /2−δ and ∆c2 = ωc −ω↑e = −∆+ω0 /2+δ.
We write the interaction Hamiltonian expressing the energy requirements in Eq. (5.3) using
complex exponentials involving these detunings as
ĤI (t) =X  Ω12j+

X  g12j−i∆1 t|eij h↓| ecΩ2+|eij h↑| e−i∆2 t + H.c.
2â |eij h↓| e−i∆1 t +




g2
c
â |eij h↑| e−i∆2 t + H.c. .
2

(5.4)

We use the effective Hamiltonian theory of Ref. [56] to derive the effective Hamiltonianin the limit where the detunings are all much greater than the Rabi frequencies. This effective
Hamiltonian has three partsĤeff (t) = ĤStark (t) + Ĥatom-atom (t) + ĤRaman (t),

(5.5)

where
ĤStark (t) =X |Ω1 |2 j+X g2 
1

j

+

X
j

4∆c1

4∆1

|↓ij h↓| − |eij he|






â† â |↓ij h↓| − |eij he| − |eij he|



Ω1 g1
c
†
|↓i
h↓|
−
|ei
he|
ei(∆1 −∆1 )t + H.c.
âjj4h(∆1 , ∆c1 )+ ↓→↑ (1 → 2),
X g2
1|ei h↓| ⊗ |↓ik he| + ↓→↑ (1 → 2)
4∆c1 jj,k6=jX
g1 g2
i(∆c2 −∆c1 )t
−
|↑i
he|
⊗
|ei
h↓|
e
+
H.c.
,
j
k
4h(∆c1 , ∆c2 )(5.6)

Ĥatom-atom (t) = −j,k6=j

(5.7)

82
and
ĤRaman (t) =

X

+

X

+

X

+

X

Ω1 Ω∗2|↑i h↓| ei(∆2 −∆1 )t + H.c.
4h(∆1 , ∆2 ) j

j

j

j

j

g1 g2
c
c
â† â |↑ij h↓| ei(∆2 −∆1 )t + H.c.
4h(∆c1 , ∆c2 )Ω1 g2
câ† |↑ij h↓| ei(∆2 −∆1 )t + H.c.
c4h(∆1 , ∆2 )
Ω2 g1câ† |↓ij h↑| ei(∆1 −∆2 )t + H.c.
c4h(∆2 , ∆1 )(5.8)

In the above expressions, h(a, b) = 2/(a−1 + b−1 ) is the harmonic mean of a and b. All
terms in the effective Hamiltonian conserve the number of excitations in |ei. This means that if
the atoms are initially in the |↓i − |↑i manifold, then the state |ei is negligibly populated and allinteractions involving this level, and consequently, Ĥatom-atom (t), can be dropped. Expressing thedifference detunings in the complex exponentials in terms of ∆, ω0 , and δ shows the presence of
rapidly oscillating terms with frequency ∼ ω0 and slowly varying terms with zero frequency or asmall frequency δ. For Ω1 ≈ Ω2 ∼ Ω, the rapidly oscillating terms can be neglected since we operate
in the regime where Ω2 /∆  ω0 . The resulting Hamiltonian consists ofĤStark (t) =X |Ω1 |2
4∆1

j|↓ij h↓| +

X g2
1

j

4∆c1

â† â |↓ij h↓|+ ↓→↑ (1 → 2)

(5.9)

and
ĤRaman (t) =

X

+

X

j

j

5.2.1

Ω1 g2
â† |↑ij h↓| e2iδt + H.c.
4h(∆1 , ∆c2 )Ω2 g1â† |↓ij h↑| e−2iδt + H.c.
4h(∆2 , ∆c1 )(5.10)

Simple picture

For ∆  ω0 , we can make the substitution ∆1 , ∆2 , ∆c1 , ∆c2 → −∆. Then, with Ω1 = Ω2 = Ω0 ,the Stark shifts from the drive lasers shift the two spin states identically and therefore lead to an
overall energy shift of −N Ω20 /4∆, where we assume Ω0 is real. Similarly, with g1 = g2 = g0 , the83frequency of the cavity mode is shifted by an amount −N g02 /4∆ on account of the atom-cavity
interaction. This can be compensated for by shifting the frequency of the drive lasers by the sameP†
amount. Introducing the collective angular momentum operators Jˆ+ ≡ j |↑ij h↓|, Jˆ− ≡ Jˆ+
, and


P
Jˆz ≡ j |↑ij h↑| − |↓ij h↓| /2, we can express the effective Hamiltonian asΩ0 g0 Ĥeff = 2δ Jˆz +â + â† Jˆ+ + Jˆ− ,4∆(5.11)

where we have let Ω0 → −Ω0 . The second term on the RHS is precisely the QND Hamiltonian,Eq. 5.1. This coarse-grained Hamiltonian is valid over time intervals ∆t  Tω0 ≡ 2π/ω0 , and therefore, we require δ  ω0 and that δ is approximately constant over the interval ∆t. Mathematically,
the latter implies d ln δ/dt  1/∆t  ω0 /2π.
5.2.2

Accounting for ω0 /∆For δ ≈ 0, h(∆1 , ∆c2 ) = ∆1 and h(∆2 , ∆c1 ) = ∆2 . To isolate the balanced cavity-assisted
Raman transitions, three requirements have to be satisfied [32]:(1) Equal drive laser Stark shifts on both spin states: Ω21 /4∆1 = Ω22 /4∆2 .
(2) Equal frequency shift of cavity mode per atom in either spin state: g12 /4∆c1 = g22 /4∆c2 .
(3) Balanced Raman transitions: Ω1 g2 /4∆1 = Ω2 g1 /4∆2 .
We note that arranging Ω1 /Ω2 and g1 /g2 to satisfy (1) and (2) above automatically resultsin satisfying requirement (3).

5.2.3

Note concerning drive laser frequencies

In practice, the frequency arrangement of the drive lasers requires their average frequency
ωav to be tuned well within the cavity linewidth, i.e. |ωc − ωav |  κ. The difference frequency
ω1 − ω2 is relatively easier to stabilize, and deviations from 2ω0 manifest as a growth of the phaseover time that can be measured and statistically modeled.

84

Figure 5.4: Free-space scattering processes considered in our model: Rayleigh scattering (blue),
Raman scattering (red) and atom loss (black).

5.3

Phenomenological free-space scattering model
The drive lasers also lead to undesirable, off-resonant free-space scattering processes withtotal rate γsc that degrade atomic coherence. We consider three such single-atom decoherence
mechanisms: (a) dephasing with probability rd : random rotation about the z-axis, (b) spontaneous
Raman spin flips : |↓i → |↑i (r↓↑ ) and |↑i → |↓i (r↑↓ ), and (c) atom loss (rl ): the atom decays
to a state |si outside the |↓i − |↑i manifold and no longer interacts with the cavity mode. The
probabilities are related by rd + r↓↑ + r↑↓ + rl = 1.
Fig. 5.4 shows the various free-space scattering (FSS) processes considered in our model. The
lifetime of the excited state is Γ = Γ↓ + Γ↑ + Γs .
The dephasing rate is set by the total rate of Rayleigh scattering [110]:
Dephasing: Γ↓

Ω21
Ω22
+
Γ
.
↑
4∆2
4∆2

(5.12)Ω2Ω2

Raman spin flips (|↓i → |↑i) occur with rate Γ↑ 4∆12 , while |↑i → |↓i occur at rate Γ↓ 4∆22[110]. Finally, atom loss occurs at a net rate given by
 2

Ω1
Ω22
Atom loss: Γs+.
4∆2 4∆2(5.13)

With Ω1 ≈ Ω2 ≡ Ω0 , the probabilities for the different FSS channels arerd =Γ↓ + Γ↑Γ↑Γ↓Γs, r↓↑ =, r↑↓ =
, rl =
.
2Γ
2Γ
2Γ
Γ

(5.14)

85
The total FSS rate isγsc = 2Γ

Ω20
.
4∆2

(5.15)

While presenting our numerical results in Chapter 6, we adopt a “symmetric loss” model
wherein the three decoherence mechanisms degrade the atomic coherence at equal rates, and spinflips in either direction occur with equal probability. This implies rd = 1/3, r↓↑ = r↑↓ = 1/6 andrl = 1/3. Such a model corresponds to Γ↓ = Γ↑ = Γs = Γ/3. In this case, from Eq. (5.14), rd = 1/3,r↓↑ = r↑↓ = 1/6 and rl = 1/3.
Our simple phenomenological model captures the FSS processes expected when a threelevel system is driven by two lasers whose difference frequency is far detuned from the two-photon
resonance. In practice, scattering from any additional excited states should be considered. Detailed
modeling of a real experiment will benefit from a rigorous first-principles derivation of the effects
of free-space scattering that also accounts for the cavity mode, as shown in Refs. [13, 115].

5.4

Stochastic master equation
Under continuous measurement, the dynamics of the density matrix ρ of the atom-cavity

system is governed by the stochastic master equation [129, 15, 131]:Pjρ̇ = −i/~[ĤQND , ρ] + κD[â]ρ + γsc N
j=1 L1 ρ√√+ ηκξ(t) iρâ† − iâρ − 2hŶ iρ ,(5.16)

with decoherence effects bundled in Lj1 ρ, given byrdjj
Lj1 ρ = r↓↑ D[σ̂+]ρ + r↑↓ D[σ̂−]ρ + D[σ̂zj ]ρ
4
i
h
i 
rl  h+D |sij h↓|j ρ + D |sij h↑|j ρ ,2(5.17)

with D[Ô]ρ = ÔρÔ† − Ô† Ôρ/2 − ρÔ† Ô/2, the Lindblad dissipator. In Eq. (5.16), η is the detection
efficiency, and ξ(t) is a white-noise process satisfying ξ(t) = 0 and ξ(t)ξ(t0 ) = δ(t−t0 ). The measured
photocurrent i(t) is √
√i(t) = Ge|αLO | η 2κhŶ i + ηξ(t) ,(5.18)

86
with detector gain G, electronic charge e, and local oscillator photon flux |αLO |2 with units ofphotons/time.

5.5

Gaussian state approximation
With no decoherence, measuring for very long times will result in preparing states arbitrarilyclose to Dicke states in the Jx basis. However, decoherence restricts the maximum achievablesqueezing well before the state begins to wrap around the Bloch sphere. This enables a Gaussian
approximation where we only track the dynamics of the means and covariances of all operators andpairs of operators of the atom-cavity system. The 5 operators X̂, Ŷ , Jˆx , Jˆy and Jˆz result in a totalof 20 dynamical equations.
To derive these equations, we first rewrite Eq. (5.16) asdρ = −i/~[ĤQND , ρ] + κD[â]ρ + γsc
+

√

N
X
j=1


Lj1 ρ dt√ηκdW (t) iρâ − iâρ − 2hŶ iρ ,
†(5.19)

where dW (t) is a Wiener increment that satisfies dW (t) = 0 and dW (t)2 = dt [120].
h i
˙The expectation value of an operator Ô is given by hÔi = Tr Ôρ and, consequently, hÔi =h i
Tr Ôρ̇ . We keep track of only the means and covariances of the five operators X̂, Ŷ , Jˆx , Jˆy andJˆz . We truncate these evolution equations at second order by factorizing third order moments ofthe type hÔ1 Ô2 Ô3 i ashÔ1 Ô2 Ô3 i ≈ hÔ1 Ô2 ihÔ3 i + hÔ2 Ô3 ihÔ1 i + hÔ1 Ô3 ihÔ2 i − 2hÔ1 ihÔ2 ihÔ3 i.

(5.20)

This procedure leads to five equations governing the means of the operators that have thetypical form of stochastic differential equations: dhÔi = adt + bdW . The time evolution of the
fifteen covariances, on the other hand, are governed by ordinary differential equations that have noterms proportional to dW . This structure is a direct consequence of the Gaussian approximation
we employ. We reproduce these equations below.

87
We use the notation h. . .i to denote means and h. . .ic to denote (co)variances evaluated using
the stochastic master equation, Eq. (5.19). The covariance is defined as hÔ1 Ô2 ic = (hÔ1 Ô2 i +hÔ2 Ô1 i)/2 − hÔ1 ihÔ2 i.
5.5.1

Operator means

The operator means obey the following set of equations.
dhX̂i =
dhŶ i =dhJˆx i =dhJˆy i =dhJˆz i =

5.5.2

r
κ
ηκ
− hX̂idt + 2
hX̂ Ŷ ic dW
2
2
r 



ΩQND ˆ
ηκκ−hŶ i +
hJx i dt +2hŶ 2 ic − 1 dW
2
2
2
r
γsc
ηκ
− hJˆx idt + 2
hŶ Jˆx ic dW
2
2
r


ΩQND 
γsc ˆ
ηκ
ˆ
ˆ
−
hJy i +hX̂ Jz ic + hX̂ihJz i dt + 2
hŶ Jˆy ic dW
2
2
2
 


ΩQND 
rl  ˆ
Nˆˆ
− γscr↑↓ + r↓↑ +hJz i + (r↑↓ − r↓↑ )−hX̂ Jy ic + hX̂ihJy i dt
2
2
2
r
ηκ
+2
hŶ Jˆz ic dW(5.21)
2

Operator covariances

Operator covariance are governed by the following sets of equations.


d
1
2
2
hX̂ ic = −κ hX̂ ic −− 2ηκhX̂ Ŷ i2c
dt
2


2
d 2
1
ηκ 
2
hŶ ic = −κ hŶ ic −− ΩQND hŶ Jˆx ic −2hŶ 2 ic − 1
dt
2
2


ΩQND
d
hX̂ Ŷ ic = −κhX̂ Ŷ ic −hX̂ Jˆx ic − ηκhX̂ Ŷ ic 2hŶ 2 ic − 1
dt2(5.22)

dκ + γscˆhX̂ Jx ic = −hX̂ Jˆx ic − 2ηκhŶ Jˆx ic hX̂ Ŷ ic
dt
2



ΩQND dκ + γschX̂ Jˆy ic = −hX̂ Jˆy ic −hX̂ Jˆz ic hX̂i + hX̂ 2 ic hJˆz i − 2ηκhŶ Jˆy ic hX̂ Ŷ ic
dt
2
2



ΩQND 
dκrl hX̂ Jˆz ic = −+ γsc r↑↓ + r↓↑ +hX̂ Jˆz ic +hX̂ Jˆy ic hX̂i + hX̂ 2 ic hJˆy idt222−2ηκhŶ Jˆz ic hX̂ Ŷ ic(5.23)

88






ΩQND ˆ2dκ + γschŶ Jˆx ic −hŶ Jˆx ic = −hJx ic − ηκhŶ Jˆx ic 2hŶ 2 ic − 1
dt
2
2



ΩQND  ˆ ˆdκ + γschŶ Jˆy ic −hŶ Jˆy ic = −hJx Jy ic + hX̂ Ŷ ic hJˆz i + hŶ Jˆz ic hX̂i
dt
2
2


−ηκhŶ Jˆy ic 2hŶ 2 ic − 1
κ

d
rl 
hŶ Jˆz ic = −+ γsc r↑↓ + r↓↑ +hŶ Jˆz ic
dt
2
2



ΩQND  ˆ ˆ−
hJz Jx ic − hX̂ Ŷ ic hJˆy i − hŶ Jˆy ic hX̂i − ηκhŶ Jˆz ic 2hŶ 2 ic − 1 (5.24)
2



d ˆ2
rl  N2
ˆhJ ic = −γsc hJx ic − r↑↓ + r↓↑ + rd +− 2ηκhŶ Jˆx i2c
dt x
2 4




N
d ˆ2
r
l
2
− ΩQND hX̂ Jˆy ic hJˆz i + hJˆy Jˆz ic hX̂i
hJy ic = −γsc hJˆy ic − r↑↓ + r↓↑ + rd +
dt
2 4

d ˆ2
hJ ic
dt z

−2ηκhŶ Jˆy i2c






N
N
N
2
ˆ
ˆ
ˆ
= −γsc (2r↑↓ + 2r↓↑ + rl ) hJz ic − r↑↓+ hJz i − r↓↑− hJz i − rl
2
2
8


2
+ΩQND hX̂ Jˆz ic hJˆy i + hJˆy Jˆz ic hX̂i − 2ηκhŶ Jˆz i(5.25)cΩQND d ˆ ˆ
hJx Jy ic = −γsc hJˆx Jˆy ic −hX̂ Jˆx ic hJˆz i + hJˆz Jˆx ic hX̂i − 2ηκhŶ Jˆx ic hŶ Jˆy ic
dt
2





r
−
r
3
3
rd
d ˆ ˆ
↑↓
↓↑
hJy Jz ic = −γscr↑↓ + r↓↑ ++ rl hJˆy Jˆz ic −hJˆy idt
2222ΩQND 
+hX̂ Jˆy ic hJˆy i + hJˆy2 ic hX̂i − hX̂ Jˆz ic hJˆz i − hJˆz2 ic hX̂i − 2ηκhŶ Jˆy ic hŶ Jˆz ic
2 




r
−
r
d ˆ ˆ
3
rd
3
↑↓
↓↑
hJz Jx ic = −γscr↑↓ + r↓↑ ++ rl hJˆz Jˆx ic −hJˆx idt
2222ΩQND 
+hX̂ Jˆx ic hJˆy i + hJˆx Jˆy ic hX̂i − 2ηκhŶ Jˆz ic hŶ Jˆx ic(5.26)
2

5.5.3

Some comments on the equations of motion

The above equations describe the conditional evolution when no external phase modulation isapplied. Such a modulation can be straightforwardly accounted for by including the contributionsof an additional Hamiltonian term ∝ Jˆz , such as the first term on the RHS of Eq. (5.11), to theequations of motion of the means and covariances.
In deriving the above equations, we have accounted for the contribution of the atom loss
terms (proportional to rl ) to the loss of coherence and the increase in diffusion. We note that atom

89
loss reduces the effective number of atoms Neff that interact with the cavity mode. However, we
simply use the total number N wherever Neff explicitly appears in these equations. That is, we
account for Ṅeff = −rl γsc Neff that leads to increased diffusion of the atomic spin components, but
approximate Neff ≈ N everywhere in the above equations.
5.5.4

Numerical evolution

We numerically evolve the dynamical equations for the means and covariances using a variation of the Improved Euler scheme for integrating stochastic differential equations [98]. Our
numerical results are obtained using a C++ program that employs linear algebra features provided
by Eigen [44], a C++ template library. The Wiener increments and random numbers required
for numerical integration are obtained using random number generators from the GNU ScientificLibrary [38]. Numerically, the infinite bandwidth of i(t) (Eq. (5.18)) means that the data samplingtime step must equal the simulation time step.

