Entanglement-Enhanced Matter-Wave InterferometrybyGraham P. Greve
B.A., University of Chicago, 2014

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

Committee Members:
James K. Thompson, Chair
Dana Z. Anderson
Eric A. Cornell
Adam M. Kaufman
Kelvin H. Wagner

ii
Greve, Graham P. (Ph.D., Physics)
Entanglement-Enhanced Matter-Wave Interferometry
Thesis directed by Prof. James K. Thompson

Using cavity quantum electrodynamics, I have explored interactions between light and matter. In one set of experiments, collective measurements of atomic ensembles of 87 Rb atoms areused to generate large amounts of spin-squeezing in a proof-of-principle quantum sensor. These
correlations between atoms are a fundamental quantum resource, capable of improving sensor
resolution beyond the limits set by individual particle wavefunction collapse. Strong atom-light
coupling and quantum non-demolition measurements enable an unprecedented factor of 60 in directly observed phase enhancement beyond the standard quantum limit. These techniques areextended to generate deterministic squeezed states using real-time feedback and homogeneously
entangled ensembles guided along the cavity.
In a second set of experiments, momentum-squeezed states are generated by the two mostsuccessful cavity-mediated approaches to entanglement: quantum non-demolition measurements
and one-axis twisting. Said states are inserted into the first Mach-Zehnder light-pulse matter-wave
interferometer with metrological enhancement due to entanglement. In this sensor, free-falling
atoms simultaneously traverse two paths through space while also entangled with each other,
exploiting the many-body nature of the system. Both Raman and Bragg transitions are used tocoherently manipulate matter-waves along the cavity axis, providing sensitivity to gravity. Theseexperiments set a path for a future generation of quantum-enhanced sensors engaging in appliedand fundamental physics.
Along the way, we have developed new techniques with potential for broad impacts on
physics. These include experiments demonstrating a novel laser cooling mechanism based on
Raman adiabatic passage and an atom-loading protocol which maximizes coupling to an intracavity standing wave; methods for driving higher-order transverse cavity modes, generating axially-

iii
smooth intracavity potentials, and narrowing laser linewidths with external optical feedback; anda proposal for continuous real-time tracking of a quantum phase – a fundamentally new capability
for precision metrology.

Dedication

To those who face the inferno of the living with constant vigilance and apprehension.

v

Acknowledgements

First and foremost, James Thompson has been the best advisor one could ask for. His scientific curiosity, lucidity, and precision have served as a model throughout my time at JILA. His
help with the experiment during these several years has been invaluable. My undergraduate advisor, Jon Simon, also deserves recognition for his passion and for forever imparting in me the vast
opportunities in this rapidly-expanding field of physics.
I was fortunate to work at times with three excellent graduate students – Josh Weiner, Kevin
Cox, and Athreya Shankar – all were exemplary scientists and, more importantly, compassionate
people. Additional credit is due for the years of support provided by the staff in the electronic shopand machine shop. Fellow graduate students Chengyi Luo and Vanessa Koh have been diligent
and quick to learn. I’m sure the experiment is well off in their steady hands.
Outside of lab, I have had a fantastic supporting cast formed mostly from fellow physics
students. Albert, Geoffrey, Ignas, Karmela, Michael, and Steve at the University of Chicago; Adam,
Andrea, Brant, Brendan, Cale, Charlie, Chris, Ian, James, Jeff, Jen, John, Katie, Lindsay, Ryan, Will,
and Zach here in Boulder – I’m thankful I found each of you.
Finally, the numerous opportunities and advantages I have had in life originate with myfamily, and I will always cherish the formative years spent with my wonderful parents, brotherand sister. Thank you for the freedom and trust, for the encouragement, and for the lessons you
have taught me. To Rachel, as well, I owe many beautiful memories and an optimism for the
future. I have been so lucky in all regards but most obviously for the unending love and supportof others.

viContents

Chapter123Introduction and background1

1.1

The second quantum revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Dissertation overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Quantum sensors, the Bloch sphere, and the standard quantum limit . . . . . . . . .

4

1.4

Spin-squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.5

Cavity quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.6

Introduction to matter-wave interferometers . . . . . . . . . . . . . . . . . . . . . . . 10

More detailed descriptions
2.1

Optical lattices and homogeneous coupling . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2

Cavity quantum non-demolition measurements . . . . . . . . . . . . . . . . . . . . . 15

2.3

One-axis twisting via cavity squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4

Description of a Raman Mach-Zehnder interferometer . . . . . . . . . . . . . . . . . . 19

Experimental apparatus
3.1

4

13

25

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Methods45

4.1

Frequency locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2

Automated relocker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

vii

5

6

7

8

4.3

Narrowing laser linewidths with external optical feedback . . . . . . . . . . . . . . . 54

4.4

Site-dependent loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5

Axially-uniform potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6

LG01 blue dipole guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Laser cooling

63

5.1

Measuring temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2

Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3

Polarization-gradient cooling into the red-detuned lattice . . . . . . . . . . . . . . . . 69

5.4

Gray molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.5

Raman sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.6

SWAP cooling with Raman transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Spin-squeezing using cavity non-demolition measurements

84

6.1

Generating large amounts of useful entanglement . . . . . . . . . . . . . . . . . . . . 86

6.2

Deterministic spin-squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3

Spatially homogeneous entanglement using time-averaged measurements . . . . . . 103

An entanglement-enhanced matter-wave interferometer

112

7.1

Setup and state preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2

Manipulating matter-waves in a high-finesse cavity . . . . . . . . . . . . . . . . . . . 118

7.3

Squeezed momentum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.4

An entangled matter-wave interferometer . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.5

Supplementary details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Conclusion and outlook

138

8.1

Summary of accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.2

Looking forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

viii
Bibliography

142

Appendix
A Acronyms

155

B Raman SWAP cooling: models and simulations

157

C Statistics of inhomogeneous coupling

166

ix

Tables

Table
3.1

System parameters and symbol glossary . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2

Transfer cavity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2

Parameters for the 18 dB squeezing experiment . . . . . . . . . . . . . . . . . . . . . . 97

6.3

Quantum efficiency summary table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.1

Spectroscopic enhancement results for Ch. 7. . . . . . . . . . . . . . . . . . . . . . . . 113

A.1 Frequently used acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B.1 SWAP cooling adiabatic transfer trajectories . . . . . . . . . . . . . . . . . . . . . . . . 163

xFiguresFigure1.1

Rubidium-87 and Bloch sphere overview . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

Entanglement on the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3

Optical and matter-wave interferometers . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1

Introduction to inhomogeneous coupling . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2

Conditional vs. deterministic squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3

Introduction to one-axis twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4

Raman transitions for interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5

Mach-Zehnder interferometer sensitivity transfer functions . . . . . . . . . . . . . . . 24

3.1

Rubidium experiment timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2

Optical cavities and spacers overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3

Cavity linewidths, transverse mode spacing, and superradiance . . . . . . . . . . . . 31

3.4

2D MOT setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5

An interference filter extended cavity diode laser . . . . . . . . . . . . . . . . . . . . . 38

3.6

Homodyne detector circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7

Homodyne detection and path length stabilization . . . . . . . . . . . . . . . . . . . . 41

3.8

Agile microwave control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1

Frequency locking chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2

Feedback fundamentals with the JILA loop filter . . . . . . . . . . . . . . . . . . . . . 47

xi
4.3

Modulation transfer spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4Beat note lock of atomic probe to cavity probe laser . . . . . . . . . . . . . . . . . . . 50

4.5

Pound-Drever-Hall setup and error signal . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6

Automated relocker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.7

Narrowing lasers with external optical feedback . . . . . . . . . . . . . . . . . . . . . 55

4.8

Site-dependent selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.9

Homogeneous lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.10 LG01 donut dipole guide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1

Thermometry: ballistic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2

Velocimetry and sideband spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3

Physical setup of Raman sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4

Landau-Zener transfers for single-photon and Raman transitions . . . . . . . . . . . 75

5.5

Experimental setup for Raman SWAP cooling in 87 Rb . . . . . . . . . . . . . . . . . . 76

5.6

Raman SWAP cooling results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.7

Adiabatic transitions and momentum transfer . . . . . . . . . . . . . . . . . . . . . . 81

6.1

Continuous real-time tracking of a quantum phase . . . . . . . . . . . . . . . . . . . . 86

6.2

Conditional squeezing overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3

Conditional spin squeezing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4

Probe-induced axial oscillations and thermal radial motion . . . . . . . . . . . . . . . 91

6.5

Atomic probe frequency stabilization scheme . . . . . . . . . . . . . . . . . . . . . . . 92

6.6

Optical block diagram for conditional squeezing experiments . . . . . . . . . . . . . 93

6.7

Electronic block diagram for conditional squeezing experiments . . . . . . . . . . . . 94

6.8

Anti-squeezing and quantum efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.9

Deterministic squeezing protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.10 Deterministic squeezing sequence and results . . . . . . . . . . . . . . . . . . . . . . . 102
6.11 Falling atoms in an optical dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

xii
6.12 Homogeneous squeezing sequence and results . . . . . . . . . . . . . . . . . . . . . . 107
6.13 Oscillations in probe coupling due to atomic motion . . . . . . . . . . . . . . . . . . . 111
7.1

Cavity interferometer system overview. . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2

Frequency hierarchy for the interferometer experiment . . . . . . . . . . . . . . . . . 115

7.3

Energy diagrams for the interferometer experiment . . . . . . . . . . . . . . . . . . . 117

7.4

Manipulating matter-waves in a high finesse cavity . . . . . . . . . . . . . . . . . . . 119

7.5

Bragg-pulse superpositions and interferometry . . . . . . . . . . . . . . . . . . . . . . 121

7.6

QND momentum squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.7

Tomography of squeezed momentum states . . . . . . . . . . . . . . . . . . . . . . . . 127

7.8

Squeezed interferometer space-time diagram . . . . . . . . . . . . . . . . . . . . . . . 128

7.9

Spectroscopic enhancement of a squeezed matter-wave interferometer . . . . . . . . 129

7.10 Interferometer fringes and phase enhancement vs. time . . . . . . . . . . . . . . . . . 130
7.11 Matter-wave interferometry timing diagrams . . . . . . . . . . . . . . . . . . . . . . . 132
7.12 Optics layout for the interferometer experiment . . . . . . . . . . . . . . . . . . . . . 133
7.13 Unwanted Bragg transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.14 Vibrometer data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.1 Simulation of momentum states during adiabatic transfer . . . . . . . . . . . . . . . . 162
B.2 Optimizing momentum transfer in Raman SWAP cooling . . . . . . . . . . . . . . . . 165

Chapter 1
Introduction and background

1.1

The second quantum revolution1
Beginning in large part with descriptions of wave-particle duality, a heuristic interpretationof quantum mechanics was developed in the early 20th century. Born, de Broglie, Schrödinger,
Heisenberg, and others worked to formalize the robust mathematical description still used today, though Einstein remained famously skeptical about the role of entanglement. The EinsteinPodolsky-Rosen (EPR) question [1] – whether a pair of particles can be entangled without local
hidden variables – disrupted the physics community until, particularly with the advent of Bell’sinequalities, the debate was affirmatively settled to the satisfaction of most physicists. In parallel,
the field greatly advanced our understanding of the structure of matter, and it led to transformative inventions including the transistor, the laser, atomic clocks, integrated circuits, and more.
But even where the rules of quantum mechanics are known, the implications are not always fully
appreciated or accessible.
It took many decades to realize experiments with verifiable amounts of entanglement. As
we shall see, such entanglement provides a resource with which to enhance precision measurement experiments. The bulk of this dissertation involves generating unprecedented amounts ofentanglement in atomic ensembles with a technique called spin-squeezing. Our contributions include lessons in using cavity-assisted quantum non-demolition measurements, deterministically
1This description is inspired by Alain Aspect’s talk at the 26th International Conference on Atomic Physics, held inBarcelona, Spain.

2creating entangled states, producing homogeneously entangled states, and combining entanglement with delocalization in a matter-wave interferometer – all of which stand to benefit current
quantum sensors. Such quantum sensors perform precision measurements that are at the forefrontof exploration into new and fundamental physics [2–7], and quantum entanglement has alreadybegun to enhance these pursuits [8, 9]. A quantum revolution, however, would not be possible
without developing new techniques to control and isolate quantum objects. As a consequence,much attention has been given in this work to the setup, techniques, and measurements that were
useful along our particular journey.
Nevertheless, the concept of entanglement remains deeply confounding when considered
in a classical framework. A simplified analogy [10] has us consider a bakery that simultaneously
produces entangled pairs of cakes, each in a separate oven. The baker finds that when both ovens
are opened early, both cakes have risen 9% of the time. Let’s suppose that, as with a soufflé, opening
the oven can ruin the cake to a point that it is not worth tasting for quality. Now, the baker also findsthat every time the left oven is opened early, the right-side cake tastes good if baked to completion.
Every time the right oven is opened early, the left-side cake tastes good. Perhaps the cakes comefrom the same batter and an early rising batter is the key to a tasty cake. Logically, we expect that
if both cakes are fully cooked, then both cakes will taste good at least 9% of the time. The great
surprise is that despite our expectation of congruent events, despite thousands of careful tests,
these two entangled cakes never taste good at the same time2 .
Now we are in the midst of a second quantum revolution marked by rapid experimental
progress. Feynman and others foresaw that entanglement was much more than an academic fact
(or a baking curiosity). Entirely new fields have developed to exploit its unique capabilities: quantum information and cryptography, quantum computation and simulation, and quantum metrol2

As explained carefully in [10], this experiment has been performed using entangled photons, where the initialqquantum mechanical polarization state is written |ψi = 12 |BL i |BR i − 38 (|BL i |GR i + |GL i |BR i). The good- (G) andqqbad- (B) tasting eigenstates are related to the rising (R) and not-rising (N ) eigenstates through |Bi = 25 |N i + 35 |Riqq
and |Gi = 25 |Ri − 35 |N i.

3ogy. The first revolution, built upon the concept of wave-particle duality, harnessed both conceptual and technological breakthroughs to change society. This second revolution utilizes the powerof entanglement and individual quantum objects in much the same way.

1.2

Dissertation overview• Chapter 1 provides an introductory description of cavity quantum electrodynamics as wellas context for spin-squeezing, quantum sensors, and matter-wave interferometers.
• Chapter 2 expands upon these topics, providing a theoretical framework for the rest of thedissertation.
• Chapter 3 describes the physical setup of the experiments.
• Chapter 4 details various experimental techniques including frequency locking, a hollow
dipole trap for guiding atoms, and a scheme for selectively loading atoms at sites of maximal probe coupling.
• Chapter 5 is a brief departure on various laser cooling methods. In particular, our development of Raman sawtooth wave adiabatic passage (SWAP) cooling is examined.
• Chapter 6 includes results of entanglement-generating experiments in the older apparatus. Using quantum non-demolition (QND) measurements, we observe a 60× sensor improvement due to spin-squeezing, deterministically-squeezed states using feedback, andentanglement between free-falling atoms using spatially-averaged measurements.
• Chapter 7 demonstrates the generation of entangled momentum states through one-axis
twisting and QND measurements. These states are then inserted into the first Mach-Zehnder
light-pulse interferometer with phase resolution below the standard quantum limit.
• Chapter 8 summarizes results one last time with a view toward the future.
• Appendix A provides a glossary of acronyms.

4

1.3

Quantum sensors, the Bloch sphere, and the standard quantum limit
The utility of quantum sensors comes from the qubit as a quantum object. It is useful todescribe a rubidium atom as a two-state (spin-1/2) system with energy eigenstates |↑i and |↓i. An
understated fact of quantum mechanics is that every 87-rubidium atom is described by this same
system, independent of when it was manufactured or where it exists. In rubidium, the labels |↑iand |↓i usually refer to the hyperfine ground state manifolds separated by ωHF ≈ 2π × 6.835 GHz.
Magnetic sublevels and optically-accessible transitions represent experimental complications orconveniences (Fig. 1.1(a and b)). The quantum superposition principle explains that each atom i
may be in a simultaneous combination |ψii = √12 (a |↑ii + b |↓ii ) with weights a and b describing
the probability for a measurement of spin to project the qubit into one eigenstate or the other.
The Bloch sphere is an incredibly powerful framework for visualizing this system. The northand south poles of this sphere correspond to |↑i and |↓i respectively. A superposition state is represented in longitude, and a and b are generally complex numbers so a phase is encoded in latitude.
We describe a collection of N identical atoms by a collective Bloch vector J~ = hJˆx x̂ + Jˆy ŷ + Jˆz ẑi oflength J~ ≤ N/2 in a fictitious coordinate space (Fig. 1.1(c)) [11]. Coherences between the states
are encoded in Jˆx and Jˆy while the population difference is Jˆz = 12 N̂↑ − N̂↓ using collectivePP
projection operators N̂↑ = i |↑ii i h↑| and N̂↓ = i |↓ii i h↓| for sums running over all atoms. For
D E
D E
brevity, we will frequently write N↑ ≡ N̂↑ and N↓ ≡ N̂↓ .
The position and momentum of an atom can be manipulated in a variety of ways, some ofwhich will be introduced later. The typical “rules of the game” for quantum sensors, however,involve a limited number of operations which are conveniently represented on the Bloch sphere.
Coherent rotations about an axis are governed by the Rabi formula,Ω2N↑ (t) = 2
sinΩ + ∆2√!2Ω2 + ∆ 2
t
2

(1.1)

for a state initialized in |↓i, Rabi frequency Ω, detuning from resonance ∆, and pulse duration

5
b)

c)

hed
stret
c

clock

cycli

ng

a)

Figure 1.1: (a) Simplified Rubidium-87 D2 -line energy diagram showing the ground hyperfine
states and one excited state |ei. (b) Level structure demonstrating excited state structure, Zeeman
sublevels (with ground states shifting mF × 0.7 MHz/G in a magnetic field), and the clock vs.
stretched transition. (c) The Bloch sphere with basis states |↑i and |↓i. The pink noise blob corresponds to a quasi-probability distribution of the orientation of the Bloch vector from trial to trial.
A typical quantum sensor works by estimating an accrued azimuthal phase φ(t).

t.3 This coupling Ω may come in the form of microwaves, as in Ch. 6, or two-photon Ramanπtransitions, as in Ch. 7. For example, a resonant π pulse with t = Ω
rotates the state to its antipodalπposition on the Bloch sphere. A π/2 pulse with t = 2Ω
with phase chosen to define a rotation axisx̂ results in a rotation from the superposition to the equator – a superposition represented by the
Bloch vector oriented along ŷ as in Fig. 1.1(c).
The most precise quantum sensors encode the sensed quantity (e.g. field, acceleration, time,etc.) in a phase φ that accrues in such superpositions. For an atomic clock, for example, the basis
states are selected to be insensitive to environmental perturbations, and an evolving phase is readout through Ramsey interferometery and compared against a stable local oscillator (LO) 4 . In an
atomic magnetometer, the Zeeman effect shifts the transition frequency between magneticallysensitive states depending on magnetic field strength, resulting in a corresponding change to the3

Adiabatic passage (Sec. 5.6) is a special case where the Bloch vector precesses about an axis which is smoothly

tuned from δ = −∞ → +∞. The elegant result is a highly-efficient transfer of population from one state to the other.
4

For now, the SI second is defined by 9,192,631,770 periods of the radiation corresponding to the unperturbed hy-

perfine frequency in cesium-133.

6
accumulated phase. An electric field detuned from the transition frequency provides an analogue
effect through the AC Stark shift (also called the light shift or Stark effect). Here, red-detuned(below resonance) vs. blue-detuned (above resonance) light results in opposite azimuthal phaseshifts on the Bloch sphere.
The accumulated sensor phase φ must be estimated from quantum measurements to extract
the desired information, typically by a rotation about a fixed axis that converts the signal to a population measurement. But projection measurements of a superposition come with an important
drawback. The number of atoms N↑ can be precisely measured but the details surrounding thecollapse of the quantum state can be considered probabilistic. Coin-flipping is a useful analogy.
Testing the probability p0 for heads vs. tails involves flipping the coin many times. Even if p0 = 0.5,
each coin flip is independent (uncorrelated) so we expect a Gaussian distribution of results: flip√
ping N = 1000 coins results on average in 500 landing heads but with standard deviation N in
the number of heads minus the number of tails. Quantum sensors using uncorrelated qubits suffer from the same issue: the independent quantum collapse of each atom leads to an rms angular
uncertainty called the standard quantum limit (SQL) [12, 13]:√∆θSQL = 1/ N rad.

1.4

(1.2)

Spin-squeezing
Quantum projection noise leads to the coherent spin-state (CSS) distribution shown on the

Bloch sphere in Fig. 1.2(a). The only options for improving fractional precision are to use more
atoms or to engineer correlations (entanglement) so that the measurement result of “coin #1” actually gives information about the state of other coins. Using N atoms in a quantum sensor isakin to running N experiments in parallel, but adding entanglement brings the sensor beyond thesingle-particle paradigm. In other words, quantum entanglement between atoms allows the atoms
to conspire together to reduce their total quantum noise relative to their total signal for phase estimation [13, 14]. Maximally-entangled Greenberger-Horne-Zeilinger (GHZ or Schrödinger “cat”)

7
states (Fig. 1.2(b)) are useful for quantum information, but such states are notoriously prone todecoherence. Gaussian spin-squeezed states (Fig. 1.2(c, d)) have gained increasing popularity inrecent years due to their accessibility and utility. As we will see, the spin-squeezing we create
is a particularly elegant form of entanglement for metrology as quantum noise is redistributedout of the sensor’s phase quadrature and into an orthogonal, unmeasured quadrature. In thisdissertation, we realize deeply-entangled states with as much as a 60-fold improvement in phaseresolution, but in principle, squeezed states can approach the fundamental quantum bound for
metrology known as the Heisenberg limit [15],∆θHL = 1/N rad.

(1.3)

The history of atomic spin-squeezing goes back many decades [13, 16]. It is closely related
to and partly motivated by the copious work done with squeezed light. Significant sensitivity
improvements were first seen in free space atomic systems by optically probing ensembles withhigh resonant optical depth [17–19]. After many years, quantum mechanics is now fulfilling its
promise to enhance sensors in the search for new physics including gravitational waves [8] anddark matter [9].
a)

b)

c)

d)

Figure 1.2: (a) A coherent spin state arising with each unentangled atom in an identical superposition. (b) A maximally-entangled GHZ state with all atoms in one of two possible eigenstates. (c)A population-squeezed state, as created by quantum non-demolition measurements. (d) A phasesqueezed state, useful for quantum phase sensing.

Quantum-enhanced atomic sensors are not far behind their optical counterparts. Collective
cavity-QED systems have now succeeded in generating large amounts [20, 21] of directly observed

8
entanglement using laser-cooled atomic ensembles [20–25], and the proof-of-principle demonstration of an entanglement-enhanced matter-wave interferometer in Ch. 7 broadens the class of accessible applications. Introducing entanglement into state-of-the-art sensors is a lively area of research, and exploiting the many-body nature of the system will be key for the next generation ofquantum devices.

1.5

Cavity quantum electrodynamics
The workhorse for creating squeezing in our experiments is the interaction between atoms

and light. For atoms in cavities, the Jaynes-Cummings model of cavity quantum electrodynamics(QED) describes the interaction of a two-level atom with a single electromagnetic mode. The TavisCummings model extrapolates to describe N atoms in a single-mode cavity:Ĥ = ~ωc â† â +

N
X
i=1

~ωa



σ̂iz
+ ~g0 âσ̂i+ + â† σ̂i− .
2

(1.4)

where ~ is the reduced Planck constant; â and â† are creation and annihilation operators for a
photon in the cavity mode with frequency ωc ; the raising and lowering operators of each atom areσ̂i+ = |↑ii h↓|i and σ̂i− = |↓ii h↑|i with transition frequency ωa ; and g0 is assumed identical amongatoms for now. In our case, we are chiefly interested in the 52 P3/2 D2 manifold which we represent
as a single excited state |ei with transition wavelength λ = 2πcωa = 780 nm. Besides the atom number
N , three system parameters emerge with importance: the full-width at half maximum (FWHM)
cavity linewidth or power decay rate κ, the excited state decay rate Γ, and the single-photon Rabi
frequency (or vacuum Rabi frequency) 2g0 .
The cavity linewidth is fundamental as information dissipates to the outside world via the
decay of fields through, in our case, two cavity mirrors. The linewidth κ = fFSR (T1 + T2 + 2Tloss ) isc
set by mirror power transmission and loss coefficients and the cavity free spectral range, fFSR = 2L

for linear cavity length L and speed of light c. The free spectral range will also reappear with importas it is the inverse of the round trip time for light which sets the longitudinal cavity mode spacing.
The atomic linewidth Γ plays an analogous role, representing coupling to the free-space con-

9tinuum through spontaneous emission. For the 87 Rb D2 line, Γ = 2π×6.065 MHz will be referred toas the linewidth of the optically-excited state |ei. The lifetime of the excited state is τ = 1/Γ = 26 ns.
This timescale is fast compared to most parameters in our system, and as a result, the fractional
population in the excited state can be conveniently neglected during many calculations throughadiabatic elimination of the excited state.
The single-photon Rabi frequency reflects the frequency at which a single excitation is exchanged between the atom and cavity field5 . We consider a TEM00 standing-wave mode at atomic
transition frequency ωa , 1/e2 waist w0 , and mode volume V =πw02 L
4 . The vacuum energy in thismode is 12 ~ωa = 0 Ev2 V for rms vacuum field Ev , and hence an atom’s dipole coupling g ≡ µEv /~(with interaction strength characterized by dipole matrix element µ) gives the vacuum Rabi frequency for our 780 nm rubidium transitions,
2g0 =

0

J = 1/2 e~r J = 3/2

r

1 ωa
.
~0 V

(1.5)

The parameters κ, Γ, and g0 combine into the cooperativity (sometimes called the Purcell
factor),
C≡

4g02
,
κΓ

(1.6)

a dimensionless parameter that characterizes the atom-photon coupling regime for a system. Somewhat surprisingly, C is a purely geometrical quantity (independent of atom or transition) thatroughly reflects the probability for an excited atom to scatter a photon into the cavity rather thaninto free-space. Even more importantly, the collective cooperativity N C is fundamentally related
to processes such as superradiance and spin-squeezing [28–31]. The quantity N C plays a role similar to the resonant optical depth in free space experiments, setting the rate at which information
can be extracted from the system relative to the rate of single-particle decoherence. This collective enhancement with N is an important phenomenon in cavity QED, leading also to strongerpatom-light coupling via the collective vacuum Rabi splitting Ω↑ = 2g0 N↑ that enables resolving
5

Conventions for electromagnetic modes and cavities can be found in Siegman’s Lasers [26] and atomic conventions

may be found in Steck’s Alkali D Line Data [27].

10projection noise. The framework for discussing cavity-mediated interactions and entanglement
continues with Ch. 2.