Chapter 6

Real-time phase tracking: Numerical experiments

In this chapter, we will use the Gaussian equations that we derived in the previous chapter
to perform numerical simulations that demonstrate real-time phase tracking below the standard
quantum limit. We will also discuss connections with the Ramsey sequence and derive an analytic
expression for the optimum metrological gain achievable when our scheme is used as a Ramsey-likesequence. We conclude with a discussion on the advantages of combining our proposal with afeedback scheme.

6.1

Estimating the phase
To estimate the phase, we average the simulated photocurrent (Eq. (5.18)) in a time window[Ti , Tf ] to obtain an estimate as
Jx(m) = −

κ
ΩQND

Y

(m)

−(Ge|αLO |)−1
= √
η Cγsc (Tf − Ti )Z Tfi(t)dt,(6.1)

Ti

where C = 2Ω2QND /κγsc is the dimensionless atom-cavity cooperativity. The phase is estimated(m)as φ(m) = (Jx /(N/2))/V(t), where the visibility V(t) ≈ e−γsc t/2 e−(Cγsc /4)t/2 accounts for the
shortening of the Bloch vector, evaluated either at the window center or end depending on where
the phase is estimated. For C  1, free-space scattering is the main reason for shortening of the
Bloch vector, whereas for C  1, measurement-induced squeezing is the dominant contribution.
While we use a simple averaging procedure for clarity, optimal filters, such as Kalman filters, canbe applied for superior phase tracking [40, 117, 14].

91
The precision of the phase estimate in a window is determined by the window duration.
A characteristic time, T0 = (ηCγsc )−1 /(N/4), is the time required to average down the photon(m)shot-noise (ξ(t) term, Eq. (5.18)) in estimating Jx(Eq. (6.1)) to the standard quantum limit

2∆Jx,SQL= N/4.

6.1.1

Note concerning the cooperativity

We have defined the cooperativity C in terms of the two-photon rates ΩQND and γsc . With√
the explicit form of γsc in Eq. (5.15), and the expression ΩQND = 2Ω0 g0 /∆, we can express C interms of the single-photon coupling strength g0 and the lifetime Γ of the state |ei as
C=2

Ω2QND
κγsc

=8

g02
.
κΓ

(6.2)

We note that our definition of C differs from the usual definition [81] by a factor of 8.
However, as Eq. (6.4) later shows, with our definition of C, the rate ηCγsc takes on the simple
interpretation as the rate at which the variance in Jx decreases as the field leaking out from thecavity is monitored. Here, η is as usual the detection efficiency of the photodetectors used for
homodyne detection. With perfect detectors (η = 1), C is therefore the ratio of the rates of thedesirable and undesirable processes.

6.2

Parameters for numerical experiments
For our numerical experiments, we use N = 105 atoms identically coupled to a cavity modewith C = 0.1, unless specified otherwise. We work in a bad-cavity regime such that N Cγsc = 0.2κ,achievable by arranging for ΩQND = 10−3 κ. We adopt the “symmetric loss” model for free spacescattering that is described in Section 5.3. However, our results are not very sensitive to the specificchoice of relative rates. The loss in visibility only depends on the total decoherence rate, while themeasurement of Jx marginally improves if the atom loss channel is dominant (see Eq. (6.11)).
Finally, the detection efficiency is assumed to be η = 0.4 [26].

92

6.3

Real-time phase tracking: A demonstration
We now demonstrate the ability of our scheme to track in real-time, a phase modulation φ(a) (t)applied for t > 0 (Fig. 6.1(a, black solid line)). At time t = −50T0 , the collective spin is initialized
to a coherent spin state (CSS) along the y-axis whose initial phase is φ0 = 0. First, measuring the(m)photocurrent in the state preparation window [−50T0 , 0] gives a phase estimate φ0(blue triangle).

This estimate is obtained at the end of this window using the procedure described below Eq. (6.1).
(m)

The value of φ0varies from trial-to-trial with a variance ∆φ2SQL = 1/N corresponding to thephase uncertainty of the initial CSS. The long state preparation window ensures strong averaging(m)down of the photon shot-noise, leading to a state with reduced phase uncertainty around φ0 , i.e.
a spin squeezed state (SSS). For the subsequent real-time tracking, two choices for the initial phase(m)reference could be used: φ0 (= 0) or φ0 .
During the time [0, 200T0 ], we average the photocurrent in windows of duration 8T0 to extract
a raw phase estimate φ(m) (j) for window j = 1, 2, . . .. We construct two estimates for the phase at
(m)

(m)

(m)

the window centers, φCSS (j) = φ(m) (j)−φ0 (hollow red squares), and φSSS (j) = φ(m) (j)−φ0(filledblue squares). The precision of these estimates is determined not just by the window duration overwhich the raw estimate is obtained, but also by the precision of the phase reference. To determine
the single-run precision of these estimates, we run 2048 trials of the experiment and histogram theerror in these estimates, an example of which is shown in Fig. 6.1(b) for the window [48T0 , 56T0 ].
(m)

The estimates φCSS use the imprecise zero phase φ0 of the initial CSS as reference, and lead to(m)a broad error histogram (red). In contrast, the estimates φSSS lead to a narrow error histogram(blue) whose spread is instead dominated by the imprecision in obtaining the raw estimates φ(m) (j)(m)

over short windows (here, 8T0 ), demonstrating the improved precision of the phase reference φ0(m)over φ0 1 . In Fig. 6.1(c) we show that the variance ∆φ2SSS of the estimates φSSS is significantly
less than ∆φ2SQL in all windows over the time we consider here, demonstrating the potential for
real-time phase tracking with precision beyond the SQL.
1
If only the form, and not the precise value, of an AC signal is to be tracked in a single run, the choice of initial
phase reference is irrelevant, making the state preparation step unnecessary.

93

Figure 6.1: Real-time continuous tracking of a time-varying phase. (a) A single experimental run:(m)
A squeezed state is prepared during [−50T0 , 0], with the initial measured phase φ0 (blue triangle)
varying in each run. Subsequently, a phase modulation φ(a) (t) = 15 mrad × sin(t/40T0 ) (blackline) is applied e.g. using a time-varying magnetic field. The blue, filled (red, hollow) markers are(m)(m)
estimates φSSS (φCSS ) of the phase using the measured photocurrent in windows of duration 8T0(m)that account for (do not account for) the initial offset φ0 . The gray shaded region indicates the1-σ SQL tolerance for this applied signal. Representative Bloch spheres for t ≤ 0 indicate the state
before and after the state preparation stage. For t > 0, Bloch spheres indicate the deflection of
the spin as a result of the phase modulation (black dots on the spheres indicate the zero phase(m)(m)
reference), as well as the equivalent spin state used for the respective estimates φCSS , φSSS . (b)(m)(m)
Histogram of phase errors φSSS − φ(a) (blue) and φCSS − φ(a) (red) over 2048 runs in one particular(m)measurement window [48T0 , 56T0 ]. (c) Single-run precision gain of the estimates φSSS relative to
the SQL at different window centers t. Here, ∆φ2SSS is the variance of Gaussian fits to histograms
such as the blue histogram in (b). Decoherence results in decreased gain over time.

94

6.4

Use as a Ramsey sequenceFirst, we note an advantage of our scheme, namely that the same photocurrent data from asingle run can be analyzed using multiple methods to extract varying information. As a demonstration, we use varying window durations TW to extract precise timing and amplitude informationfrom a sudden jump in phase (at TJ = 50T0 in Fig. 6.2(a)). Starting with an initial CSS at t = 0, we
continuously estimate the phase by averaging the photocurrent over moving windows of durations
TW = 2T0 (red) and TW = 20T0 (blue). Clearly, the shorter window reproduces the time variationof the phase more precisely. To estimate the amplitude of the jump φJ , we compute the difference
(m)

φJ

(m)

(m)

in the estimates φW1 , φW2 in the two windows W1 ≡ [TJ − TW , TJ ] and W2 ≡ [TJ , TJ + TW ]that border the jump time TJ (Fig. 6.2(b))2 . While the shorter window results in faster response,
the longer window gives a more precise estimate of the jump amplitude (Fig. 6.2(c)).
Alternatively, the sudden phase jump in the protocol depicted in Fig. 6.2(b) can be replacedwith a “dark” phase accumulation time of duration TD where no measurements are performed. The
scheme can then be identified as a Ramsey-like sequence where a squeezed state is prepared in W1 ,phase accumulates in an interrogation time TD , and finally, phase is read out in W2 , without everconverting the phase information into a population difference. In this Ramsey mode, the achievablegain in phase resolution using the prepared squeezed state compared to a CSS is∆φ2SQL
∆φ2
(m)

(m)

=

2
∆Jx,SQL
2
 V ,
(m) 2
∆Jx,diff

(6.3)

(m)

where Jx,diff = Jx,W2 −Jx,W1 and V is the visibility at the end of the first window [11, 26]. Fig. 6.2(d)plots the numerically extracted gain (markers) versus the window duration TW for different values
(m)

(m)

of cooperativity C. Gaussian fits to histograms of Jx,diff were used to extract values for (∆Jx,diff )2 .
2

For simplicity, we assume that the jump time of our test signal is known in advance. In practice, the jump timeand amplitude can be simultaneously estimated using tools from optimal filtering theory.

95

Figure 6.2: (a) A sudden jump in the phase with amplitude φJ = 40 mrad at TJ = 50T0 is tracked
in the same run using moving windows of durations TW = 2T0 (red) and TW = 20T0 (blue), showing
the faster response of the shorter window. (b) Protocol to estimate φJ . (c) Histograms, over 2048
(m)runs, of φJ for TW = 2T0 (red) and TW = 20T0 (blue), demonstrating the greater precision of
the longer window. For TW = 2T0 , W2 was offset by a small time 0.2T0 to allow transients ontimescales of κ−1 to decay. (d) Gain in precision over a CSS in Ramsey mode as the duration ofW1 and W2 is varied, for fixed Cγsc and different values of C. Analytic results (lines) calculatedusing Eqs. (6.3) and (6.11) are in excellent agreement with simulations (markers).

96

6.5

Analytic expression for variance in the difference measurement
To understand the numerically observed trend, we work with the equations derived in Sec(m)tion 5.5 to obtain an analytic expression for (∆Jx,diff )2 . The time evolution of the conditionalvariance hJˆx2 ic satisfies the Riccati equation


d ˆ2
N
2
− ηCγsc hJˆx2 i2c ,hJ ic = −γsc hJˆx ic − β
dt x
4

(6.4)

where we have used the bad-cavity limit to adiabatically eliminate hŶ Jˆx ic and substitute it with√
the quantity −(ΩQND /κ)hJˆx2 ic in the equation for hJˆx2 ic . In the regime where N C  1 and√βηN C  1, the solution to Eq. (6.4) simplifies to
s
s
"s
#
4β
βN
C
4β
N
hJˆx2 (t)ic =coth
γsc t +.
(6.5)
4 N Cη4η
N Cη
The conditional mean hJˆx i is given by
Z t
p
γsc
0
dt0 hJˆx2 (t0 )ic e− 2 (t−t ) ξ(t0 ).
hJˆx (t)i = − ηCγsc

(6.6)

0

From Eq. (5) and the bad-cavity relation hŶ i = −(ΩQND /κ)hJˆx i, the instantaneous photocurrent i(t) carries information about hJˆx i but is corrupted by photon shot noise. Using Eq. (6), the(m)estimate Jx

from the photocurrent measured in an interval [Ti , Tf ] is related to the conditionalmean hJˆx i as
Jx(m) =

1
Tf − Ti

Z Tf
Ti



1
ˆ
dt hJx (t)i − √ξ(t) ,
ηCγsc

(6.7)

(m)
where hJˆx i satisfies Eq. (6.6). In deriving Eq. (8), we perform the first measurement Jx,1 (TW )(m)over the window [0, TW ], and the second measurement Jx,2 (TW ) over the window [TW , 2TW ]. The(m)variance in the difference measurement (∆Jx,diff )2 (TW ) is given by

2
(m)
(m)
(m)
(∆Jx,diff )2 (TW ) = Jx,2 (TW ) − Jx,1 (TW ) ,(6.8)

where the overbar indicates averaging over all possible realizations of the noise ξ(t), which has theproperties ξ(t) = 0 and ξ(t)ξ(t0 ) = δ(t − t0 ). We use the approximation that γsc TW  1 for the

97measurement windows we consider, so that the exponential factor in Eq. (6.6) can be set to unity.
The resulting integrals can be evaluated analytically, resulting in


T0
4β TW
1+
+
4
TW
3ηN C T0N
(m)(∆Jx,1 )2 (TW ) =(6.9)

for the variance in the first measurement, andT016β TW
1++
4TW
3ηN C T0N
(m)(∆Jx,2 )2 (TW ) =(6.10)

for the variance in the second measurement. For the variance in the difference measurement, weget(m)(∆Jx,diff )2 (TW )2∆Jx,SQL

=2

8β TW
T0
+
,
TW
3ηN C T0

(6.11)

popt=
where β = rd + r↓↑ + r↑↓ + rl /2, giving a minimum normalized variance of 8 β/3ηN C at TWp
T0 3ηN C/4β. The expression for β shows that the normalized variance is not very sensitive to therelative probabilities of the decoherence mechanisms. For typical values of C ∼ 0.1 and N ∼ 105 ,Fig. 6.2(d) shows that a gain upwards of 11 dB can be achieved. The (N C)−1/2 scaling of the(m)minimum normalized variance in Jx,diff leads to an optimal phase resolution scaling as ∆φ ∼ N −3/4
compared to ∆φSQL = N −1/2 radians.

6.5.1

Physical explanation for optimum measurement window

Although the photon shot noise is averaged down as the measurement window TW is increased,
undetected photons emitted via free-space scattering (FSS) lead to increased ignorance about theactual state of the collective spin. The photocurrent measurements in the initial parts of themeasurement window are no longer as reliable in estimating the current value of Jx as those in(m)

the latter parts, since FSS has significantly affected the collective spin state. Since Jx,diff is thedifference of measurements in two such windows, for very large TW , the correlation in these two
measurements decreases as a result of FSS. The upshot: The window has an optimum duration
below which the measurement suffers from photon shot-noise, and above which it is affected by
FSS.

98
6.5.2

Absence of noticeable Zeno effectA natural question concerns the quantum Zeno effect: Does the real-time phase measurement
inhibit the evolution of the phase? For the parameters used in the simulations, we do not observe any
noticeable Zeno effect. Here, we describe a theoretical situation where it can likely be observed. In
the absence of free-space scattering, Eq. (6.9) implies that measuring for very long times suppresses
the photon shot noise arbitrarily, and that the variance of the measurement is eventually limited
only by the spread of the initial CSS over the eigenstates of Jˆx . Although our Gaussian theory
cannot describe this extreme limit, we can use Eq. (6.9) to qualitatively estimate the time required
to resolve such eigenstates by assuming that the photon shot noise contribution N T0 /TW ∼ 1,
leading to TW ∼ N T0 . The Zeno effect can be observed if the phase evolves by 1/N radianson a timescale & N T0 , translating to a phase modulation rate ∼ 1/(N 2 T0 ) ∼ Cγsc /N . In thesimulations, free-space scattering takes over well before the Jˆx eigenstates can be resolved and weonly prepare squeezed states with variance a factor of O(1 − 10) below the SQL. Furthermore, thepeak modulation rate, for example, in Fig. 6.1, is ∼ 107 times greater than 1/(N 2 T0 ). These factors
probably suggest why the Zeno effect is not noticeable in our simulations.

6.6

Advantage of a feedback loop
While Ramsey sequences only measure phase changes unambiguously in the interval [−π/2,π/2], our scheme readily extends to tracking large excursions |φ(t)|  π: The measured current i(t)can be used in a feedback loop [128, 2, 130, 127, 120] to adjust the differential phase offset ψ1 −ψ2 ofthe drive lasers such that i(t) is continuously driven back to zero. The feedback loop continuously
adjusts the spin component probed by the cavity mode such that it is always perpendicular to themean spin direction, while mapping the phase φ(t) onto the feedback signal as φ(t) = (ψ1 − ψ2 )/2.
This way, large phase excursions can be tracked while remaining in the small angle measurement
limit. By encoding the spins in hyperfine levels that have an intrinsic splitting, our scheme has
the unique capability to greatly increase the unambiguous interval of phase evolution that can be

99
0
1%
5%
10%
15%
(a)
φ (t)

6

4

15
10
5
0

φ(t) (mrad)∆φ2SQL /∆φ2SSS (dB)

8

−5

2

−10
0
0

50

100
time t/T0

150

−15
200

Figure 6.3: Effect of number fluctuations on real-time phase tracking. Single-run precision gain
computed using error histograms of 2048 experimental runs (compare to Fig. 6.1(c)). Parameters
are from Fig. 6.1, except that the number of atoms in each run is variable, with a mean of N̄ = 105
and ∆N/N̄ indicated by the percentages. Data points (markers) extracted from the numericalexperiments are in very good agreement with semi-analytic results (solid lines) obtained using thesimple expression in Eq. (6.13) that accounts for number fluctuations. The black dashed line plots
the applied phase modulation, for reference.

continuously tracked, for example in atomic clocks. While feedback schemes using intermittent
non-demolition population measurements have been used to extend this interval in a Ramsey-likesequence [63], our scheme continuously tracks the phase and removes the need for state rotations
altogether. Another benefit of feedback is that it greatly suppresses sensitivity to variations or
uncertainties in scale factors relating i(t) to φ(t), including uncertainties in atom number, as wedemonstrate below.

6.6.1

Atom number fluctuations

Here, we show why the number fluctuations are not important in a practical realization of
our proposal that implements feedback.
(m)

In the small angle limit, the measured phase is linearly related to Jx

as

(m)

φ(m) =

2Jx,
VN(6.12)

where V is the visibility and N is the number of atoms. So far, we have assumed fluctuations only

100
(m)

in Jx . In addition, when N is not precisely known, the error in φ(m) is given by
(∆φ

(m) 2

) ≈



2
VN

2

(∆Jx(m) )2 + (φ(m) )2



∆N
N

2
.

(6.13)The expression in Eq. (6.13) is approximate because we neglect the covariance of fluctuations
in Jx and N . The error from ∆N is negligible compared to the error from ∆Jx when
!
(m) 2
4
∆J
x
(φ(m) )2  2.
V∆N(6.14)

In other words, the presence of number fluctuations only sets an upper bound on the dynamical range of the phase that can be tracked, and does not impose a fundamental restriction.
In order to validate this argument, we introduce number fluctuations in our numerical experiments for the situation depicted in Fig. 6.1, see Fig. 6.3. When ∆N = 0, we recover the results ofFig. 6.1(c). As we increase ∆N , the precision gain sharply falls in regions where the tracked phasehas large amplitude. To verify the simple formula, Eq. (6.13), we take the numerically obtained(m)variance in the absence of number fluctuations (black markers) as representing the (∆Jx )2 termin Eq. (6.13) and add, by hand, the contribution of the (∆N )2 term, to obtain the solid lines shownin Fig. 6.3. The values of φ(m) that enter this expression are approximated to be the values of theapplied phase modulation (black dashed line) at the window centers. The very good agreement
between the numerically extracted (markers) and semi-analytic (solid lines) results validate theexpression, Eq. (6.13), for the error in φ.
Figure 6.3 and Eq. (6.13) imply that, in a practical realization, the effect of number fluctuations can be suppressed if the measured phase is always maintained close to zero. Feedback
enables continuous tracking of large phase excursions in such a small angle measurement limit, andtherefore greatly suppresses sensitivity to atom number fluctuations. In fact, by the same argument, a feedback loop will also suppress variations or uncertainties in any scale factors relating themeasured current to the phase φ(t), and not just the atom number.

Chapter 7

Squeezing on momentum pseudospins: Formalism

Current Bragg interferometers operate in free space with state-of-the-art technology enabling
control of large numbers of atoms [21, 55, 54, 20]. This technical progress has now achieved highsignal-to-noise ratios for determining the relative phase shift. However, despite the use of large
atom numbers, their operation can be completely described in terms of single-atom physics since
the atoms are uncorrelated. As a result, regardless of whether further technical improvements
are realized, the phase sensitivity of these interferometers in the near future will be fundamentally√
constrained by the SQL of ∆φSQL = 1/ N radians, where N is the number of atoms. Monotonicallyincreasing N to improve precision suffers from problems such as practical limitations in trappingand cooling [122, 18], and uncontrolled phase changes that arise from atomic collisions [99]. Schemes
to produce squeezed states of momentum pseudospins are therefore attractive as a means to achieveprecision beyond the corresponding SQL for a given N . A major hurdle to producing such states
in Bragg interferometers is that squeezing requires a channel for the atoms to controllably interact
with each other, and such a channel is unavailable in current interferometer designs.
The recent demonstration of a hyperfine-changing Raman gravimeter operating inside an
optical cavity [47] motivates us to envisage a similar operation of Bragg interferometers in cavities in
the near future. The availability of a cavity mode naturally opens up a channel for mediating atomatom interactions. Previous proposals for cavity-based squeezing on momentum spins [103] requiresignificant experimental overhead dedicated to achieving squeezing while the actual interferometer
itself operates in free space. Here, we propose an alternative approach that marries the generation of

102
cavity-mediated spin squeezing [76, 53, 88] with the well known advantages of operating the entire
interferometer inside a cavity [47]. Importantly, our scheme does not require any experimental
overhead to generate interactions beyond what is already needed to run a Bragg interferometer ina cavity. In fact, we show how all-to-all atomic interactions are generated by simply switching off
one of the two Bragg lasers and suitably adjusting the frequency and power of the other.
The use of momentum pseudospins in Bragg interferometers necessitates two unique considerations. First, the atomic cloud will always have a non-zero momentum width even after velocity
selection. This width can typically be neglected in the analysis with uncorrelated atoms. Second,
momentum pseudospins cannot be considered as closed two level systems since the same pair ofcounterpropagating electromagnetic fields couples the pseudospin states to other momentum states,
albeit with varying detunings. As a result, leakage to undesirable momentum states is unavoidable
even while applying efficient Bragg pulses for spin rotations, and also when attempting to engineer interactions for spin squeezing. In our work, we account for the momentum width as well asleakage to undesirable states and show that they can be important when considering the efficiency
of a spin squeezing protocol applied to momentum pseudospins. Nevertheless, as we demonstrate,
appreciable spin squeezing can still be achieved under suitable and potentially realizable operating
conditions.
In the process of accounting for the effects of momentum width and losses to undesirable
states, we show how to extend modeling techniques originally developed for spin systems to interacting atoms in matter-wave interferometers where information is encoded in external degrees of
freedom. This ability to map the continuous momentum variable onto a discrete quantum pseudospace allows us to directly employ methods developed for finite dimensional systems [106, 92, 70]. The techniques we use to study our system are widely applicable for investigations of beyond
mean-field physics in a broad range of schemes involving interacting atoms whose momentum states
are coupled by electromagnetic fields.
In this chapter, we will introduce a scheme to generate squeezing on momentum pseudospins
and also develop the necessary framework to study the dynamics of this system. After describing

103
the system under consideration, we will first provide an intuitive explanation for why one can expect
squeezing in this system. Next, we derive a master equation describing the atom-cavity interactions.
Further, we also adiabatically eliminate the cavity mode to arrive at a second effective masterequation for the atoms only. We also describe approximate numerical methods for each of the twomaster equations that enable us to obtain complementary insights into the squeezing dynamics.

7.1

Setup
We consider a collection of N atoms with mass M in a ring cavity with resonance frequency

ωc as shown in Fig. 7.1(a). Each atom consists of two electronic levels |gi and |ei with transition
frequency ωa . A laser with frequency ωl drives one mode of the ring cavity. The cavity resonanceis red detuned from the atomic transition by a detuning ∆c = ωa − ωc > 0, while the drive
laser is detuned by ∆l = ωa − ωl > 0. The relative detuning of the laser and the cavity is∆cl = ωl −ωc  ∆c , ∆l . Upon absorption or emission of a photon with wavevector k, the momentumof an atom is shifted by ~k, where k = |k|.

Figure 7.1: Experiment setup and working principle. (a) A cloud of atoms interacts with twocounterpropagating modes of a ring cavity. One mode (mode 1) is driven at frequency ωl , while
the counterpropagating mode (mode 2) is in vacuum, i.e. not pumped. The scheme enables cavitymediated interactions between every pair of atoms. (b) The excitation or de-excitation of a single
atom is off-resonant. However, the exchange of excitation between two atoms is a resonant process.

104

7.2

Basic working principle
The underlying principle for how squeezing is generated in our scheme is summarized inFig. 7.1(b) for the case when the pseudospin is encoded in the states |↓i ≡ |g, 0i and |↑i ≡ |g, 2~kiwith transition frequency 4ωr , where ωr = ~k 2 /2M is the atomic recoil frequency. We denote thedriven mode as mode 1 and the counterpropagating mode as mode 2. The drive laser frequency isarranged such that ∆cl = ωl − ωc = 4ωr + δ, where δ is a two-photon detuning typically assumed
to be > 0 in this work. The excitation of an atom from |↓i to |↑i (green circle in Fig. 7.1(b)) isfacilitated by the absorption of a drive photon and subsequent emission into mode 2. The energy
imbalance between the photon exchange and the spin excitation is∆E↓→↑ = (~ωl − ~ωc ) − 4~ωr = ~δ.

(7.1)

Similarly, the de-excitation of a second spin (magenta circle in Fig. 7.1(b)) is accompanied by theabsorption of a photon in mode 2 and subsequent emission at the drive frequency, leading to anenergy imbalance∆E↑→↓ = (~ωc − ~ωl ) − (−4~ωr ) = −~δ.

(7.2)

However, from Eqs (7.1) and (7.2), the simultaneous excitation of one atom and de-excitation of theother is resonant, facilitated by the four-photon process consisting of absorption of a drive photon,emission and absorption of a virtual cavity photon and subsequent return of the photon to thedrive laser. Assuming that the cavity mode couples identically to all the atoms, the cavity modecannot distinguish which atom was excited and which one was de-excited, leading to an effective
Hamiltonian of the formĤeff ∝ Jˆ− Jˆ+ ,where Jˆ± =

(7.3)

 †
±
+
−
σ̂
,
with
σ̂
=
|↑i
h↓|
and
σ̂
=
σ̂j+ . The effective Hamiltonian, Eq. (7.3),
j=1 j
j
j
j

PN

can be expressed as
Jˆ− Jˆ+ = Ĵ · Ĵ − Jˆz Jˆz − Jˆz ,

(7.4)

105
where Ĵ = Jˆx x̂ + Jˆy ŷ + Jˆz ẑ and Jˆi for i = x, y, z are the Cartesian components of the collective
angular momentum Ĵ formed by the momentum pseudospins.1

The second term is the familiar

one-axis twisting (OAT) interaction that gives rise to spin squeezed states useful for quantum
metrology [76]. The first term, on the other hand, opens a many-body energy gap that has been
experimentally observed, for example, using spins encoded in optical clock transitions [88]. We
briefly discuss how the latter effect manifests in our system in Section 8.7.

7.3

Atom-cavity interactions
We now proceed to derive a master equation that reflects the underlying atom-cavity inter-action at the heart of the resonant spin exchange intuitively described in the previous section. The
Hamiltonian governing the dynamics of the atom-cavity system is
!
N
2
X
X
p̂2j
~ωa
Ĥ =
~ωc â†s âs+(|eij he| − |gij hg|) +
2M
2
s=1

j=1

N X
2
X


~g  iks ẑj|eij hg| + â†s e−iks ẑj |gij he|âs e2
j=1 s=1
√+ ~ κ αe−iωl t â†1 + α∗ eiωl t â1 ,
+

(7.5)

where g is the atom-cavity vacuum Rabi frequency, κ is the cavity decay rate and α is the amplitudeof the drive laser with |α|2 the photon flux in units of photons/time. The operators â†s , âs for s = 1, 2
respectively describe the creation and annihilation of photons in modes 1 and 2 whose wavevectors
satisfy k1 = −k2 = kẑ (see Fig. (7.1)(a)). The operator ẑj represents the position of atom j along
the cavity axis. The decay of the cavity fields is accounted for by the standard Lindblad dissipatorof the type D[Ô]ρ = ÔρÔ† − Ô† Ôρ/2 − ρÔ† Ô/2 for jump operator Ô and density matrix ρ. The
resulting master equation is
2
i X
1 h
ρ̇ =
Ĥ, ρ +
κD[âs ]ρ.
i~

(7.6)

s=1

We neglect free-space scattering in our analysis since superradiant decay (Section (8.4)) is typicallythe dominant dissipation mechanism (see Appendix B.3 and discussion in Section 8.8).
1

The symbols x̂, ŷ and ẑ refer to unit vectors along the coordinate axes and are not to be confused with position
operators.

106
7.3.1

Adiabatic elimination of the excited state

First, we adiabatically eliminate the excited state |ei based on the large detuning of the drive
lasers and cavity modes from the |gi ↔ |ei transition (Section 7.3.1). We work in an interaction
P
picture rotating at the drive frequency ωl with free evolution Hamiltonian Hf = j ~ωl /2(|eij he| −P|gij hg|) + s ~ωl â†s âs . The resulting interaction picture Hamiltonian is
(1)
ĤI

=

N
X
j=1

!
2
X
p̂2j
~∆l
+
(|eij he| − |gij hg|) −~∆cl â†s âs
2M
2
s=1

N X
2
X


~g  iks ẑjâs e|eij hg| + â†s e−iks ẑj |gij he|2
j=1 s=1
√+ ~ κ αâ†1 + α∗ â1 .

+

(7.7)

The coherence operator |eij hg| satisfies the equation2dg X † −iks ẑj |eij he| − |gij hg| .
|eij hg| = i∆l |eij hg| − i
âs e
dt
2

(7.8)

s=1

In a far-detuned regime, we can set |eij he| − |gij hg| ≈ −1. We then transform to the cavity frame†,(c) i∆cl t

by substituting â†s = âse, |eij hg| = |eij hg|(c) ei∆cl t and adiabatically eliminate |eij hg|(c) toget
2|eij hg|(c)g X †,(c) −iks ẑj
≈−
.
âs e
2∆c

(7.9)

s=1

In the drive frame, the annihilation operator for a mode s satisfies the equation
N

κ

√
d
g X −iks ẑjâs = −
− i∆cl âs − ie|gij he| − i καδs,1 + F̂s ,dt
22(7.10)

j=1

where F̂s is the noise operator associated with coupling to the modes outside the cavity. Using thehermitian conjugate of the expression, Eq. (7.9), leads to
N
2
κ

d
g2 X Xâs ≈ −
− i∆cl âs + iâs0 e−i(ks −ks0 )ẑjdt
24∆c
j=1 s0 =1√− i καδs,1 + F̂s .

(7.11)

107
These equations can be obtained from the effective Hamiltonian
(2)

ĤI

=

N
X
p̂2j
j=1

−

N
X

2M



2
X
N g2
â†s âs~ ∆cl +
4∆c−~g 2 

s=1

â†1 â2 e−ikeff ẑj + â†2 â1 eikeff ẑj



4∆c

√ 
+ ~ κ αâ†1 + α∗ â1 .
j=1(7.12)

Here keff = k1 − k2 = 2k is the effective wavevector. The cavity resonance is now shifted by−N g 2 /4∆c because of the presence of the atoms. Modifying the drive frequency ωl → ωl −N g 2 /4∆creturns the detuning to ∆cl .

7.3.2

Replacing mode 1 by a c-number

Further, on long timescales, the upwards propagating mode (mode 1) is composed of a macroscopic steady state amplitude with small fluctuations around this value. The macroscopic amplitude
β (|β|  1) is found from the mean-field equationβ̇ = −κ2
√− i∆cl β − i κα.

(7.13)√For t  κ−1 , the steady-state value is β = −i κα/(κ/2 − i∆cl ). We displace mode 1 by the
amplitude β by making the transformation â1 → β + â1 . Apart from introducing some constantterms that can be neglected, the resulting Hamiltonian isĤa−c =

N
X
p̂2j
j=1

−

2M

−

N
X
~g 2 
j=1

4∆c

2
X

~∆cl â†s âss=1

β ∗ â2 e−ikeff ẑj + βâ†2 eikeff ẑj ,(7.14)

with keff = k1 − k2 = 2k the effective wavevector along ẑ. The dissipative part of the masterequation remains the same. The second line of Eq. (7.14) reflects the dominant photon exchange
between the macroscopic field in mode 1 and the vacuum of mode 2. In writing Eq. (7.14), we
have neglected the small exchange process between the vacuum fields of the two modes. Thisapproximation allows us to keep track of only mode 2 and ignore the other terms containing â1 in

108
the master equation since mode 1 only interacts with the atoms and mode 2 through the c-number
β.

7.4

Momentum width using the |n, qi notation
The momentum shift operator e±ikeff ẑj appearing in Eq. (7.14) can only shift the momentum

in units of ~keff . For simplicity, we consider initial atomic states that are clustered around |↓i ≡|n↓ ~keff i, i.e. hp̂j (0)i = n↓ ~keff , where n↓ is an integer. (A superposition of |↓i and |↑i ≡ n↑ ~keffwith n↑ = n↓ + 1 can be subsequently obtained by a π/2 Bragg pulse.) We introduce two labelsn, q to represent a momentum state as |pi ≡ |n, qi. The label n denotes the momentum center andis defined as
n=

p
~keff

(7.15)

where ||x|| denotes the nearest integer to x. The label q quantifies the deviation from a center andis defined as
q = p − n~keff .

(7.16)

We note that an initial offset from an integer multiple of ~keff , i.e. hp̂j (0)i = n↓ ~keff + poff can betrivially accounted for by denoting states as |pi ≡ |n, q, poff i so that p = n~keff + q + poff . Note that|poff | < 1/2 since larger values can be modeled as an offset about a shifted initial center n↓ → n↓ ±1.
Without loss of generality, we assume poff = 0.
The momentum width is characterized by a spread σq . We assume that the initial momentum
spread σq is small compared to the difference between subsequent centers, i.e.
σq
 1.
~keff

(7.17)

Combined with the fact that the dynamics under Eq. (7.14) does not change the spread but only
shifts the center, this assumption ensures that we can assume the orthogonality relation
hn0 , q 0 |n, qi = δn,n0 δ(q − q 0 ),for all momentum states that significantly participate in the dynamics.

(7.18)

109

7.5

Numerical solution: Semiclassical Langevin equations
The master equation governing the atom-cavity interactions is:ρ̇a−c =i1 hĤa−c , ρa−c + κD[â2 ]ρ.
i~(7.19)For an atom in the initial state |n, qj i, the momentum shift operator eikeff ẑj only shifts the centern but does not change qj . As a result, it can be expressed as
eikeff ẑj =

∞
X
n=−∞

|n + 1, qj ij hn, qj | .