1.6

Introduction to matter-wave interferometers
Interferometery makes use of the superposition principle to extract phase information froma system. Two or more parts parts of this superposition are sent along different conceptual paths,
spatial or otherwise, which differ in terms of each trajectory’s susceptibility to a sensed quantity. The ever-growing list of applications includes automotive, sea, air and space navigation; flow
visualization; biological research; and detection of gravitational waves. Interferometers are particularly suited for inertial sensing, either as gyroscopes detecting rotations through the Sagnac
effect6 or as accelerometers. Mechanical, resonating, and optical accelerometers and gyroscopes
now comprise mature technologies with established markets.
Optical interferometers have been employed for well over a century. Just as optical interferometers can be improved using non-classical correlations, so too can their atomic counterparts. In
this work, the term atom interferometer is used to include all atom-based interferometers including
Ramsey interferometers such as atomic clocks. Matter-wave interferometers will refer to a subset of
sensors characterized by the use of external momentum states to distinguish the interferometer trajectories. These sensors combine particle delocalization and interference to enable extraordinary
measurement precision, accuracy, and bandwidth for a broad and ever-growing range of applications including: searches for dark matter [32, 33], dark energy [34], and gravitational waves [35, 36];
measurements of the fine structure constant [4, 5]; tests of fundamental physics [6, 7, 37, 38]; andinertial sensing [39–41]. Such devices come in many packages and topologies, employing a wealthof clever techniques for improving precision and avoiding external noise. The basis states, the Raman and Bragg transitions, and the effectively infinite-range interactions demonstrated in Ch. 7
are scalable and compatible with today’s most precise experiments. With the direct observation of
6

The Sagnac effect will not be discussed further, but a rotation at rate Ω can be detected by a matter-wave interfer-ometer as Φ = 2mΩ · A/~, where m is the atomic mass and A is the physical area enclosed by the paths.

11phase resolution below the standard quantum limit, we have entered an era of quantum control
that aims to truly get the most out of our systems.
Light-pulse matter-wave interferometers operate in direct analogy to optical interferometers,
except the role of light and matter are reversed. A Mach-Zehnder interferometer splits light with abeam splitter, redirects the beams, and interferes the waves at a final beam splitter (Fig. 1.3). This
language is re-appropriated for the (π/2 − π − π/2) operations of the matter-wave interferometer
relevant for this dissertation [42]. For atomic basis states |↑i = 10 and |↓i = 01 , the light-pulse
beam splitters B̂ and mirror M̂ are modeled by unitary matrices




1
i
1
0
1 



B̂ = √  ; M̂ = 

2 i 1
0 eiΦ
where a relative phase Φ has been inserted by hand. The probability to measure each output
D
E2
D
E2
2
2
mode is P (↓) = ↓ B̂ M̂ B̂ ↓= sin Φ2 , P (↑) = ↑ B̂ M̂ B̂ ↓= cos Φ2 . If that phaseis smoothly varied through multiple radians, the result is a canonical sinusoidal interferometer
fringe.
a)

b)
Time

Figure 1.3: (a) A Mach-Zehnder optical interferometer. Light traverses both paths (red and blue)
and interferes at the output beam splitter. (b) A Mach-Zehnder matter-wave interferometer. Atoms
traverse both paths, accelerating under gravity and receiving momentum kicks from π/2 (beamsplitter) and π (mirror) light pulses.

To first-order, the total interferometric phase scales with the space-time area enclosed by thematter-wave interferometer,2Φ = a · keff · Tevol
,

(1.7)

12
where ~keff is the momentum imparted by the beam splitter pulses, a is a uniform acceleration to
be measured, and the interrogation time Tevol is ideally half of the sequence duration. Considerable engineering has gone into magnetic shielding, vibration isolation, and designing miniature,
low power, and portable instruments for use in noisy environments. The stability of matter-wave
interferometers can ultimately be improved by increasing the momentum separation, increasing
the evolution time, and maximizing the duty cycle for interrogation, but the emphasis of Ch. 7
is on decreasing the single shot noise below the standard quantum limit via entanglement. Moreframework for Mach-Zehnder matter-wave interferometry is found in Sec. 2.4.

Chapter 2
More detailed descriptions

2.1

Optical lattices and homogeneous coupling
Optical lattices are periodic potentials resulting from the interference of two or more beams.

In this dissertation, they play a role in trapping, guiding, and cooling atoms and they determine
the statistics of quantities such as spectroscopic enhancement. 3D and 2D lattices are excellent fororganizing dense arrays of atoms [43], but for our purposes, we are concerned with the propertiesof a 1D lattice1 . The conservative interaction between the light field and the atom’s induced dipole
moment causes the AC Stark shift (also frustratingly called the light shift) of potential energy. For afield with detuning ∆ from the |↑i → |ei atomic transition frequency ωa , the dipole approximation
and rotating wave approximation give an interaction Hamiltonian [44]

ĤI ≈


~Ω
~Ω∗
i∆t
† −i∆t|ei h↑| âe +|↑i he| â e
.
2
2

(2.1)

If the detuning ∆ is large compared to the Rabi frequency Ω, second order perturbation theory
gives the shifts in energy

ES = ±D
E2↑ ĤI ↓
~∆

= ±~

Ω2
4∆

(2.2)

where the positive and negative signs correspond to the AC Stark shift on |↑i and the excited state|gi respectively. The sign of the energy shift therefore depends on the sign of the detuning. By
1

A 2D triangular lattice is used during Raman sideband cooling in Sec. 5.5. The presence of a 1D intracavity latticetechnically constitutes the only 3D lattice used in this work.

14
convention, blue detuning refers to ∆ > 0 and red detuning refers to ∆ < 0. The conservative
potential for a blue-detuned trap is repulsive and atoms seek intensity minima. A red-detuned
trap is attractive and atoms seek intensity maxima.
Consider an electric field E1 = E0 ei(ω0 t+kL z) with wavenumber kL = 2π/λL incident along
a two-mirror cavity axis Ẑ. Within the cavity, the counter-propagating field interferes to form a~Ω2standing wave intensity pattern, the axial potential U (z) = 4∆0 sin (kL z)2 . The peak intensity isthus four times larger than the intensity for a single beam running wave. The total AC Stark shift
is proportional to intensity and determines the trap parameters, but often in rubidium we are also
concerned with the differential shift between |↑i and |↓i as both states are perturbed by the same
field. Clearly, the differential Stark shift becomes less relevant as the laser is further detuned.
One more helpful approximation is adopted throughout this dissertation. The bottom of thissinusoidal standing wave looks like a quadratic harmonic oscillator potential, sin(kL z)2 ≈ kL2 z 2 forq
2U0Ω2
small z. To good approximation, the axial trap frequency is ωax = kL m
for trap depth U0 = 4∆
Rb
and rubidium mass mRb . For temperature units, we simply rescale the trap depth by ~/kB for the
Boltzmann constant kB . Here we have been discussing the axial potential, but the Gaussian modes
supported within the cavity also provide a radial trapping potential [26]. Fortunately, a Gaussian− 2rw

is also expanded as a quadratic, e

2

0

2

≈ 1 − 2r
w0 , so the above picture still applies. Little more will

be said except that the timescale for radial trapping frequency differs by a few orders of magnitudein our experiment. Typical axial trap frequencies are of order ωax ∼ 2π × 100 kHz and typical radial
trap frequencies are of order ωrad ∼ 2π × 1 kHz.
Finally, trapping atoms at periodic “pancakes” along the cavity axis leads to a recurring topic
within this dissertation: inhomogeneous coupling between the standing wave lattice and the standing wave probe modes. In general, atoms are spaced at λL /2 intervals and they couple to a probewith unrelated wavelength λp . Atoms at positions with no probe light do not contribute to a measurement. Because atoms are spread out over 103 lattice sites spread over 1 mm, the distributionof atom-probe couplings is essentially uniform. More will be said in Ch. 4, and there are a numberof ways to avoid this including site-dependent loading (Sec. 4.4), spatially-averaging the position

15
a)

b)

Inhomogeneous atom-probe coupling

Homogeneous atom-probe coupling=

Figure 2.1: Equivalence between (a) an inhomogeneously-coupled ensemble of N0 atoms and(b) slightly fewer N homogeneously-coupled atoms and a smaller Bloch sphere [45]. In this case,spatial inhomogeneity arises from a mismatch between probe and the periodic trapping of the 1D
intracavity lattice (not shown).

of the atoms (Sec. 6.3), probing with a running wave in a ring cavity, or using a commensurate lattice [21]. For several of our experiments, atoms are trapped over many wavelengths of the probe
standing wave such that the atom-probe couplings form a nearly uniform distribution. In this case,2 = 3 g 2 , and C = 3 C [20, 45–47] relative to a
we simply define effective quantities N = 23 N0 , geff
4 0
4 0

system with completely homogeneous coupling. This ensures consistency between the measured
projection noise and the size of the observed vacuum Rabi splitting discussed in the next section.

2.2

Cavity quantum non-demolition measurements
Quantum non-demolition measurements are used to gain information about atoms throughtheir effect on a probing field. In the weak-excitation limit, where the number of cavity photons or
atomic excitations is small compared to N , the atom-cavity Hamiltonian (Eq. 1.4) in the |↑i → |ei
rotating frame becomes [29]p
ĤQND = ~δc â† â + ~ N↑ g0 ĵâ† + ĵ † â

(2.3)

p
with cavity detuning δc = ωc − ωa and atomic raising and lowering operators ĵ † ≈ Jˆ+ / N↑ andpp
P ±ĵ ≈ Jˆ− / N↑ modifying the collective raising and lowering operators Jˆ± =N↑i σ̂i . The
collective enhancement factor that emerges is crucial for resolving projection noise.
Without atoms, the cavity has a single Lorentzian transmission profile associated with ωc .
Reference [29] uses input-output theory to derive the eigenfrequencies of the new normal modes

16
for the coupled atom-cavity system,
δc ±
ω± =

q
δc2 + Ω2↑
2

(2.4)

.

The coupling of light to atoms in |↑i modifies the round-trip phase of light through repeated absorption and emission, giving rise to a separation of the normal modes by the collective vacuumpRabi frequency Ω↑ = N↑ 2g0 at δc = 0. The transmission or reflection dip associated with barecavity resonance shifts (and broadens) as a result of atoms in |↑i. Because the cavity does not reveal
which atoms are in |↑i, scanning a probe laser across the dressed cavity resonance amounts to acollective population measurement, and the degree to which the measurement can be considered
non-demolition is determined by the number of free-space scattered photons per atom required toresolve the frequency shift.
We now restrict ourselves to the problem of measuring an unentangled coherent superposiQ~√1tion |ψi = N
i=1 2 (|↑i + |↓i) described by the Bloch vector J = J x̂. On average, N↑ = N↓ = N/2,
but there are anti-correlated projection noise fluctuations in the populations, ∆N↑ = ∆N↓ =√N /2. In terms of the pseudospin projection operators (Jˆz = 12 N̂↑ − N̂↓ , etc.) the variances are(∆Jz )2 = (∆Jy )2 = N/2. The ratio of the rms spin projection noise amplitude to the length of the
Bloch vector sets the standard quantum limit for angular resolution (Eq. 1.2), ∆θSQL = ∆Jz /|J| =√√1/ N and ∆φSQL = ∆Jy /|J| = 1/ N .
Using Eq. 2.4, the rms fluctuations of the mode frequency ω+ due to ∆N↑ give quantum
projection noise in frequency units,∆ωQPN =Ω↑
dω+g0∆N↑ = √ q.
dN↑
2 2 Ω2 + δ 2
↑

(2.5)

c

The experiments of this dissertation are restricted to the far-detuned limit, δc  Ω↑ , so we mayqg02 N↑
N↑ g02further simplify ω+ = δc + δc and ∆ωQPN =2 δc . More commonly, we write the cavity
frequency shift ωc0 −ωc ≈g02 N↑
δc . Experimentally, these quantities must be compared to fundamental

and technical limitations on the detection system. The problem then reduces to measuring a cavity
frequency shift with better than 10 – 100 kHz resolution without inducing too much single-particle

17
wavefunction collapse.
The non-demolition measurements described here localize a squeezed state with reduced
fluctuations ∆Jz , but the trial-to-trial Jz can be considered to be drawn from the original coherent
state distribution. With conditional squeezing, the value of a pre-measurement Jzp is used to subtract
quantum noise from a final measurement Jzf [16, 17]. Fluctuations in the quantity (Jzf − Jzp ) arethen compared to the standard quantum limit (Fig. 2.2(a)). It is crucial that the pre-measurement
preserves the coherence or contrast of the quantum state. With high quantum efficiency and highcollective cooperativity N C, cavity QND measurements have realized the largest amounts of spinsqueezing to date [20, 21]. The pre-measurement can also be used to deterministically drive thestate to a target Jz value such that Jzf alone demonstrates fluctuations below the standard quantum
limit (Fig. 2.2(b)). The treatment of QND measurements here has glossed over details of spin-flips,
excited state structure, atomic motion, inhomogeneous coupling, and other issues, but these details
are discussed in the corresponding experiment chapters and in App. C.
a)Conditional squeezingb)

Deterministic squeezing

Figure 2.2: (a) In conditional squeezing, the pre- and final measurements are strongly correlated (black) but either measurement on its own has resolution worse than the standard quantum limit. Fluctuations of the difference (Jzf − Jzp ) reveal sub-SQL statistics (gold). (b) With
deterministically-squeezed states, the final measurement alone has sub-SQL resolution. Data isfrom the experiment in Sec. 6.2.

18

2.3

One-axis twisting via cavity squeezing
The linear Hamiltonian Ĥ = ~ω Jˆz causes a precession on the Bloch sphere at a constant rate

about ẑ. For example, the differential AC Stark shift between two states modifies the transition
frequency to this effect. More interestingly, the nonlinear one-axis twisting Hamiltonian
ĤOAT = χOAT Jˆz2

(2.6)

describes a precession at a rate proportional to Jz with resultant shearing of a coherent state intoa metrologically useful spin-squeezed state [14]. Such states are also deterministically generated,
independent of detector performance. This Hamiltonian has been realized with optical cavities
and in other physical systems including collisional interactions [48–55]. Achieving these dynamics through cavity-mediated interactions has been described [30, 31] and demonstrated [25, 56]
through an approach sometimes called cavity feedback. As with other squeezing mechanisms, there√
is a scaling of the observable squeezing W ∝ 1/ N C set by dissipation. Much work has gone
into characterizing the role played by curvature of the Bloch sphere, which sets a theoretical limit
2of W ∼ 1/N 3 [31]. In contrast, interest in achieving the two-axis countertwisting Hamiltonian


ˆ
ˆ
ˆ
ˆ
ĤTAT = χ Jz Jy + Jy Jz is driven by its Heisenberg-limited scaling, W ∼ 1/N .
In the cavity-mediated one-axis twisting scheme, a fixed frequency drive tone is offset on theslope of a dressed cavity resonance by δp & κ/2 (Fig. 2.3(a)). The spin populations tune the cavity
closer to or further from resonance with the drive tone such that the intracavity probe intensity islinearly dependent on Jz . This light creates an AC Stark shift on the states that therefore depends
on the population of atoms in |↑i. In practice, the coherent state of an ordinary laser leads tophase fluctuations imparted by photon shot noise and a Hamiltonian Ĥ ∝ â† âJˆz . Still, the unitaryinteractions drive shearing of the atomic quantum noise distribution with a resulting squeezed
state minimum noise projection oriented at a small angle α0 from ẑ (Fig. 2.3).
Experimentally, the protocol is simplified if one works symmetrically-detuned so as to beinsensitive to atom number fluctuations, and with δp = κ/2 to suppress free-space scattering. In
our system, the cavity is near the |↑i → |ei transition. Operating at larger detunings both enables

19
a)

b)

c)

OAT

-100

0

100

200

300

400

Frequency offset fromdressed cavity resonance [kHz]Figure 2.3: (a) For one-axis twisting, a tone is applied at a fixed frequency δp from the dressedcavity resonance. The amount of light entering the cavity interacts with the quantum state as adifferential light shift, driving shearing of the quasi-probability distribution (pink) seen (b) in the
Jy -Jz plane or (c) on the Bloch sphere.

linearized regime even in the presence of shot-to-shot total atom number fluctuations and reduces
the QND interactions that can compete with the desired one-axis twisting interaction [57]. In general, as N C increases, one finds an optimal balance between scattering and QND broadening with
δp further from cavity resonance. Finally, it is worth mentioning that one-axis twisting interactions are also amenable to extensions such as two-axis twisting for more squeezing [31] and signal
amplification schemes [58–60].

2.4

Description of a Raman Mach-Zehnder interferometer
The toolkits of modern interferometers include various techniques: Bragg pulses (especiallyfor large momentum transfer), Bloch oscillations, conjugate geometries, etc. In this dissertation, we
opt to build a relatively simple light-pulse Mach-Zehnder interferometer using stimulated Raman
transitions. This technique was first explored in Ref. [42] using moving beams of sodium atoms.
This interferometer’s sensitivity to gravity g originates from either Doppler shifts of the Raman
tones or from laser phases imprinted at the position of a wavepacket. There are many excellent
resources and reviews for matter-wave interferometry that explore the origins and equivalence of
these ideas in further detail [61–64].
For our Raman transitions, there are two counter-propagating beams with wavevectors k~1

20
and k~2 ≈ −k~1 and optical frequencies ω1 and ω2 (Fig. 2.4). The beams have a difference frequency
ωL = ω2 − ω1 near ωHF , resonant with the ground states of a three-level atom. For simplicity, we
write k ≡ k1 = −k2 . The electric fields of the lasers are
0

E1 (z, t) = ˆ1 E0,1 ei(ω1 t−k1 z+ϕ1 (t)) + c.c.

(2.7)

0

E2 (z, t) = ˆ2 E0,2 ei(ω2 t−k2 z+ϕ2 (t)) + c.c.

(2.8)

e
~

D
E
~ 0,j ↓ eϕ0j and initial↑ ~r · E

with polarizations ˆ1 , ˆ2 , single-photon Rabi frequencies Ωj =phases ϕ01 , ϕ02 . The phase difference between the two Raman lasers at the center of the wavepacket
at a certain point in time will be denoted with φ. The intermediate state is adiabatically eliminatedto reduce the problem to that of a two-level atom (see App. B or [61, 65]). The resulting dynamics
are described by the Rabi formula using two-photon Rabi frequency Ω↓↑ =Ω∗1 Ω22∆ and the two-

photon detuning δ of the lasers from the atomic transition frequency. When the lasers are not quiteqon resonance, the effective Rabi frequency is ΩR = Ω2↓↑ + δ 2 .
a)b)

Figure 2.4: (a) Energy-level diagram for Raman transitions for interferometry. The excited state |ei
has negligible population and will be adiabatically eliminated. (b) Beam configuration for velocitysensitive Raman transitions.
In the presence of the lasers, the energies are ~ω↓ and ~ω↑ for the atomic states |↓i and |↑i.
Consider a cloud of 87 Rb atoms cooled, prepared in |↓i, and selected with the same momentum
p = mv0 along the cavity axis. Absorption from one beam and emission into the other necessitates
2~k momentum kicks, and the internal spin state and the external momentum state of the atom

21
are correlated as a result. In particular, atoms in |↓, pi are coupled to |↑, p + 2~ki. Other states are
neglected for this discussion as non-resonant, unpopulated, or adiabatically eliminated, and hence
we will omit the momentum labels. Note that in an atom’s frame of reference, the frequency ofeach Raman tone is Doppler shifted kv for an atom at velocity v = gt after accelerating for time t,
and hence the Raman resonance is shifted by (k1 − k2 )gt. The total detuning from resonance δ isdetermined not only by the laser difference frequency ωL but also by the atom’s velocity, the recoil
shift, AC Stark shifts, etc.,δ = ωL − ωHF − (k1 − k2 )v − ~(k1 − k2 )2
2m

(2.9)

Mach-Zehnder interferometers are characterized by a π2 − π − π2 pulse sequence separated
by free evolution time Tevol as described in Sec. 1.6. The π/2 and π pulses have durations of τ and
2τ , respectively. The effects on the probability amplitudes C↑ and C↓ from each pulse and freeevolution are given byC↓ (t)
C↓ (t0 )

=M
C↑ (t)C↑ (t0 )with evolution transfer matrix [66, 67]



0
ΩR t0
cos
e−iω↓ t2M (t0 , t, ΩR , φ) = 00−ie−i(ωL t0 +φ) sin ΩR2 t e−iω↑ t(2.10)−iei(ωL t0 +φ) sin
cos



ΩR t0
2





ΩR t0
2



e−iω↑

0
e−iω↓ tt0
(2.11)

with t0 ≡ t − t0 . The free evolution matrix simplifies with ΩR = 0 to
−iω↓ Tevole
M (Tevol ) = 00e−iω↑ Tevol.

(2.12)

The full transfer matrix for the Mach-Zehnder interferometer is the product of five matrices,
MMZ = M (2T + 3τ, φ3 , Ω, τ )M (T + 3τ )M (T + τ, φ2 , Ω, 2τ )M (τ )M (0, φ1 , Ω, τ )(2.13)

where the phases φ1 , φ2 , φ3 are the Raman tone phase differences imprinted at the position of
the center of the wavepacket during each of the Raman pulses. For population beginning in |↓i,22C↓ (0) = 1, the probability of measuring an atom in |↑i after the interferometer gives the interferometer fringeP↑ (2T + 3τ ) =1 − cos (φ1 − 2φ2 + φ3 )2

(2.14)and the interferometric phase Φ = φ1 − 2φ2 + φ3 comes from the relative phase accumulated
between the two arms of the interferometer as before. These phases can be evaluated with φ(t) =Rt00
t0 δ(t ) dt at φj = φ(tj ) where t0 is the time of the first π/2 pulse.
As atoms accelerate due to gravity, the differential Doppler shift of the counter-propagating
tones chirps linearly, and the difference frequency is ramped at b = 2kg to remain on resonance.
Evaluating Φ with ω1 − ω2 − (k1 − k2 ) · gt gives, to lowest order, the Mach-Zehnder interferometer
phase shift:2Φ = (2kg)Tevol.

(2.15)

Experimentally, the acceleration g is found by determining the value of b required to cancel thephase shift induced by gravity,2δΦ = (2kg − b)Tevol,(2.16)In reality, the interferometer is also sensitive to phase noise from the Raman pulses, mirror vibrational noise, and to all mechanisms that modify the transition frequency during the sequence.
Differentiating between a signal that arises from gravity versus some noise is a significant challenge, so to understand the sensitivity of this interferometer to gravity, laser phase noise, vibrations, and other effects, we consider δΦ, the change in interferometric phase due to an infinitesimalphase jump δφ at time t:δΦ(δφ, t)
.
δφ→0
δφ

gs (t) = lim(2.17)

An interferometer is ordinarily operated on the side of fringe, Φ = π/2, for maximum sensitivity.
For infinitesimally short Raman pulses, the effect of a phase excursion during the evolution timeyields δP (|↑i) ≈ ±δφ/2 where the positive and negative responses differ for the first and second

23
windows. To model the effect of an excursion during a finite pulse, we break the transfer matrix
into two composite pulses which occur with phases φ and φ + δφ. Following [66], we shift the time
origin to be at the middle of the π pulse so that we can write the transfer function, odd and given
only for t > 0 (Fig. 2.5(a)),
sin (ΩR t)
0<t<τ





gs (t) = 1
τ < t < Tevol + τ







− sin (ΩR (Tevol − t)) Tevol + τ < t < Tevol + 2τ

(2.18)The interferometric phase shift can then be calculated for an evolving differential phase between
the Raman tones φ(t) as
Z ∞
δΦ =

gs (t)
−∞

dφ(t)
dt.
dt

(2.19)

The frequency-dependent transfer function of the interferometer H(ω) comes from considering its
response to a sinusoidal modulation φ(t) = A0 cos (ω0 t + ϕ). The rms response, averaged over aR∞
random distribution of ϕ, gives δΦrms = |A0 ω0 G(ω)| where G(ω) = −∞ e−iωt gs (t) dt is the Fouriertransform of the sensitivity function. The transfer function H(ω) = ωG(ω) yields the rms interferometric phase noise,2

Z ∞(σΦ,rms ) =|H(ω)|2 Sφ (ω) dω

(2.20)

0for some power spectral density of Raman phase noise Sφ (ω) (typically to be compared to thestandard quantum limit in radians). Explicitly, the Mach-Zehnder transfer function is
4iΩR
H(ω) = ω 2sinω − Ω2R

ω (Tevol + 2τ )
2




cos

ω (Tevol + 2τ )
2



ΩR
+
sin
ω



ωTevol2(2.21)

which is plotted for our experiment in Fig. 2.5(b). The next steps in the procedure depend on theeffect Sφ considered and are continued in [64, 66]. In our system, Raman laser frequency noise issuppressed by symmetrically detuning the tones from two cavity modes. Mirror vibrations alter
the reference frame and appear as real inertial forces (Sec. 7.5.5) and start to become relevant close
to the longest evolution times we use.

24

b)

Sensitivity function

+1

0

-1
-1.0

-0.5

0

Time [ms]

0.5

1.0

Mach-Zehnder transfer function

a)

101
100
10-1
10-2
102

103

104

105

Frequency [Hz]Figure 2.5: The Mach-Zehnder sensitivity transfer functions (a) gs (t) and (b) |H(ω)|2 plotted forour experimental parameters (ΩR = 2π × 10 kHz, Tevol = 1 ms, τ = 25 µs). For the overall transfer
function |H(ω)|2 , oscillations are evident at 1/(Tevol + τ ) and the low-pass effect is set by 1/(ωΩ2R ).
Oscillations above 1 × 104 Hz are smoothed with rms values over a cycle.

Chapter 3
Experimental apparatus

3.1

Overview
The electronic structure and moderate mass of rubidium make it one of the simplest atoms

to study, and the maturity of related technologies, e.g. optical coatings, laser diodes, commercial
Bose-Einstein condensate sources, etc., make it among the most practical. In atomic physics, the element remains among the most active scientific testbeds for exploring fundamental physics throughquantum information and precision measurements. Since first light in 2007, the Thompson lab
rubidium apparatus has intentionally facilitated flexible, principally proof-of-principle projects.
Apart from a common preparation paradigm – collect atoms in a 3D magneto-optical trap (MOT),
cool into a red-detuned intracavity lattice, optically-pump into an initial state – the projects havevaried significantly.
First spin-squeezing
17.7 dB, deterministic,
Calculations,
Spin-squeezing theory,
experiments,and homogeneous
construction, setup
10.2 dB squeezing
3.4 dBsqueezing
Calculations, setup,
Rb superradiant laser,
Entanglement-enhanced
Supercooling, Raman
Synchronization andtrapping, cooling,magnetometer,
atom interferometry,
SWAP coolingdephasingmicrowaves
dephasing physics
site-dependent selection

2007

First light

2008

2009

Vacuum Rabi
splittings

2010

2011

Sr superradianceproject begins

2012

2013

2014

2015

“First generation”
apparatus

2016

2017

2018

“Second generation”
apparatus

2019

2020

2021

Figure 3.1: A timeline of Thompson lab rubidium experiment developments. The top labels attempt to delineate distinct experimental directions. The bottom labels mark a few important eventsand mark the first and second generation apparatus described in this work.