(7.20)j
We define generalized population and coherence operators σ̂nm
as
j
σ̂nm
= |n, qj ij hm, qj | .

(7.21)j, since an initial value of qj remains
We have dropped the label qj in defining the operators σ̂nm

constant during the subsequent dynamics governed by the master equation. The procedure to
sample the initial value of qj will be discussed shortly in Section 7.5.1.
The free energy term can be expressed as
∞
X
p̂2j
j
=
~ωnj σ̂nn
,
2M
n=−∞

(7.22)

where ~ωnj = (n~keff + qj )2 /2M . The frequency ωnj can be better expressed asωnj = 4ωr n2 + 2nq̃j σ̃q + q̃j2 σ̃q2 ,(7.23)

where we have introduced the dimensionless quantities q̃ = q/σq and σ̃q = σq /~keff . The Hamiltonian, Eq. (7.14), can now be expressed as
Ĥ =

XX
j

n

j
~ωnj σ̂nn− ~∆cl â†2 â2 −X X ~g 2 
j

n

4∆c


j
j
β ∗ â2 σ̂n,n+1+ βâ†2 σ̂n+1,n.

(7.24)

Expressed this way, the atom-cavity interaction is reminiscent of the detuned Tavis-Cummings
model that is at the heart of cavity-based spin exchange schemes considered for optical clocktransitions [53, 88]. The Hamiltonian governing the dynamics is dependent on the initially sampledvalue of qj which enters through the frequencies ωnj .

110
jWe write down the dynamical equations for the corresponding c-numbers sjnm ↔ σ̂nmand

ζ ↔ â2 within the truncated Wigner approximation (TWA) framework. The TWA techniqueinvolves evolving the classical equations of motion associated with the Hamiltonian subject to initialconditions sampled from the Wigner distribution of the initial state [94, 92]. While the sampling
procedure accounts for the quantum and statistical uncertainties in the initial conditions, the time
evolution governed by simple classical equations is a good approximation in a variety of situations
since, in leading order, these equations remain unaffected by quantum fluctuations [94, 9, 106, 116,113]. In the case of open systems, the coupling to the reservoir not only gives rise to additional
damping terms in the classical equations but also introduces Langevin noise terms [9] with diffusion
coefficients satisfying the Einstein relations [82]. The expectation value of any symmetrized productof operators is obtained directly by computing the corresponding c-number product, called the Weyl
symbol, and averaging this quantity over several trajectories with appropriately sampled initialconditions and noise realizations [92]. The expectation values of operator products in a different
ordering can be subsequently obtained through the use of appropriate commutation relations.
We introduce the effective coupling strength geff = g 2 |β|/2∆c and without loss of generality
assume that β is real. The c-number equations are



d j
geff  ∗  jj
snm = −i(ωm− ωnj )sjnm + i
ζ sn,m−1 − sjn+1,m + ζ sjn,m+1 − sjn−1,m ,
dt
2
r


X
X j
d
geff
κ
κ
ζ=−
− i∆cl ζ + i(ξ1 (t) + iξ2 (t)) ,(7.25)
sn+1,n +
dt
2
2
4
n
j

where ξl (t) for l = 1, 2 are white-noise processes satisfying ξl (t) = 0 and ξl (t)ξl0 (t0 ) = δl,l0 δ(t − t0 ).
The bar indicates averaging over several trajectories with different noise realizations and initialconditions (see Section 7.5.1). The equation for ζ is a stochastic differential equation because of
the noise arising from coupling to modes outside the cavity [9, 121]. We refer to this model as
the multi-center model (MCM) because of its ability to track an arbitrary number of momentum
centers.

111
7.5.1

Initial Conditions

The mode amplitude ζ = a+ib is initialized according to the Wigner distribution of a vacuum
state asP (a, t = 0) = P (b, t = 0) = N (0, 1/2),(7.26)

so that |ζ|2 = hâ†2 â2 i + hâ2 â†2 i /2 = 1/2. Here, N (µ, σ) denotes a normal distribution with mean
µ and standard deviation σ.
For the atoms, we first consider each atom to be in a state described by the density matrix
ρ

(1)

1
(0) = √
2πσq

Z ∞
−∞

2

2

dq e−q /2σq |n↓ , qi hn↓ , q|(7.27)

where the restriction Eq. (7.17) ensures that states with |q| ∼ ~keff do not contribute significantly
so that the limits of integration can be extended to ±∞.
We note that by using two labels n, q to characterize the momentum, we have effectively split
the momentum phase space distribution into one for the discrete label n and one for the continuouslabel q. To sample q, we note that the momentum space distribution of the state described by
Eq. (7.27) is Gaussian with spread σq . Therefore, in each trajectory, the value of qj for any atom
j is drawn as
P (qj , t = 0) = N (0, σq ) =⇒ P (q̃j , t = 0) = N (0, 1).

(7.28)

In general, our approach in its present formulation is valid for any initial atomic density matrix
that is diagonal in the momentum basis. The density matrix for atom j can then be interpreted
as a probability distibution for the initial value of qj from which this value can be sampled ineach trajectory. As already noted, the value of qj remains constant during the subsequent timejevolution and the only effect of qj is to modify the frequencies ωnj , ωm
, . . . that enter the Hamiltonian,Eq. (7.24), for each trajectory.
To appropriately sample the n-space distribution corresponding to the state described by
Eq. (7.27), we note that the discrete levels n, m, . . . are reminiscent of the different mJ levels in a2J +1 spin manifold. Here, the choice of J depends on the number of discrete levels that participate

112
significantly in the dynamics. We initialize the c-numbers sjnm according to the DTWA (discrete
truncated Wigner approximation) prescription [106, 92], namely,sjn↓ ,n↓ = 1,
1
P (2 Re{sjn↓ ,m6=n↓ } = ±1) = P (2 Im{sjn↓ ,m6=n↓ } = ±1) = ,2∗
sjm6=n↓ ,n↓ = sjn↓ ,m6=n↓ ,sjn6=n↓ ,m6=n↓ = 0.

(7.29)

We note that our choice of initial conditions is consistent with a formal generalization of the
Truncated Wigner Approximation technique to systems with D discrete states on a given site [70].
Prior to implementing our squeezing protocol, a Bragg pulse rotates the state of each atom
to an equal superposition of the n↓ , n↑ centers. Starting with the initial conditions in Eq. (7.29),
we obtain the c-number values corresponding to such an equal superposition by numerically implementing a fictitious instantaneous state rotation that rotates each spin to lie on the equatorial planeof the Bloch sphere formed by n↓ , n↑ (Appendix B.1). The observables from the MCM simulations
are averaged over 2000 trajectories in order to sample the initial conditions and noise realizations.

7.6

Effective atom-atom interactions

7.6.1

Elimination of the cavity field â2

The spin exchange dynamics anticipated in Section 7.2 is confirmed when mode 2 is adiabatically eliminated to obtain a master equation describing the effective atom-atom interactions.
When mode 2 is negligibly excited, it can be considered as a reservoir in a vacuum state withdensity matrix R0 = |0i h0|. We use the superoperator formalism to adiabatically eliminate mode
2 [19]. We split the master equation, Eq. (7.19), into system, reservoir as well as system-reservoir

113
Liouvillians. These terms are given by


N
2
X
p̂j
LS ρa−c = −i , ρa−c  ,
2M ~
j=1
h
i
LR ρa−c = −i −∆cl â†2 â2 , ρa−c + κD[â2 ]ρa−c ,


N


2
X
g
β ∗ â2 e−ikeff ẑj + βâ†2 eikeff ẑj , ρa−c  .
LSR ρa−c = = −i −4∆c(7.30)

j=1

We first transform to an interaction picture with L0 = LS + LR . We then haveρ̃˙ a−c = L̃SR ρ̃a−c ,(7.31)

where ρ̃a−c = e−L0 t ρa−c and L̃SR = e−L0 t LSR eL0 t . We integrate Eq. (7.31) and substitute theformal solution for ρ̃a−c (t) in the same equation to getρ̃˙ a−c = L̃SR (t)ρ̃a−c (0) +Z t
0dt0 L̃SR (t)L̃SR (t0 )ρ̃a−c (t0 ).

(7.32)

We assume that mode 2 acts as a reservoir in the vacuum state, i.e. the reservoir density matrix
is R0 = |0i h0|. At t = 0, the initial uncorrelated state is ρ̃a−c (0) = ρ̃a (0)R0 , where ρ̃a (0) is
the density matrix for the atomic ensemble. We then use a decorrelation approximation to writeρ̃a−c (t) ≈ ρ̃a (t)R0 for later times, and trace out mode 2 ashi Z t˙ρ̃a = TrR L̃SR (t)ρ̃a (0)R0 +dt0 TrR [L̃SR (t)L̃SR (t0 )ρ̃a (t0 )R0 ].

(7.33)

0

The first term vanishes because hâ2 i = hâ†2 i = 0 in the vacuum state.
Next, we find the time evolution equations governing the superoperators associated with
˜2 ⊗ I,˜† ⊗ I,˜2 )T and Iˆ ⊗ (â˜† )T . Here Iˆ is the identity
ˆ â
ˆ Iˆ ⊗ (âmode 2 that enter L̃SR , namely â22
operator, i.e. Iˆ |ni = |ni for any Fock basis vector |ni. The notation Â ⊗ (B̂)T is to be understoodas the operation Â |ni hm| B̂ for a vector |ni hm| in the Liouville space of mode 2 [110]. Theseequations are found to beκd˜
˜2 ⊗ Iˆ− i∆cl ââ2 ⊗ Iˆ = −
dt
2

κ


d ˆ
˜2 )T =˜2 )T − κ â
˜2 ⊗ Iˆ .
I ⊗ (â+ i∆cl Iˆ ⊗ (âdt2(7.34)

114
The solution to this coupled set of differential equations isκâ2 ⊗ Iˆ e−( 2 −i∆cl )thi
˜2 )T (t) = Iˆ ⊗ (â2 )T − â2 ⊗ Iˆ e( κ2 +i∆cl )t + â2 ⊗ Iˆ e−( κ2 −i∆cl )t .
Iˆ ⊗ (â˜2 ⊗ I(t)
ˆ
â
=



(7.35)˜† ⊗ I(t)˜† )T (t).
ˆ and Iˆ ⊗ (â
Hermitian conjugation of these two equations yields the expressions for â22
For brevity, we denote Ŝj ≡ eikeff ẑj . From Eqs. (7.33) and (7.35), we arrive at 2 2 X
N Z t
h
g |β|
0
˜
˜
ρ̃˙ a = −dt0
Ŝj† (t)Ŝj 0 (t0 )ρ̃a (t0 )e−(κ/2−i∆cl )(t−t )4∆c00j,j =10˜
˜− Ŝj (t)ρ̃a (t0 )Ŝj†0 (t0 )e−(κ/2+i∆cl )(t−t )0˜˜− Ŝj 0 (t0 )ρ̃a (t0 )Ŝj† (t)e−(κ/2−i∆cl )(t−t )

+

i
0
˜
˜
ρ̃a (t0 )Ŝj†0 (t0 )Ŝj (t)e−(κ/2+i∆cl )(t−t ) .

(7.36)In arriving at Eq. (7.36), we have used the fact that the reservoir is approximately in thevacuum state to set hâ2 â2 i = hâ†2 â2 i = 0 and hâ2 â†2 i = 1. The time evolution of the system operator˜Ŝj (t) is given by

!
!
p̂2j
p̂2j
˜
ikeff ẑjŜj (t) = exp i
exp −i
t e
t .
2M ~
2M ~

(7.37)

Once again, we introduce generalized population and coherence operators, but with an extralabel q, asj,qσ̂nm= |n, qij hm, q| ,(7.38)and expand the momentum shift operator eikeff ẑj as
∞ Z ∞
X
j,q
eikeff ẑj =dqσ̂n+1,n.

(7.39)

n=−∞ −∞We then have
∞ Z ∞
X
˜
j,q
Ŝj (t) =dqei∆ωn (q)t σ̂n+1,n,(7.40)

n=−∞ −∞

where we have introduced ∆ωn (q) = 4ωr (1 + 2n + 2q̃σ̃q ). As an example, we explicitly write down
the first term in Eq. (7.36):
 2 2 X
g |β|
−
4∆c
0
j,j

Z t

0

dt
0

XZ ∞
n,n0

−∞

Z ∞
dq
−∞

0

0

j,q
dq 0 σ̂n,n+1
σ̂nj 0,q+1,n0 ρ̃a (t0 ) ×

h κ
 i
h κ
 i
exp −
− iδn (q) t exp− iδn0 (q 0 ) t0 ,22(7.41)

115
where δn (q) ≡ ∆cl − ∆ωn (q).
The restriction σ̃q  1 ensures that only operators associated with q  ~keff contribute to
the dynamics. We further assume that for the momentum centers that significantly participate in
the dynamics, the corresponding |κ/2 − iδn (q)| is sufficiently ‘large’. We will quantify this criterion
self-consistently later on (see Section 7.6.1.1). Then, the integral over t0 can be performed under aMarkov approximation by setting ρ̃a (t0 ) ≈ ρ̃a (t) to get−
2 |β|/4∆ 2
0 0
g
0
c
j,q
dq
dq 0 κ
e−i(∆ωn (q)−∆ωn0 (q ))t σ̂n,n+1
σ̂nj 0,q+1,n0 ρ̃a (t).
0
0
−∞
−∞
2 − iδn (q )XXZ ∞j,j 0 n,n0

Z ∞(7.42)We repeat this calculation for the remaining three terms. We define the coherent and dissipative
coupling strengths as 2 2 2 2g |β|g |β|δn (q)
κ/2χn (q) =, Γn (q) =
,
2
2
2
4∆c
κ /4 + (δn (q))
4∆c
κ /4 + (δn (q))2(7.43)and perform the reverse interaction picture transformation with L0 = −LS to obtain an effective
master equation governing the dynamics of ρa :


N
∞ Z ∞
X
X
1 
j,q
ρ̇a =
dq ~ωn (q)σ̂nn, ρa 
i~−∞
j=1 n=−∞
Z ∞

XXZ ∞
0 0
0 0
j,q
j,q
σ̂nj 0,q+1,n0 ρa − ρa σ̂nj 0,q,n0 +1 σ̂n+1,n− idq
dq 0 χn0 (q 0 ) σ̂n,n+1
j,j 0 n,n0

−∞

−∞


0 0
0 0
j,q
j,q
+σ̂n+1,nρa σ̂nj 0,q,n0 +1 − σ̂nj 0,q+1,n0 ρa σ̂n,n+1
Z ∞

XXZ ∞
0 0
0 0
j,q
j,q
+
dq
dq 0 Γn0 (q 0 ) σ̂n+1,nρa σ̂nj 0,q,n0 +1 + σ̂nj 0,q+1,n0 ρa σ̂n,n+1
j,j 0 n,n0

−∞

−∞


0 0
0 0
j,q
j,q
−σ̂n,n+1
σ̂nj 0,q+1,n0 ρa − ρa σ̂nj 0,q,n0 +1 σ̂n+1,n.

(7.44)

We make the simplifying assumption that χn (q) ≈ χn (0) ≡ χn , Γn (q) ≈ Γn (0) ≡ Γn , thatallows to pull χn , Γn outside the integrals. We find that this requirement constrains
σq
 min
n
~keff



δn
16ωr


,

(7.45)

where δn ≡ δn (0) and the values of n considered correspond to the centers that significantly
participate in the dynamics. In deriving the simple expression in Eq. (7.45), we have assumed

116
that the dispersive interaction dominates, i.e. δn  κ/2 for participating centers. For detunings
δn↓  4ωr , Eq. (7.17) is clearly a more stringent requirement than Eq. (7.45).
Further, the simultaneous excitation and de-excitation of a pair of atoms is near-resonant
only when the same centers are involved, which corresponds to terms with n = n0 in Eq. (7.44).
For terms with n = n0 ± 1, the exchange process is energetically detuned by 8ωr .2

From these

considerations, the effective master equation, Eq. (7.44), can be written asZ
1 X X ∞j,qρ̇a =dq ~ωn (q)σ̂nn, ρa 
i~
−∞
n
j
Z ∞

XX Z ∞j,q
j 0 ,q 0
j 0 ,q 0j,qσ̂n+1,nρa − ρa σ̂n,n+1
σ̂n+1,n− i
χndq
dq 0 σ̂n,n+1
n

j,j 0

−∞

−∞

0

0

0

0

j,q
j ,q
j,q
j ,q
ρa σ̂n,n+1− σ̂n+1,n+σ̂n+1,nρa σ̂n,n+1
Z ∞XX Z ∞j,q
j 0 ,q 0
j 0 ,q 0j,qρa σ̂n,n+1+ σ̂n+1,nρa σ̂n,n+1+
Γndq
dq 0 σ̂n+1,n
j,j 0

n

−∞



−∞


j,q
j 0 ,q 0j,qj 0 ,q 0
σ̂n+1,n.
ρa − ρa σ̂n,n+1−σ̂n,n+1
σ̂n+1,n(7.46)The master equation can be considerably simplified now because the integrals over q, q 0 no
longer involve χ and Γ. By interchanging the dummy variables (j, q) ↔ (j 0 , q 0 ), terms in the thirdline cancel. Also, the two terms on the second line can be cast in a Hamiltonian form. We thentransform to an interaction picture with free evolution Hamiltonian
Ĥf =

XXZ ∞
j

n

−∞

 j,q
dq 4~ωr n2 + q̃ 2 σ̃q2 σ̂n,n,(7.47)and denote the interaction picture density matrix by ρ̃a , to arrive at the effective master equationρ̃˙ a =i XX
1 hĤeff , ρ̃a +
Γn
i~
0
n
j,j

Z ∞

Z ∞
dq

−∞
j 0 ,q 0−∞j,qj 0 ,q 0
dq 0 2σ̂n+1,nρ̃a σ̂n,n+1

j,qj 0 ,q 0j,q−σ̂n,n+1 σ̂n+1,nρ̃a − ρ̃a σ̂n,n+1
σ̂n+1,n,(7.48)

2
We note that ignoring terms with n 6= n0 amounts to assuming that rates of the order of 8ωr are ‘rapidly
oscillating’. Therefore, the model we derive here is strictly speaking only valid for squeezing rates N χn↓  8ωr , andcannot be expected to predict all features seen in the MCM in the strong driving regime (such as in Fig. 8.2, also see
Appendix B.2) even when more than two centers are tracked.

117
where the Hamiltonian Ĥeff is given byĤeff =

XXZ ∞
n

j

+

XX
j,j 0

7.6.1.1−∞j,qdq (8n~ωr ) (q̃σ̃q ) σ̂n,n

Z ∞

Z ∞
dq

~χn
−∞

n

−∞

0

0

j,q
j ,q
dq 0 σ̂n,n+1
σ̂n+1,n.

(7.49)

Validity of the Markov approximation

In Section 7.7, we will consider the evolution under the action of a truncated Hamiltonian,Eq. (7.57), derived from the more general Eq. (7.49). Additionally, when σ̃q ≈ 0, the truncatedHamiltonian results in squeezing at a rate Q ∼ N χn↓ . The Markov approximation used in Eq. (7.41)
involves retaining only the leading term in the integration-by-parts expansion of the integrand.
Neglecting the next-to-leading term amounts to approximating that
1 dρ̃a (t)/dt
 1.
ρ̃a (t) κ/2 − iδn↓(7.50)

Since the atomic dynamics proceeds at rate ∼ N χn↓ , the Markov approximation requires that|κ/2 − iδn↓ |  N χn↓ .
7.6.2

Collective angular momentum operators

The resulting master equation can be compactly expressed in terms of operators analogous tocollective angular momentum operators. We define operators Jˆn± , Jˆnz acting on any two consecutive
momentum centers n, n + 1 asJˆn+ =

N Z ∞
X
j=1

−∞

j,q
dq σ̂n+1,n, Jˆn− =

N Z ∞
X
−∞

j=1

1
j,q
dq σ̂n,n+1, Jˆnz =
2

N Z ∞
X
j=1

−∞

dq




j,q
j,q
σ̂n+1,n+1− σ̂n,n.
(7.51)



With Jˆnx = Jˆn+ + Jˆn− /2 and Jˆny = Jˆn+ − Jˆn− /2i, the operators Jˆnx , Jˆny , Jˆnz satisfy the usual
angular momentum commutation relationshiJˆnj , Jˆnk = ijkl Jˆnl ,(7.52)

where jkl is the usual Levi-Civita symbol for the right-handed coordinate system formed by thex, y, z axes. Once again, the restriction on initial states, Eq. (7.17), ensures that the limits of118
integration over q can be extended to ±∞ while still allowing the use of the orthogonality relation
Eq. (7.18) in deriving the commutation rules in Eq. (7.52). Specifically, the collective spin consistingof the pseudospin-1/2 systems formed by the two centers n↓ , n↑ are characterized by the operatorsJˆn±↓ , Jˆnz↓ .
The master equation, Eq. (7.48), for the reduced density matrix ρ̃a can be expressed as
∞
i
X
1 h
ρ̃˙ a =Ĥeff , ρ̃a +2Γn L[Jˆn+ ]ρ̃a ,i~n=−∞(7.53)

with the effective HamiltonianĤeff =

N
∞ Z ∞
X
X
j=1 n=−∞ −∞j,qdq (8n~ωr ) (q̃σ̃q ) σ̂n,n
+

∞
X
n=−∞

~χn Jˆn− Jˆn+ ,(7.54)

where the coherent and dissipative coupling strengths, χn and Γn , are defined as 2 2 2 2
δn
g |β|
κ/2
g |β|
, Γn =
,
χn =
2
2
2
4∆c
κ /4 + δn4∆cκ /4 + δn2(7.55)

with δn ≡ ∆cl − 4ωr (1 + 2n). (Section 7.6.1).

7.7

Numerical solution: Cumulant theory for one and two-atom operators
To make computations tractable, we assume that the n↓ , n↑ centers form a closed two-levelsystem while studying the collective spin dynamics using the master equation, Eq. (7.53). To thiseffect, we truncate Eq. (7.53) asi
1 h Tρ̃˙ a =Ĥeff , ρ̃a + 2Γn↓ L[Jˆn+↓ ]ρ̃a ,i~(7.56)

where the truncated Hamiltonian is
T
Ĥeff
=

N
X
X Z ∞
j=1 n=n↓ ,n↑−∞j,qdq (8n~ωr ) (q̃σ̃q ) σ̂n,n+ ~χn↓ Jˆn−↓ Jˆn+↓ .

(7.57)We recall that, with σ̃q = 0, the effective Hamiltonian, Eq. (7.57), is analogous to the standard spin
exchange/one-axis twisting model studied for closed two-level systems coupled to a cavity [76, 53,
88, 72] (also compare with Eq. (7.3)), and provides a reference model against which complications
arising from the nature of momentum states can be contrasted.

119
Exact solutions even for the truncated master equation, Eq. (7.56), are computationallyintractable because of the exponential scaling of the Liouville space with atom number. We use
an approximate method where we only keep track of expectation values of single atom and two
0

2,q
1,q
atom operators, of the type hσ̂n1,q
a ,nb i and hσ̂na ,nb σ̂nc ,nd i, where the n values can take either n↓ or

n↑ . Since we are ignoring the other momentum centers, we refer to this model as the two-centermodel (TCM). Furthermore, for the numerical simulation, we consider 2L + 1 discrete q̃ values
to sample the Gaussian wavepacket within rσq from center, where r is a small natural number,
typically r = 3. As a result, we have

q̃j =


j
− 1 r, j = 0, 1, . . . , 2L.
L(7.58)For the one-atom operators, the evolution of the expectation value is given by the following
equation.
d 1,q̃
hσ̂
i= −dt na ,nbΓn↓ δna ,n↓ + δnb ,n↓ + iχn↓ δnb ,n↓ − δna ,n↓ + 8iωr (nb − na )q̃ hσ̂n1,q̃ia ,nb+ δna ,n↑ δnb ,n↑ 2Γn↓ hσ̂n1,q̃
i↓ ,n↓+ δnb ,n↑ (N − 1)λ∗n↓X− δnb ,n↓ (N − 1)λn↓

X

j

j

2,q̃

hσ̂n1,q̃σ̂ j ia ,n↓ n↓ ,n↑2,q̃hσ̂n1,q̃σ̂ j ia ,n↑ n↑ ,n↓X2,q̃+ δna ,n↑ (N − 1)λn↓hσ̂n1,q̃σ̂ j i↓ ,nb n↑ ,n↓
j

X
2,q̃
− δna ,n↓ (N − 1)λ∗n↓hσ̂n1,q̃σ̂ j i,↑ ,nb n↓ ,n↑(7.59)jwhere λn = Γn + iχn and the index j runs from 0 to 2L.
The expectation values of two-atom operators are governed by the following equation.
d 1,q̃
0hσ̂
σ̂ 2,q̃ i = −dt na ,nb nc ,ndΓn↓ δna ,n↓ + δnb ,n↓ + δnc ,n↓ + δnd ,n↓+iχn↓ δnb ,n↓ − δna ,n↓ + δnd ,n↓ − δnc ,n↓+8iωr (nb − na ) q̃ + (nd − nc ) q̃ 0
0




0

hσ̂n1,q̃σ̂ 2,q̃ i
a ,nb nc ,nd0− δnb ,n↓ δnd ,n↑ λn↓ hσ̂n1,q̃
σ̂ 2,q̃ i − δna ,n↑ δnc ,n↓ λ∗n↓ hσ̂n1,q̃
σ̂ 2,q̃ ia ,n↑ nc ,n↓↓ ,nb n↑ ,nd

120
0

0

σ̂ 2,q̃ iσ̂ 2,q̃ i − δna ,n↓ δnc ,n↑ λ∗n↓ hσ̂n1,q̃
− δnb ,n↑ δnd ,n↓ λn↓ hσ̂n1,q̃
a ,n↓ nc ,n↑↑ ,nb n↓ ,nd
0

0

0

0

σ̂ 2,q̃ iσ̂ 2,q̃ i + δnc ,n↑ δnd ,n↑ 2Γn↓ hσ̂n1,q̃+ δna ,n↑ δnb ,n↑ 2Γn↓ hσ̂n1,q̃
a ,nb n↓ ,n↓↓ ,n↓ nc ,ndσ̂ 2,q̃ iσ̂ 2,q̃ i + δna ,n↑ δnd ,n↑ 2Γn↓ hσ̂n1,q̃
+ δnb ,n↑ δnc ,n↑ 2Γn↓ hσ̂n1,q̃
a ,n↓ n↓ ,nd↓ ,nb nc ,n↓+ δna ,n↑ (N − 2)λn↓

X
0
3,q̃
hσ̂n1,q̃σ̂ 2,q̃ σ̂ j i↓ ,nb nc ,nd n↑ ,n↓
j

X
0
3,q̃
hσ̂n1,q̃σ̂ 2,q̃ σ̂ j i− δna ,n↓ (N − 2)λ∗n↓↑ ,nb nc ,nd n↓ ,n↑
j

0

3,q̃

0

3,q̃

+ δnc ,n↑ (N − 2)λn↓X−δnc ,n↓ (N − 2)λ∗n↓X+δnb ,n↑ (N − 2)λ∗n↓

X
0
3,q̃
hσ̂n1,q̃σ̂ 2,q̃ σ̂ j i
a ,n↓ nc ,nd n↓ ,n↑jjhσ̂n1,q̃σ̂ 2,q̃ σ̂ j i
a ,nb n↓ ,nd n↑ ,n↓hσ̂n1,q̃σ̂ 2,q̃ σ̂ j i
a ,nb n↑ ,nd n↓ ,n↑

j

X
0
3,q̃
− δnb ,n↓ (N − 2)λn↓hσ̂n1,q̃σ̂ 2,q̃ σ̂ j i
a ,n↑ nc ,nd n↑ ,n↓j+ δnd ,n↑ (N − 2)λ∗n↓X− δnd ,n↓ (N − 2)λn↓

X

j

j

0

3,q̃

0

3,q̃

hσ̂n1,q̃σ̂ 2,q̃ σ̂ j i
a ,nb nc ,n↓ n↓ ,n↑hσ̂n1,q̃σ̂ 2,q̃ σ̂ j i
a ,nb nc ,n↑ n↑ ,n↓(7.60)To close the set of equations, we factorize the three-atom expectation values as
000

hσ̂n1,q̃σ̂ 2,q̃ σ̂ 3,q̃ i ≈a ,nb nc ,nd ne ,nf

0

00

0

00

hσ̂n1,q̃σ̂ 2,q̃ ihσ̂n1,q̃
i + hσ̂n1,q̃
σ̂ 2,q̃ ihσ̂n1,q̃i
a ,nb nc ,nd
e ,nf
c ,nd ne ,nf
a ,nb
00

0

0

00

σ̂ 2,q̃ ihσ̂n1,q̃
i − 2hσ̂n1,q̃
ihσ̂n2,q̃ihσ̂n1,q̃i.
+ hσ̂n1,q̃
e ,nf na ,nb
c ,nd
a ,nb
c ,nd
e ,nf

(7.61)To speed up computation, we identify “partial sums” which are recurring summations that appear
in the evaluation of the right-hand-side of Eq. (7.59) and Eq. (7.60) for each q̃, q̃ 0 , and evaluate
these partial sums only once per time step (See Appendix A in Ref. [111]).
As in the MCM, the initial conditions are determined by the state described in Eq. (7.27).
The one-atom expectation values are initialized as2

hσ̂n1,q̃(0)i =↓ ,n↓1 e−q̃ /2
√
∆q̃,
N
2π

(7.62)

where ∆q̃ = r/L is the spacing between adjacent q̃ values and the normalization constant2L −q̃j2 /2
X
e
√
N =
∆q̃
2π
j=0

(7.63)

121
ensures that the norm of the initial density matrix is unity even with a finite number of samples.
The two-atom expectation values are initialized as00hσ̂n1,q̃σ̂ 2,q̃ (0)i = hσ̂n1,q̃(0)ihσ̂n1,q̃(0)i.
↓ ,n↓ n↓ ,n↓
↓ ,n↓
↓ ,n↓

(7.64)

Next, an instantaneous rotation transforms these quantities to correspond to a state that is
an equal superposition of n↓ , n↑ (Appendix B.1). The identical initial conditions for each atom andthe permutation symmetry of the master equation enable us to avoid separate indices for every
atom in the system, with the number of atoms N explicitly appearing in the equations for the
0

1,q
2,q
quantities hσ̂n1,q
a ,nb i and hσ̂na ,nb σ̂nc ,nd i.

Chapter 8

Squeezing on momentum pseudospins: Results

We are now ready to investigate the efficiency of our proposed momentum-pseudospin squeezing scheme. As a concrete example, we consider parameters relevant for Bragg transitions on the
1 S − 3 P transition in Strontium [31].
01

We show that appreciable spin squeezing can be demon-

strated using modest laser powers. First, we introduce the figure of merit for spin squeezing andoutline the considerations leading to our choice of parameters. Then, we study the interplay ofsqueezing and superradiance, the dynamics under very fast squeezing, and the effect of a non-zero
momentum width. We also discuss the manifestation of an experimentally observable many-bodyenergy gap. We conclude this chapter with comments on the results and possible extensions of this
work.

8.1

Spin squeezing: Figure of merit
The Jˆnz↓ Jˆnz↓ term implicit in Eq. (7.57) (see Eq. (7.4)) can be exploited to prepare spin

2 [76] is defined as
squeezed states. We recall that the Wineland squeezing parameter ξR
2ξR=

Here, the contrast C is given byC=

1 Vmin.
C 2 VSQL(8.1)|hĴn↓ i|
,
N/2

(8.2)

where Ĵn↓ = Jˆnx↓ x̂ + Jˆny↓ ŷ + Jˆnz↓ ẑ. For a given state, Vmin is the variance in a spin component in theplane perpendicular to the mean spin direction (characterized by the unit vector n̂MSD ), minimized

123
over all axes in this plane. Mathematically,
Vmin =minn̂⊥n̂MSDĴn↓ · n̂

2 

.

(8.3)

VSQL = N/4 sets the corresponding SQL for unentangled atoms and is the variance of any spincomponent in this plane for a coherent spin state [76].

8.2

Considerations for choosing parameters
First, we note that the single atom-cavity vacuum Rabi frequency can be expressed as g =√Cκγ, where C is the cooperativity of the cavity and γ is the inverse lifetime of the excited state.

Our model imposes two constraints that limit |β| to the range∆c1  |β|  √,Cκγwhere we have used g =

√

(8.4)

Cκγ. The lower bound |β|  1 allows us to treat mode 1 as a classicalfield represented by the c-number β. The upper bound ensures that the excited state |ei is negligibly
populated, i.e.
g 2 |β|2
 1,
4∆2c

(8.5)

thereby ensuring that the adiabatic elimination of |ei is valid. We work with |β| values such that|β| ≥ 100 and the excited state population is ≤ 0.01.
The constraint imposed by Eq. (8.4) translates to requirements on the laser power and limits
on the squeezing rate. We express these requirements in terms of a ratio R characterizing therelative strength of the dissipative and dispersive interactions, defined as
R=

Γn↓
κ
=
.
χn↓
2δn↓

(8.6)

Experimentally, the steady-state photon number |β|2 in mode 1 is set by the power in thedrive laser asκ|β| = 2
κ /4 + ∆2cl
2



P
~ωl


≈

4R2 P
,
~ωl κ

(8.7)

124
where the approximation assumes that δn↓  ωr so that ∆cl ≈ δn↓ , and that the interactions arein the dispersive regime i.e. R2  1. From Eq. (8.4) the required laser power range is
2500κ
P
∆2c
≤
≤
.
R2
~ωl
100CγR2(8.8)

In standard one-axis twisting with closed two level systems, squeezing proceeds at a characteristic rate Q = N χn↓ [53, 72, 76]. From Eq. (7.55), the rate of squeezing Q is proportional to|β|2 , and consequently, the input power P , asγ2κQ ≈ |β|2 2 N C 2 R ≈
8∆c



P
~ωl



N C 2 γ 2 R3
,
2∆2c

(8.9)

where we have assumed R2  1. Therefore, Q is constrained to the range

1250 R

8.3

κCγ
∆2c


N Cγ ≤ Q ≤

R
N Cγ.
200

(8.10)

Parameters for the 1 S0 − 3 P1 transition in 88 Sr
Although our scheme is applicable to a wide variety of atomic species, here we consider

its efficiency when it is implemented on the 689 nm 1 S0 − 3 P1 transition of 88 Sr. Our choice
is motivated by the advantages of using ground-state 88 Sr in Bragg interferometers [31], suchas its extremely small scattering cross-section, insensitivity to stray magnetic fields and ease ofexperimental manipulation, including accessing the parameter regimes required for our scheme.
The inverse lifetime of the excited state is γ/2π = 7.6 kHz while the single photon recoil frequency
is ωr /2π = 4.74 kHz. The spin-1/2 system is encoded in |↓i ≡ |1 S0 , 0~ki and |↑i ≡ |1 S0 , 2~ki
implying that n↓ = 0, n↑ = 1. We consider N = 103 atoms in a cavity with decay rate κ/2π = 100
kHz, and with either of two cooperativities, C = 1 (C = 10). The single atom-cavity vacuum Rabi√
frequency g = Cκγ then takes the value g/2π ≈ 27.6 kHz (87.2 kHz). We assume that the cavity
resonance is detuned from the atomic transition such that ∆c /2π = 200 MHz.
Squeezing by one-axis twisting occurs when the dispersive interactions dominate, corresponding to the regime R  1. We consider R in the range 0.025 − 0.2 in our study, corresponding to125δn↓ /2π ≤ 2 MHz. We operate in the regime ∆c  N CRγ, which ensures that the squeezing induced fields in both the modes, 1 and 2, are small compared to the macroscopic field |β| in mode
1. Further model-enforced constraints (Section 8.2) restrict the photon number in mode 1, |β|2 ,to the range 1 × 104 − 2 × 106 (1 × 104 − 2 × 105 ). Experimentally, these constraints translate
to varying the power P in the drive laser in a range 10 nW − 150 µW (10 nW − 15 µW), andachievable squeezing rates Q/2π in the range 5 Hz − 7.6 kHz (0.5 kHz − 76 kHz) (Section 8.2). We
only consider squeezing rates such that Q/δn↓  1 (< 1/50 in all simulations), allowing for the
adiabatic elimination of mode 2 in deriving the two-center model (see Section 7.6.1.1). Even in thisregime, while very slow rates are undesirable from a technical perspective, very fast squeezing with
Q & ωr leads to coupling with momentum states outside the pseudospin manifold and degrades thesqueezing, as we will demonstrate.
Finally, to account for the momentum width of the atomic cloud, we consider values σ̃q ≤ 0.1√
to satisfy the requirement, Eq. (7.17), of our model. The dephasing rate µd = 4 2ωr σ̃q associatesa characteristic timescale to the momentum width. Specifically, for a collection of atoms initialized
in the same, equal superposition between the two centers n↓ , n↑ and undergoing free evolution,
2 2

the contrast C decays as C(t) = e−µd t . With σ̃q ≤ 0.1, the corresponding maximum rate isµd /2π = 2.7 kHz.