26
A timeline of the history of the apparatus is provided in Fig. 3.1. Two distinct systems wereused during the work of this dissertation. The “first generation” apparatus detailed below was
moved and improved during the summer of 2014. It was used beginning with various spinsqueezing experiments and retired at the end of the SWAP cooling experiment in 2017. For the“second generation” apparatus, the cavity was redesigned, a 2D MOT was added, and the vacuum system was rebuilt with future experiments in mind.
In this chapter, experiment subsystems are described with particular attention towards improvements unique to the second generation apparatus. A comparison of the typical experimental
regimes is provided in Table 3.1. The rest of the chapter elaborates on the control system, vacuum
chambers, glass cells, science cavities, atom sources and 2D MOT, magnetic coils, lasers, homodyne
detector, microwaves, and transfer cavity.

Description

SymbolFirst gen.

Second gen.

Typical atom number
Rb transitionN

5 × 105
Stretched/cycling

1 × 103
Clock

Cavity linewidth (FWHM)
Cavity finesse
Cavity length
Cavity free spectral range
Mode waist
Cooperativity

κ/2π
F
L
FSR/2πw0C3.15(10) MHz
2532(80)
1.85 cm
8.105 GHz70(1) µm
0.056(7)56.2(2) kHz
132000(1000)
2.21 cm
6.788 GHz
72(1) µm
2.76(5)

Table 3.1: System parameters for both cavity QED systems given for 780 nm light. Symbols are used
4g 2

throughout the dissertation. Cooperativities C = κΓ0 are given for the optical cycling transition
assuming homogeneous atom-cavity coupling.

3.1.1

Control system and software
All of the experiments here are coordinated with software written in LabVIEW. The main

program allows easy modification of control sequences, compiling spreadsheet-style input into
instructions for several analog and digital input/output boards (PN: National Instruments PCIe

27
6259, PCI 6233, PCI 6733), direct digital synthesizers (DDSs, PN: Analog Devices AD9959), andCCD or CMOS cameras (PN: Basler scA640-70gm, scA750-60gm; FLIR Chameleon3 CM3-U3-1352M).
The program also interfaces with several commercial instruments and provides a host of customtailored monitoring and data analysis tools. Sequences typically run with 1 Hz repetition rates
synchronized with a 60 Hz line voltage trigger. Additional analysis is performed with Igor Pro,
Mathematica, and Python.

3.1.2

Experiment chamber and vacuum system
The structural foundation of the apparatus is a spherical cube from Kimball Physics with 4.5”

top/bottom, 2.75” side CF flanges. The top port supports the glass cell where the atoms are trapped
and manipulated. The side and bottom ports are used for attaching electrical feedthroughs, vacuum pumps, and for the first generation, a rubidium source. One side port is reserved for an
AR-coated window, and beams guided through this window bounce off a 45◦ mirror to couple
into the cavity.
An ultra-high vacuum pressure is maintained within the experiment chamber as is necessary
for long trap lifetimes. During the initial pump down with a dry scroll pump and turbo pump, it
is imperative to wrap the apparatus with resistive heater tapes and “bake out” contaminants formany hours. After the initial pump down, low pressure is maintained within the science cell byan ion pump. For the first generation, a Varian VacIon Starcell (50 L/s, PN: 9191340) ion pumpran continuously to achieve typical pressures of 10−9 Torr. For generation two, it was replacedwith a 75 L/s pump from Gamma Vacuum (PN: 75S-DI-2V-SC-N-N) that reaches pressures in the
10−8 Torr range, limited by the low-temperature bakeout necessitated by the construction of thescience cell. Pressure P is read out by the pump controller’s current and corroborated with thealkali-atom rule of thumb P ≤ 2 × 10−8 Torr · s/τ for MOT 1/e loading time τ [68]. A titanium
sublimation pump is occasionally activated to little overall effect. Vacuum pressure inside the2D MOT cuvette is maintained with a non-evaporable getter (NEG) for differential pumping as
described in Sec. 3.1.5.

28
3.1.3

Glass science cell
Different glass cells are used to house the first and second generation cavities, providing

ample optical access without the eddy currents associated with a metal chamber. Both cells are
fused to a glass-to-metal transition and 4 12 ” CF flange. The flange is attached to the top port of thespherical cube. The dimensions of the cells are approximately 2” × 2” × 5”. These dimensions are
kept tight to the cavity spacer for vacuum considerations and so that a microwave antenna can beplaced fairly close to the atoms for fast hyperfine transitions.
The first generation apparatus uses a borosilicate glass (Borofloat) cell without anti-reflective(AR) coatings. As a result, each beam passing through the cell loses about 4% of power per surface, creating inconvenience when retroreflection is desired. Nonuniform wall thickness leads to“fringes” in the MOT beams that cause fluctuations in the MOT position. The design for the second generation cell has two significant changes. First, fused-silica (“quartz”) is used for its smaller
thermal expansion coefficient (5 × 10−7 K−1 ) compared to Borofloat (3 × 10−6 K−1 ). This left thepossibility of heat fusing AR-coated plates without cracking the coating. Second, a 16 mm outer
diameter quartz tube protrudes from one corner of the cell to attach a 2D MOT.
In the end, the JILA shop used a sodium-silicate bonding technique [69] to construct the second cell from 5” × 2” × 0.12” AR-coated plates from Specialty Glass Products. The AR coating is
specified to have less than 0.2% (0.6%) reflectance at 0◦ (45◦ ) angle of incidence at 780 nm. Additional right-triangular prisms from Esco Optics buttress the interior corners. Initial pump downs
repeatedly revealed vacuum leaks along one edge that could not be filled by vacuum-compatible
silicone sealer, so a TorrSeal-like epoxy was heavily applied in an act of desperation. Although theleaks stopped, the final pressure was limited by a bakeout that only reached 80◦ C.

3.1.4

Science cavity and spacer
An optical cavity is a resonator for light waves consisting of mirrors fixed to a spacer. Suchcavities are vital to our experiments in a number of ways. The descriptions of a cavity field as a har-29
monic oscillator and an atom as a two-level system form a versatile playground for modeling manyother physical systems. The two mirrors set boundary conditions on optical fields that confine thegeometry of the experiment. High finesse is akin to high optical depth, increasing the strengthof infinite-range atom-light interactions. Collective effects arising from the indistinguishability ofatoms in the cavity form the basis of our non-demolition measurements and sensor readout.
The squeezing experiments of Chapter 6 and the interferometry experiments of Chapter 7
rely on separate optical cavities. In both cases, two super-polished dielectric coated mirrors with
7.75 mm diameter, 4 mm thickness, and 5 cm radius of curvature are fixed to a rigid spacer toform a vertical, nearly confocal Fabry-Pérot resonator. Both mirrors are epoxied with TorrSeal
to Piezomechanik HPCh 150/12-6/2 piezos for feedback up to kHz frequencies limited by the resonances of the loaded piezos. The transmission of the bottom (input) mirror is designed to besignificantly higher than the top mirror, allowing ∼ 95% of incident photons to return towards a
homodyne detector. The dominant difference between the cavities is that the interferometry cavity
had 50× higher finesse. A summary of the other relevant experimental parameters is included inTable 3.1.
The first generation science cavity was used for spin-squeezing demonstrations between August 2014 and June 2016. The spacer was designed with optical access and rejection of acoustic
noise in mind [70]. It was machined by hand out of a single block of Zerodur (Fig. 3.2(a)). A Macor
jig was used for attaching the mirrors and piezos used in stabilizing the cavity length. Mirrors
were supplied by Advanced Thin Films.
For the second generation cavity, a new spacer was needed that would not impede atoms
coming from the 2D MOT. This spacer was machined out of Macor on a CNC machine (Fig. 3.2(b)).
An “adjustable sled” design allows tuning the cavity length such that the free spectral range (FSR)
is nearly equal to the rubidium hyperfine transition frequency, ωHF ≈ 6.835 GHz, L ≈ 2.19 cm.
The piezos have a maximum throw of about 4 microns, and our target FSR is constrained to about
20 µm so that a reasonable amount of Raman light can enter the cavity non-resonantly. Although it
would be possible to machine within this tolerance, accounting for the width of piezo, glass, epoxy,

30
a)

b)

c)
piezo

cavitymirroradjustablesled piececavitymirrors

tripodguide holes

g2

planarour cavities

g1
piezo
concentric

confocalFigure 3.2: Optical cavities and spacers. (a) The first generation Zerodur spacer houses a moderate finesse cavity for generating large amounts of spin-squeezing. (b) The second generation
Macor spacer supports a high finesse cavity. Ceramic screws are used to fine-tune the length fora free spectral range near ωHF . Although the spacer is pictured horizontally here, it is mounted
vertically on a tripod of Macor rods for free-fall interferometery. Both images were rendered in
Solidworks. (c) A cavity-stability diagram shows both the first and second generation cavities arewithin the shaded island of stability: 0 ≤ g1 g2 ≤ 1 where gi ≡ 1 − RLi for mirror radius of curvature
Ri and cavity length L [26].

and mirror coatings seemed risky. The original cavity length set using calipers was off by roughly
700 µm, but this was easily corrected and the ceramic screws and nuts were tightened to fix the free
spectral range at ωFSR = 6.788 GHz. Although the vibrational modes of an asymmetric spacer areconcerning, finite element modeling in COMSOL predicts the lowest resonant eigenfrequencies
to be around 5.2 kHz (bending) and 6.1 kHz (torsional), significantly above the lowest piezo resonance frequency. The cavity is mounted vertically on a tripod of Macor rods which are themselves
fixed on a Macor base supported by four alumina posts. Small Viton balls dampen the vibrational
couplings between parts. The rest of the complex vibration isolation system described for previous iterations of the experiment was abandoned when moving the first generation apparatus – theheavy hanging mass had a tendency to act as a pendulum.
The mirrors, which come from FiveNine Optics, are coated for high reflectivity between
780 nm and 795 nm for high finesse. The coating also has moderate transmissivity at the 760 nm
and 820 nm lattice wavelengths for ease of use and to avoid parametric heating. The linewidth ofthe cavity is measured by the ring-down (exponential decay) of transmitted light after switching off

31
a resonant laser. In Fig. 3.3(a), a photodiode (PN: Hamamatsu S5971) with 10 kΩ transimpedance
resistor collects the decaying field from the sideband of the 780 nm Raman laser after the EOM
modulation has been disabled with a fast switch (PN: Mini-Circuits ZASWA-2-50DR+). The exponential fit gives κ = 1/τ = 2π × 56(2) kHz. This should be compared to the measurements two
years prior (2π×51.2(2) kHz) and to the design linewidth (2π×50.5 kHz, assuming 2 ppm loss per
mirror). From the mirror specifications (Fig. 3.3(b)), the linewidths at {760 nm, 795 nm, 813 nm} areTransmission
photodiode signal
[mV]

a)

b)

150
100
50
0

0

5

10

15

transmissivity[ppm]

approximately 2π×{160, 60, 170} kHz.

1000
100
10

20

760

Time after turning off laser [µs]c)-FSR/2

HG1,1

HG0,16

-3

-1

-2

800

820

Wavelength [nm]
HG0,0

HG0,1

1

0

2

3

FSR/2

Frequency from TEM00 [GHz]d)Atoms loaded
into lattice (×103)

780

400
300
200
100
0

400

500

600

700

800

900

1000

Cavity detuning from F=2 to F’=3 [MHz]Figure 3.3: Second generation cavity linewidth, transmissivity, and atom loading fluctuations.
(a) Cavity linewidth ring-down measurement for 780 nm light, after guiding rubidium ensembles
into the bottom mirror for over 2 years. (b) Specified mirror transmissivity for the lossy mirror inparts per million (ppm). The transmission coefficient of the other mirror, and loss coefficients forboth mirrors, are close to 2 ppm. (c) The predicted transverse mode frequencies of the 100 lowestorder HGm,n modes for our second generation cavity. For a given HG0,0 mode, the frequency
difference with the closest HGm,n mode is plotted. Modes are clustered with characteristic spacingnear 110 MHz. (d) The number of atoms loaded into the lattice as a function of atomic detuning(i.e. cavity length). Pink rectangles are separated by about 110 MHz as the spacing of transverse
modes coincides with inhibited loading.

32With the high-finesse second generation cavity, we observe superradiance-inhibited atom
loading at certain cavity lengths. For mirrors with R1 = R2 = 5 cm radius of curvature separatedby L = 2.2 cm, the HGm,n mode frequencies are found by solving1
ωL√− (n + m + 1) arccos( g1 g2 ) = qπ
c

(3.1)

with longitudinal mode number q and gi = 1 − RLi . Although an HG0,0 mode and HG0,1 modeare split by 2.1 GHz, the nearest HG8,8 mode is only 110 MHz away from HG0,0 (the longitudinalmode numbers differ by 5). It turns out that the transverse modes of this cavity are all clustered ingroups with spacings of about 110 MHz (Fig. 3.3(a)). One consequence is that we must be wary ofaccidentally inserting tones on higher-order modes when using the cavity as a frequency filter.
The other practical impact of the higher-order transverse modes is that certain cavity detunings facilitate synchronization within the cavity that lead to superradiant emission. If a sufficientnumber of atoms overlap with the HGm,n cavity mode while loading a MOT such that the system enters the collective strong coupling regime, the dynamics of the collective excitation dominate over single-particle decoherence and the result is a pulse of light out of the cavity. Theselight pulses were seen on a CCD and correlated with reduced and fluctuating numbers of atoms
trapped in the lattice for our experiments (Fig. 3.3(b)). Our solution was to operate the experiment
in between the detunings that support this phenomenon.

3.1.5

Atom source and the 2D MOT
Prior to the interferometer experiment, the atom source was a rubidium ampoule located in a

glass arm at the bottom of the vacuum chamber. Resistive bands warm the container to anywhere
between 30◦ C - 100◦ C, creating enough background vapor pressure in the main chamber to loada MOT. One piece of lab lore holds that it is always difficult to see a MOT in the first 24 hours ofbreaking vacuum. Perhaps by the time rubidium has sufficiently coated the inside of a chamber,enough vapor pressure has accumulated to support a MOT.
1

By inverting these equations, the measured transverse mode splitting and L (from the free spectral range) provide

a very accurate check on the radius of curvature, assuming the mirrors are symmetric.

33
It should also be mentioned that in one incident, we flooded the chamber with far too muchrubidium vapor. Within an hour, the cavity finesse at 780 nm appeared about three times lower.
A makeshift cryopump, formed by cooling the bottom of the chamber with dry ice, did not help
restore the finesse, so we cleared the vapor with a turbo pump. However, the finesse of the cavity
remained spoiled in a detuning-dependent way. As a last ditch effort, we ran current through a pairof heater coils wrapped around the mirrors. This worked for two weeks before the wires failed due
to excessive current, after which the cavity finesse was reduced in a detuning-independent way.
The 45◦ mirror was also coated with what was presumably rubidium.
The second generation system is designed specifically with a spin-squeezed interferometerin mind. To preserve the high-finesse mirrors, we opt to load from a 2D MOT rather than to fill themain chamber with a background vapor [71]. The 2D MOT setup is shown in Fig. 3.4. Atoms arecollected inside a 1” × 1” × 2” rectangular Pyrex cuvette and pushed through a 1.5 mm aperture
towards the main chamber. There are two Rb dispensers (PN: SAES RB/NF/3.4/12 FT10+10) and
one neutral evaporative getter (NEG, PN: SAES St 171 LHI/4-7) to provide differential pumping
within the cuvette. One Rb dispenser is typically run at 3.0 A during the day, the other kept readyin case of emergency, and the NEG runs continuously at 2.3 A after activation. The electrical leads
are welded to stainless steel wires and isolated with a ceramic spacer disk and glass tubing. A
polished (reflective) copper disk with 0.8 mm aperture is welded to a stainless steel tube to allow
for differential pumping. The extensive experience of Hans Green and Dana Anderson’s groupwere very helpful while designing this system.
The 2D MOT and repumper beams use roughly 50 mW and 2 mW, respectively. Cylindrical lenses create an elliptical beam profile that is sent to the long faces of the cuvette along twopaths. The polarization is then circularized, and another quarter-wave plate in conjunction with aretroreflective mirror ensures that the counter-propagating beams also have the requisite polarization. A small amount of light from the MOT laser is split off to form a “push beam” incident alongthe 2D MOT axis, accelerating the atoms toward the main chamber by radiation pressure force.
Light reflecting off the copper disk also provides some transverse cooling to the atoms which is34b)

Repumperf=5.08 cm

λ/2

f=2.54 cm

cyl.

PBS

f=20 cm

PBS

λ/2

10

20

30

40

1

[cm]

MOT

0

λ/2

0

λ/4
-1

λ/4

λ/2

f=20 cmcyl.

Magnetic field [Gauss]

-2

Push
beam

1

2

[cm]

2D MOT
beams

PBS

0

-1

c)
to sciencechamber

1

[cm]

a)

λ/4

0

-1

vacuumfeedthrough

λ/4

-1

0

1

[cm]

d)

Pyrex cuvette

Rb dispenser

copper aperture
NEG

glass spacers
stainless steel tube

to science chamberceramic spacer

Figure 3.4: The 2D MOT setup. (a) Beams from the MOT and Repumper DBR lasers are overlapped to form the “2D MOT beams” and the “push beam”. Most of the power is sent to the 2D
MOT beams transverse to the rectangular Pyrex cuvette. These beams are shaped by cylindricallenses into ellipses with approximate 1/e2 waists 2 cm × 0.75 cm. Some of the available power isalso sent to a push beam through the end of the cuvette, “pushing” atoms along ẑ towards thescience chamber through a small aperture in a copper disk. Light from the push beam and its
retroreflection on the copper disk also provide necessary cooling. (b) As described in Sec. 3.1.6,
the 2D MOT coils surrounding the cuvette form a quadrupole field with a 24 G/cm gradient at thecenter. The magnetic vector field and magnitude, assuming 1.5 A current, are modeled along the
cuvette and (c) looking down the cuvette towards the chamber. (d) A scaled cross-section of the 2D
MOT setup shows the main chamber vacuum is separated from the cuvette by a copper apertureat the end of a stainless steel tube. Inside of the cuvette region are two rubidium dispensers andone NEG. Carefully placed spacers keep the getter leads electrically isolated.

35
instrumental in observing the 2D MOT.

3.1.6

MOT and bias coils
Magnetic fields are established by running current through magnet coils according to Am-

pere’s law and the Biot-Savart law. The resulting fields are superposed, of course, with static magnetic fields from nearby optical isolators, the ion pump, and the Earth’s interior. All coils describedin this section are outside vacuum, wound with 20 AWG polyester-insulated copper wire (PN:
Belden 8076), and stabilized with Araldite 2011 or 2012 epoxy.
A pair of elongated MOT coils provides the quadrupole field for the 3D MOT. Up to 3 A of
current is run through 128 turns of anti-Helmholtz coils placed in electrical series. Both coils are
10 cm × 4 cm and mounted 5 cm from the center of the cavity. The field gradient produced at thecenter of the cavity is about 34 G/cm, conventional for cold atom experiments.
Three pairs of Helmholtz bias coils are used for positioning the MOT at the center of thecavity, zeroing the magnetic field during polarization gradient cooling, and setting the Zeeman
splitting and quantization axis during various other tasks. To facilitate adding the 2D MOT to the
new science cell, it was necessary to wind new bias coils with more optical access. We chose to
3D print a lightweight set of six square bias coil frames, each approximately 26 cm × 26 cm, fixed
together into a cube by nylon screws. Acrylonitrile butadiene styrene (ABS) plastic is reliable up
to 80◦ C but the coils do not heat significantly. Each coil uses 75 turns of wire. The maximum 1.5 A
current provides a nearly uniform 3 G field. For rubidium, this corresponds to a 2.1 MHz Zeeman
splitting between adjacent magnetic sublevels.
The 2D MOT coils are 3D printed and wound in a similar way. These coils are poweredwith up to 1.5 A in a “racetrack configuration” to produce an elongated quadrupole field [71]. The
frames are roughly 2.5 cm×6.7 cm to fit tightly around the cuvette, and the 2D MOT retroreflection
optics are directly attached to the frame. Each coil is 120 turns, allowing a magnetic field gradient
up to 24 G/cm at the center of the cuvette.

36
3.1.7

LasersNearly all aspects of an atomic physics experiment revolve around the description of a laser

as a coherent frequency source. Lasers prepare ensembles in low-entropy states, carrying away
momentum after addressing atomic transitions. The fields produce potentials to trap, guide, manipulate, and mediate interactions between atoms. The ability to adjust laser frequencies, phases,
and amplitudes offers enormous flexibility. Nevertheless, practical details such as a laser’s spectral
properties, stability against environmental perturbations, and output power also play a role in thefunctioning of an experiment.
Here we distinguish three separate laser architectures: distributed Bragg reflectors (DBRs),
extended cavity diode lasers (ECDLs) with diffraction gratings, and “cateye” interference filter
lasers (a class of ECDLs). The last two technologies have seen significant design and assembly
within our lab. All of the lasers output 60 – 120 mW of near-infrared light. They are driven by
low-noise laser current and temperature controllers built by the JILA electronics shop. Section 4.1
includes an overview of frequency stabilization techniques.
Distributed Bragg reflector lasers generate our MOT, repumper, and optical pumping light.
Commercial DBR architectures are extremely robust against acoustic noise, and the ∼ 500 kHz freerunning linewidths do not inhibit trapping or certain laser-cooling schemes. With the external
feedback scheme described in Sec. 4.3, we use DBRs with linewidths narrowed to about 1 kHz for
probing the atoms, mediating atom-cavity interactions, and driving transitions for interferometery.
All of our DBR diodes are nominally centered near 780 nm (PN: Photodigm PH780DBR080T8 and
PH780DBR120TS).
Extended cavity diode lasers (ECDLs) provide the light for stabilizing the cavity, forming the
far-detuned optical lattices, manipulating the atoms, probing their state, and observing SWAP cooling. These ECDLs use the Littrow design detailed in Justin Bohnet’s dissertation [70]. A diffraction
grating provides optical feedback to the diode. Wideband wavelength selection is made by adjusting a fine-threaded screw to pivot the flexure-mounted grating. A piezo also adjusts the grating

37
angle of incidence with feedforward from the laser current to increase the mode-hop free tuning
range into the multiple-GHz range. The diodes were nominally centered at 780 nm (PN: DL-7140201S) or 795 nm (PN: Eagleyard EYP-RWE-0840-06010-1500-SOT02). The free-running ECDLs have
FWHM Lorentzian linewidths around 200 kHz but the relative noise between the probe laser anda cavity mode was reduced to the equivalence of a laser with 5 kHz linewidth (or lower) using anintermediary locking chain [72]. To make such measurements, the measured phase noise of thelocked probe laser is approximated as a flat power spectral density of instantaneous frequency
noise Sν (f ) ≈ S̃ν which is itself related to the Lorentzian FWHM ∆ν through ∆ν = π × S̃ν [72].
An interference filter laser concept has gained recent popularity, promising improved stability against acoustic noise at similarly narrow linewidths [73, 74]. We designed and built a version
used for the 760 nm blue-detuned lattice laser, and strontium-project labmates have built additionalvariations. Our implementation, shown in Fig. 3.5, forms a linear cavity between the rear reflecting
surface of a diode centered at λ0 = 760 nm (PN: Eagleyard EYP-RWE-0760-02010-1500-SOT12) anda 10% reflective output coupler placed at the focus of a pair of lenses. This “cateye” configurationof lenses results in reduced sensitivity to processes that would cause misalignment. A Fabry-Pérot
etalon (0.3 nm FWHM interference filter centered at λ0 = 767.9 nm, PN: Iridian GX000004) was
held in a micrometer-adjustable rotation mount. The angle of incidence θ between the optical pathq0
and the beam selects a single external cavity mode according to λ = λ0 1 − (sin(θ)/neff )2 for index of refraction neff ≈ 2. The temperature is stabilized with two thermoelectric coolers (TECs, PN:
LairdTech UT8-12-25-F2) placed in series between the aluminum base and heat sink block, but acommercial diode holder with integrated TEC may be a better option for the future. When designing for maximal stability against perturbations, care should be taken to avoid springs, especiallyin mounting the output coupler and diode.
The lasing mode is the result of a competition between several gain factors: the semiconductor gain profile, the interference filter’s transmission window, the diode cavity from rear to front
facet (L = 1.5 mm), and the external cavities from each diode facet to output coupler (L = 65 mm)√√[75]. Each cavity has an associated finesse F = 4 r1 r2 /(1 − r1 r2 )2 for amplitude reflection coef-

38
a)

b)

Diode gain

Diode cavity

DiodeCollimating
lensInterference
filter

f=4.5 mm16°

Cateyelens

Outputcoupler

f=11 mm

R=10%

Piezo

Cateyelens

f=18 mmInterference filter

External cavities

394.2

6.5 cm

394.4

394.6

394.8

Frequency [GHz]

Figure 3.5: Interference filter lasers. (a) Schematic for the 760 nm interference filter extended
cavity diode laser. (b) An example interference laser mode spectrum (black) is the product of thediode gain (green, assumed to be a relatively broad Gaussian), diode cavity (red, L = 1.5 mm, antireflective coated facet r2 = 0.017 specified,√r1 = 0.92 estimated), external cavities (gold, L = 65 mm
and 66.5 mm, r1 = 0.017 and 0.92, r2 = 0.1), and interference filter (blue, 0.3nm FWHM). The
highest peak determines the laser oscillation frequency. Adapted from [75].

ficients r1 , r2 , giving rise to an Airy function mode structure,
T (ω) =

1,
1 + F sin (2nωL/c)
2

(3.2)

with length L and index of refraction n. The laser oscillates at the frequency for which the product of these contributions, which may be unstable in time or temperature, is largest. As seen in
Fig. 3.5(b), the long external cavity gave narrow resonances with depth determined by the amountof feedback, but a change in the external cavity length is liable to pull the frequency or mode-hop
the laser. The output coupler is therefore mounted on a ring-actuator piezo (PN: Piezomechanik
HPCh 150/12-6/2) to stabilize the cavity length and to feedforward on current, extending the mode
hop-free tuning range to over 10 GHz. The linewidth is measured to be near 40 kHz using the power
spectral density of transmitted light when locked on side of fringe to a broad cavity. This linewidthis also corroborated by measuring the transmission lineshape as the laser is swept through a narrow cavity resonance. Tests on the strontium-project interference filter lasers have shown longer
external cavities are correlated with lower linewidths, as is to be expected.
Tapered amplifiers (TAs) (Eagleyard gain chips PN: EYP-TPA-0780-00500-3006-CMT03-0000
and EYP-TPA-0780-01000) are required to reach the optical powers needed for a MOT and for39
Raman sideband cooling. The lifetime of lasers and TAs can be extended considerably by limiting
the operating current and powering off overnight. Our older TA has operated at 1500 mA out of
3000 mA for over 10 years, which seems to exceed other reported lifetimes.

3.1.8

Homodyne detector
We use balanced homodyne detection [76] to obtain information about the atom-cavity sys-tem. In homodyne detection, a strong local oscillator (LO) field is overlapped with the signal field
at a non-polarizing beam splitter cube and sent into two photodiodes. The difference of photocurrents is a direct DC measurement record of one quadrature of the signal field. By definition, theLO and signal are at the same frequency, and the measured quadrature is selected by adjusting thephase of the LO. Measuring the I quadrature gives information about the signal field’s amplitude,
but the Q quadrature contains all of the field’s information when the probe is on cavity resonance.
Compared to heterodyne detection, which can generally measure the useful quadrature only 50%of the time, homodyne detection has no fundamental reduction in quantum efficiency. In addition,
noise that is common-mode to both photodiodes cancels out. The main drawback for the technique
is its sensitivity to relative path length fluctuations between the LO path and the signal path, so
the relative phase between these two fields must be stabilized.
We use the circuit in Fig. 3.6 to measure the response of the probing field’s Q quadrature to
the presence of atoms in a cavity. This detector was designed, built, and tested by Elissa Picozzi,
Kevin Cox, and James Thompson. The photocurrents of two photodiodes (PN: Hamamatsu S5972)
are combined to form an AC output, a DC DIFF output, and a DC SUM output. To stabilize the relative
phase against path length fluctuations, light near the Rb D2 -line is split into a probing path andan LO path. The probing path is incident on the cavity and includes both a carrier path lengthstabilization tone (rejected at the cavity) and small fs = 81.1 MHz EOM sidebands, the lower ofwhich is the atomic probe sideband tone near cavity resonance. The LO path is shifted to be near theatomic probe tone’s frequency with an 81.1 MHz AOM before being sent to the detector. Using the81.1 MHz beat note from the detector’s AC output, the relative phase is stabilized using a phase-

40

3pF

50kΩ

S5972

0.1uF

1kΩ

100Ω

18H

+15V

AD8015

10uF

--

+

1kΩ

OPA656UAC
output

3pF

3kΩ

3kΩ

2490Ω

+
+

2kΩ

-15V

S5972
100Ω

1kΩ
18H

2kΩ

OPA656U

×3 Gain
SUM output (DC)
50Ω

+

+

50Ω

6kΩ

50kΩ
0.1uF

AD825AR1kΩ

×1 Gain
DIFF output (DC)AD825AR6kΩ

Figure 3.6: Homodyne detector circuit diagram, omitting power supplies and bypass capacitors.
The AC output uses a high frequency transimpedance amplifier (PN: AD8015) for the 80 MHz path
length reference tone. The DIFF path is used for probing the number of Rb atoms in |F = 2i with abandwidth from DC to 1.5 MHz. The SUM path is used in calibrating probe power. A potentiometeron the DIFF stage allows cancellation of intensity noise on the homodyne reference tone to better
than 3 × 10−4 .

frequency detector that feeds back on the VCO that drives the LO path AOM. This setup is shownin Fig. 3.7(a). The bandwidth of the path length phase lock is about 50 kHz.
The phase φs of the path length stabilization reference frequency selects the detected quadrature. Reference [29] introduces an invaluable model for the full complex field response normalized
to the incident field when probing the atom-cavity system. A useful result is the bare cavity reflected field response for linewidth κ and loss rate from the transmissive mirror κ1 , written as afunction of the detuning from resonance δp :0 atoms

F (δp ) ≡ Ir + iQr −−−−→ F (δp ) ≈ 1 −κ1.
iδp + κ/2

(3.3)

We detect the reflected field of an asymmetric (mostly-reflective) cavity, so the phase quadrature
Qr appears as a dispersive lineshape when sweeping through resonance. As described in Sec. 1.5,
the presence of atoms in |↑i shifts the dressed cavity frequency, and the challenge is to estimate
this frequency shift to infer the atom number (Fig. 3.7(b)). The phase quadrature thus gives maximal sensitivity for resolving small frequency shifts, and hence atom number, when parked on

41a)

Homodyne detection and path length stabilization

Atomic probe

Hold probe frequency if probe carrier is off
IN PFD

DC out(×2)

REF

AC outMonitorϕs

DDS

detector

VCO

LF

90-110 MHz

fs

Probe carrier
CPL

IN
OUT

AOM

EOM

Homodyne LO path
Path length stabilization tone, reflecting from cavity
Atomic probe tone
Cavityb)
DC DIFFsignal [Q/I0]

0 atomsbare cavity

0

1000
atoms
0

0.2

0.4

0.6

0.8

Time [ms]Figure 3.7: Homodyne detection. (a) Setup for probing and path length stabilization. The atomic
probe sideband tone (red) carries atom-cavity information at the homodyne LO tone frequency(orange). The path length stabilization tone (blue) encodes information about relative path length
fluctuations which is fed back to the LO frequency using a phase-frequency detector (PN: Analog Devices HMC440), an AOM, and a DDS reference that is phase coherent with the sideband
frequency. (b) Typical DC DIFF measurement of the reflected Q quadrature. A dispersive signal
measures the bare cavity resonance frequency (dashed brown) or the shift from N↑ = 103 atoms(green). Here the atomic probe is swept linearly from a known frequency at 1.5 MHz ms−1 . The
reduction of amplitude and linewidth broadening are explained in [29].

resonance as in Chapter 6.
This detector was designed for 50 µW of LO power incident on each port. The path length stabilization tone is only 50 nW and the signal fields are better described in terms of detected photons
per probing window (from 5 × 101 to 5 × 104 photons per 100 µs). We estimate the intensity modeoverlap between the cavity TEM00 and the LO beam is about 95(3)%. The technical noise from thedetector (the portion of the noise not attributable to photon shot noise of the LO) sets another quantum efficiency of Qelec = 86(2). The photodiode sensitivity of 0.54 A/W reflects the probability to42turn a photon into photocurrent and corresponds to a quantum efficiency of QPD = 0.86.

3.1.9

6.8 GHz microwaves
A stub-tuned dipole antenna is used to drive microwave transitions between 87 Rb hyperfine

ground states near ωHF = 2π × 6.835 GHz. Frequency and phase noise must be extremely well√controlled to perform rotations below the θSQL = 1/ N radian standard quantum limit for N
atoms. As described in [77] and [78], a low-noise 100 MHz crystal oscillator (PN: Wenzel Sprinter501-04517D or ULN 501-16843) is phase-locked to a 10 MHz signal (PN: SRS FS725 or a JILA-shared
Cs reference clock). This oscillator drives a nonlinear transmission line from which all tones but the68th harmonic are filtered out by copper pipe-cap filters. The 6.8 GHz signal is mixed at a singlesideband mixer with a tunable DDS signal near 35 MHz, allowing for control of microwave phase,frequency, and amplitude. A high-power amplifier (PN: Mercury Systems L0607-42) delivers up to
16 W of microwaves to drive hyperfine transitions with up to Ω = 2π × 50 kHz Rabi frequency. A
similar microwave frequency chain is described in Ref. [79], though this report misstates the phasenoise in our system.
After modulation, single sideband phase noise is −130 dBc/Hz at the relevant spectral components (10 kHz to 1 MHz offset from the 6.835 GHz carrier). Clock-transition spectroscopy lineshapes were broadened as little as 100 Hz at full width half maximum (FWHM) for atoms in a shallow potential. We estimate that this frequency source would limit the resolution of a basic MachZehnder interferometer with Tevol = 0.3 ms to the standard quantum limit for 2.5 × 106 atoms [80].

Agile frequency control for the interferometer experiment Additional frequency control
is required for the interferometer experiment. As the atoms accelerate due to gravity g, the relative Doppler shift for light propagating upwards versus downwards chirps linearly in time. The
frequency difference between the Raman tones must therefore be swept at b(t) = 2gtk = 2π ×25.11 kHz to keep a transition on resonance. In addition, we need amplitude control to set the Rabi
frequency during velocity selection and beam splitter operations, and phase control to set the ap-

43

10 MHzatomicclock~135 MHz

DDS

6.800 GHz

LPN7110-SMT

LF

100 MHz
Wenzelcrystal osc.

ϕϕ + 90°Nonlineartransmission line,
68th harmonic

100 MHz + chirp

DDS

Frequency
doubler

EOM

Cavity probebeat note locks,
13.6 GHz

Variousfilters

I

LO

Singlesidebandmixer

Q

10 Watt
amplifierFrequency
divider

Microwave
antenna,
6.835 GHz

EOM

Raman tones,
3.417 GHz

Figure 3.8: Agile frequency control. The 6.8 GHz low phase-noise microwave source (green box)
remains virtually unchanged from [78]. The output is sent to a single sideband mixer (PN:
HMC496LP3); the I and Q ports of this device must be 90◦ out of phase, but a DDS (PN: AD9959)
controls the overall phase and amplitude. A second DDS near 100 MHz adds a common-mode 25
kHz/ms chirp associated with gravitational acceleration. The frequency divider was used so thatfirst-order EOM tones would be separated by the hyperfine transition frequency. The doubler wasused to create high-order EOM tones for bridging considerable frequency gaps between lasers.

propriate rotation axes during the sequence. This is accomplished with the setup in Fig. 3.8, where
two phase-coherent DDS channels near 135 MHz manage the phase, amplitude, and frequencychanges. These channels are mixed down with a single channel at 100 MHz + b(t)/2π and upconverted at the 6.800 GHz single sideband mixer mentioned earlier. Because the Raman tones are
generated as first-order sidebands on an EOM, we use a low noise frequency divider (PN: Analog
Devices HMC862A) to modulate the EOM near 3.417 GHz.

3.1.10Transfer cavity

The transfer cavity spacer was built from Zerodur in 2008, resting in a metal canister withglass windows. A number of incidents reduced the finesse below the design specifications. One
piezo wire shorted in 2011, but the remaining piezo allowed feedback for stabilizing the cavity
length, e.g. to a spectroscopy setup. The original plan was to pump out the canister for additional
thermal stability, but the middling vacuum pressures were ideal for creating a plasma when highvoltages were applied to the piezo. As a result, the pressure is kept at atmosphere and the canister44
provides some temperature stability and protection against air currents. The second piezo shorted
before the atom interferometer experiment, and thus the length of the cavity could no longer betuned reliably. Nevertheless, the transfer cavity is still useful as a passive frequency reference.
Parameters for the cavity at the end of its lifetime are found in Table 3.2. A new ultra-low expansion(ULE) glass reference cavity was commissioned from Stable Lasers in 2021. The 10 cm ULE cavity
has finesse F = 1.8 × 105 and linewidth near κ = 2π × 10 kHz over the 760 – 830 nm range of
wavelengths.
Transfer Cavity Parameter

Value

Cavity length
Free spectral range
Linewidth at 795 nm
FinesseMirror radius of curvature
Mirror transmission
Round-trip loss (measured)
Maximum input power

3 cm
5 GHz
840 kHz
6000
5 cm
117 ppm
850 ppm
100 mW

Table 3.2: Cavity parameters for the long-used, home-built transfer cavity.

Chapter 4
Methods

This chapter aims to clarify a few disjoint experimental methods. Section 4.1 reviews usefulfrequency stabilization techniques. Section 4.2 describes an electronic solution for automaticallyrelocking servos. Section 4.3 outlines a scheme to narrow laser linewidths using feedback fromexternal cavities. Section 4.4 presents a site-dependent atom selection approach to the inhomogeneous coupling problem. Section 4.5 covers the creation of axially-uniform “optical dipole traps”,
and Sec. 4.6 introduces the blue-detuned “donut” dipole guide used for interferometry.

4.1

Frequency locking
Low-noise lasers and cavity frequency stabilization are crucial for cold-atom experiments.

Maintaining phase coherence [81] is vital for processes including laser cooling techniques and homodyne detection. Despite our significant laser linewidths, we are able to manipulate and resolve
the state of atoms by ensuring that noise on the cavity was common-mode with noise on the probing laser [72]. A variety of feedback techniques are used for setting and maintaining laser frequencies and cavity lengths. The aim of this section is to provide an overview of technologies that were
useful for us and references for further exploration. A summary of the locking chains used for thework in this dissertation is provided in Fig. 4.1.
Feedback as a formal concept is covered in many helpful references including Feedback forphysicists: a tutorial essay on control [82] and Feedback Systems (particularly the chapter on PID Control) [83]. Some of these ideas are used in virtually every lab but language can differ considerably.

46
Forming a MOT
MOT
780 REF

Spin-squeezing locking chain
795 REF

Transfercavity820
lattice

SciencecavitySpectroscopy lock
Cavityprobe

Atomicprobe

REP

Beat note lockMatterwave interferometry locking chain
Transfercavity760 bluelatticePound-Drever-Hall lock

SciencecavityCavityprobe

820
lattice

Raman
laser

Atomicprobe

SWAP cooling locking chain

Raman sideband
cooling

F2 Laser

RSBC
laser

795 REFF1 Laser

Figure 4.1: The locking chains used in the experiments of this dissertation. The dashed arrow in the
spin-squeezing chain represents the atomic probe’s “active lock” to the dressed cavity resonancewhile it is also phase-locked to the cavity probe (Ch. 6).

Conceptually, the challenge is to design a system that creates an error signal voltage e indicating
the deviation from a setpoint s; the controller’s output voltage u must act on some physical object
to guide the error signal back to the setpoint. With proportional-integral-derivative control, theoutput voltage is a sum of a term directly proportional to e, an integral term which accumulates
past errors, and a derivative term which anticipates future errors:Z tu = kp e + kie(τ ) dτ + kd
0

de
.
dt

(4.1)

JILA high-speed loop filters (LFs) allow fast PID control onto laser currents, VCO frequencies,
variable attenuation, EOM modulation, piezo voltages, and more. The differential error signal
inputs on these devices are labeled A and -B. This design uses two integrators and a few gain andspan settings to tailor the complex, frequency-dependent loop gain L(f ) detailed in Fig. 4.2(b).
Importantly, real systems such as lasers, cavities, and electronic devices have gains and poles thatmust also be included in the full loop transfer function. Besides the output port, there is an optional
PZT OUT with an additional bandwidth limit for driving piezos. Similar products are now available
commercially.
Let us consider the model in Fig. 4.2(a). A noisy input voltage Vin (t) is summed with theloop filter output which feeds back on Vset − Vout (t). Creating a stable feedback loop amounts to

47
a)

b)

Feedback model

Loop filter transfer function

Vin(t)
u

e

-

-B

A

Vout(t)
Vset

Loop gain [dB]

-1

Gain
boost
Lead
span

-2

0

0

-1

+1
0
Gain 1, 2, finefPI 1 fBW fPI 2
fD
Frequency (log scale)Figure 4.2: Feedback and loop filters. (a) Toy model for stabilizing a generic voltage. (b) JILA
double-integrator loop filter transfer function. In the frequency domain, each integrator gives aslope of 1/f , giving large gain at low frequency but causing a 90◦ phase lag. Each differentiatoradds a slope of f , and the 90◦ phase advance can correct for some loss of phase margin at highfrequency. Slopes of 1/f and 1/f 2 are colloquially referred to as −1, −2, and so on.

designing a transfer function such that the output voltage approaches the setpoint in the recursive
relationship
Vout = Vin + (Vout − Vset ) × L(f )(4.2)

i.e. L(f ) should be made large for all frequencies in
Vout =

Vin − Vset L(f ).
1 − L(f )(4.3)

For large L, the output voltage can approach the setpoint and noise can be properly suppressedat the frequency f . Problems occur near the unity gain frequency fBW defined by |L(fBW )| = 1,
where noise will be amplified unless Re [L(fBW )] < 0. The challenge then is typically to maximize
the gain and unity gain bandwidth while ensuring that the system remains stable. In particular,
adding each integrator results in greater gain under an additional 1/f slope but incurs a 90◦ phaselag. To ensure closed-loop stability, the phase margin, defined as the difference between the open
loop phase lag vs. −180◦ , must remain positive. If the transfer function crosses the unity gain
frequency fBW with a slope of 1/f 2 , the double-integrator phase lag leads to positive feedback anduncontrolled oscillations. However, small amounts of excess phase margin loss may be tolerated

48– the resulting unity gain noise bumps visible at fBW are even helpful for characterizing the servo
bandwidth.

4.1.1

Spectroscopy locks
Two separate spectroscopy setups were used during experiments involving the cavity. Bothsystems involved a room temperature rubidium vapor cell containing both 85 Rb (72.2% naturalabundance, stable) and 87 Rb (27.8%, half-life 5 × 1010 years). The rough idea behind this techniquecomes from scanning a laser frequency across a particular resonance of some atomic species; aphotodiode detects decreased transmission due to absorption in the cell and feedback can in principle stabilize the laser frequency to this wavelength. Because modulation spectroscopy is robust
against temperature changes and laser amplitude or polarization drifts, and because the transition
frequency is set by nature, the technique is useful as an “absolute” frequency reference at the startof a locking chain. Frequency precision is limited, however, by a broad excited state linewidth Γ.

PBS
EOM

λ/2
LF

b)

Modulation transfer spectroscopy

780 REF

5.2 MHz0.3
Rb vapor cell

PBS

Voltage [V]

a)

0

-0.3
0

Time [ms]50

Figure 4.3: Modulation transfer spectroscopy [84]. (a) Setup for locking the 780 REF laser tothe D2 -line |F = 3i ↔ |F 0 = 4i transition in 85 Rb. Laser light (blue) is detected by a photodiode(brown) to generate a photocurrent. The loop filter feeds back to the grating piezo voltage andthe laser diode current. (b) A representative spectroscopy error signal for the 780 REF laser, savedon an oscilloscope. As the loop filter output voltage (yellow) is swept, the laser frequency crosses
through the |F = 3i ↔ |F 0 = 4i transition resonance.

In practice, our spectroscopy locks are built around phase-modulating a laser, and it is help-

49
ful to expand the electric field Ei in terms of Bessel functions Jn (β) for modulation index β:
Ei = E0 sin (ω0 t + β sin(ωm t))
= E0

∞
X

(−1)n Jn (β) sin(ω0 t − nωm t).

(4.4)
(4.5)

n=−∞

When detected on a high-speed photodiode, rf currents nominally cancel due to the opposite signsof the sidebands in Eq. 4.5. In the presence of a medium like atomic vapor, however, absorption
and dispersion modify the response to each sideband separately, resulting in a dispersive-like error
signal (Fig. 4.3(b)). In addition, a ∼ 1 mW counter-propagating saturation beam is used to pump
the atoms for Doppler-free spectroscopy. The backbone of our frequency chain is a 780 nm modulation transfer spectroscopy system [84] which locked the 780 REF laser frequency to the D2 -line|F = 3i ↔ |F 0 = 4i transition in 85 Rb. A 795 nm FM saturation spectroscopy lock [85] also sets the795 REF laser frequency to the “crossover” feature halfway between the D1 -line |F = 3i ↔ |F 0 = 2i
and |F = 3i ↔ |F 0 = 3i transitions in 85 Rb.

4.1.2

Beat note locks
A beat note here refers to a heterodyne beating of laser tones at two frequencies ω1 and ω2and electric field amplitudes E1 and E2 . The polarization, divergence, and spatial alignment of
two beams are tuned to maximize the mode overlap at a high-speed photodiode (typically PN:
Hamamatsu GaAs 4176-03). Eliminating components at frequencies outside of the photodiodebandwidth reveals a signal at the difference frequency,
I ∝ |E1 cos (ω1 t) + E2 cos (ω2 t)|2→

IBN ≈ E1 E2 cos [(ω1 − ω2 ) t] .

(4.6)

Our beat note locks are implemented with the help of a phase-frequency detector (PFD, PN: Analog
Devices HMC440) which is chosen for its low phase noise floor and convenient frequency range.
The PFD takes a reference input 20 – 1000 MHz and a VCO input 20 MHz < ω1 − ω2 < 2 GHz
which is divided internally by a fixed integer frequency division ratio. This device acts as a sortof comparator with charge pumps outputting an error signal voltage that indicates which signal

50
is crossing zero first (phase) or more often (frequency). Negative feedback is used to drive theerror signal to zero, at which point the signals are equal in phase and frequency. Feeding back on
one laser to stabilize a beat note to a reference frequency ensures the lasers are phase locked andcoherent.
a)b)

Beat note lock

Cavity probe

0

LF

25 dBm
13.6 GHz-20

Atomic probe~50 MHz

PFD
(×7)

[dBc]

EOM

-10
-30

fBW

-40
-50
-60
-70

348

349

350

351

352

Beat note frequency [MHz]Figure 4.4: Offset beat note locking. (a) Phase lock between the atomic probe and cavity
probe lasers. A fiber-coupled EOM modulated at 13.6 GHz adds 9th order sidebands to the cavity probe. The 50 MHz reference frequency and ×7 PFD division ratio ensure the lasers are locked
122.4 GHz + 350 MHz apart. (b) The beat note between the atomic probe carrier locked to a cavity
probe sideband, as measured by a spectrum analyzer. Unity gain bumps are seen at fBW ≈ 400 kHz
but the vast majority of the power is concentrated in the carrier.

This type of lock is tremendously useful for setting a frequency difference between lasers (aswith the 1.3 GHz offset between MOT and REF lasers) and for controlling changes in homodyne
phase due to path length fluctuations (see Sec. 3.1.8). For bridging large frequency gaps between
lasers, multiple techniques are used. As long as the beat note is within the bandwidth of a GaAs
photodiode, it can be mixed down, as with the 6 GHz REP to REF beat note. If the frequency gap islarger, a fiber EOM on one laser path is modulated, and we stabilize a beat note created between
a high-order sideband and the laser to be locked (as with the 12th-order 13.6 GHz cavity probe
sideband used in locking the atomic probe). Finally, although feeding back on the VCO input on
the PFD is more common, it is also possible to feed back on the reference port, as with the “activelock” used to keep the probe on cavity resonance in Sec. 6.
The linewidth of a beat note on a spectrum analyzer is related to the laser linewidth butwith some important caveats. First, it is a relative measurement, so the feedback gain should be51
decreased as much as possible in order to make accurate statements about free-running (intrinsic)
linewidths. Second, the beat note is a convolution of both lasers so a narrow reference laser (orsome simplifying assumption) is needed to resolve the linewidth of the laser under test. Finally,
the Lorentzian linewidth of a laser may be obscured by the Gaussian-shaped noise near the carrier,so it can be more useful to fit a Lorentzian to only the wings of the lineshape. Fortunately, there are
more robust ways to measure laser linewidths such as self-heterodyne techniques or measurements
of instantaneous frequency fluctuations.

4.1.3

Pound-Drever-Hall locks
Locking a laser frequency relative to a cavity resonance is one of the most important ex-

perimental techniques available to us. Pound-Drever-Hall (PDH) locking adapts a technique forstabilizing microwave resonators [86] to the optical domain [87, 88]. Importantly, the technique decouples laser frequency noise from intensity noise. As seen in Fig. 4.5, a laser is phase-modulatedby an EOM, Ei = E0 sin (ω0 t + β sin(ωm t)), and detected in reflection on a high-speed photodiode.
For a purely phase-modulated beam, there would be an exact cancellation of rf currents due tothe sign reversal of the equal-intensity sidebands, but the reflected electric field incorporates thedetuning-dependent cavity response. In a typical application, sidebands at ωm thus act as phasereferences in stabilizing the frequency of the carrier. Mixing the photodetector signal with an LO
at ωm gives access to the familiar dc error signal lineshape as a function of the detuning from cavity
resonance δp ,1e(δp ) = −2 |E0 |2 J0 (β)J1 (β)Im [F (δp ) F ∗ (δp + ωm ) − F (δp ) F ∗ (δp + ωm )] ,(4.7)

using the cavity response function F (δp ) from Eq. 3.3.
PDH locking in this form is used for stabilizing the red and blue lattice lasers to the transfer
cavity. It is used for locking the cavity probe with 50 nW detected by avalanche photodiode (APD,
1

Assuming small modulation index β < 1 so the Bessel expansion is valid and modulation frequency ωm  κ so

the sidebands are completely rejected by the cavity.

52b)Pound-Drever-Hall locking

0.1

LF

TransfercavityVoltage [V]Red lattice
laser

EOM

a)

10 MHz0fm
10 MHz-0.1
0Time [ms]50

Figure 4.5: Pound-Drever-Hall locking. (a) A typical PDH setup for locking an optical frequency
to a cavity resonance, including modulation of an EOM and demodulation at a mixer. (b) A PDH
error signal for locking the red lattice to the science cavity. As the loop filter output voltage (yellow) is swept, the sidebands and carrier cross through the cavity resonance [87]. In this case,
fm = 10 MHz sidebands are added by modulating the laser current rather than phase modulation
through an EOM, but the end result is the same.

gain ≈ 150) during the squeezing experiments. Feeding back on cavity piezos also allows forlocking the science and transfer cavities to PDH error signals, although the bandwidth is limited
to a few kHz by the loaded piezo resonances. Finally, we began locking to a weak sideband featureof the cavity probe during the interferometer experiment because little intracavity power could be
tolerated. This technique allowed using about 50 pW in the resonant sideband with only 50 pW of
the 3 µW carrier tone leaking into the cavity non-resonantly.

4.2

Automated relocker
The standard JILA loop filter design includes a relay for switching between engaging a lockand repeatedly ramping the output voltage. When a lock drops, the output voltage typically rails,
and in most setups the relay needs to be manually re-engaged. Dropped locks may be incidental,
arising from acoustic perturbations, interrupted signals, or drifting parameters; but they may also
be intentional. During the interferometer experiment, for example, we adiabatically release theatoms from a red-detuned lattice stabilized to the cavity resonance. Because the lattice light is
temporarily turned off, there is no error signal to maintain the lock, and each run of the experimentnecessitates relocking the lattice laser frequency upon completion. The “relocker” described here

53
is our solution.
The relocker can be considered a machine with three states: calibrating, listening, and relocking (Fig. 4.6(a)). It takes as input the unmodified error signal and the output of a loop filter(Fig. 4.6(b)). During the calibration process, average VES and peak-to-peak error signal ∆VES voltages are sampled and stored. The unmodified error signal is sent to the loop filter and lock is engaged; the in-lock output voltage VOUT is stored, and the device begins listening for a drop event.
If the output voltage deviates from VOUT beyond a hard-coded threshold, the device enters the relocking subroutine: the error signal is replaced by a linear voltage ramp which causes the output
voltage to sweep as well. Meanwhile, the error signal voltage is monitored. Once the error signal
voltage crosses above and then back below a programmable threshold related to VES 2 , control isreturned to the loop filter (Fig. 4.6(c)). If the output voltage indicates a successful relock, the device
begins listening again, but if the relock attempt has failed, the relocking procedure quickly repeats.
The relocker is built around a PSoC5LP programmable microcontroller on a FreeSoC2 development board. The CPU interfaces with a printed circuit board that houses a few op-amps forvoltage conversion and the digital switches for enacting state logic. The same functionality could
be achieved with an Arduino, albeit with relaxed technical specifications. Relocking speed was nota primary concern, but after fine-tuning, the device achieved 100% relock rate with typical relock
times of 10 ms.
Interfacing with a serial port is useful in reporting calibration settings and passing time traces
to a monitoring program written in Python. A TTL is used to temporarily block relocking attempts
in cases where the error signal is intentionally disrupted. There are numerous opportunities tofurther improve upon the design. Calibration settings could be stored in persistent memory or sent
serially. The range of output voltage sweeps could be adaptive. Instead of setting the thresholdand sign of the error signal programatically, these could also be adjusted through a user interface.
2

For locking to a PDH signal, we monitor for a voltage below VES − 0.4 × ∆VES and then a voltage above the same

threshold, which is sufficient for ensuring the loop filter attempted to relock near the central slope even with the presenceof noise.