8.4

Limits set by superradiance
We first consider the case of σ̃q ≈ 0, i.e. negligible momentum width. Figure 8.1(a) plots theevolution of the spin squeezing parameter in the C = 1 case for values of R in the range 0.025 − 0.2,
and with |β|2 ≈ 5.4 × 105 . Modest laser powers, up to 40 µW, are sufficient to maintain thisintracavity photon number for the range of R considered here (Section 8.2). In this parameter2 reaches its minimum
regime, the TCM (dashed) and MCM (solid) results agree excellently until ξR
2 arises as a trade-off between the twisting dynamics that decreases
value. The minimum value of ξR

Vmin (Eq. (8.1) and fluctuations in superradiant decay from n↓ to n↑ that increase this quantity[53, 72]. For smaller R, the larger value of δn↓ strongly suppresses dissipation relative to dispersive

126
2 . However, forinteractions (Eq. (7.55)), leading to improved squeezing, i.e. smaller values of ξR

fixed |β|2 , the absolute squeezing rate N χn↓ also decreases with larger δn↓ (Eq. (7.55)), leading toslower squeezing dynamics. Therefore, as summarized in the inset, smaller R values enable greater
metrological gain, but the time taken for squeezing also increases when |β|2 is fixed.

Figure 8.1: Interplay of squeezing and superradiance for different R = κ/2δn↓ values. (a) Evolution
2 for R = 0.025, 0.05, 0.1, 0.2. Inset: Maximum metrological gain (in dB) and time takenof ξRto achieve this gain. (b) Population in n↓ , n↑ for R = 0.2, with total population in all centersadding up to N = 103 . (c) Population in n+1 for different R values. In this panel, N = 103 , C = 1,
|β|2 ≈ 5.4×105 . Solid (dashed) lines represent MCM (TCM) results. Four centers, n↓ , n↑ , n+1 , n+2 ,were tracked in the MCM simulations, with negligible population in n+2 .

The population dynamics at the different momentum centers reveal the effect of superradiance. Figure 8.1(b) shows the evolution of populations in n↓ , n↑ for the case of R = 0.2. The rapid
decrease (increase) in n↓ (n↑ ) population reflects superradiant decay on the n↓ → n↑ transition.
Further, the MCM enables an investigation of the leakage to centers outside the spin manifold,highlighting the power of this technique. We denote the first k centers higher than n↑ as n+k , andthe first k centers lower than n↓ as n−k . The MCM reveals that a small number of atoms (< 10)
are lost to n+1 during the squeezing dynamics, as seen in Fig. 8.1(c) for the various R values.
However, the excellent agreement between the TCM and MCM results in Fig. 8.1(a) indicates thatin this parameter regime, the centers n↓ , n↑ can be effectively treated as a closed two-level spin-1/2

127
manifold.

8.5

Squeezing faster and fasterA simple two-level model, such as the TCM, would predict that the squeezing rate can bearbitrarily increased by simply pumping in more laser power so that |β|2 is increased. Figure 8.2(a)2 in the case C = 10, R = 0.05 (δ /2π = 1 MHz) for different values
explores the evolution of ξRn↓
2 attains the sameof |β|2 /104 in the range 2 − 16. As expected, the TCM (dashed) predicts that ξR

minimum value faster when |β|2 is increased. However, the MCM results (solid) present a different
2 indeed attains its minimum faster, but this value also increases,
narrative: As |β|2 increases, ξR−2drops by ∼ 3 dB (factor ofsignaling a degradation of squeezing. In fact, the metrological gain ξR2) as |β|2 increases from 2 × 104 to 16 × 104 .
Large oscillations in the MCM curves as |β|2 is increased indicates the breakdown of the twocenter model. A study of the population dynamics at the different centers confirms this breakdown.
As seen in Fig. 8.2(b), although the populations in n↓ (n↑ ) follow the general decreasing (increasing)trend expected from n↓ → n↑ superradiant decay, the TCM and MCM population transientssignificantly differ in the case of strong driving (|β|2 /104 = 16). Further, the MCM transientsdisplay pronounced oscillations with a frequency ∼ 8ωr , corresponding to the relative detuning
between the n↓ ↔ n↑ and n↑ ↔ n+1 , n−1 ↔ n↓ transitions.
Giant population oscillations in n±1 , shown in Fig. 8.2(c-d), confirm the significant participation of these centers in the dynamics as |β|2 increases. A simple Rabi oscillation model qualitatively
explains the occupation of these states: The coherent superposition of the n↓ , n↑ centers serves asa large collective spin that sources mode 2. Both cavity modes, 1 and 2, are now macroscopicallyoccupied and drive two-photon Rabi oscillations between n↓ ↔ n−1 and n↑ ↔ n+1 with approximate two-photon detuning 8ωr . We find that the maximum population Pnmax
in n±1 predicted by±1
this model is given by (see Appendix B.2)
N
Pnmax
≈
±1
2



N χn↓
8ωr

2
.

(8.11)

128

2 for |β|2 /104 = 2, 4, 8, 16. (b) Population
Figure 8.2: Squeezing faster and faster. (a) Evolution of ξR
in n↓ , n↑ for |β|2 /104 = 16, with total population in all centers adding up to N = 103 . Solid(dashed) lines represent MCM (TCM) results. (c-d) Population in, respectively, n−1 and n+1
centers, for various drive strengths. (e) Comparison of simulated n±1 populations to analytic resultof Rabi oscillation model (see Text). In this panel, N = 103 , C = 10 and R = 0.05. Six centers,
n−2 , n−1 , n↓ , n↑ , n+1 , n+2 , were tracked in the MCM simulations with very low populations in n±2 .

Figure 8.2(e) compares the first oscillation peak in the n±1 populations with the analytic formulaEq. (8.11). For small occupations (small |β|2 ), the formula agrees very well with the simulations,whereas the discrepancy becomes about a factor of 2 at the largest occupation (|β|2 = 16 × 104 ).
In this strong driving regime, the coherence that develops between n↑ , n+1 and n↓ , n−1 is no longer
negligible and modifies the field in mode 2 considerably, leading to the breakdown of the simple
Rabi oscillation picture presented here (Appendix B.2).
Finally, we note that Fig. 8.2(c) (Fig. 8.2(d)) displays trends such as the decrease (increase)in the amplitude of population oscillations in n−1 (n+1 ) over time and the gradual deviation of
the oscillation troughs from zero. These features likely arise from the complex interplay of the129
Rabi flopping and cavity mediated superradiant decay on the n↓ → n↑ , n−1 → n↓ and n↑ → n+1
transitions.

8.6

Effect of momentum width
We now consider the case when the atomic cloud has non-zero momentum width. For thisstudy, we use the parameters from Fig. 8.2, i.e. C = 10 and |β|2 /104 = 2, 4, 8, 16. Fig. 8.3(a) shows
2 for σ̃ = 0, 0.025, 0.05 and 0.1 in the case when |β|2 /104 = 4. In this panel,
the evolution of ξRqthe solid and dashed curves respectively indicate the MCM and TCM models. Three trends canbe observed from this figure: (T1) When the rate of squeezing is fast relative to the dephasing
2 transient is similar (blue) to the zero width case (red) while the minimum value(∝ σ̃q ), the ξRattained is greater indicating slight degradation of squeezing. (T2) For larger momentum width, the2 transient displays oscillatory behavior signifying competition between squeezing and dephasing
ξR
2 initially decreases(orange). (T3) As the width increases further and dephasing dominates, ξRslightly but then steeply increases to values well above unity, signaling rapid loss of squeezing(black).
These trends are summarized in Fig. 8.3(b), where the maximum metrological gain achievable
is plotted as a function of |β|2 for different values of σ̃q . The σ̃q = 0 case (red) reflects the study
performed in Fig. 8.2 and shows that very strong driving lead to loss of squeezing as a resultof coupling to other momentum centers. At the other extreme is the case of σ̃q = 0.1 (black),
where rapid dephasing leads to a complete loss of squeezing for weak driving, and barely observablesqueezing (∼ 2 dB) even for very strong driving. For intermediate widths σ̃q = 0.025, 0.05 (blue,
orange), the squeezing suffers at both ends, with dephasing restricting the squeezing at weak driving,
and coupling to other centers serving as a limitation at very strong driving. For these widths, an
optimum drive strength therefore exists where the metrological gain is maximized, as reflected by
the variation of the gain for the four cases of |β|2 considered here.
As Fig. 8.3(a) exemplifies, we observe that the TCM (dashed) typically qualitatively reproduces the features seen in the MCM (solid) when studying the effect of momentum width. Except at

130

2 in the case of
Figure 8.3: Squeezing in the presence of momentum width. (a) Evolution of ξR|β|2 /104 = 4 for σ̃q = 0, 0.025, 0.05, 0.1. Solid (dashed) lines represent MCM (TCM) results. (b)Maximum metrological gain as a function of drive strength for different σ̃q values. (c) Evolution of
2 in the TCM for σ̃ = 0.1 and |β|2 /104 = 2 when N = 0, 1, 2, 4 echo pulses are inserted. The
ξRqE
NE = 1, 2, 4 cases evolve identically to the NE = 0 case until the first echo is applied (at different
times in the three cases). The gray broken line shows the σ̃q = 0 case with no echoes. (d) Evolution
2 in the TCM when N = 2 echo pulses are inserted. Otherof the constituents, C and Vmin of ξRE
details are the same as in Fig. 8.2.

very strong driving, the TCM and MCM agree reasonably well in the (T1) cases until the minimum
squeezing time, after which the MCM rises very steeply compared to the TCM. In the (T2) cases,
both models capture the oscillatory behavior but can be very different quantitatively. Finally, bothmodels agree very well in the (T3) case. The difference in the two models is not only because of
the extra momentum centers tracked by the MCM, but also because of the approximations used insolving for the dynamics in these models. In the TCM model, we force all non-trivial three-atom131
correlations to zero using a systematic truncation scheme (Section 7.7). However, the MCM is a
TWA-style approach that can, in general, capture the build-up of non-trivial three-atom correlations, which should be anticipated in an interacting system such as the one considered here. As
an example, the general steep increase of the MCM curves after the minimum squeezing time inthe (T1) cases is a manifestation of the effect of three-atom correlations, also visible in the cases
plotted in Fig. 8.2(a). On the other hand, the superposed oscillations at frequency ∼ 8ωr are a
result of coupling to the n±1 momentum centers.
The dephasing-induced degradation of squeezing can in fact be reversed. To elucidate thispoint, we consider the case of |β|2 /104 = 2 and σ̃q = 0.1, a situation where achieving squeezing
is seemingly hopeless because of weak driving and rapid dephasing (red curve in Fig. 8.3(c)). As
a minimal toy model to illustrate our protocol, we consider the TCM and interrupt the squeezing
dynamics with a series of ‘instantaneous’ echo pulses (Appendix B.1.2). In a frame rotating at 4ωr ,the axis of rotation for these echoes is the same as that of the initial π/2-pulse used for preparing2
the equal superposition of the n↓ = 0, n↑ = 1 centers. Figure 8.3(c) shows the evolution of ξR

when NE = 0, 1, 2, 4 echo pulses are inserted during the course of the squeezing dynamics. The
2 when σ̃ = 0. The timing of the N > 0 echo pulses aregray broken line shows the evolution of ξRqE
2 in the σ̃ = 0 case (∼ 0.3
such that they approximately divide the time to achieve the minimum ξRqms) into a sequence of T, 2T, . . . , 2T, T segments, where the number of 2T segments is NE − 1.
2 as it periodically attains minima < 1 as the
The insertion of echo pulses leads to a revival of ξR
2 fromspins re-phase after an echo pulse is applied. Increasing the number of such echoes prevents ξR

blowing up to very large values at any point during its evolution and also maintains the periodicallyattained minima close to the σ̃q = 0 transient.
The applicability of such a protocol to revive the squeezing parameter goes beyond only
momentum pseudospins, and is useful on a variety of platforms where squeezing is desired in thepresence of unavoidable on-site disorder, for example, in the case of NV centers. For a practical implementation using momentum pseudospins, the non-zero echo pulse duration ( & 2π/4ωr to avoid
leakage to centers outside n↓ = 0, n↑ = 1) and the effect of momentum width on pulse efficiency

132[118] have to be considered. Nevertheless, with suitable choice of parameters, we anticipate partial
2 to be observable despite these deviations from our toy model.
revivals in ξR

Finally, we investigate the constituent observables of the spin squeezing parameter to better2 comprises of two observables,
understand this strong revival phenomenon. From Eq. (8.1), ξR

namely, C (Eq. (8.2)) and Vmin (Eq. (8.3)). Figure 8.3(d) plots the evolution of these observables as
2 for the case of N = 2. The re-phasing of the spins after each echo leads to the expected
well as ξR
E

increase of C. However, Fig. 8.3(d) shows that this increase alone is not responsible for the strong
2 . As the spins re-phase, V
revival of ξR
min also reaches its minima close to the times when C peaks,2.
thereby leading to sharp dips in ξR

8.7

Collective physics with a many-body energy gap
Apart from squeezing, yet another type of collective behavior manifests as a result of thecavity mediated atom-atom interactions. We consider the observable C⊥ , defined as the normalized length of the projection of the Bloch vector on to the equatorial plane of the Bloch sphere.
Mathematically,C⊥ =qhJˆnx↓ i2 + hJˆny↓ i2
N/2.

(8.12)

Figure 8.4(a) plots the evolution of C⊥ in the case σ̃q = 0.05 for different values of |β|2 /104 = 2, 4, 8.
The TCM (dashed) and the MCM (solid) are in qualitative agreement in all cases and in quantitative
agreement when dephasing dominates, i.e. for weak driving (red). The gray broken line shows
the corresponding decay of C⊥ for freely evolving atoms, i.e. with no interactions, which obeys√
2 2
the analytical expression C⊥ (t) = e−µd t , where µd = 4 2ωr σ̃q . Clearly, interactions lead to an
observably slow decay of contrast compared to the free evolution case.
The effective Hamiltonian, Eq. (7.57), provides insight into the slow decay of C⊥ in thepresence of interactions. We note that for any n,Jˆn− Jˆn+ = Ĵn · Ĵn − Jˆnz Jˆnz − Jˆnz .

(8.13)

133

Figure 8.4: Manifestation of a many-body energy gap. (a) Evolution of C⊥ for σ̃q = 0.05 for different
values of |β|2 /104 = 2, 4, 8. (b) TCM results using the same parameters as in (a), but with the gap
Hamiltonian ĤG turned off. The gray broken line in each case shows the decay of C⊥ under freeevolution. Solid (dashed) lines represent MCM (TCM) results. Other details are the same as inFig. 8.2.

We introduce the many-body gap Hamiltonian, ĤG = ~χn↓ Ĵn↓ · Ĵn↓ . The initial uncorrelated
many-body state can be visualized as a coherent spin state in the equatorial plane of the Bloch
sphere corresponding to the maximum quantum number Jn↓ = N/2 associated with the operatorĴn↓ · Ĵn↓ . In other words, this initial state satisfies
NhĴn↓ · Ĵn↓ (0)i =
2




N
+ 1 , C⊥ (0) = 1.
2

(8.14)

The first term of Ĥeff in Eq. (7.57) is not collective, causing dephasing of individual spins that
leads to shortening of the mean spin length and populates shells of lower Jn↓ . The presence of ĤG
introduces an energy penalty for populating shells of lower Jn↓ . Specifically, ĤG dictates that
ĤG |Jn↓ , MJn↓ i = ~χn↓ Jn↓ Jn↓ + 1 ,(8.15)

implying that the transition to a lower shell, Jn↓ → Jn↓ − 1, incurs an energy penalty|∆EJ | = 2~χn↓ →Jn↓ −1n↓ Jn↓ .