54
Calibratingb)c)

Relocker

User completed
routineMicrocontroller

Error signal

Voltagea)

Output

ΔVES

voltage

Listening
Detecteddrop
RelockingRelock failed

Relock successful

A
-B

VES

Voltage

No drop

LF

2
1

VOUT

Actuator
0

Time [ms]15

Figure 4.6: Automated relocker. (a) State diagram. (b) Connection diagram. The microcontroller
switches loop filter inputs (green) and outputs a frequency ramp (red) according to its state. The
actuator converts the loop filter output voltage to an error signal. (c) Theory of operation usinga PDH error signal. A full sweep of the output voltage produces the entire lineshape (top), but if
the relocker measures an error signal voltage below 1 then above 2 , control is passed back to theloop filter, re-engaging the lock (bottom). After re-engaging lock, the output voltage must settle
within the tolerated region (green), i.e. must not rail, to be considered “successful”.

4.3

Narrowing laser linewidths with external optical feedback
Stabilizing a laser to a cavity is invaluable for removing low-frequency noise due to thermalor mechanical fluctuations, but in some situations, a laser with intrinsically narrow linewidth isdesired instead. The history of narrowing diode lasers with optical feedback from an external cavity(as opposed to the diode’s internal cavity) goes back several decades [89]. Optical feedback froman electric field returning to a diode familiarly leads to instability which is typically circumvented
with optical isolators. The same effect can also be harnessed to narrow the power spectrum by
several orders of magnitude. Experimentally, the effect of the feedback depends on the strengthand on the phase of the reflected field relative to the lasing field. An analytic expression for the
linewidth has multiple solutions for most feedback strengths, leading to complex spectral properties [89], but for a narrow range of feedback strength, the linewidth is narrowed or broadened
depending on the round-trip phase of the light.
We use optical feedback on our DBR lasers to reduce the ∼ 500 kHz intrinsic Lorentzian
linewidths to around 1 kHz. The temperature and vibration stability of a DBR makes it an attractive

55
candidate for the technique. The approach is similar to previous schemes involving electronic
feedback onto a piezo [90] and onto an EOM [91] to control the round-trip external cavity phaseshift.
b)

R=10%
Beamsplitter

DBR Laser

Isolator

To experiment

Piezo Mirror

EOM

λ/4

EOM

150 cmPBS

Sν [Hz2/Hz]

a)

1014

1011

107

104
101

102

103

Frequency

104

105

Figure 4.7: Narrowing a laser with external optical feedback. (a) About 10% of the DBR output
power is picked off and sent to a 3 m long (round-trip) external cavity formed by a piezo-mounted
mirror. Additional optics reduce the fraction of light returning to the laser to 10−3 to 10−4 relative
to the output power. The lasing frequency is controlled by the piezo, and mode stability requires
feeding back on the piezo and EOM. (b) Sν (f ) measured for a free-running DBR before (blue) andafter (red) narrowing. Because the lasers are unlocked, there is a lot of low-frequency (Gaussian)noise, but the Lorentzian noise at high frequency is reduced several orders of magnitude for the
narrowed laser.

In our setups, the current and temperature of the DBRs is left free-running. The lasing
frequency is then pulled by a combination of the length of the external cavity as set by a piezomounted mirror and by the phase modulation of a free-space lithium niobate EOM. The feedback
ratio for each path is set near 10−3 to 10−4 by a 10% pick-off plate, an ND filter, and a polarizing
beam splitter cube which rejected a fraction of power set by a λ/4-wave plate. Rather than measuring the feedback ratio accurately, we adjust the ratio to narrow the width of a laser beat note. The
external cavity round-trip lengths are about 3 m in free space, and longer cavities are correlatedwith narrower achievable linewidths. The description of laser mode competition in Sec. 3.1.7 may
also be helpful for intuition.
These narrowed lasers are locked, as usual, to beat notes or PDH error signals. The transfer
function of the loop filter stabilizing the frequency of each laser is carefully designed to maintain56
stability in the presence of resonances and filter poles. Electronic feedback is provided by two
separate paths: slow feedback drives the piezo to change the length of the external cavity, primarily
setting the laser frequency and providing dynamic range. The resonances of the piezo mounts arepushed above 40 kHz with guidance from Ref. [92]: a 2 mm × 2 mm piezo is epoxied between acomparably-sized thin mirror and the tip of a lead-filled copper cone. The piezo path contains
additional low-pass filtering. Fast feedback is handled by the EOM, but the crystal has a sharp
resonance at 900 kHz set by its geometry. The unity gain bandwidths are limited near 500 kHz as aresult.
We measure Sν (f ), the power spectrum of the relative frequency between two narrowed
lasers, by beating the lasers against each other at a high-speed photodiode. The photodiode signal
is sent through a frequency-to-voltage converter (two coaxial cables and a mixer which form aninterferometer) and into a network analyzer. The frequency of the lasers is biased to the side of
the interferometer fringe and the interferometer slope is used to convert a voltage noise spectrum
to frequency units. In Fig. 4.7(b), the high-frequency (Lorentzian) components of the spectrum
are reduced by three orders of magnitude, to less than 1 kHz after occasional tune-ups. A truly
Lorentzian laser of FWHM ∆ν has corresponding instantaneous frequency noise Sν = ∆ν/π. The
observed low-frequency noise in Fig. 4.7 can be attributed to Gaussian processes including thermal
and mechanical fluctuations which are ideally negligible e.g. after locking to a cavity.

4.4

Site-dependent loading
As we saw in Sec. 2.1, the standing wave modes of a two-mirror cavity lead to inhomogeneity

in the coupling of individual atoms to intracavity light. Atoms are localized to lattice sites but the
local probe intensity at each site may differ considerably. As a result, trapping atoms in a latticethat is incommensurate with the probe standing wave can lead to reduced effective atom number,degraded coherence, optomechanical oscillations, and complicated analysis, all of which obscure
underlying physics.
One solution is to use a commensurate trapping wavelength λL that is a multiple of the prob-

57
ing wavelength λp [93]. However, the trapping wavelength may be desired as a system parameter,
for example with state dependent or magic wavelength traps. Available mirror coatings may also
inhibit this option. Another possibility in certain experiments is to use the running wave modesof a ring cavity. In Sec. 4.5, we describe a dipole trap configuration based on interference of cavity modes, maintaining radial confinement while allowing atoms to move and time-average away
probe inhomogeneity [94].
Here we demonstrate one more technique for homogenizing the coupling between latticeand probing lasers. Atoms are held in an incommensurate 813 nm standing wave lattice andpumped into |↑i ≡ |F = 2, mF = 0i. A Stark shift beam blue-detuned one free spectral range fromthe atomic probe beam is incident on the cavity, providing a spatially-dependent frequency shift
to the atoms (Fig. 4.8). Due to the opposite parity of adjacent longitudinal modes near the centerof the cavity, atoms with high coupling to the atomic probe have no coupling to the Stark beam. A
microwave π-pulse transfers the atomic probe-coupled atoms to |↓i ≡ |F = 1, mF = 0i and a blowaway beam transverse to the cavity removes the remaining atoms with radiation pressure force.
Atoms can be optically-pumped back to |↑i and the process repeated for improved selectivity.
Nearly all of the physics here is captured by the Rabi formula, as the probability for each
atom to undergo a transition depends on gi , the position-dependent coupling of the ith atom to theprobe, throughΩ2
P (gi ) = 2
sinΩ + ∆2π2√Ω2 + ∆ 2
Ω

!2
,

(4.8)

for ∆ ≡ δm − (gi /g0 )2 δs , where Ω is the microwave Rabi frequency detuned δm from resonance andδs is the peak AC Stark shift. A significant detuning due to the AC Stark shift beam makes a transition very improbable for any atoms not at antinodes, and only atoms that undergo a transition
will survive.
Although the site-dependent selection approach involves discarding many atoms, this may
not inhibit certain experiments, and the degree of homogeneity scales favorably with the number ofretained atoms. With a completely incommensurate lattice, the ensemble-averaged coupling nor-

58

Atomic
probe

b)
|e⟩

Stark
beam

1

|↑⟩

Stark beam
Atomic probe
2

|↑⟩
|↓⟩

c)

atoms
1

Energya)

Blow-away
beam

Position along cavity|↓⟩

1

Position along cavity

Figure 4.8: Site-dependent selection for homogeneous atom-cavity coupling. (a) Level diagram.
The atomic probe and Stark beam are split by one free spectral range so that their standing waves
are out of phase near the center of the cavity. (b) The Stark beam can be thought of as dressing
the energy levels of |↓i in a position-dependent way. Atoms at the maxima of the atomic proberemain resonant and still undergo microwave transitions, but the rest will be blown away. (c)Experimental setup and microscopic view showing the phase mismatch of the atomic probe (red),
Stark beam (blue dashed), and lattice (yellow). Atoms are initially localized to the lattice sites but
only survive if they are uncoupled to the Stark beam as at 1 .

DEmalized to the maximal coupling is (gi /g0 )2 = 0.5. Using the site-dependent selection scheme,DE
we observe an increase in the normalized coupling to (gi /g0 )2 = 0.88(1) after one site-selective
cycle [95]. At this point, 11% of atoms remain compared to 18% from a theoretical model. For thisdemonstration, δs = 2π × 32.7 kHz, Ω = 2π × 0.17 kHz, and δm = −2π × 2.7 kHz. Because theremaining atoms are localized to the quadratic part of the probing potential, they are less susceptible to optomechanical oscillations associated with the force induced by the gradient of the probe
standing wave. After two site-selective cycles, fractional amplitude of probe-induced oscillations
are reduced to 32% of the oscillations measured without site-dependent selection [95].
This scheme allows atoms to remain tightly trapped (i.e. in the Lamb-Dicke regime) andleaves the lattice wavelength as a degree of freedom. Shaping the microwave pulses might improve
performance in the future. While we do not rely on the scheme for the upcoming squeezing orinterferometry experiments, it remains a broadly-useful tool for physics based around atom-cavity
interactions.

59

4.5

Axially-uniform potentials
Another approach to the inhomogeneous coupling problem involves transforming a stand-ing wave lattice into an axially-smooth potential by interfering multiple modes within the cavity. This technique is useful in creating homogeneous coupling for spin-squeezing [94] and interferometry, but the same idea was explored much earlier in Ref. [96]. For creating this axiallyuniform lattice, multiple TEM00 longitudinal modes of a cavity are driven by a laser modulated
at the free spectral range ωFSR . The opposite parity of fields injected on adjacent modes ensuresthat near the center of the cavity, the combined axial potential is approximately uniform, U (z) ∼sin(k0 z)2 + cos(k0 z)2 = 1, but radial confinement remains strong (Fig. 4.9). A typical optical dipole
trap (i.e. optical tweezers) relies on the field gradient at the tightly-focused waist of a runningwave beam. The same optical dipole force is at play in our standing-wave configuration, hence
our occasional description for this technique as a “dipole trap”.
a)

b)

c)

Standingwave
lattice

Dipoleguide

Carrier±1 sideband

-2

-1

0

1

2

Laser frequency [FSRs]

-2

-1

0

1

2

Laser frequency [FSRs]

Axial
potential
depth±2 sideband++=
Position along cavity

Figure 4.9: Creating axially-uniform intracavity potentials. (a) Ordinary standing wave lattice,with axial fringes shown as red disks. The number of “pancakes” and spacing are not to scale. (b)Adding sidebands at multiples of the cavity free spectral range creates a dipole trap with uniform
axial potential at the center of the cavity. (c) The superposition of longitudinal modes leads todestructive interference at the cavity center and constructive interference at the cavity mirrors.
The axial potential depths shown here have exaggerated wavelengths but correspond to axial trap
depths calculated for β = 1.21.

More carefully, the axial component of the potential at a distance z from the cavity center can

60
be written"
U (z) = U0#X

Jn2 (β) cos2 ((k0 + δkn ) z) +

n odd

X

2
Jm
(β) sin2 ((k0 + δkm )z) ,(4.9)

m evenwhere Jn (β) is the nth Bessel function for modulation index β, δkn = nωFSR /c is the change inwavevector for sidebands offset by n cavity free spectral ranges. Interference terms between the
sidebands oscillate at ωFSR and can safely be ignored. A trade-off in modulation index is made between minimizing the potential at z = 0 vs. extending the region with potential below a minimum
threshold. In Ref. [94], the atomic ensemble can be a few mm long, so we opt for a modulationindex that roughly maximizes the region where |U | < 0.1U0 . Tuning the modulation frequency
away from the free spectral range or changing the modulation depth with time could offer additional degrees of freedom. In Fig. 4.9(c), the first few sidebands are plotted with exaggerated
wavelength for β = 1.21.

4.6

LG01 blue dipole guide
Atoms in the interferometer experiment undergo free-fall along the cavity axis but require

radial confinement. The dipole trap configuration of the previous section comes close to achieving
this, but atoms are susceptible to Bragg scattering from the residual axial potential. The axiallyuniform lattice technique is combined with a 760 nm blue-detuned Laguerre-Gaussian LG01 -like“donut” mode to create the blue dipole guide. The LG01 field is truly zero on-axis, allowing atoms toaccelerate freely under gravity, but the rms radial position due to finite temperature still necessitates using modulation at ωFSR to reduce axial potentials. The repulsive blue-detuning ensures that
atoms are confined to the cavity mode (rrms  w0 ) for the entirety of the interferometer sequence.
The donut-mode LG01 profile is constructed from the ±1st diffraction orders of a binarymask fork-pattern hologram plate [97]. Mechanical stress-induced birefringence on the cavity
mirrors breaks cylindrical symmetry, splitting the Hermite-Gaussian modes in the decompositionπLG01 = HG01 + ei 2 HG10 by an amount δHG = 100 – 500 kHz depending on the cavity piezo voltage [98]. Prior to entering the cavity, the ±1-order diffraction patterns from the forked phase plate

61
are sent along two separate paths. A δHG offset is added using two AOMs on one path (typicallyδHG = 350 kHz for work presented). An EOM on the other path provides sidebands for locking
the cavity to this laser via Pound-Drever-Hall. The two paths are overlapped at a polarizing beam
splitter and sent into the cavity (Fig. 4.10(c)). Because δHG is much greater than the radial trap
frequency, the atoms experience the time-averaged radial trapping potential of an LG01 mode.
The diffraction plate was created following Ref. [97], via chromium deposition of 133 lines/mm
on an AR-coated 1” wedged blank. Other techniques including spatial light modulators are available. The pattern for these LG0l plates can be written∞ nπ h 
 y i1 X
P0l (x, y) =sinccos n kx x − l arctan,
2 n=−∞
2x(4.10)

where kx is the frequency of lines far from the dislocations at the center of the plate (Fig. 4.10(a)).
The same phase pattern is produced by interfering a TEM00 mode with the desired LG mode. Optimal performance requires careful placement of the plate at a beam waist and carefully centering
the beam on the fork pattern. The total measured transmission efficiency is about 46%, with 24% of
the transmitted power going into each of the ±1-order modes and 44% remaining in the 0th-order
mode.

62

c)

Fiber EOM
(for dipole trap)
LG01 incavityHG01=
eiπ/2 HG10

LG01

-1
+1
Isolator

EOM

EOM for PDH
sidebands
PBS

PBS

AOMs offset~200 kHz

AOM

+

P04

PDH lockto cavity

λ/2

b)

P01

Diffraction
plate

760nm

EOM

AOM

a)

Figure 4.10: Creating an intracavity LG01 donut dipole guide. (a) Close up (×200) of the diffraction plate fork patterns used for creating the LG01 blue-detuned dipole guide and LG04 modes fortesting. (b) First-order HG modes are combined into an LG01 -like mode within the cavity. (c) Relative tilt and stress on the cavity mirrors breaks cylindrical symmetry, so we use a diffraction plate
to create two LG01 modes and offset their frequency by δHG . The modes are recombined to forma donut-like mode within the cavity which is verified by a CCD measuring in transmission. The
total power is controlled by an AOM placed prior to the fiber EOM (not shown).

Chapter 5
Laser cooling

Advances in AMO physics are predicated on the ability to cool and control particles. Lasercooling techniques have revealed new frontiers for neutral atoms, ions, and mechanical resonators[99–101]. Although the “final book” on the subject has not been written, our lab frequently finds
use for Metcalf and van der Straten’s Laser cooling and trapping [102]. Understanding methods andlimitations for removing entropy from a system is a fundamental goal of AMO and continues tobe widely explored [101, 103–108].
Some notable cooling mechanisms have been explored but never used in our system, including velocity-selective coherent population transfer (VSCPT) [109], cavity cooling [104], delta-kickcooling [110], and electromagnetically-induced transparency (EIT) cooling [111]. Other methods
such as interferometric cooling [108] and microwave projection cooling [112, 113] are also valuablefor the inspiration and insight they provide. In 2016, we briefly investigated the prospects for a collective cavity-mediated cooling mechanism called supercooling [114]. The scheme involved engineering superradiance using a Raman transition within the cavity QED weak-coupling regime. We
were able to observe superradiance with a single-photon counting module (PN: Excelitas AQRH10-FC) and extend it for a few milliseconds with repumping, but ultimately, we became concernedthat heating due to repumping photon recoils would exceed cooling in our implementation.
The rest of this chapter summarizes the laser cooling and thermometry techniques that were
implemented at various points in our experiment. Basic descriptions and parameters are provided for Doppler cooling, polarization-gradient cooling (PGC), and lambda-enhanced gray mo-64lasses. Particular attention is given to Raman sideband cooling (RSBC, Sec. 5.5), which required
subtle changes from typical setups, and Raman sawtooth-wave adiabatic passage (SWAP) cooling(Sec. 5.6), which was a new technique developed within our lab. With SWAP cooling, we achieve a
1D equilibrium temperature of 5.9(3) µK – far below the Doppler cooling limit. The prospects forusing SWAP cooling to avoid spontaneous emission while laser cooling new atomic or molecularspecies remains very exciting.

5.1

Measuring temperature
Temperature is an important diagnostic for our system, uniquely helpful for understand-ing the potentials and pulses experienced by the atoms. Typically, we are concerned with relative
improvements rather than absolute measurements, though during the SWAP cooling experiment,
lowering the temperature was the explicit goal. Here we will explore four distinct techniques formeasuring temperature: fluorescence time-of-flight, dispersive cavity measurements of radial expansion, Raman velocimetry, and lattice sideband spectroscopy. More sophisticated temperature
measurements are available, for example through obtaining the Doppler width from g (2) correlations or through heterodyne measurements to extract the spectrum of emitted light [104].

Fluorescence measurements. With fluorescence imaging, an atom absorbs a resonant photon and re-emits it for collection on a detector. Because the universe can in principle detect whichatom emitted the photon, this is a destructive measurement that leads to quantum collapse. In
practice, we snap off the confining lattice at t = 0, allow the cloud some dark time to expand fromits initial rms width σi , and then turn on the MOT beams for fluorescence during a typical 100 µs
exposure (Fig. 5.1(a)). It’s possible to measure only atoms in |↑i, but adding a repumper for |↓iatoms improves the signal-to-noise significantly. By integrating over the vertical distribution of
atoms trapped in a lattice, we have sufficient signal-to-noise to measure radial temperature, but
axial temperatures can be difficult (unless we simply assume that the atoms have equilibrated to65
equal axial and radial temperatures). We rely on a simple Gaussian ballistic expansion fit model,skB Trad 2
σ(t) = σi2 +
t
(5.1)
mRb
for Boltzmann constant kB = 1.38 × 10−23 J K−1 and rubidium mass mRb = 1.443 × 10−25 kg. Our
rudimentary imaging systems are based around inexpensive CCD cameras (PN: FLIR CM3-U313S2M-CS) and 2” optics placed anywhere from 3” to 18” from the atoms. Technical noise limits
the precision of the close-in camera to 80 atoms/shot, and we see signal-to-noise limited temperature fits after about 10 ms of ballistic expansion. Other complications such as the presence of
background atoms and finite exposure times make fluorescence a very approximate technique forus.

Radial ballistic expansion. We can instead leverage our cavity dispersive measurements tomonitor the average coupling of the atomic cloud as it expands out of the cavity mode (Fig. 5.1(b)).
Here, atoms are held in a lattice with radial trap frequency ωrad until the lattice is turned off, allowing atoms to disperse on a timescale fast compared to their falling. The probability to find an
atom at position x and y after time t is
Z ∞


 

py t
px t
P (x, y, t) =Px x −
Px y −
Pp (px )Pp (py ) dpx dpymm−∞with 1D position and momentum distributions (normalized with Ax and Ap )
2 x2 mωrad
Px (x) ≡ Ax exp −
;
2kB T


Pp (p) ≡ Ap exp−p2 /2mkB T

.

Each atom in |↑i contributes to the shift from atoms in N↑ with a weighting that accounts for its
radially-dependent cavity coupling. With atomic probe beam waist w0 = 70 µm, the number ofatoms in the cavity in N↑ inferred from the cavity shift is modeled asZ ∞N↑ (t) ∝


exp

0

−2r2
w02


P (x, y, t) 2πr dr =2
mRb w02 ωrad2 + 4k T (1 + ω 2 t2 ) .
mRb w02 ωrad
B
rad

(5.2)

In Fig. 5.1(b), the radial expansion timescales are compared for a PGC-cooled ensemble at 5 µK
and a Raman-sideband cooled ensemble around 0.5 µK. Even after several milliseconds, a large

66
fraction of the ensemble is within the cavity mode. For this data, the red lattice and blue-detuned
dipole guide must be temporarily disabled during expansion. An automated relocker is used on
the lattice, and for durations less than 10 ms, the cavity PDH lock could re-engage automatically.
b)

Axial ballistic expansion (fluorescence)
Axial waist [mm]

2.5
2.0
1.5

20(1) μK

1.0
0.5
0

0

5

10

15

Expansion time [ms]20

Radial ballistic expansion (QND)
Atoms in cavity [arb.]a)1

0.5(1) μK

5.1(2) μK
0

0

2

4

6

8

10

Expansion time [ms]Figure 5.1: (a) Ballistic expansion of the axial distribution after polarization gradient cooling(PGC), measured with fluorescence on a CCD. (b) Radial temperature measurements of atoms
after gray-molasses (red) and Raman sideband cooling (RSBC), using dispersive measurements of
the atomic coupling to the cavity mode.

Raman velocimetry. Two-photon (Raman) velocimetry is a useful thermometry techniquewhen the axis of interest is colinear with a set of Raman beams ω1 and ω2 (Fig. 5.2(a)). This was
the preferred technique for our one-dimensional SWAP cooling and for measuring the axial temperature during the interferometry experiment. The idea is to spectroscopically excite a portionof atoms using the counter-propagating (velocity-selective) pair of Raman tones, transferring only
a narrow velocity class from |↓i to |↑i where the relative population is subsequently measured
with fluorescence or dispersive population measurements sensitive to the number of atoms in |↑i[115]. A full velocimetry scan requires many trials, scanning the two-photon detuning between
the Raman tones to fully cover the Doppler width of the ensemble. Power-broadened or Fourierbroadened distributions must be avoided at low temperatures by using a sufficiently low-power,
long-duration excitation pulse.

Sideband spectroscopy. A final temperature diagnostic, sideband spectroscopy, measures67
excitations of axial lattice modes (Fig. 5.2(b)). Sideband spectroscopy assumes atoms are in a harmonic potential with quantized motional states n described by a thermal Maxwell-Boltzmann distribution
Pn =

n̄n
(n̄ + 1)n+1(5.3)

for the average number of motional excitations n̄. The red and blue sideband transitions remove
and add one motional quanta, respectively. Because |n = 0i has no lower energy state to transition
to, an atom in the motional ground state is unaffected by the red sideband. Consider an atom inPan electronic ground state but motional superposition, |ψi = ∞n=0 cn |ni |↓i. The probability toexcite the atom with the red or blue sideband using a resonant pulse of duration t is
∞
X

n̄n
2
Pred =
n+1 sin
(n̄
+
1)
n=1



Ωn;n−1 t
2


;

∞
X

n̄n
2
Pblue =
n+1 sin
(n̄
+
1)
n=0



Ωn;n+1 t
2


(5.4)

with motional state-changing modified Rabi frequency Ωn,n+1 ≡ Ωn+1,n as in Ref. [116]. The takeaway (explored in Ref. [116] and elsewhere) is that the ratio of the probabilities R = Pred /Pblue =n̄/(n̄ + 1) is quickly learned from comparing the red and blue sideband peak heights after a spectroscopic scan. This ratio is independent of experimental parameters like drive time and Rabi
frequency. Inverting the relationship yields the mean occupation of the motional state,
n̄ =

R
.
1−R

(5.5)

As an example, we measured n̄ = 0.7 after cooling atoms in a lattice with PGC and a shortstage of lambda-enhanced gray molasses. We were also able to verify the adiabaticity of lowering
the lattice from 20Erec to 3Erec by then raising it back up and measuring the same n̂ and atom number. In a deeper trap, we measured n̄ = 0.4 after Raman sideband cooling. Sideband spectroscopy
has the added bonus of re-affirming the axial trap frequency, and the asymmetric lineshape reflects
the radial distribution as atoms off-axis experience shallower trapping frequencies.

68b)

Atoms [arb. units]

Raman velocimetry
1
11 μK

42 μK
0
-0.3

0

0.3

Detuning of velocity-selective beams [MHz]Sideband spectroscopy
Atoms [arb. units]

a)

1
blue
sidebandred
sideband
0

-100

-50

0

50

100

Two-photon detuning [kHz]Figure 5.2: (a) Velocimetry results from the SWAP cooling experiment. The red data corresponds
to the 42 µK initial distribution, cooled with PGC. The blue distribution corresponds to 11 µK after
four cycles of SWAP cooling. (b) Sideband spectroscopy with n̄ = 0.8 in a red-detuned lattice withpeak ωax = 2π ×88 kHz. The radial distribution of atoms leads to inhomogeneous trap frequencies.

5.2

Doppler cooling
Doppler cooling is the workhorse in most AMO systems, and it forms the basis of the magneto-

optical trap (MOT) that begins all experiments in this dissertation. For a laser with wavelength λred-detuned slightly below an atomic transition frequency, the Doppler shift of an atom moving
towards the source increases the probability to absorb a photon compared to that of an atom moving away. Each photon absorption induces an ~k ≡ ~ 2π
λ momentum change. Heating from the~Γspontaneous emission of the excited state sets the Doppler cooling limit TD = 2k
= 146 µK in
B
87 Rb, where Γ = 2π × 6.066 MHz is the decay rate of the optically-excited 52 P
3/2 state.