(8.16)As a result, individual atom dephasing is slowed down, leading to slower decay of C⊥ .
We verify this qualitative explanation in Fig. 8.4(b), where we study the dynamics of C⊥
under the TCM with the gap Hamiltonian ĤG turned off. The decay of C⊥ is then in excellent

134
agreement with the free evolution case, although interactions are present through the remaining
terms in Eq. (7.57) and the dissipative term of Eq. (7.56).
Investigations with the TCM indicate that the presence of the gap Hamiltonian ĤG is anadvantage from a metrology perspective. The slow decay of contrast leads to a smaller value for the
2 compared to the case when Ĥ is turned off. Further, the subseminimum squeezing parameter ξRG
2 after the minimum value is attained is slowed down when Ĥ is present. We notequent rise of ξR
G

that the non-zero momentum spread is an intrinsic source of dephasing in a Bragg interferometer,
and the cavity-mediated interactions we engineer naturally provide a many-body gap protectionthat suppresses this dephasing.
In general, our results are consistent with other examples that confirm that the presenceof a many-body gap arising from correlations can supresses adverse effects of single-atom decoherence [88] and potentially contribute to extending the coherence time for precision metrology.
This ability to engineer many-body correlations driven either by mediated interactions or particle
statistics represents an emerging paradigm for advanced metrology [74].

8.8

Final comments and possible extensions
For studying various aspects of the problem, we have focused on the 1 S0 − 3 P1 transition in

88 Sr as an example, working in parameter regimes where < 10 dB of metrological gain is achievablein a few hundred microseconds to a few milliseconds based on the driving strength. While more thansufficient for a proof-of-principle experiment, we expect that with suitable choice of parameters—
small momentum width, small ratios of dissipative to dispersive interactions (R = κ/2δn↓ ) andmoderately strong driving strengths, ∼ 10 dB of metrological gain can be achieved. Such parameters
are within the reach of current technology: State-of-the-art cooling and velocity selection techniques
are able to provide samples with σ̃q . 0.01 while still ensuring appreciable atom numbers [66, 54].
The R value can be tuned to smaller values by detuning the drive laser farther away from the cavity
resonance. Strong driving at large detunings is not a problem since modern lasers are able to deliver
orders of magnitude more power than the hundreds of microwatts required in our proof-of-principle

135parameter regimes. In addition to squeezing, the same experimental setup can also be used todemonstrate and explore collective physics associated with the opening of a many-body energy gap
by measuring a different observable, namely the contrast C⊥ (Eq. (8.12)).
While in principle the R value can be made arbitrarily small to suppress superradiance andgreatly improve the squeezing, with fixed atom number the power required to squeeze at a specified
rate Q rapidly increases as 1/R3 (Eq. (8.9)), motivating considerations of elegant related schemesthat are not as sensitive to superradiance. Recent schemes developed for squeezing on optical clocktransitions circumvent this problem by either squeezing faster using a twist and turn mechanism
achieved by introducing a resonant drive [53] or by an unconventional choice of initial state thatdrives the squeezing in a spin component orthogonal to that affected by superradiant decay [72].
The former can be implemented on momentum pseudospins using an additional pair of resonant
Bragg lasers injected, for example, one free spectral range away from the cavity mode used forsqueezing. The latter scheme requires an initial state with two ensembles pointing along opposite
directions in the equatorial plane of the Bloch sphere. It can be implemented by launching two
clouds with equal number of atoms which are initially in the n↓ and n↑ states respectively andapplying a common π/2-pulse to rotate them to the equatorial plane. However, in either case, acareful study of the effects of momentum width and potential leakage to other momentum centers
has to be performed. The techniques developed here can be readily used to undertake such a study.
The latter scheme, combined with differential rotations on the two ensembles [31], can potentiallybe used to implement an entangled atom Bragg gradiometer.
Since our scheme relies on emission and absorption of a cavity photon, it is only applicable tostates separated by 2~k. Nevertheless, the squeezing can be transferred to higher diffraction orders
by subsequently applying large momentum transfer pulses [21, 93]. Finally, our scheme has naturalextensions [50] to circumvent situations where the detection noise limits the utility of the preparedspin squeezed states for metrology. By adjusting the frequency of the drive laser, the sign of thesqueezing interaction can be reversed, thereby making our scheme amenable to interaction-based
readout schemes such as twisting echoes [30] that achieve precision below the standard quantum

136
limit even in the presence of detection noise [87, 37].
From a modeling standpoint, while simulations with N = 103 atoms only require modest
computational resources, the favorable linear scaling of the multi-center model nevertheless allows
a straightforward extension to simulating atomic sources with larger number of atoms. Moreover,
our results are directly relevant to potential proof-of-principle experiments with small atomic clouds.
The unraveling of the dynamics into phase space trajectories is enabled by considering initial atomic
density matrices that have a momentum width but are still diagonal in the momentum basis.
The finite spatial extent of realistic atomic sources implies the presence of some degree of initialcoherence between the various momentum components. In future work, we will explore possible
extensions of our model to study the effects of such initial coherences on the squeezing dynamics,
which may be particularly important for sources such as Bose-Einstein condensates launched fromstrongly confining traps.
In addition to superradiant decay, single atom free-space scattering (FSS) also degrades thesqueezing. Superradiance, being collectively enhanced, is the dominant source of degradation inmost of the parameter regimes we have considered and therefore we have only focused on thisdissipation mechanism. The parameter regime where superradiance dominates FSS is R2  1/N C(Appendix B.3), and therefore, FSS is not important when large atom numbers are used suchthat this inequality is satisfied. Nevertheless, FSS can be straightforwardly included in both thesimulation models demonstrated here with very little computational overhead by accounting forthe corresponding Lindblad terms. The scaling of the multi-center model remains linear in atom
number since FSS occurs independently for each atom.
From a broader perspective, several mature atomic and atom-like platforms are beginningto demonstrate exotic many-body phenomena such as discrete time crystals [135, 22], many-body
localization [61, 75] and dynamical phase transitions [59, 114]. Bragg interferometers operating incavities open avenues for engineering interactions, and the theoretical techniques we have developed
here can be used to explore the complex interplay of interactions, losses, disorder and global state
rotations in other configurations involving momentum pseudospins.

Chapter 9

Conclusion

In Chapter 1, we listed three focus areas for improved quantum metrology. Accordingly,
we first discussed the modeling and successful demonstration of near ground-state cooling of large
trapped ion crystals with more than 100 ions. Powered by the robust technique of laser coolingby electromagnetically induced transparency, large ion crystals in Penning traps are now one step
closer to becoming improved motion sensors that can potentially aid in future dark matter searches.
Second, we presented a new technique to continuously and precisely track the relative phase of aspin superposition. Our scheme was based on atoms interacting with a lossy cavity mode, which
was in turn monitored using homodyne detection. Such a scheme is expected to be useful for singleshot tracking of irreproducible signals and for continuous, uninterrupted operation of microwave
atomic clocks. Finally, we proposed a scheme to engineer squeezing on momentum pseudospins foruse in atomic Bragg interferometers, a platform traditionally operating on single-atom physics andwhere controllable atom-atom interactions have been hard to engineer till date. Our scheme canhelp improve the measurement precision of such interferometers and also aid in studies of collective
quantum dynamics using momentum pseudospins. We also developed a numerical framework formodeling interacting momentum pseudospins, which can be used to study a range of quantum
metrology and simulation protocols that are based on variants of Bragg interferometers. We hopethat the tools discussed and demonstrated in this thesis will soon contribute towards improving
quantum metrology on atomic platforms.

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Appendix A

Supplementary Material for Chapters 3 and 4

A.1

Equations of motion for first and second order moments
In Table A.1, we introduce a set of “partial sums” that not only simplify the notation, butalso speed up the computation time by identifying recurring summations and evaluating them only
once per time step. In addition, we split the equations of motion for each moment into three parts
that correspond respectively to L0 , L1 and L2 contributions from the master equation, Eq. (3.11).
A.1.1

Internal moments

The dynamics of the internal moments are governed by the following equations.
L0 contribution


d j
i
hσg1 g1 i = − Ω∗1,j hσgj1 e i − c.c. + Γ1 1 − hσgj1 g1 i − hσgj2 g2 idt
2Symbol
X
Pµ,j

XX
Pµ,j

bX
Pµ,jn

dX
Pµ,jn

Xσα
Pµ,j

Xσα
Pµ,[qj]Definitionµm λjm hXm iPµ µl,m λjl λjm hXl Xm iP µm λjm hbn Xm iPµ†m λjm hbn Xm i
P µjm λjm hXm σα iPµjm λqm hXm σα iPTable A.1: Definition of partial sums to simplify notation and speed up computation. The symbols
Xm and λµjm are defined after Eq. 3.13.

149
iΩ1,j j ∗ iΩ∗2,j j
d jhσg1 g2 i = i (∆1,j (t) − ∆2,j (t)) hσgj1 g2 i +hσg2 e i −hσg1 e idt22 iΩ2,j j
iΩ1,j
Γ
d j
hσg1 e i = −− i∆1,j (t) hσgj1 e i −2hσgj1 g1 i + hσgj2 g2 i − 1 −hσg1 g2 i (A.1)
dt
222
L1 contribution

1
d j
Xσg e
hσg1 g1 i + = − Ω∗1,j P1,j 1 + c.c.
dt2
Ω1,j Xσeg2 Ω∗2,j Xσg1 ed jhσg1 g2 i + = −
P
−
P
dt
2 1,j
2 2,j
 Ω
Ω1,j  Xσg1 g1
d j
Xσg g
2,j Xσg1 g2
X
hσg1 e i + =2P1,j+ P1,j 2 2 − P1,j
+
P
dt
2
2 2,j

(A.2)

L2 contributionoi n ∗  XX jd j
X Xσg1 e
X 2 j
P1,j− 2(P1,j) hσg1 e i − c.c.
hσg1 g1 i + =Ω1,j P1,j hσg1 e i + 2P1,j
dt
4
iΩ1,j  XX j ∗
d j
X 2 j
X Xσeg2− 2(P1,j) hσg2 e i∗hσg1 g2 i + = −P1,j hσg2 e i + 2P1,j
P1,j
dt
4

iΩ∗2,j  XX j
X 2 j
X Xσg1 e+− 2(P2,j) hσg1 e i
P2,j hσg1 e i + 2P2,j
P2,j4
iΩ1,j  XX
d j
Xσg g
Xσg g
X
X
hσ i + =P1,j (2hσgj1 g1 i + hσgj2 g2 i − 1) + 2P1,j)(2P1,j 1 1 + P1,j 2 2 − P1,jdt g1 e4X 2
− 2(P1,j) (2hσgj1 g1 i + hσgj2 g2 i − 1)iΩ2,j  XX j
X 2 j
X Xσg1 g2
P2,j hσg1 g2 i + 2P2,j− 2(P1,j) hσg1 g2 i
P2,j
(A.3)
+
4

A.1.2

External moments

In the following equations, the index µ takes on values 1, 2 to account for the two EIT lasers.
L0 contributiondhbn i = −iωn hbn idtd
hbn bk i = −i(ωn + ωk )hbn bk idtd †hb bk i = −i(ωk − ωn )hb†n bk i
dt n
L1 contribution
X λµjn dhbn i + = −Ω∗µ,j hσgjµ e i − c.c.
dt
2
µ,j

(A.4)

150

X λµ j,nd
hbn bk i + = −dtd †hb bk i + =
dt n

2


j
Ω∗µ,j hbk σgjµ e i − Ωµ,j hbk σeg
i
+ n↔k
µ

µ,j

X λµj,n jΩ∗µ,j hbk σgjµ e i − Ωµ,j hbk σeg
i
µ
2
µ,j

X λµj,k j∗j∗−Ω∗µ,j hbn σeg
i
−
Ω
hb
σ
i
µ,j
n
gµ e
µ
2
µ,j





(A.5)

L2 contribution
X iλµj,n dXσg e
hbn i + =Ω∗µ,j Pµ,j µ + c.c.
dt
2
µ,j

X
d
sc
j
j
hbn bk i + = −Γµ hu2 iegµ λscµ,jn λµ,jk (1 − hσg1 g1 i − hσg2 g2 i)dtµ,jX iλµ 
n
Xσg e
j,n
dX ∗ jXΩ∗µ,j (Pµ,jk) hσgµ e i + Pµ,jhbk σgjµ e i + Pµ,j µ hbk i
+

2
µ,j
)
o
n
o
X
− 2Pµ,jhσgjµ e ihbk i + Ωµ,j gµ e → egµ
+ n↔k
X
d †
j
sc
j
hbn bk i + =Γµ hu2 iegµ λscµ,jn λµ,jk (1 − hσg1 g1 i − hσg2 g2 i)
dt
µ,j

−

X iλµj,n 
µ,j

2

n
Xσg e
X
dX ∗ j) hσgµ e i + Pµ,jhbk σgjµ e i + Pµ,j µ hbk iΩ∗µ,j (Pµ,jk
o

n

o

X
− 2Pµ,jhσgjµ e ihbk i + Ωµ,j gµ e → egµnX iλµj,k 
Xσg e
dX
X
j
+
hbn σegi∗ + Pµ,j µ hbn i∗Ω∗µ,j Pµ,jnhσgjµ e i + Pµ,j
µ
2
µ,j

o
n
o
X
− 2Pµ,jhσgjµ e ihbn i∗ + Ωµ,j gµ e → egµ

A.1.3

(A.6)

Hybrid moments

Table A.2 introduces some additional partial sums, now over the ions instead of the modes,that will further aid in compact presentation and faster computation by identification of recurringsummations.
L0 contribution
dijhbn σgj1 g1 i = −iωn hbn σgj1 g1 i −Ω∗1,j hbn σgj1 e i − Ω1,j hbn σeg
i
1
dt
2

Symbol1
QLjn−

L (1),σα

Qjn2

L (2)
L (3)

Qjn2

P

µ,q6=jλµq,n2

Ω∗µ,q hσgqµ e i − c.c.

P

iλµ
Xσα
qn
2 Pµ,[qj]

P

iλµ
qn
X
2 Pµ,qΩ∗µ,q hσgqµ e i + c.c.


P
iλµ
qn
∗ P Xσgµ e + c.c.
Ω
µ,q
µ,q
µ,q6=j 2µ,q6=jQjn2

151

Definitionµ,q6=jΩ∗µ,q hσgqµ e i + c.c.

Table A.2: Additional partial sums, over the ions rather than modes, to simplify notation andspeed up computation.

+Γ1 hbn i − hbn σgj1 g1 i − hbn σgj2 g2 idhbn σgj1 g2 i = i (∆1,j (t) − ∆2,j (t) − ωn ) hbn σgj1 g2 i
dtiΩ∗2,jiΩ1,jj
hbn σeg2 i −hbn σgj1 e i
+
2
2


d
Γ
hbn σgj1 e i = −− i (∆1,j (t) − ωn ) hbn σgj1 e idt2
 iΩ2,jiΩ1,j−2hbn σgj1 g1 i + hbn σgj2 g2 i − hbn i −hbn σgj1 g2 i
2
2


d
Γ
j
j
hbn σeg
+
i
(∆
(t)
+
ω
i
=
−
i
)
hbn σeg
1,j
n
1
1
dt
2
iΩ∗1,j iΩ∗2,j
2hbn σgj1 g1 i + hbn σgj2 g2 i − hbn i +hbn σgj2 g1 i
+
2
2

(A.7)

L1 contributionΩ∗1,j λ1j,n jd
1 n ∗  dX ∗ j
Xσg e
j
X
hbn σg1 g1 i + = −hσg1 e i −Ω1,j (P1,jn ) hσg1 e i + P1,jhbn σgj1 e i + P1,j 1 hbn i
dt
2
2


o
X
j
1
− 2P1,jhbn ihσgj1 e i + Ω1,j g1 e → eg1+ QL
jn hσg1 g1 iΩ∗2,j λ2jn jdj
hbn σg1 g2 i + = −hσg1 e i
dt
2


Ω1,j
Xσeg2
dX ∗ jXjX
j∗−(P1,jn) hσg2 e i∗ + P1,jhbn σeg
i
+
P
hb
i
−
2P
hb
ihσ
i
n
1,j n
g2 e
1,j
2
2
∗


Ω2,j
Xσg e
dX ∗ j
X
X−(P2,jn) hσg1 e i + P2,jhbn σgj1 e i + P2,j 1 hbn i − 2P2,j
hbn ihσgj1 e i
2
j
1
+QL
jn hσg1 g2 iΩ1,j n  dX ∗ j
d
Xσg g
X
X
hbn σgj1 e i + =2 (P1,jn ) hσg1 g1 i + P1,jhbn σgj1 g1 i + P1,j 1 1 hbn i − 2P1,j
hbn ihσgj1 g1 idt2o Ω1,j λ1Ω2,j λ2jn j
jndX ∗+ g1 g1 → g2 g2 − (P1,jn) +hσgj1 g1 i +hσg1 g2 i
2
2

152
+


Ω2,j  dX ∗ j
Xσg g
X
X
(P2,jn ) hσg1 g2 i + P2,jhbn σgj1 g2 i + P2,j 1 2 hbn i − 2P2,j
hbn ihσgj1 g2 i
2

j
1
+QL
jn hσg1 e iΩ∗1,j n  dX ∗ jd
Xσg1 g1jX
jX
jhbn σeg
i
+
=
2
(P
)
hσ
i
+
P
hb
σ
i
+
P
hb
i
−
2P
hb
ihσ
i
n
1,jn
g1 g1
1,j n g1 g1
1,j n
g1 g1
1,j
1
dt
2


o Ω∗ λ 11,j jndX ∗hσgj1 g1 i + hσgj2 g2 i − 1+ g1 g1 → g2 g2 − (P1,jn) +2Ω∗2,j  dX ∗ j
Xσg g
X
X
+
(P2,jn ) hσg1 g2 i∗ + P2,jhbn σgj2 g1 i + P2,j 2 1 hbn i − 2P2,j
hbn ihσgj1 g2 i∗
2
j
∗
1
+QL
jn hσg1 e i