Doppler cooling techniques have been widely applied to produce atoms and molecules at
mK to sub-µK temperatures and high phase-space density [117–123]. However, standard Doppler
cooling is limited in both final temperature and maximum force by the linewidth and wavelengthof the available optical transitions – properties that are provided by nature and not under controlof the experimentalist. Extending Doppler cooling to molecules is also notoriously difficult. After each photon absorption event, a molecule needs to return to the initial state via spontaneousemission for Doppler cooling to proceed, but it may decay instead into the abundance of accessible
vibrational and rotational degrees of freedom [124, 125]. This challenge has motivated the devel-

69
opment of alternative techniques that rely on reducing the role of spontaneous emission relative
to Doppler cooling.
To form a MOT, a spherical quadrupole magnetic field is generated by running current
through anti-Helmholtz MOT coils. The atomic energy levels of atoms far from the trap center are
shifted closer to resonance [102, 126]. Light red-detuned from the cycling transition (|F = 2i →|F 0 = 3i) leads to repeated photon absorption and emission. The two ingredients together entail aspatially- and velocity-dependent damping force that typically traps an ensemble of ∼ 108 atoms
with a density of 1 × 1010 cm−3 at a temperature near TD .
In our system, six beam launchers send 1 cm diameter beams towards the center of the cavity, each containing 6.5 to 10 mW of MOT power and 100 µW of repumper power. In early setups,
we used polarization optics to divide the beams along six separate pathways. After the squeezing experiments, we instead overlapped the MOT and repumper into an OZ Optics six-way fiber
splitter, reducing the complexity of the optics table and the tendency for accidental misalignment.
While accumulating atoms, the MOT laser is about 1.5Γ red-detuned from F = 2 → F 0 = 3 anda repumper laser (REP) at the F = 1 → 20 transition restores atoms that scatter into |F = 1i. The
quadrupole field of a MOT enhances trapping via the spatially-dependent Zeeman shift which, together with photon polarizations, ensures outward-moving atoms are kicked in towards the centerof the trap.

5.3

Polarization-gradient cooling into the red-detuned lattice
Additional atomic structure beyond hyperfine levels allows for a range of sub-Doppler cool-ing techniques [27, 102]. Such methods are needed to load our red-detuned lattice with significant
atomic density. We create a sort of optical molasses via polarization-gradient cooling (PGC) that
requires no additional lasers or polarization optics beyond those used for the MOT [127]. We simply disable the quadrupole field, set our bias coils to zero the magnetic field, tune our MOT laser
frequency to be 7Γ blue-detuned from F = 2 → 20 , and adjust intensities. The six beam launchers
typically send about 6.5 mW of MOT power and 5 µW of repumper power for PGC.

70
The σ + and σ − polarizations of the six beams form a polarization gradient and hence a
spatially-varying AC Stark shift to each atom’s ground state energy levels. When an atom travels
to the maximum of the potential “hill”, it has higher probability to scatter, optically-pumping intoa state with a potential minimum. With careful tuning, we see this form of “Sisyphus cooling”
achieve ensemble temperatures of 10 µK to 20 µK, but in principle, the limit is the photon recoil
2 2limit, Tr = m~RbkkB ≈ 360 nK for rubidium mass mRb and k/2π = 780 nm.
The red-detuned lattice is present during this process as an intracavity standing wave with
“far-detuned” wavelength λL = 813 – 823 nm. During the 5 – 20 ms loading period, a few hundred
atoms are loaded into each of ∼ 2500 lattice sites spread over 1 mm. Empirically, we see that thetemperature after PGC is limited to about 10% of the red lattice trap depth. The ratio of temperature
to trap depth sets the rms radial position of the atoms, so additional cooling is needed to improve
the radial homogeneity of the atoms.

5.4

Gray molasses
Atoms are further cooled by a short stage of lambda-enhanced gray-molasses cooling. Simi-

lar to PGC, gray molasses relies on two-photon Raman transitions and a potential energy landscapecreated by counter-propagating σ + and σ − beams [128]. The local polarization forms a “dark” linear superposition of |F = 2, mF i states which decouples from the electric fields. Slow moving
atoms remain in a dark state adiabatically, but faster moving atoms undergo diabatic transitions
into “bright” states which are decelerated by light blue-detuned from the |F = 2i → |F 0 = 2i transition. The Λ-enhancement refers to the presence of an additional repumper tone which forms a
Λ system as |F = 1i ↔ |F 0 = 2i ↔ |F = 2i, turning the dark state into a superposition involving
F = 1 states with decreased scattering rate.
We use a fiber-coupled EOM to add phase-coherent ωHF = 2π × 6.835 GHz sidebands to theD2 -line MOT laser. While the atoms were held in a moderately-deep λL = 813 nm lattice (80 µK
2

2

~ 4π
or 250 Erec where Erec = 2m), the incoherent repumper laser is turned off. The MOT laser
λ2
lremains 7Γ blue-detuned of F = 2 → 20 with 2.5 mW per beam and 100 µW in a sideband near

71
F = 1 → 20 . After applying lambda-enhanced gray-molasses cooling for 5 ms, we measure radial
and axial temperatures as low as 5 µK.

5.5Raman sideband cooling
In Raman cooling, a series of velocity-selective Raman pulses between ground-state hyper-

fine levels remove momenta [129, 130]. In one dimension, counter-propagating beams with frequencies ω1 and ω2 nearly complete the |ai to |bi Raman transition. When these frequencies are
sufficiently detuned from |ii, the excited state may be eliminated and the system is well-described
as a two-level system. A detuned Raman pulse excites the atom from |ai to |bi, while an opticalpumping pulse tuned to |bi ↔ |ii allows decay back into |ai (with momentum change spanning±~k). By tuning a pulse using ω1 − ω2 ≡ ∆ 6= 0, some velocity class becomes resonant due tothe Doppler shift and receives a 2~k kick towards v = 0. The two photon resonance condition isδ = ω1 − ω2 − ωHF = 2k(v + vrec ) with k = ω1 /c ≈ ω2 /c. The chosen pulse states and detuning prevent atoms with v ≈ 0 from being excited. A zero magnetic field is desired so that shiftedmagnetic sublevels do not broaden the Raman resonance. Extending the technique to additional
dimensions works similarly.
Degenerate Raman sideband cooling (RSBC) is a clever extension that can achieve subrecoil
temperatures [131, 132]. RSBC here works by creating a degeneracy between optical lattice vibrational modes and Zeeman sublevels. The technique is robust against experimental parameters andcan even cool directly to degeneracy given enough laser power [133]. We began using RSBC during
the interferometer experiment because small radial temperatures allowed more velocity selected
atoms (i.e. a higher phase-space density) and higher pulse fidelity. We reached radial temperatures of 0.7(3) µK. Expanding RSBC to the axial dimension using the intracavity lattice may be astraightforward improvement for the future.
In our setup (Fig. 5.3), Helmholtz bias coils set a 0.11 G magnetic field along the cavity axis
Ẑ so that the Zeeman splitting between adjacent states within a hyperfine manifold is nominally
2π × 75 kHz. Three RSBC beams form a triangular lattice in a plane perpendicular to Ẑ with axial

72
trap frequency ωax = 2π × 75 kHz. The polarizations of the three beams are twisted 10◦ from vertical to create the Raman coupling for driving the vibrational mode transition |F = 1, mF , ntri i →|F = 1, mF − 1, ntri − 1i. It was vital to clean up the polarization at the input and output of thefibers using polarizing beam splitter cubes. A combination of σ + and π optical pumping light initializes atoms in |F = 1, mF = 1i and this optical pumping light is kept on at a low intensity forcontinuous cooling. During the cooling, background radial confinement is still provided by theblue dipole trap and atoms are supported against gravity by the red intracavity lattice with trap
depth of ∼ 250Erec
We are interested in the 2D potential for three co-planar vertically-polarized beams k1 , k2 , k3
with amplitudes E1 , E2 , E3 and frequency ω,
~

~

~

2

U (r) = E1 ei(k1~r+ω0 t) + E2 ei(k2~r+ω0 t) + E3 ei(k3~r+ω0 t) .

(5.6)

A series expansion near the bottom of the potential reveals quadratic traps with approximately√√
degenerate frequencies for E1 = 2E2 = 2E3 and beams separated by θ12 = θ13 = 113◦ . The
introduction of a horizontal polarization component breaks the orthogonality for driving vibrational transitions between |ni → |n − 1i. Without this component, the matrix element for the
Raman coupling would be very close to zero [132].
The RSBC laser is an ECDL blue-detuned 2π×50 GHz to the blue of the |F = 1i ↔ |F 0 = 2i D2transition so that atoms are trapped at the nulls of the lattice, suppressing scattering off the coolingbeams. The laser frequency does not need to be stabilized because the cooling method can toleratemany GHz of drift. About 10 mW is sent into a tapered amp (gain chip: Eagleyard EYP-TPA0780-01000) built expressly for this setup, and the output power is divided into a 100 mW fiber
launcher and two 50 mW launchers. All three launchers focus to beam waists of approximately
4 mm × 0.1 mm extended elliptically along the cavity mode. The launchers are about 45 cm fromthe atoms.
The RSBC potential is ramped up exponentially over 0.4 ms, cooling happens for 1.2 ms, andthe potential is ramped down over 0.3 ms. Smoothly transitioning the trap depth was crucial for73
observing cooling (including using a 13 kHz low pass filter to smooth the VGA controlling AOM
limiting the RSBC power). We then wait a quarter of the radial trap period from the combined
red and blue lattices, 0.225 ms, for atoms to oscillate back to the center of the cavity, converting
potential energy back to kinetic energy, at which point we repeat the entire cycle. The cycle is also
repeated a third time, except after the cooling window, the red lattice is smoothly lowered from80 µK to 3 µK over 3 ms. The RSBC potential is gradually lowered for the last time, and the redlattice is turned off entirely to allow the atoms to fall under gravity.
a)b)

Top view

Side viewc)

100 mW

Microwave
antenna

113°

50 mW

2D

M

O
T..
.

50 mW

d)
RSBC
MOT
beam

RSBC
beam
SpacerRepumper
σ+

Coil

RSBC
beam
mF=-1

mF=0

mF=1

Figure 5.3: Raman sideband cooling. Physical setup with RSBC beams (blue): (a) top view and(b) side view, showing the cavity spacer (white), the MOT coils (orange) and the necessity for nonplanar and nonequilateral beam geometry when launching into the glass cell. (c) The pyramidal
lattice was calculated for a 2:1:1 power ratio and 113◦ between the strong beam and each weak
beam. (d) A single Raman sideband cooling cycle. Atoms are held in an RSBC lattice with axial trap frequency equal to the Zeeman splitting, ωax = ωZ . An atom in |mF = 1, n = 2i (yellow)
undergoes a two-photon Raman transition to |mF = 0, n = 1i, and repumping brings it back to|mF = 1i with the net result of reducing the motional excitation n by one quanta.

5.6

SWAP cooling with Raman transitions
A new laser-cooling mechanism called Sawtooth Wave Adiabatic Passage (SWAP) cooling

was demonstrated by the strontium side of the Thompson lab using a narrow-linewidth transitionin 88 Sr [134]. SWAP cooling uses counter-propagating lasers that are slowly swept in frequency

74
through an atomic resonance to facilitate Landau-Zener adiabatic transfers [135, 136]1 . Repeatedtransfers enable the removal of many photon-recoils of momentum from atoms or molecules before
spontaneous emission occurs. This technique is robust against laser frequency noise and smallchanges in atomic transition frequency. SWAP cooling has now been used for slowing dysprosium[137] and for rapidly forming denser Sr MOTs [138], and further exploration continues in various
groups [139–142].
The proof-of-principle experiments described in the remainder of this chapter use 87 Rb to
demonstrate that the SWAP cooling mechanism does not require a closed, single-photon transition [143]. Instead, it is amenable to atoms and molecules with at least two long-lived groundstates bridged by a two-photon transition. Cooling molecules remains a significant challenge, andwithout dwelling on the dissimilarities between molecules and atoms, any cooling scheme thatsuppresses the probability to scatter into undesired states is potentially of interest for the field. In
Sec. 5.6.2, some arguments are provided for the merits of SWAP cooling in a regime where little tono free-space scattering is permitted.
For SWAP cooling with a single-photon transition, counter-propagating lasers address theatoms just as in standard Doppler cooling. The frequency is ramped in a sawtooth pattern throughthe transition frequency ωa that separates a ground state |ai and a long-lived optically excited state|bi with lifetime τ = 1/Γ (Fig. 5.4). For an atom in motion, the relative Doppler shift of the two laser
beams sets an important time ordering: the beam counter-propagating to the atom’s motion firstpasses through resonance, driving an adiabatic transition from |ai to |bi along with a momentum
kick associated with photon absorption that opposes the atomic motion. The co-propagating beam
next drives a transition back from |bi to |ai via stimulated emission that, again, slows the atom. In
net, each sweep ideally removes 2~k of momentum regardless of the direction the atom is moving.
In our demonstration of SWAP cooling using two-photon transitions, there are two metastableground states, |ai and |bi (Fig. 5.4(b)). Two lasers tuned off-resonance from an intermediate optically1

From numerous discussions, it may be helpful to explicitly state that the Landau-Zener adiabatic process differsfrom stimulated Raman adiabatic passage (STIRAP), although the techniques are intimately related.

75
excited state |ii engineer an effective optically-excited state with tunable lifetime (or, equivalently,
linewidth) instead of relying on the properties of the original optical transitions. The laser dressing|ai is swept with an asymmetric sawtooth waveform through the two-photon resonance, driving
adiabatic Raman transitions between |ai and |bi. At the end of a cycle, some atoms may erroneously
remain in |bi, so optical pumping if briefly applied to return them to |ai. In this manner, the limitations set by spontaneous emission can be bypassed: the Doppler temperature associated with
the intermediate state no longer applies, and population loss to undesired states is avoided. Afterabout 10 ms, we achieve a 1D equilibrium temperature 25 times lower than the usual TD = 146 µK
cooling limit for standard Doppler cooling in 87 Rb.
b)c)Population

a)

1.0
0.5
0.0
0.5
0

2-level system(narrow transition)

3-level system(Raman transition)

-0.5
0.0

0.25

0.50

0.75

1.0

Time

Figure 5.4: Landau-Zener transfers for single-photon and Raman transitions. SWAP coolinglevel diagrams for (a) a single-photon transition and (b) a two-photon Raman transition. In the
Raman configuration, a large detuning ∆ from the intermediate state ensures the lifetime of theexcited state |ii is effectively infinite except when it is used in returning atoms to |ai following each
sweep. (c) Landau-Zener adiabatic transfers are simulated in the presence of scattering. As ω2 isswept through resonance (bottom, δ(t) = 0), population is swapped between the atomic states(top). In units of the scattering rate, the sweep range is ∆swp = Γ/3, ∆ = 300Γ, Ωab = Γ/150. The
scattering rate Γ causes the decay of coherence.

5.6.1

Laser cooling rubidium with adiabatic transfers
Figure 5.5 shows our setup for demonstrating Raman SWAP cooling in 87 Rb. The two groundstates are the hyperfine “clock” states: |ai ≡ |F = 2, mF = 0i and |bi ≡ |F = 1, mF = 0i. Thesestates are coupled by lasers at frequencies ω1 and ω2 , both far-detuned from the intermediate state

76|ii by ∆ ≈ 2π × 2 GHz. The east/west bias coils establish a uniform magnetic field to set thequantization axis along X̂, the propagation axis of the cooling beams. The cooling beams incident
from +X̂ are vertically polarized and the beams incident from the −X̂ are horizontally polarized.
Dipole selection rules disallow two-photon transitions with pairs of photons of the same linear
polarization, i.e. the absorption and stimulated emission of pairs of orthogonal linearly polarized
photons. As a result, only counter-propagating photons with difference frequency of ωHF = 2π ×6.835 GHz drive the transitions, and internal state changes are always accompanied by a net photon
recoil momentum of 2~k where ck ≈ ω1 ≈ ω2 .
a)

c)
OP

sweep

Frequency inintoatom rest frame
0.5
[MHz]

OP

sweep

into

0
-0.5

d)
b)

1

Atomicpopulation
Magnetic field andquantization axis

Atom, velocity

0e)
Averagemomentum0

Polarization

1

Time [ms]Figure 5.5: Raman SWAP cooling scheme. (a) 87 Rb level diagram. Two-photon transitions between |ai and |bi are induced by ω1 (blue) and ω2 (red). (b) Experimental layout. A moving atom
(green) interacts with two pairs of counter-propagating laser beams (blue, red) with ω2 swept ina sawtooth manner. (c) The four unique laser frequencies observed in the rest frame of the atom
after considering the Doppler shifts that separate counter-propagating lasers in frequency by 2kv.
Offsets have been subtracted so that points marked with circles correspond to allowed two-photon
resonances which involve pairs of orthogonally-polarized, counter-propagating laser beams. (d)Two-photon Landau-Zener transitions transfer an atom from |ai to |bi and back to |ai during each
sweep. (e) The counter-propagating lasers ideally remove 4~k momentum per cooling sweep.

The two independent lasers at ω1 and ω2 are phase-locked and their outputs are split into
the ±X̂ (Fig. 5.5(b)). We arbitrarily choose to hold the frequency ω1 fixed while the frequency

77
ω2 is swept linearly in time, downward in frequency through the two-photon atomic resonanceat ω1 − ω2 ≈ ωHF . The Landau-Zener formula describes the probability to undergo a diabatic
transition between two energy levels during the sweep [135, 136],
Pd = e−ξ

where

ξ=

π 2
Ω /α,
2 ab

(5.7)

1 Ω2
with sweep rate α ≡ dω2 /dt. Here, the two-photon Rabi frequency is Ωab ≈ Ω2∆, and Ω1 ≈ Ω2 are

the single photon Rabi frequencies of the respective frequency components ω1 and ω2 .
If the sweep rate is kept low, the system undergoes an avoided crossing and adiabatically
evolves into the opposite atomic state. The two-photon resonance condition δ(t) ≡ ω1 −ω2 −ωHF =±2kv is met twice due to the Doppler shifts of the laser frequencies as seen by an atom moving
at velocity v (Fig. 5.5(c)). Each adiabatic passage through imparts a net momentum change of 2~k
per transition, i.e. 4~k per ideal sweep. The direction of the frequency sweep determines whether
momentum is added or subtracted.
The Doppler shift becomes of order Ωab as the speed of an atom approaches zero. At thispoint, the time-ordering breaks down for the adiabatic transfers detailed previously and the slowing mechanism begins to fail [134]. The eventual failure of a transfer, either due to broken timeordering, scattering, or imperfect adiabatic transfer efficiency, means there could be considerable
probability to find atoms in |bi at the end of a sweep, and the next cycle would result in heating. We
therefore apply π-polarized optical pumping light transverse to the cooling beams to return any
misplaced atoms to |ai before each frequency sweep. Optical pumping in this scheme plays a roleanalogous to spontaneous emission in Ref. [134] for reducing entropy, breaking time reversibility
and allowing for the compression of phase space. The optical pumping was applied for 100 µs,
though this is much longer than necessary given the fast decay rate for the excited state.
In our SWAP cooling experiment, the ω1 laser is locked to a spectroscopy setup, and ω2 islocked to the first-order sideband of a fiber EOM modulated by an rf signal generator (PN: SRS
SG384) at the 6.8 GHz offset frequency plus a triggered sawtooth frequency ramp. Around 107
atoms are loaded into a 3D MOT and cooled to 42 µK with polarization gradient cooling to nar-

78
row the initial velocity distribution, reducing the required frequency sweep range. The atoms are
initially pumped into |ai with π-polarized light applied by the transverse optical pumping beams.
The cooling lasers are turned on with AOMs and ω2 is swept for of order 1 ms and over 1 MHz (tocover the Doppler width of the distribution) as shown in Fig. 5.5(c) with the intended atomic state
populations and momentum changes shown in Fig. 5.5(d and e). The SWAP sequence of opticalpumping and sweeping is repeated several times, until a significant fraction of atoms are physicallydisplaced from the cooling/velocimetry beams due to an imbalance in repumping beam power.
The final 1D temperature of the atoms is measured by velocimetry as described in Sec. 5.1.
Atoms are returned to |ai as part of the final SWAP cycle and then transferred to |bi with a microwave π-pulse so that a small background of atoms in |ai can be blown away. Velocity-selective
Raman transitions are applied with the cooling beams for 90 µs to drive atoms within a small velocity range back to |ai [115]. The population in |ai is finally determined using fluorescence andthe resulting Voigt profiles2 are fit to extract the temperature (Fig. 5.6(a)). Velocimetry allows better
precision than time-of-flight measurements of the cloud’s ballistic expansion in this setup, but thetwo methods returned consistent results.
From an initial temperature of 42(3) µK, atoms are cooled in 1 dimension to 10 µK after theapplication of 9 sweeps with fixed two-photon Rabi frequency Ωab = 2π × 22 kHz and sweep
range of ∆swp ≡ 2π × 0.8 MHz (purple points in Fig. 5.6(b)). The sweep range ∆swp is chosen to
include 95% of the atoms when accounting for the Doppler shifts of the initial velocity distribution.
Because the initial sweeps narrow the velocity distribution, the cooling rate and final temperature
can be optimized by progressively decreasing the two-photon Rabi frequency and sweep range.
√
For the green points in Fig. 5.6(b), after three sweeps, the Rabi frequency is reduced to 0.5 × Ωaband the sweep range is reduced to 0.5 × ∆swp such that the Landau-Zener adiabaticity parameter ξ(Eq. 5.7) remains unchanged. After two additional cycles, sweeps are performed with two-photon
2

The Maxwell-Boltzmann temperature distribution G(v, T ) =qΩab /2k
Lorentzian L(v) = v2 +Ω
2 /4k2 , resulting in a Voigt profile V (v, T ) =
ab

2

− mv
m
e 2kB T
2πkB TR∞−∞

convolves with the excitation pulse

G(v 0 , T )L(v − v 0 ) dv 0 . In our parameter

regime, the resulting lineshape has only minor deviation from a Gaussian, but we easily correct for this effect.

79
Rabi frequency√

0.4 × Ωab and sweep range 0.4 × ∆swp . In this manner, the measured temperaturereached 5.9(3) µK after 9 total sweeps.
b)

50

40

30

1.0
0.5
0

0

2

4

6

8

10

8

10

Number of sweepsc)20

5.9(3) µK

10

0
0

1

2

3

4

5

6

Number of sweeps

7

8

9

Velocity [mm/s]

Temperature [μK]1.5Relative phasespace density

a)

50
0
-50
0

2

4

6

Number of sweeps

Figure 5.6: Raman SWAP cooling results. (a) Reduction in temperature, determined by velocimetry. The atomic ensemble is cooled as low as 5.9 µK in one dimension over the course of several
sweeps. For the purple points, all sweeps used a two-photon Rabi frequency Ωab ≈ 2π × 22 kHz
and sweep range ∆swp = 2π × 0.8 MHz. For the green points, Ωab and ∆swp were successively
reduced between sweeps as described in the text. (b) The relative one-dimensional phase-spacedensity for the corresponding purple or green points in (a). The phase-space density begins todecrease after four sweeps despite the decreasing temperature because atoms begin to leave thevelocimetry beams. (c) To cool into a moving reference frame (blue points), we apply a frequency
offset δAOM = 2π ×150 kHz in the lab reference frame between the beams from opposite directions.
By sweeping ω2 downwards, the atoms cool and equilibrate into a moving reference frame which
has a velocity vF ≡ δAOM /2k (dashed orange line). Reversing the sweep direction (pink points)
caused the atoms accelerate in the opposite direction without bound.

As atoms are cooled, the entropy is reduced and the relative 1D phase space density ρ/ρ0 =∆x0 ∆v0 /(∆x∆v) in Fig. 5.6(c) is increased modestly. Here ∆x0 and ∆v0 (∆x and ∆v) are the measured cloud size and velocity spread before (after) cooling. Given that phase space compression
cannot occur without dissipation, the lack of spontaneous emission in an ideal sweep is initiallyalarming. But one can achieve phase space compression without actually scattering a photon – itis sufficient that the atom could have scattered a photon were it within a particular region of phasespace. When an atom’s Doppler shift satisfies kv . Ωab , there is significant probability of fin-

80
ishing a sweep in the wrong internal state so a scattered optical pumping photon signals that an
atom has a small velocity. In the early cooling stages, when kv  Ωab , the atom is always moved
towards lower velocity without scattering a photon, so even not scattering a photon acts as information to the environment that reduces the probability that the atom is in the portion of phasespace with kv < Ωab . The loss of atoms from the effective cooling/imaging region limited the observed phase-space density after a few sweeps. Larger beam diameters or better balanced opticalpumping beams would mitigate this issue.
To emphasize the critical role of the frequency sweep direction, a fixed relative offset frequency δAOM = 2π × 150 kHz is applied between the counter-propagating beams such that atoms
are cooled into a moving reference frame with velocity vF = δAOM /2k. When the laser frequency
is swept downward as before, the atoms are accelerated into and then equilibrate into the moving
frame (blue points in Fig. 5.6(c) saturating at the predicted velocity, orange dashed line). In contrast, reversing the frequency sweep direction without changing δAOM accelerates the atoms in the
opposite direction (pink points) even though the direction of the moving reference frame remains
unchanged. As expected, the atoms are accelerated to speeds larger than |vF |.

5.6.2

A model for Raman SWAP cooling with scattering
The choice of sweep rate α is optimized by considering two effects. First, adiabatic transfer

can be extremely efficient, but the laser frequency must be swept slowly such that the LandauZener diabatic transition probability (Eq. 5.7) is small. Second, spontaneous emission from theexcited states results in heating. To avoid scattering from the applied cooling beams, the probability to have not scattered a photon during the total sweep time should be close to unity, P̄sc =e−Rsc ∆swp /α ≈ 1, and hence the sweep cannot be excessively slow. Rsc is the total spontaneous
scattering rate from the far-detuned intermediate state(s). For a single laser and excited state, wehave [27]
Rsc =

 
Γ
2 (Ω/Γ)2
Γ Ω2
≈
.
2
2
2 1 + 4 (∆/Γ) + 2 (Ω/Γ)
2 2∆2

(5.8)

81
In Fig. 5.7(a, circles), we explore the probability for successful adiabatic transfer from |ai to|bi using σ + -polarized beams from a single direction. The two-photon Rabi frequency is kept thesame as above. This configuration allows Doppler-free, dipole-allowed transitions. The optimum
sweep rate α near 1 MHz/ms maximizes the transfer efficiency as desired for efficient cooling. The
measurements closely follow a prediction from numerically integrating optical-Bloch equations including scattering rates and measured Rabi frequencies (orange). For comparison, the red dashedline is the Landau-Zener 1 − Pd theoretical transfer efficiency which ignores free space scattering.
A simple probability-based model of a full cooling sweep, assuming 79% chance to transfer eachway while allowing for the possibility of a state change by free-space scattering, then predicts a
2.2~k expected momentum reduction, the same as was measured for the first sweep of Fig. 5.6(b).
Later sweeps are less efficient.