L2 contributiondhbn σgj1 g1 i + =dtiΩ∗1,j λ1jn Xσg1 e
P1,j2
i n ∗  XX
dX ∗ Xσg1 e
X 2+Ω1,j P1,j hbn σgj1 e i + 2(P1,jn) P1,j− 2(P1,j) hbn ihσgj1 e i
4



o
L (1),σg1 g1
L (2)
L (3)
+ Qjn2 − Qjn2
hσgj1 g1 i
− Ω1,j g1 e → eg1+ Qjn2iΩ∗2,j λ2jn Xσg1 e
P2,j
2


iΩ1,j
XX
j
X 2
j
∗
dX ∗ Xσeg2−
P1,jhbn σeg
−
2(P
)
hb
ihσ
i
i
+
2(P
)
P
n
1,j
g2 e
1,jn
1,j
2
4
∗


iΩ2,j
X 2
XX
dX ∗ Xσg1 e+− 2(P2,j) hbn ihσgj1 e i
P2,j
hbn σgj1 e i + 2(P2,jn) P2,j4
L (1),σg1 g2
L (2)
L (3)
hσgj1 g2 i
+Qjn2
+ Qjn2 − Qjn2
iΩ1,j n  XXd
X 2
dX ∗ Xσg1 g1
hbn σgj1 e i + =2 P1,j hbn σgj1 g1 i + 2(P1,jn− 2(P1,j) hbn ihσgj1 g1 i
) P1,j
dt
4

 
o
XX
dX ∗ X
X 2+ g1 g1 → g2 g2 − P1,j
hbn i + 2(P1,jn) P1,j − 2(P1,j) hbn idhbn σgj1 g2 i + =dt

iΩ1,j λ1jn Xσg1 g1 iΩ2,j λ2jn Xσg1 g2
P1,j
P2,j
+
2
2

iΩ2,j  XX
dX ∗ Xσg1 g2
X 2+
P2,j hbn σgj1 g2 i + 2(P2,jn) P2,j− 2(P2,j) hbn ihσgj1 g2 i4
L2 (1),σg1 e
L (2)
L (3)
+Qjn
+ Qjn2 − Qjn2
hσgj1 e iiΩ∗1,j n  XXdjjdX ∗ Xσg1 g1
X 2
j
i
+
=
−
hb
σ
i
+
2(P
)
P
−
2(P
)
hb
ihσ
i
hbn σeg2
Pn g1 g1
n
1,j
1,jn
1,j
g1 g1
1,j
1
dt
4

 
o
XX
dX ∗ X
X 2+ g1 g1 → g2 g2 − P1,j
hbn i + 2(P1,jn) P1,j − 2(P1,j) hbn i+iΩ∗1,j λ1jn  Xσg1 g1
Xσg g
X
P1,j
+ P1,j 2 2 − P1,j2
iΩ∗2,j  XX
dX ∗ Xσg2 g1
X 2−
P2,j hbn σgj2 g1 i + 2(P2,jn) P2,j− 2(P2,j) hbn ihσgj1 g2 i∗
4
−

(A.8)

153
L (1),σeg1

+Qjn2

A.2



L (2)
L (3)
+ Qjn2 − Qjn2
hσgj1 e i∗

(A.9)

Analytic formula for the bright fraction: Derivation outline
The bright-state fraction is P (|↑i) = 1/Nzz
j 1/2(1 + hσ̂j i), where hσ̂j i, the population differ-Pence after the final π/2 pulse, is equal to −hσ̂jx i before that pulse. The problem therefore amounts
to computing the quantity hσ̂jx i before the final state rotation. For the Hamiltonian, Eq. (4.3), thepropagator Û (t) factorizes as Û (t) = ÛSM (t) × ÛSS (t), where the spin-motion propagator ÛSM (t)and the spin-spin propagator ÛSS (t) are given by [123, 34]



Y
X
∗
ÛSM (t) =exp 
αnj (t)â†n − αnj(t)ân σ̂jz  ,njÛSS (t) = exp −iX

Jij (t)σ̂iz σ̂jz  .

(A.10)

i6=j

Before detailing the coupling constants, we note that these propagators have to be adapted tothe thermometry sequence depicted in Fig. 4.2. For the purpose of computing the bright fraction,
the effect of the spin-echo pulse can simply be viewed as a change in the sign of F in the second armof the ODF sequence. Further, an arbitrary phase offset φ0 can be introduced in the second-armof the ODF sequence, which modifies the lineshape. These features can be accounted for by the
replacement cos(µr t) → g(t) in Eq. (4.3), where
cos(µr t),
t<τ





g(t) = 0,
τ < t < τ + tπ







− cos(µr t + φ0 ), τ + tπ < t < 2τ + tπ .

(A.11)

With this adaptation, the coupling constants in Eq. (A.10) are given by
r
Z t
~
F
0
dt0 g(t0 )eiωn t ,αnj (t) = −i Mjn~
2M ωn 0
Z t
Z t1
X F2
~
Jij (t) = Imdt
dt2 g(t1 )g(t2 )eiωn (t2 −t1 ) .
M
M
in
jn
1
2
~
2M
ω
n 0
0
n

(A.12)

154
We evaluate these expressions within the rotating-wave approximation that |δn | = |µr −ωn | µr , µr + ωn , where the frequency δn = µr − ωn is the detuning of the ODF difference frequency
from mode n. Specifically, at time 2τ + tπ , the expressions evaluate to
r

F Mjn
~  −iδn τ
αnj (2τ + tπ ) =e+ e−iφ0 e−iδn (τ +tπ ) − e−iφ0 e−iδn (2τ +tπ ) − 1
2~δn
2M ωn
X F 2 Min Mjn ~(2δn τ + sin(δn (2τ + tπ ) + φ0 ) + sin(δn tπ + φ0 )Jij (2τ + tπ ) =4~2 δn22M ωnn−2 sin(δn τ ) − 2 sin(δn (τ + tπ ) + φ0 )) .

(A.13)For µr close to the COM mode (n = 1) frequency, here denoted ω1 , the contribution of the othermodes is negligible because of the large detunings δn for n 6= 1. As a result, αnj ≈ 0 for n 6= 1
and the symmetric coupling of the c.m. mode to all the ions results in α1j (2τ + tπ ) ≡ α andJij (2τ + tπ ) ≡ J, independent of the ion numbers i, j, with expressions
α =
J

=

r

F
~  −iδ1 τ√e+ e−iφ0 e−iδ1 (τ +tπ ) − e−iφ0 e−iδ1 (2τ +tπ ) − 1
2~ N δ1 2M ωz
F2~(2δ1 τ + sin(δ1 (2τ + tπ ) + φ0 ) + sin(δ1 tπ + φ0 )2
2
4~ N δ1 2M ωz−2 sin(δ1 τ ) − 2 sin(δ1 (τ + tπ ) + φ0 )) .

(A.14)The lineshapes in Fig. 4.3 are obtained with a phase offset of φ0 = 0.
With the modes initially in thermal states characterized by mean occupations n̄n , and thespins initialized along the x direction, the evolution of any observable can be computed using the
propagator Û (t) whose form is detailed in Eq. (A.10). Specifically, the expression for hσ̂jx i evaluatesto [102]hσ̂jx i =


Y


i6=j


"
#

X
cos(4Jjk ) exp −2|αnj |2 (2n̄n + 1) .


(A.15)n

A convenient way to account for thermal motional states is to first compute hσ̂jx i for initial Fockstates of the modes and then sum over these initial states with appropriate coefficients.
For µr close to the COM mode hσ̂xj i becomes independent of j and reduces tohσ̂ x i = (cos(4J))N −1 exp −2|α|2 (2n̄1 + 1) .

(A.16)

155
The ODF lasers also lead to free-space scattering that results in decoherence at rate Γ. This results
in an additional factor of exp (−2Γτ ) in Eq. (A.15) and Eq. (A.16) that decreases the value of hσ̂jx i.

Appendix B

Supplementary Material for Chapters 7 and 8

B.1

Implementing instantaneous state rotations

B.1.1

Multi-center model

In the multi-center model, we implement an instantaneous rotation in order to initialize thec-numbers in accordance with the initial state being in an equal superposition of the n↓ , n↑ centers.
We adopt a pragmatic approach to implement such a rotation: In the lab frame, we consider afictitious HamiltonianN~Ω X  jĤ =σ̂n↓ ,n↑ e−iθ + σ̂nj ↑ ,n↓ eiθ
2

(B.1)

j=1

to act on the collection of atoms for a time T = π/2Ω so that the pulse area is A = π/2. Hereθ specifies the orientation of the axis of rotation on the equatorial plane of the Bloch sphere.
j
By ignoring the energy difference ~(ωm− ωnj ) between any pair of states n, m, we are making

the assumption that the pulse is ‘instantaneous’. While in practice any state preparation pulse
requires a finite amount of time, here we assume such instantaneous pulses for simplicity and to
avoid complications associated with pulse efficiencies and momentum widths [118].

B.1.2

Two-center model

In the two-center model, instantaneous state rotations are used for state initialization andfor probing the effect of echo pulses on the evolution of the squeezing parameter. To implement
perfect, instantaneous rotations, we consider a Bloch sphere for each q̃ value with the North and157
South poles represented by the states |n↑ , q̃i and |n↓ , q̃i respectively. The perfect rotation pulses
are assumed to not couple states with different q̃. The transformation of this pair of states undera rotation with axis n̂ and pulse area A (∈ [0, 2π]) is,




0
|n↑ , q̃i|n↑ , q̃i , = U (n̂, A) 





0
|n↓ , q̃i|n↓ , q̃iwhere the matrix U (n̂, A) is given byAAz cos 2 − in sin 2
U (n̂, A) = −i(nx − iny ) sin A2

(B.2)



−i(nx + iny ) sin A2 
cos A2 + inz sin A2

.


(B.3)

Since we track expectation values, we need to recast this transformation in terms of the means of
one and two-atom operators. In what follows, we label n↑ , n↓ using binary digits, i.e. n↑ ≡ 0 andn↓ ≡ 1. For one-atom operators, we define v1q̃ with elements v1q̃,j = hσ̂n1,q̃
j ,mj i, where j = 0, . . . , 3
and nj (mj ) is the second (first) digit from the right in the binary decomposition of j. The vectorv1q̃ transforms under the Bragg pulse to v̄1q̃ = M1 (n̂, A)v1q̃ , where|2

 |U11 ∗U11 U12
M1 (n̂, A) = U ∗ U11
 12
|U12 |2

∗ U
U11
21

∗ U
U21
11

∗ U
U11
22

∗ U
U21
12

∗ U
U12
21

∗ U
U22
11

∗ U
U12
22

∗ U
U22
12

0


|U21

|2





∗
U21 U22 .

∗
U22 U21 
2|U22 |

(B.4)

0

2,q̃
For two-atom operators, we similarly define v2q̃,q̃ with elements v2q̃,q̃ ,j = hσ̂n1,q̃
j ,mj σ̂rj ,sj i, where j =0, . . . , 15 and nj , mj , rj , sj are respectively the fourth, third, second and first digits from the right in00

the binary decomposition of j. This vector transforms as v̄2q̃,q̃ ,j = M2 (n̂, A)v2q̃,q̃ ,j where M2 (n̂, A) =M1 (n̂, A) ⊗ M1 (n̂, A) is a 16 × 16 matrix obtained as the Kronecker product of M1 with itself.

B.2

Rabi oscillation model for population leakage
We consider the case when σ̃q ≈ 0. The two spin states correspond to |n↓ i = |0~ki andn↑ = |2~ki. We assume that mode 2 is dominantly sourced by the coherence between n↓ and n↑

158
and neglect the fluctuating terms to simplify the equation for ζ (Eq. 7.25) toκdgeff X jζ=−− i∆cl ζ + isn↑ ,n↓ .
dt
2
2

(B.5)

j

We transform to the rotating frame sjn↑ ,n↓ = s̃jn↑ ,n↓ e4iωr t , ζ = ζ̃e4iωr t . At short times (N Γn↓ t  1),assuming that the state is prepared along the x-axis of the Bloch sphere in the n↓ , n↑ manifold,s̃jn↑ ,n↓ ≈ 1/2 for all j. Then, using the fact that |κ/2−iδn↓ |  N χn↓ , we can adiabatically eliminate
ζ̃ as
ζ̃ ≈

igeff /2 X jgeff Ns̃n↑ ,n↓ ≈ −,κ/2 − iδn↓
δn↓ 4

(B.6)

j

where in the last approximation we have assumed that R = κ/2δn↓  1. As an example of
population leakage, we consider the n↑ ↔ n+1 transition. By symmetry, the same arguments
hold true for the n↓ ↔ n−1 transition. Assuming sn+1 ,n+1 is negligible, sn↑ ,n↑ ≈ 1/2, and zero
populations and coherences associated with n+2 , the equation for the coherence sn+1 ,n↑ reads
N  geff 2 4iωr t
d j
sn+1 ,n↑ = 12iωr sn+1 ,n↑ + ie,dt
4δn↓ 2

(B.7)

2 /4δ
where we have used the expression for ζ from Eq. (B.6). From Eq. (7.55), the combination geffn↓can be immediately identified as χn↓ for R  1. Solving for sjn+1,n↑ gives
sjn+1 ,n↑ = −N χn↓ 4iωr te− e12iωr t .
32ωr

(B.8)

Further, still neglecting the n+2 center, we can arrive at an equation for the dynamics of the
population in n+1 asgeff ∗
d j
sn+1 ,n+1 = iζ sn+1 ,n↑ − ζsn↑ ,n+1 ,
dt
2

(B.9)

which can be solved using Eq. (B.6) and Eq. (B.8) to give1
sjn+1 ,n+1 ≈

4



N χn↓
8ωr

2
(1 − cos 8ωr t) .

(B.10)

This expression explains the oscillations at frequency ∼ 8ωr that can be seen in the populations
at the n±1 centers in Fig. 8.2(c-d), while the peak value scaled to the number of atoms gives theanalytic expression for Pnmax(Eq. (8.11)) plotted in Fig. 8.2(e).
±1159From Eq. (B.8), the maximum magnitude of the coherence sjn+1 ,n↑ is N χn↓ /16ωr . In estimating the intracavity field, we assumed that it is sourced only by the sn↑ ,n↓ coherence. Thisapproximation is valid as long as
sn+1 ,n↑ 1 =⇒ N χn↓  8ωr .
sn↑ ,n↓(B.11)The breakdown of the approximation, Eq. (B.11), signals the strong driving regime, i.e. it is theregime where the squeezing rate N χn↓ becomes comparable to the relative detuning 8ωr between
the n↓ ↔ n↑ and n↑ ↔ n+1 , n−1 ↔ n↓ transitions.

B.3

Relative importance of free-space scattering
Here, we analyze the relative importance of single-atom free-space scattering and collective

superradiant decay in increasing the variance Vmin that enters Eq. (8.1). Since the squeezing isdriven by a term ∼ Jˆz Jˆz , the axis corresponding to the minimum variance orients towards thez-axis over time [76]. As a result, we can estimate the degrading effect of various diffusive processes
by estimating the corresponding increase in (∆J z )2 .
Free-space scattering: We assume that once a photon is scattered into free-space, the atom
recoils in a random direction and is lost from the atomic cloud. The rate of emission for a single
atom is γ g 2 |β|2 /4∆2c , where the term in parenthesis is the effective population in |ei as a resultof the drive laser. Starting with an equal superposition of |g, n↓ i and |g, n↑ i, each such photon
could have been scattered equally likely from these two states, and so we have (assuming γt  1)
γ g 2 |β|2Ṅn↓ /N = Ṅn↑ /N = −.
(B.12)
24∆2c
Scattering from the n↓ (n↑ ) state of any single atom increases (decreases) J z by 1/2, therefore, theincrease in variance in a time t is(∆J z )2
N/4

  2 2
4γt g |β|22N(−1/2)
+
(1/2)
N
2
4∆2c
2 !
  2 2γt g |β|−(−1/2 + 1/2)24∆2c
 2 2
g |β|
= γt
.
4∆2c
=

(B.13)

160
Superradiant decay: The Lindblad term ∝ Γn↓ in Eq. (7.53) contributes the following time
evolution for hJˆz i:d ˆzhJ i = 2Γn↓ hJˆ− Jˆ+ i = 2Γn↓ hĴ · Ĵi − hJˆz Jˆz i − hJˆz i ,dt(B.14)

where we have used Eq. (7.4). For our initial state, we have hĴ · Ĵi = N/2(N/2 + 1), hJˆz Jˆz i = N/4
and hJˆz i = 0, so that,Ṅn↑ = −Ṅn↓ = Γn↓ N 2 /2,(B.15)

where Nn↑ ≈ N/2 + hJˆz i and Nn↓ ≈ N/2 − hJˆz i. The above rates are valid for times such thatN Γn↓ t  1. We can identify a per-atom rate of emission as Γn↓ N/2. Each such photon increases
J z by 1, therefore, the increase in variance in a time t is(∆J z )2
4
= N
N/4
N

N Γn↓ t(+1)2 −
2N Γn↓ t2

2 !
≈ 2N Γn↓ t.

(B.16)

From Eq. (B.13) and Eq. (B.16), the contribution of free-space scattering can be neglectedcompared to that of superradiant decay when 2 2
g |β|1γt 2N Γn↓ t =⇒ R2 
.
2
4∆c
NC

(B.17)

Here, R = κ/2δn↓ is assumed to be  1. As a result, when R2 becomes comparable to theinverse collective cooperativity, free-space scattering can no longer be neglected. In the simulations
presented in this paper, N = 103 , C = 1, 10, giving N C = 103 , 104 . As a result, R  0.032, 0.01
respectively for the two values of C. The values of R we consider are in the range 0.025 − 0.2,
and therefore some of our parameter regimes (e.g. R = 0.025, C = 1) do not satisfy the preceding
requirement. A more precise estimate of the squeezing parameter for such regimes requires theinclusion of free-space scattering. Nevertheless, in an experiment, increasing the total number ofatoms leads to a larger product N C and reduces the relative importance of free-space scattering atfixed R.