0.8
0.6
0.4
0.2
0
0.1

1

10

Laser frequency sweep rate

100[MHz/ms]

Maximum Δp per sweep
for SWAP cooling [ħk]Population inafter
one adiabatic sweep

b)

1.0

4

80

3

60

2

40

1

20

0

1

10

Quality parameter Q

100

0

Momentum transferred when 1/eparticles remain [ħk]a)Figure 5.7: Adiabatic transitions and momentum transfer. (a) Population in |bi after one adiabatic transfer attempt (black points), measured using fluorescence detection. To drive a single transfer, we use co-propagating σ + -polarized light with fixed ω1 and ω2 swept through resonance at rate α = dω2 /dt. Adiabatic transfers are performed with (Ωab , Γ, ∆, ∆swp ) = 2π ×
(20 kHz, 6 MHz, 2 GHz, 0.8 MHz). The familiar Landau-Zener prediction (red dashed) fails at low
sweep rate due to off-resonant scattering from the intermediate state |ii. A numerical simulation
including off-resonant scattering and the experimental parameters (orange) shows better agreement with the data but some room for better modeling. (b, left) The achievable momentum transfer per SWAP cooling sweep (blue dashed) approaches the ideal 4~k at large quality parameterΩ2
Q ≡ π2 Γ∆. (b, right) With the worst-case scenario in mind where a particle is lost if it undergoes
swp
a single spontaneous emission event, we plot the predicted achievable momentum transfer when
1/e particles remain versus the quality parameter Q (green).

82
To further generalize predictions for how scattering will limit SWAP cooling, consider the|ai ↔ |ii ↔ |bi Λ-system shown in Fig. 5.4(b). For simplicity, we will assume that the intermediate
state |ii decays with equal rates of Γ/2 into both |ai and |bi and that the large detuning limit ∆ Γ, Ω is satisfied. For convenience, we also assume equal single-photon Rabi frequencies Ω1 = Ω2 =22ΩΩ. The two photon Rabi frequency is then Ωab = 2∆
, and the total scattering rate is Rsc = ΓΩ.
4∆2

Details supporting the following summary of results are provided in Appendix B.0.3.
For this toy model, we first find the optimum momentum transferred ∆p during a coolingsweep after optimizing for sweep rate α. In Fig. 5.7(b, blue), ∆p is plotted against a dimensionlessquality parameter,Q≡ln (Pd )π Ω2≈.
2 Γ∆swp
ln P̄sc(5.9)

The sweep range ∆swp is not entirely free to change as it must sufficiently cover the Doppler widthof the velocity distribution. For Q  1 then, the momentum transfer per sweep saturates to theideal value 4~k, and the effective force is 4~k/tswp , where the time to complete each sweep is tswp .
2∆
At the optimized sweep rate, tswp = 8 ln(2Q)∆ . The quality parameter Q scales with laserπΩ4 swpintensity I and wavelength as Q ∝ Iλ3 , but interestingly, it does not depend on the dipole matrix
element M between the states |ai, |bi and the intermediate state |ii. With these parameters, the2sweep time scales as roughly tswp ∝ M∆4 I 2 ∆swp .
Avoiding spontaneous emission is of primary importance for cooling molecules. In theworst-case scenario, every molecule that spontaneously emits a photon is completely lost into unaddressed “dark states”. In Fig. 5.7(b, green), we evaluate ∆p|1/e , the average momentum removed
when 1/e molecules remain, by numerically optimizing the sweep rate. For Q  1, the numerical
2.1Q
result is well approximated by ∆p|1/e ≈ ln(4(Q+14))~k and one can remove many photon recoilsof momentum from a molecule before it is likely to be lost. The optimized sweep time is well2∆
approximated by tswp ≈ ln(4(14+Q))∆ .
3.2Ω4 swp

In principle, 3D cooling should be possible using additional pairs of cooling beams along Ŷ
or Ẑ (where the 1D cooling above has been along X̂). Experimentally, it may be difficult to quickly

83~ for cooling one dimension at a time, so the available
change the direction of the magnetic field B
laser polarizations become a subtle obstacle. One possibility is to set the magnetic field axis alongX̂ + Ŷ + Ẑ such that some polarization projection always facilitates two-photon transitions – but thelight projected into π-polarization does not contribute to cooling and instead results in additional~ may be kept along X̂ while ω2 is sent along
free-space scattering. In a second scheme, ω1 and B
each direction to be cooled. In this case, sweeps along Ŷ and Ẑ maximally reduce 2~k (rather than4~k) momentum per sweep, thus leading to a reduced cooling rate.
In the future, it may also be interesting to explore the manipulation of ensembles via accelerations and decelerations. The initial temperature, which constrains the required sweep range∆swp , was tens of µK in this experiment but it could potentially be orders of magnitude higher if
the available laser power permits large Q. More complex waveforms for the laser intensity anddetuning might also improve the cooling rate by decreasing the required sweep range [144].

Chapter 6
Spin-squeezing using cavity non-demolition measurements

Entanglement is a fundamental quantum resource, capable of improving precision measurements and required for all quantum information science. Advances in the creation, manipulation,and characterization of entanglement will be required to develop practical quantum computers,
quantum simulators, and enhanced quantum sensors. Spin-squeezed states could be used to improve a variety of quantum sensors, with today’s best atomic clocks [43, 145–147] being particularly
promising candidates. In this chapter, we demonstrate creating large amounts of entanglement oninternal degrees of freedom, the basis states of such sensors.
Collective (“joint”) measurements with atom-cavity systems have been the most effectivemeans for generating highly-entangled squeezed states to date. With the first-generation apparatus described in Chapter 3, we directly observed up to W −1 = 59(8) times (17.7(6) dB) reductionin quantum phase variance relative to the standard quantum limit [20]. The methodologies andexperimental details that led to this result are present in Sec. 6.1. A separate atom-cavity systemachieved up to 18.5 dB of spin squeezing using a similar non-demolition measurement scheme[21]. These are the largest reports of metrological enhancement in any system to date. In Sec. 6.2,
a microcontroller is introduced to our system for generating deterministically squeezed states atparticular Jz values. In Sec. 6.3, the standing wave lattice is transformed into an axially-smooth
dipole trap so that atoms are free to fall. Using the atom’s time-averaged coupling to the atomic
probe to maintain squeezing was a large step towards cavity-based, spin-squeezed matter-wave
interferometery. Many details of this chapter overlap with Kevin Cox’s dissertation [148].

85
6.0.1

Real-time tracking of a quantum phase
As a detour to these topics, we had the pleasure to work with theorists Murray Holland

and Athreya Shankar on a proposal for continuous real-time tracking of a quantum phase below
the standard quantum limit [149]. Nearly all quantum sensors operate with the Ramsey sequenceparadigm, estimating a quantum phase φ(t) between states of a superposition that evolves during
an interrogation time. Unfortunately, the destructive readout of a Ramsey sequence requires resets
and repeated trials to infer the phase from population measurements. In the scheme of Ref. [149],
a single run gives continuous information about a time-varying signal. As an added bonus, phasesqueezed states develop as a natural result of the quantum non-demolition probing scheme. A
sensor such as this is in some sense the holy grail of precision metrology, achieving sub-SQL resolution and dramatically improving bandwidth by avoiding state resets, not to mention the lack ofnoise from avoiding rotations and the unambiguity when accumulating multiple radians of phase.
In this proposal, nearly-balanced cavity-assisted Raman transitions interfere to create atomcavity interactions that reveal the collective state of the atoms through a leaking cavity field I(t).
Homodyne detection of that field yields information about the environment of the sensor, and inprinciple allows for tracking a signal over time. Two lasers ω1 and ω2 are symmetrically-detuned ω0from the cavity mode and the cavity frequency completes both Raman transitions. By controlling
the laser phases, balancing the Rabi frequencies Ω1 = Ω2 ≡ Ω0 , and operating far-detuned from theatomic transition, the atom-cavity Hamiltonian is the sum of a Jaynes-Cummings and anti-JaynesCummings interaction,
ĤT =

~ΩT ˆ
x̂Jx .
2

(6.1)

 √
 √
Here x̂ = â + â† / 2 is the amplitude quadrature, ŷ = â − â† / 2i is the phase quadrature,[x̂, ŷ] = i, and â, â† are cavity annihilation and creation operators. The interaction strength is√ΩT = 2Ω0 g0 /∆.
The two drives can be considered to initialize counter-acting superradiant pathways thatexactly cancel when Jx = 0 (φ(t) = 0). For a small deviation in φ, however, the pathways add

a)

b)

c)

86

Time

Frequency

Figure 6.1: Continuous real-time tracking of a quantum phase. (a) Energy-level diagram and (b)frequency diagram for the quantum phase-tracking proposal. Cavity-assisted Raman transitionsbalance and interfere to drive a QND interaction. (c) Spin-squeezed states form naturally in thephase quadrature. The emitted cavity field I(t) ∝ φ(t) is detected in homodyne.

D E
D E
constructively, giving rise to an output field. Even with Jˆx = 0, Jˆx2 6= 0 and quantum fluctuations source the Y quadrature of the cavity field. Measurement back-action in Jz arises due tothe indistinguishability of these processes, as is required for a spin-squeezed state.
This scheme has not been realized thus far, and using rubidium poses some practical difficulties. One possibility is to use the F = 1 manifold and map the spin-1/2 system described in [149]onto a spin-1 system. Even so, one must worry about compensating the quadratic Zeeman shift
while still lifting the degeneracy of the mF levels. The scheme also has stringent requirements onphase noise and the driving laser linewidth but a proof-of-principle demonstration could be withingrasp in current systems.

6.1

Generating large amounts of useful entanglement
In our spin-squeezing experiments [20], collective measurements of the atomic population

are used to project an ensemble of rubidium atoms into an entangled state and verify this entanglement. Quantum noise of a pre-measurement cancels noise in the final measurement in this techniquecalled conditional squeezing [16–19, 46, 47, 150–152]. It is worth saying explicitly that conditionalsqueezing refers to the use of a pre-measurement, not a post-selection of data or some other dubious data analysis technique.

87
The initial demonstrations of squeezing via cavity-based quantum non-demolition measurements [46, 47] have been followed by rapid gains in achievable metrological enhancements.
High-finesse cavities set new records in the amount of useful entanglement by exploiting the√W ∝ 1/ N C fundamental scaling with cooperativity C in QND schemes [20, 21, 29]. Using an
optically-cycling transition, the fundamental scaling improves further, to W ∝ 1/N C [20, 151].
The basis states of this chapter follow this insight. Although the magnetic sensitivity of the |↑i ≡|52 S1/2 , F = 2, MF = 2i and |↓i ≡ |52 S1/2 , F = 1, MF = 1i states makes them unsuitable in a clock,they are good magnetometers and are useful for exploring the limits of our understanding. In
truth, suitability for quantum sensing can be a subtle question to answer. For example, all of the
population-squeezed states described here require additional operations for insertion in an atomic
clock. After accounting for the impact of non-unitary state evolution on the performance of an
atomic clock, our states still compare quite favorably to others [153]In this section, probe light is sent into the cavity detuned δc = 2π × 400 MHz to the blue of
the |↑i to |ei ≡ |52 P3/2 , F = 3, MF = 3i optical atomic transition as shown in Fig. 6.2(a). Atoms in|↑i behave like small pieces of glass, imparting a phase shift to the reflected probe light which isseen as a shift from the empty cavity resonance frequency ωc to ωc0 . The collective population N↑ isinferred from this frequency shift. Because the |↓i ↔ |ei transition is so far from laser resonance,
atoms in |↓i hardly interact with the intracavity light. It is crucial that the collective, dispersive
measurement of N↑ does not collapse individual atom wavefunctions.
At the start of every experimental trial, a 3D MOT is formed from the background rubidium
vapor pressure and cooled to about 10 µK with polarization-gradient cooling (PGC). A far-detuned
823 nm intracavity lattice then traps N = 4 × 105 atoms tightly on axis over a 1 mm extent. This
allows for considerable overlap with the probe TEM00 mode and it prevents atoms from moving significantly during the sequence. The cavity finesse is F = 2532(80) and the linewidth isκ = 2π × 3.15(10) MHz. A 1.1 Gauss bias magnetic field along the cavity axis sets the quantization
axis. The spatially inhomogeneous coupling of atoms to the cavity mode is handled by considering
an effective coupling and effective atom number, as in [29, 45, 47, 151]. Neglecting small correc-

88
a)

b)

d)

Mi

photons

T

Time

2
1

Time

e)
×103

c)

400

CBG

300
C

200

-1
-2
0

30

Trial number

60

Atoms in

0

100
0

0

π/2

π

3π/2

2π

Readout phase ϕ [rad]Figure 6.2: Conditional squeezing overview. (a) The relevant 87 Rb energy levels, bare cavity resonance frequency ωc , and dressed cavity resonance frequency ωc0 (N↑ ). (b) Conditional squeezing
sequence. Probe windows (teal) measure the population in N↑ or N↓ as labelled. Microwave pulses(black) prepare the initial superposition and enable a spin-echo protocol. (c) Spin noise reduction
R at Mi = 30 × 103 photons. The corresponding noise distribution (red) is drawn next to the initialcoherent state distribution (gray). (d) Contrast measurement sequence. The phase φ is scanned
to measure the fractional length of the Bloch vector after the pre-measurement. (e) Contrast datawith CBG = 0.98 (red, measured with Mi = 0 photons during the pre-measurement) and C = 0.75
(blue, for Mi = 30 × 103 photons).

tions for radial inhomogeneity, the number of atoms N0 , the single-atom Rabi frequency 2g0 at an
antinode of the probe mode, and the cooperativity parameter C0 at an antinode of the probe mode2 ≈ 3 g 2 , and C ≈ 3 C respectively.
are related to the effective quantities by N ≈ 23 N0 , geff
4 0
4 0

The conditional squeezing sequence is shown in Fig. 6.2(b). At the beginning of each trial,
atoms are optically pumped into |↓i with over 90% purity, and the remaining atoms in F = 2
are blown away with a beam transverse to the cavity. A microwave π/2-pulse places each atom
into an equal superposition of spin states, preparing the collective Bloch vector along ŷ. At thispoint, a measurement of the spin projection Jz is made1 , followed by a spin-echo π-pulse, andthen a pre-measurement with outcome labeled Jzp = N↓p /2. The final measurement Jzf = N↓f /2
immediately follows. Although a measurement of N↑ at this point allows additional cancellation
1

The results of this measurement are not used in constructing a noise estimator here. Rather, the first measure-ment probe photons only serve to spin-echo probe-induced inhomogeneous light shifts. The window is labelled N↑p indiagrams.

89of quantum noise in principle, it was advantageous to avoid added noise from state rotations instead. The variance of the difference of the two population measurements normalized to quantum
projection noise (QPN) is the spin noise reduction,

R≡

∆Jz
∆Jz,QPN

2


=∆(N↓f − N↓p )
√
N /2

2
.

(6.2)

Each population measurement outcome is obtained by measuring the Q quadrature of the reflected
atomic probe field in homodyne and averaging the signal over a 40 µs window.
The Wineland parameter for enhancement of phase resolution is [13]

W =

∆θ
∆θSQL

2
=

R
CBG
C2

(6.3)

i.e. to observe an enhancement in phase resolution, one must also evaluate the fractional atomic coD
Eherence or “contrast” that remains after pre-measurement, C ≡ 2 Jˆ /N . The coherence is determined in a separate set of experiments by using microwave rotations after the pre-measurement
to rotate the Bloch vector such that its total length can be determined. CBG = 0.96 is the backgroundcontrast as determined from measurements of the contrast at Mi = 0 probe photons. This small
correction represents normalizing W with respect to the length of the Bloch vector in the absenceof pre-measurement, giving a 0.2 dB improvement in the reported squeezing compared to what
would be calculated with W = R/C 2 .
In Fig. 6.3(a), the directly-observed noise reduction R is plotted against the average numberof photons Mi incident upon the cavity during a single measurement window. Photon shot noise
scaling is seen at low photon numbers, but at high photon numbers, other noise sources begin tocontribute including scattering and optomechanically-induced decorrelations. Also apparent in
Fig. 6.3(a), the contrast remaining after the pre-measurement decreases with increasing Mi , primarily due to undesired free space scattering causing collapse of individual atomic wavefunctions
into |↑i. The directly-observed squeezing is optimized at W = 17.7(6) dB near Mi = 3.8 × 104 .
The maximum quantum noise reduction R−1 = 92(9) or 19.6(4) dB below QPN is limited
partly by the technical noise between the probe laser and the cavity which set a floor about 25 dB

90
a)

b)

10 log10(C2, R, W) [dB]

0
-5
-10

W=17.7(6) dB

Time

c)

50

-15

25
-20
4

5

6 7 8 9

10 4

2

3

4

5

0
-π/2

6

Incident probe photons, Mi

d)

5
-5

-200

-100

0

-π/4

0

π/4

π/2

Aligned pulse rotation ψ [rad]
100

200

Conditional probability distribution [mrad]

Figure 6.3: Conditional spin squeezing results. (a) Squared contrast C 2 (blue), spin noise R(red), and spin squeezing W (black) are plotted versus the average number of incident photons
Mi in a single measurement window. The solid lines are fits to the data, the blue band is thepredicted loss of contrast from free-space scattering, and the gray band indicates the total error bar
for spin squeezing W . (b) A sequence with an additional microwave pulse aligned with the Blochvector rotates the state through ψ to map out the conditional probability distribution. (c) The spinprojection R is plotted versus ψ and fit to an ellipse. (d) The reconstructed conditional probability
distribution of the quantum state (blue) and the corresponding coherent spin state distribution(red) drawn with 1:1 aspect ratio.

below QPN (see Ref. [72] which used a precursory heterodyne detection system). Spin noise reduction was also limited significantly by optomechanics induced by toggling the probe light. Dueto the incommensurate probing and trapping potentials, some atoms are located on the slope of
the probe standing wave and thus experience a force that drives axial oscillations in the trap. In
addition, the minima of the trapping potentials are spatially offset by the presence of the probe
field. The ensemble average of each atom’s time-dependent coupling to the probe mode results in
the ringing shown in Fig. 6.4(a) (red) over a 40 µs probing period.
To partially cancel the optomechanical ringing, we employ a short, impulsive turn-on sequence. A 2.5 µs half power pulse induces ringing, and a full strength pulse applied one quarter ofan axial oscillation period later coherently zeroes the initial axial ringing such that the atoms cometo rest at the new trap minimum. As shown in Fig. 6.4(a), this technique significantly reduces theamount of ringing, but it only improved the optimal squeezing in Fig. 6.3 by an estimated 0.6 dB.

91b)

Axial oscillations

Radial oscillations
0.2

400

Spin noise R

[kHz]

a)

0

-400
0.00

0.02

0.04

Time during measurement window [ms]

0.1

0.0

0.0

0.5

1.0

1.5

2.0

Delay time T after pre-measurement [ms]Figure 6.4: Optomechanics. (a) Probe-induced axial oscillations are apparent from turning on
the probe for 40 µs (red, 43 traces averaged) but can be greatly reduced with a half-power 2.5 µs
kick followed by a 2.5 µs delay (blue, 30 traces averaged). The 2.5 µs kick duration corresponds
to a quarter of the axial trap oscillation period. An 80 MHz offset has been subtracted from thevertical axis. (b) Thermal radial motion reduces correlations during a time T between pre- andfinal measurements. Oscillations correspond to twice the radial trap frequency.

Tighter trapping or homogeneous coupling of atoms to the atomic probe (Sec. 6.3) are possible
avenues toward reducing optomechanical effects.
Oscillations from radial motion are also important when considering quantum sensors, especially those operating with longer timescales. In Fig. 6.4(b), spin noise reduction R is evaluatedversus delay time T between measurements using the sequence of Fig. 6.3(a). The results without
atoms were subtracted to remove contributions from technical noise sources. The spin noise R oscillates at twice the radial frequency of the trapping potential (ωrad ≈ 2π × 900 Hz) due to thermal
motion. The monotonic increase in R was not explored further, but a 3D optical lattice or a smaller
ratio of atomic temperature to lattice depth could help reduce oscillations in the noise due to radial
motion.

6.1.1

Experimental details
Figure 6.5 explains the frequency locking chain and the use of multiple longitudinal cavity

modes for this work. The UHV cavity is stabilized to the red lattice laser with 1.5 kHz servo bandwidth. A fiber EOM sideband of the lattice laser was stabilized to an independent transfer cavity
with 1 MHz servo bandwidth, allowing 5 to 9 GHz of continuous tuning relative to the transfer

92
cavity. The transfer cavity is locked with 1.5 kHz PDH servo bandwidth to a 795 nm reference
spectroscopy system to prevent long-term drift. All state rotations in this experiment were performed with microwaves broadcast by a stub-tuned dipole antenna placed just outside of the glasscell. Close to 2 W of microwaves near ωHF produced a Rabi frequency of 2π×50 kHz. Microwaves
are gated by square-wave pulses of a fast TTL switch (PN: Mini-Circuits ZASWA-2-50DR+).
The atomic probe is used to determine the population in N↑ through the shifted cavity mode
frequency ωc0 − ωc . Relative frequency noise between the atomic probe laser and the empty cavity
is mitigated by stabilizing the atomic probe to a cavity probe 122.4 GHz away. The atomic probe is
σ + polarized to take full advantage of the cycling transition for strong light-atom coupling as wellas to avoid spontaneous Raman transitions to other ground states caused by spontaneous emission

Cavity
modes

[29].

122 GHz

Laser
modes

PDH lockto cavity

AP phase locked to CP

Homodyne
LO
ProbingsidebandRed lattice
823 nm

AP locked
to cavity

Path lengthstab. tone

81.1 MHz

Atomic probe laser
780.25 nm

Cavity probe

Frequency

Figure 6.5: Experimental frequency diagram including the locking scheme of the atomic probe(green) and cavity probe (gray) lasers. The two longitudinal resonances of the cavity that these
two lasers probe were separated by 122.4 GHz. The unshifted cavity mode at ωc is detuned δc blue
from the atomic resonance ωa . The presence of atoms in |↑i shifts this cavity mode to ωc0 . Frequency
stabilization techniques and the homodyne local oscillator beam (purple dashed) are also shown.

The light reflected by the cavity is detected in homodyne to determine the detuning of theatomic probe from ωc0 , as described in Sec. 3.1.8. Near resonance, the reflected Qr quadrature re√sponse of the field is related to the incident field Ii ∝ Mi and the detuning δp between the probe
light and the cavity resonance by [29]:4δp  κ1 
Qr
=
Ii
κ
κ

1+

(κ/κ0 )2
p
2
N↑ 2g
2δc0

(6.4)

93
where δc0 = ωc0 − ωa ; ωa is the optical atomic transition frequency, and the dressed cavity linewidth
2
2
p
p
is κ0 = (κ+ΓN↑ g/δc0 )/(1+N↑ g/δc0 ). Experimental diagrams of the optics and electronics
are shown in Fig. 6.6 and Fig. 6.7. Cavity and trap parameters are given in Table 6.2
Atomic
Probe
Laser
From
Fig. 6.7
6.8 GHzOff

full power
half power

kick

EOM

φs

81.1
MHz

AOM

λ/2

Arduino

λ/2

λ/4

rot

Isolator

Rb

9.7 MHz PDF
DemodCavityProbe
Laser

Homodyne
LO phase

EOM

9.7 MHzPDH mod

PBS

RF
50:50
BS

DC

To
Fig. 6.6

Homodyne detector

Figure 6.6: Optical block diagram. The resonance frequency of the optical cavity ωc0 is detectedusing homodyne detection of the atomic probe laser (red). Homodyne detection is performed on an
fs = 81.1 MHz sideband on the atomic probe laser. This sideband can be applied at half power by
the kick switch to provide an extra impulsive kick to the atoms in order to cancel optomechanicalringing. The carrier of the atomic probe laser is detected in heterodyne (RF port) to provide apath length reference (see Fig. 6.7) for stabilizing the homodyne detection phase. The cavity probe
laser (blue) is P.D.H. locked, via the Lcav loop filter, to another longitudinal mode of the opticalcavity, unshifted by atoms, and provides stabilization of the atomic probe laser’s frequency to thecavity frequency. The atomic probe and cavity probe are separated optically via polarization. Realtime feedback is applied using an Arduino microcontroller that controls the sign and duration of
6.8 GHz µ-wave pulses. More details are given in Fig. 6.7.

The atomic probe’s red sideband is turned on for approximately 40 µs for each measurement,
and the DC homodyne signal is used to actively lock the sideband’s frequency to ωc0 . This “activelock” is achieved by feedback to the 90 MHz VCO that provides the frequency reference fca towhich the cavity/atomic probe beat note is phase-locked. The frequency of the atomic probe andall derived tones are phase-locked via the loop filter LSHA . The characteristic settling time of theservo is 1 µs for a unity gain frequency of 160 kHz. The first few µs of each measurement are94
To
Fig. 6.5

fs /2=40.55 MHz

From
Fig. 6.6

RF

VCO

φs

Feedback microwave
pulse for targetingJz states

AOM

81.1 MHz

Homodyne
LO phase

Path length stabilization

sign

PFD

(×4-16)

6.8 GHz: ϕ ϕ+π

Homodyne detector

sampleVariable gain

hold

SHA

Remove DC offset

Sample/Hold
Cavity
Probe
Laser

DAQ

ArduinoDC-

13.6 GHz

EOM
Atomic
Probe
Laser

SHA

Hold probe
frequency
when probe
is offVCO

90-100 MHz

PFD

REF

Phase-lock atomic probe to cavity probe

fca
sweep

Freq. offset
70 MHzDDS

Figure 6.7: Electronic block diagram. The homodyne detection phase is stabilized by detecting
the carrier of the atomic probe beam with the signal appearing at 81.1 MHz at the RF port. The
phase of this signal is locked to a DDS frequency reference by applying feedback through the Lstab
loop filter to a VCO controlling the homodyne AOM. The homodyne difference signal (DIFF) is
used to stabilize the atomic probe laser to the atom-shifted cavity mode at ωc0 . The signal is highpass filtered at 1 Hz to remove slowly drifting DC offsets and then passed through a variable
gain amplifier (used to maintain constant loop gain as Mi is varied) before entering the loop filter
LSHA . The output of LSHA is used to control a VCO which provides a phase reference to a phaselock between the atomic probe laser and the cavity probe laser using loop filter La . The cavity
frequency ωc0 is detected by sampling the output of LSHA . When the atomic probe is off, a sample
and hold circuit is used to hold the output of the loop filter. A separate synthesizer (DDS) can beused to perform sweeps of the atomic probe. Real-time feedback is applied by the Arduino basedon the sampled output of LSHA . The Arduino can control the sign of the feedback by switching(sign) between two 6.8 GHz sources that are 180◦ out of phase.

discarded for this reason. To convert ωc0 to a voltage, we record the output of the LSHA loop filter,directly sampling at 2.5 MHz with the data acquisition computer (DAQ). This active lock improves
the dynamic range of the detection system, removes sensitivity to scale-factor noise, creates moreconsistent optomechanical effects, and removes nonlinearities associated with the dispersive error
signal. For diagnostics, it was occasionally useful to eschew the active lock and instead sweep thelower sideband frequency across the cavity resonance. To accomplish this, the atomic probe laser’s

95
beat note with the cavity probe laser was phase-locked to a direct digital synthesizer (DDS) thatprovides a programmable reference frequency fca in place of the usual VCO.
Because the atomic probe lower sideband is turned off between measurements, its frequency
must be held fixed using the sample-and-hold circuit shown in Fig. 6.7. Not only does this prevent
railing but it’s also necessary to avoid slewing. The circuit samples the loop filter voltage provided
to the VCO that generates fca whenever the sideband is on. When the sideband is turned off, thecircuit holds the output voltage of the loop filter so that fca remains at its previous value.
Trial-to-trial fluctuations in total atom number are significantly larger than fluctuations due
to projection noise. To reduce the range over which the probe laser has to slew during the firstN↑p measurement, a “pre-centering” measurement is performed 1.5 ms before each experimental
squeezing trial: a π/2 microwave pulse rotates atoms to a superposition of |↑i and |↓i and theprobing sideband is centered by the feedback loop at ωc0 . The atomic probe frequency is then held
while the probe light is switched off. The atoms are then optically pumped back to |↓i for the actual
spin squeezing measurements described earlier.
To vary the power in the probing sideband (as quantified by the number of incident probe
photons in a single measurement window Mi ), we adjust the phase modulation index for the EOM
at fs . For reference, a typical sideband/carrier power ratio for Mi = 36500 incident photons was
0.004. The power in the path length stabilization tone, and hence the open loop gain of the path
length phase stabilization for homodyne detection, were relatively unaffected as Mi changed. By
contrast, a variable gain amplifier (VGA; Analog Devices AD8337) had to be inserted after the homodyne detector to compensate for the open loop amplitude gain of the active lock which scales√with Mi . When the data acquisition computer changes the rf power that set Mi , it also simultaneously scales the VGA gain to keep the net loop gain fixed. DC offsets on the homodyne difference
port are removed using a differential amplifier to subtract a low pass filtered trace (τ = 1 s) fromthe fast 40 µs measurement windows.
With this approach, the atomic probe sideband could be locked to the dressed cavity resonance with a large dynamic range, from Mi = 150 to Mi = 3 × 105 . When Mi . 100 in a 40 µs

96window, the average number of detected photons within the servo’s time scale of 1 µs approaches
unity. Photon shot-noise would then impose rms fluctuations on the atomic probe’s frequency
that are comparable to the cavity half-linewidth, fundamentally reducing the signal to noise for
estimating ωc0 .
Mi can be understood as an estimate of the average number of photons in the atomic probe
lower sideband crossing a plane directly in front of the cavity input mirror during a 40 µs window,
counting only those that are spatially mode matched to the cavity mode. Although there is up to
25% uncertainty in this calibration owing to the uncertainty in spatial mode-matching of the atomic
probe mode and the homodyne reference beam, the only effect is on the prediction of contrast lostin Fig. 6.3 – it does not lead to any uncertainty in the amount of squeezing.

6.1.2

Quantum efficiency and the anti-squeezed quadrature
Compared to earlier work [47, 151], the quantum efficiency of the system is improved bynearly an order of magnitude. The cavity finesse is increased by a factor of 3.5, so the linewidth κ issmaller by the same factor. This cavity is primarily transmissive at a single end, i.e. the input mirror
transmission coupling rate κ1 is much greater than the output mirror’s transmission coupling rate
κ2 . As a result, measurement of the probe light in reflection accounts for a quantum efficiency
Qcav = κ1 /κ = 0.83(3). This eliminated the need for detecting transmitted cavity light.
To determine the probe detuning δp , we estimate the ratio Qr /Ii from the detected fields
Qd /Id . Vacuum or photon shot noise that appears in the detection of the Qd quadrature limits theresolution on our ability to determine ωc0 . We express the noise in the ratio as(∆Qd )2
1=+ (∆fl )2 + rMin ,
(0)
Id2
4Mi Q

(6.5)

1

(0)

where the one-window quantum efficiency Q1includes fundamental losses of signal to noise

resulting from both photon losses and technical noise floors shown in Table 6.3. The additionalterms represent noise contributions from the technical noise floor (∆fl )2 associated with residual
frequency noise on the atomic probe laser relative to the cavity mode frequency, and noise from

97
Cavity & probing parameters (probe λ = 780 nm)
4g 2

Single-atom cooperativity C = κΓeff
Single-atom vacuum Rabi splitting geff
Input coupling κ1
Output coupling κ2Internal losses κL
Linewidth κ
Dressed-cavity linewidth (4 × 105 atoms, δc = 400 MHz) κ0
Q.E. due to internal losses κ1 /κ
FinesseFree spectral range
TEM00 waist size w0
Cavity length
Mirror radius of curvature

0.044(6)2π × 0.44(3) MHz
2π × 2.60(5) MHz
2π × 0.17(1) MHz
2π × 0.38(8) MHz
2π × 3.15(10) MHz
2π × 3.6(1) MHz
0.83(3)2532(80)
8.105(5) GHz
70(1) µm
1.849(1) cm
4.999(5) cm

Trap parameters (lattice λ = 823 nm)
2π × 5.8(6) MHz
1400(150)
115 µK
0.30(3) W
800(130)181(20) kHz
900(50) Hz
71(1) µm

Linewidth
FinesseTrap depth
Circulating power Pcirc
Power Buildup (Pcirc /Pinc )Axial trap frequency
Radial trap frequency
TEM00 waist size w0

Table 6.2: Relevant cavity parameters at the atomic and cavity probe laser wavelength λ = 780 nm
and at the lattice laser wavelength λ = 823 nm. The symmetric, standing wave cavity’s mirror
transmission coefficients, T1 on the probed end (1) and T2 on the closed end (2), are expressed interms of coupling rates κ1,2 = T1,2 × (free spectral range). The atomic decay linewidth of |ei isΓ = 2π × 6.065 MHz. The dressed cavity linewidths κ0 include broadening of the cavity resonanceat ωc0 due to spontaneous scattering from the atoms.

optomechanical ringing rMin , which we model by scaling Mi with an arbitrary polynomial of ordern 6= −1.
We define a new effective quantum efficiency Q1 which includes the effects of the technicalnoise floor and optomechanics and write the noise in homodyne detection as(∆Qd )2
1=
2
4Mi Q1
Id

(6.6)

98
SourceQ

Path efficiency, Qpath
Cavity-mode/homodyne overlap, Qo
QE of cavity (κ1 /κ), Qcav
Technical noise from detector, Qelec
Detector QE, QP D
Probe on-time, Qturn on

0.75(3)0.95(3)0.83(3)0.86(1)0.86(2)0.86(1)(0)

Total, Q1

0.37(5)

Table 6.3: Quantum efficiency losses come from sources of signal loss and added noise floors.
Qturn on comes from finite laser turn-on time, active lock settling, and ring-canceling “kicks” during(0)which the probe is on but we do not collect information. The total quantum efficiency Q1 =0.37(5) is the product of all the measured contributions.

where Q1 is given by
(0)

Q1 =

Q1
(0)

1 + 4Mi Q1 ((∆fl )2 + rMin )

.

(6.7)

This effective quantum efficiency provides a useful figure of merit for the experiment andcan be compared to measurements of the increase in area of the Bloch vector’s noise distribution.
Just as the back-action from gaining information about a particle’s position causes uncertainty inits momentum, increased phase resolution comes at the expense of increased uncertainty in theanti-squeezed quadrature. We measure this directly by including a microwave rotation about anaxis parallel to the Bloch vector through angle ψ. By scanning ψ over separate trials, a visualizationof the state is reconstructed (Fig. 6.3(c)). The state has ∆Jz ∆Jx /(∆Jz,QPN )2 = 6.1 and is therefore
no longer a minimum uncertainty state, reflecting the finite quantum efficiency Q)1 in extracting√
information. In particular, the increase of area is proportional to 1/ Q1 . The measured increaseof area and its scaling with Mi indicates the quantum efficiency of a joint measurement of a single
population was Q̃1 = 38(14)%, in agreement with an independent prediction of 37(5)% from measuring path efficiencies, cavity loss, detector efficiency, technical noise floors, and laser turn-on
times.
To reconstruct the probability distribution describing our state, we measure the normalized
probability distribution P (Jzf (ψ) − cos(ψ)Jzp ) for obtaining a differential measurement outcome

99
Jzf (ψ) − cos(ψ)Jzp . The weighting of the pre-measurement by cos(ψ) ensures that we only condition the final measurement to the degree that the two spin projections overlap. An inverse Radontransform [154] on the measured P (Jzf (ψ) − cos ψJzp ) yields the conditional probability distribution shown in Fig. 6.3(d).
The magnitude of noise in the back-action quadrature grows anomalously with the numberof probe photons Mi . We generalize the spin noise reduction to be a function of rotation angle ψ:π2
2
R(ψ) = ∆ (Jzf (ψ) − cos ψJzp )2 /∆Jz,QP
N . The anti-squeezing is defined as A ≡ R( 2 )CBG /C ,in direct analogy to the Wineland squeezing parameter, W = R(0)CBG /C 2 . The anti-squeezing
parameter describes the variance in the azimuthal phase of the Bloch vector relative to the standard
quantum limit A ≈ (∆φ/∆φSQL )2 , up to the small correction for the background contrast CBG .
The anti-squeezing A is plotted versus Mi in Fig. 6.8. The data is fit to a model that includes
three contributions A = A0 + A1 Mi + A2 Mi2 . The constant term A0 is associated with projection noise as well as noise in the rotations. Quantum back action A1 rises linearly with Mi , with
contribution shown by the blue shaded region. Noise scaling as Mi2 , for example due to classicalintensity noise on the circulating probe laser power, contributes to the total back action as shownby the red shaded region.

10
20

2000

30
40

1000

50
0

10000

20000

Incident photons Mi

Quantum efficiency

b)
3000

Squeezing

Anti-squeezing

a)

5

1
Area of noise
distribution

0.8

4

0.6
Predicted

0.4

2

0.2
0

3

0

10000

20000

30000

1

Incident photons Mi

Figure 6.8: Anti-squeezing and quantum efficiency. (a) The anti-squeezing A is plotted versus
Mi (brown circles). The linear contribution to the rise in A, A1 , is shown in blue and the quadratic
contribution A2 in red. The squeezing (black diamonds and fit) is plotted on the right axis. (b) The
area of the noise distribution is calculated from the data in (a) and plotted in red. The measured
effective quantum efficiency Q̃1 is plotted in gold with an error bar shown as a gold band. At low(0)
Mi , Q̃1 is consistent with the prediction (dashed line) Q1 from Table 6.3.

100
Squeezing data W is also taken at the same experimental settings (brown points and line
in Fig. 6.8(a)). This allows us to infer the normalized area of the quantum noise distribution,q2
2 , shown in red in Fig. 6.8(b). The increased area of the noise dis∆φ∆θ/∆θSQL= W A1 /CBG
tribution can be used as an alternate, global measurement of the effective quantum efficiency Q1 .
Specifically, the total quantum efficiency of the entire measurement sequence is proportional to
the square of the increase in the angular area of the noise distribution and can be written Q̃1 =2 ). The factor C 2 comes from the angular momentum uncertainty relation and ac4/(A1 W C 2 /CBG

counts for the fact that the SQL increases as the Bloch vector shrinks. The factor of four arises due
to finite measurement strength and an unused pre-measurement. Q̃1 as measured by the area of
the noise distribution is plotted in Fig. 6.8(b) in gold. The gold shaded region represents the considerable uncertainty in the extrapolation of Q̃1 due to uncertainty in the fit of the anti-squeezing(0)

data of Fig. 6.8(a). At low photon number, Q̃1 agrees with the composite value of Q1 from Table
6.3. At higher photon number, Q̃1 rises due to the effective quantum efficiency losses from thetechnical noise floor and optomechanics discussed earlier.

6.2

Deterministic spin-squeezing
Real-time feedback on the results of collective measurements opens the path to even more ac-cess for improved quantum technologies, including Heisenberg-limited atomic sensors [155], atominterferometers with reduced mean field shifts [48, 156], studies of topological physics [157], quantum teleportation [158], and error correction [159, 160]. Quantum noise suppression with real-time
feedback has been considered theoretically [161, 162] and demonstrated in previous experiments[163, 164] but without an enhancement in phase resolution that explicitly signifies entanglement.
Following the protocol of Fig. 6.9, we use the outcome of a dispersive measurement to actively steer the collective spin-projection of an ensemble of 50 × 103 laser-cooled and trapped 87 Rb
atoms to a target entangled quantum state. Except for the reduced atom number and the additionof the microcontroller to control the feedback, the setup was identical to the previous section (seeFigs. 6.5, 6.6, and 6.7). With real-time feedback, we generate target states with enhanced angular

101
resolution as much as (∆θSQL /∆θ)2 = W −1 = 5.5(8), or 7.4(6) dB below the SQL, comparable atthe time to the best enhancements using only unitary evolution.
a)b)
2. Feedback to desired Jz

Initialize Arduino
Timeout

TTL inN↑p: Store 40μs of analog input
TTL inN↓p: Store 40μs of analog input

1. Measure to create(conditional) squeezing

Repeatn timesSet W, microwave signfrom N↑p-N↓p; turn on pulse
No op (12ns)
After n cycles
Turn off pulse

FinishFigure 6.9: Deterministic squeezing via feedback. (a) Bloch sphere depiction of the deterministicsqueezing protocol showing deliberate feedback to the “purple” Jz value. (b) Pseudocode for the
microprocessor routine. Some care was taken in initialization so that we could interface directlywith the hardware, for example by counting processor cycles and writing directly to ports rather
than using slow-but-convenient wrapper commands.

In each run of this experiment, atoms are optically pumped into |↓i and then prepared in a su√
perposition of (|↑i + |↓i) / 2 by a microwave π/2-pulse. The spin projection Jzp = (N↑p − N↓p ) /2
is measured, where N↑ and N↓ are averages of the probe homodyne signal over a 40 µs window(with a π-pulse in between to swap the measured populations). To steer the Bloch vector to a tartarget

get Jztarget, the microcontroller gates an additional rotation through polar angle θf b ≈ 2 × (Jz−Jzp )/N . After the feedback, a final measurement of the spin projection Jzf = (N↑f −N↓f )/2 demon√
strates a reduced variance compared to ∆Jz,QPN = N /2. As before, the remaining fractional
atomic coherence (contrast) after pre-measurement and feedback must also be measured to verify entanglement. The phase of a π/2-rotation after the feedback step is scanned through 2π tomeasure the reduced length of the Bloch vector. At spin-noise reduction R−1 = 9.5(4) dB, lossof contrast reduced squeezing by 2.1 dB for a direct observation of W −1 = 5.5(8) (7.4(6) dB) of
deterministic squeezing via feedback on a pre-measurement.

102
targetFeedback toward Jz= 0 is evident in the time trace of Fig. 6.10(a), as the final two dressedcavity frequency ωc0 measurement windows that provide N↑f and N↓f are consistently closer involtage than is the case for the two pre-measurement windows. The outcomes Jzp and Jzf are plotted versus trial number and collated into histograms for five different target states in Fig. 6.10(b).
Projection noise for this data (independently confirmed by measuring the scaling of ∆Jz with N ) is
∆Jz,QPN = 218(10), consistent with the measured ∆Jzp = 235(24). By implementing the feedback
protocol, each target state is reached with noise below the original projection noise.

[kHz]

a)

b)
80

0 μs

100 μs200 μs

0

1000

1000

500

500

0

0

-500
-1000

Time

-500
0

50

100

0

50

100

-1000

Trial number

Figure 6.10: Deterministic squeezing sequence and results. (a) Measured cavity resonance frequency during a single trial, with the corresponding pulse sequence displayed below. A constant
12 MHz frequency offset has been subtracted. The deterministic squeezing sequence is shown withmeasurement windows (green), fixed microwave rotations (black) and the feedback rotation (orange). Feedback has made the result of the final measurement windows more equal, as intended
targetfor Jz= 0. (b) Pre-measurements Jzp (left) and final measurements Jzf (right) are plotted versus trial number and accumulated into histograms. The pre-measurement shows noise statistics
near QPN. Five different Jz states are targeted (five distinct colors on right) and each exhibitednoise below QPN. The maximum deterministic squeezing here, W = 7.4(6) dB, corresponds to thered dataset.

An Arduino Due microcontroller with 84 MHz internal clock sets the sign of the θf b rotation
by digitally toggling between two microwave sources that are 180◦ out of phase. The magnitude
is controlled by varying the duration tf b = n × 12 µs for which the microwaves are applied, with adiscrete timing resolution of 12 µs set by the internal clock. Measuring Jzp amounts to sampling theprobe’s loop filter output voltage at the Arduino’s analog-to-digital converter with 18 points per
40 µs window. Atom number fluctuations are small enough to ignore. The input technical noise
floor, timing jitter, and timing resolution of the microcontroller are sufficient to allow up to 20 dB

103
of squeezing.
Although we deliberately used fewer atoms to relax the noise requirements on our microwave
source, the amount of resolvable entanglement is still primarily limited by amplitude and frequency noise during the rotations. To characterize the noise added from feedback, we measureconditional spin noise ∆ (Jzf − Jzp ) in a sequence without feedback and find R−1 = 12.4(7) dB,
i.e. feedback leads to approximately 2.9 dB of added noise. Next, we perform the sequence withno microwave rotations of any kind, measuring the same spin population N↑ four times. In thissequence we attain R−1 = 14.0(5) dB, 1.6 dB less than the sequence with rotations but no feedback.
This measurement suggests a rotation noise floor due to microwave amplitude and frequency noisethat is approximately 17.5 dB below projection noise for 50×103 atoms. Rotation noise on the feedback pulse is particularly problematic as certain rotation errors and light shifts which cancel after
two π pulses no longer cancel when feedback is applied. Improving the precision of microwave
rotations remains a major obstacle in working with atomic spin states with extreme phase resolution.

6.3

Spatially homogeneous entanglement using time-averaged measurements
In previous cavity-based squeezing experiments (in this dissertation and elsewhere) stand-ing wave probe modes perform collective measurements, but the atoms are trapped in 1-dimensionallattices defined by a standing wave cavity mode with a significantly different wavelength. Although entanglement can be generated without using a cavity [17, 49, 51, 152, 165], free spaceexperiments have not yet achieved the degree of squeezing observed using optical cavities. The
release of spin-squeezed atoms into free space poses a major challenge for cavity-based atom interferometry and other applications.
In the experiments described earlier, some atoms are positioned in sites near nodes and somenear anti-nodes of the entanglement-generating probe light. As a result, atoms contribute to thecollective measurement with different strengths. In this case, the large degree of squeezing exists
only for this specific coupling configuration and would be largely lost after releasing the atoms into

104
the arm of an interferometer, since each atom’s contribution to the final signal will be different fromthe original configuration [45]. Recent experiments have demonstrated information retrieval after
free-space release by recapturing atoms in the lattice [166]. Other solutions to this inhomogeneous
coupling problem are described in Sec. 2.1, but they are generally incompatible with cavity-based
interferometry.
In Ref. [94], the conditional squeezing of Sec. 6.1 is expanded with a method for creatinghomogeneous spin-squeezed states by allowing the atoms to traverse many wavelengths of thestanding wave probe during each collective measurement. Atoms experience a time-averaged
coupling to the cavity so that every atom is measured with the same strength, ensuring homogeneous entanglement. This is accomplished by creating an effective dipole trap (as opposed to thestanding wave lattice) with a uniform axial potential described carefully in Sec. 4.5. The dipole trap
maintains transverse confinement of the atoms while allowing free movement subject to gravity
along the vertical cavity axis and facilitating W = 11(1) dB of directly observed squeezing. Fluorescence images and noise scalings further show that the generated squeezing is homogeneously
shared among the atoms to a large degree, in principle allowing significant amounts of squeezing
for guided matter-wave interferometry. This work serves as a stepping stone for achieving theentanglement-enhanced matter-wave interferometer in Ch. 7.
As before, we trap 87 Rb atoms in the central 2 mm of a 2 cm long optical cavity, and cavity
mode we probe is tuned δc = 2π × 400 MHz to the blue of the |↑i to |ei ≡ |52 P3/2 , F = 3, mF = 3itransition for dispersive QND population measurements of Jz = 12 (N↑ − N↓ ). To avoid extra
rotation-added noise, we use only N↑ measurements and Jz = N↑ − N2 because N↓ = N − N↑ .
√Quantum projection noise leads to fluctuations ∆Jz,QPN = N /2. Experimental parameters grms ,δc , and N are used to scale between cavity frequency measurements ω and Bloch sphere projecp
2 / 4g 2 N + δ 2 . The atoms are initially cooled to approximately 10 µK
tions Jz , ∂ω/∂Jz = grms
rms ↑cwith PGC and trapped in a far off resonance red-detuned optical lattice at λL = 823 nm with corresponding wavevector k0 = 2π/λL .
At this point, however, we convert the standing-wave lattice into the effective dipole trap.

105
Multiple adjacent TEM00 longitudinal modes of the cavity are driven by phase modulating thelattice laser at the free spectral range (FSR), ωFSR = 2π × 8.1050(5) with modulation index β ≈ 1.3.
Phase-modulation is provided by a fiber-coupled EOM (PN: EOSpace PM-0S5-10-PFA-PFA-800).
Adjacent modes have opposite symmetry with respect to the cavity center, so to lowest order, thecarrier creates a cos2 (k0 z) standing-wave intensity profile while the sidebands create sin2 (k0 z)/2
intensity profiles, and the sum of the standing waves cos2 (k0 z)+sin2 (k0 z) = 1 causes a net uniform
intensity profile along the cavity axis. Please see Sec. 4.5 for more details.

5

Distance from cavity center [µm]c)0.4

ß=1.200.3

ß=1.32

0.2
0.1
0

0

1

2

3

4

5

Distance from cavity center [mm]Fraction of atoms incavity mode0

Axial potential envelope
Ures / U0b)Gravity and axial
potentiala)tdrop [ms]1

0 5 10 15 20

0.5

0

0

40

80

120

Fall time, tdrop [ms]Figure 6.11: Falling atoms in an optical dipole trap. (a) The optical dipole trap axial potential
combined with a gravitational potential (exaggerated) at a distance z from cavity center. Near thecenter of the cavity where the dipole trap is uniform, an atom is free to fall; but for an atom initiallyconfined in a deep well, it remains trapped unless it can tunnel or “boil” out. (b) The envelope ofthe residual lattice potential Ures (z) normalized to the peak lattice potential depth U0 is plotted near
the cavity center, optimized for a minimum at z = 0 (gold, β = 1.20) and for the minimal fractionof trapped atoms determined experimentally (red, β = 1.32). (c) Fraction of atoms remaining in
the cavity mode vs. fall time, fit to a model R(t) described in the text. This data has been modified
to roughly account for a background loss of atoms resulting in underestimates before 10 ms.

The increase in residual lattice depth away from the center of the cavity is generally not
sufficient to stop a falling atom; rather, we expect the atom to be guided by the optical dipole trap
until it collides with the lower mirror. For falling atoms, the residual lattice looks like an amplitude
modulation and has the capacity to drive Bragg scattering. The tilted potentials also provide a richlandscape for Wannier-Stark physics [167, 168]. In Fig. 6.11(c), the number of atoms coupled to thecavity mode is measured as a function of time by continuously monitoring the dispersive shift ofthe cavity resonance frequency. The data was roughly normalized to account for background atom

106
loss and is reasonably described by a fit,Z ∞1√
R(t) =
gt
L
2π
− 2t
2

r


r

2
Rb v0
mRb −m1
mRb L − gt2
2kB Tax√e
dv0 =1 + erf,kB Tax2
kB Tax 2 2t

(6.8)

which assumes atoms accelerate due to gravity g for time t before being lost at the position of thelower mirror L/2. The integral gives the fraction of atoms remaining from a velocity distribution
with Boltzmann constant kB , mass mRb , and axial temperature Tax . For comparison, ballistic radial
expansion out of the cavity mode would occur in only 2 ms had we completely turned off the
optical lattice. Free fall and guiding are corroborated by fluorescence images taken after various
drop times Tfall as shown in Fig. 6.11(c, inset).
Figure 6.11(b) shows the depth of the transformed potential Ures (z) as a function of distancefrom the center of the cavity for two different values of β. At β = 1.20, the potential is completely
uniform at the center of the cavity. We instead overdrive the dipole trap with β ≈ 1.32 to maximize
the fraction of freely falling atoms. This drive results in a wider overlap between small values for
Ures (z) and the atomic spatial distribution. Overdriving also causes the lattice potential wells to
be converted into small potential peaks, giving additional potential energy. Fluorescence images
indicate that at long times only 5(1)% of the atoms remain trapped in a residual lattice while themajority of atoms accelerate along the cavity axis.
For a fixed total atom number, the projection noise induced fluctuations of the cavity resonance frequency ∆ωQPN is smaller in the dipole trap than in the lattice. In the lattice, the atoms contributing most to the dispersive shift are those near the probe antinodes, with Jaynes-Cummings
coupling parameter near the maximum value g0 = 2π × 0.519(5) MHz. In the ideal time-averaged√
situation, each atom couples with rms coupling strength grms = g0 / 2, leading to smaller frequency fluctuations.
In App. C, we define a model where fractionally, ζ of the atoms release into the dipole trap
and are assumed to have perfectly homogeneous coupling. 1 − ζ of the atoms remain fixed inposition and maintain their original inhomogeneous coupling. In this model, the projection noise
induced fluctuations of the cavity resonance frequency come from adding the weighted variances

107
2of the two ensembles, ∆ωQPN = grmspp
N (3 − ζ)/ 8(g02 N + δc2 ).

This difference in the scaling of projection noise for the two configurations is confirmed versus total atom number with the measurement sequences of Fig. 6.12(a). The N↑ and N↓ windows
represent the outcome of a measurements ω↑ and ω↓ of the cavity resonance frequency sensitive
meas = ∆(ω − ω ) areto the respective population. The observed projection noise fluctuations ∆ωQPN↑↓

plotted. For the purposes of this comparison, a small amount of technical noise that does not havethe proper scaling with atom number was subtracted from this data. The in-lattice data is used asa calibration of g0 with the theoretical scaling plotted in green. The dipole trap data is fit to themodel 2 × ∆ωQPN (since the measurement sequence includes two anti-correlated windows, ω↑ andω↓ ) with ζ = 1.0(2) as the only free parameter.
a)

b)
Tdrop

100

Time
in lattice80
60

in dipole trap

40
20
0

0

20

40

60

Total atom number N ×10

380

Spin noise reduction
and contrast loss [dB]

[kHz]

120

Tdrop

Time

0

contrast loss

-5
-10

spin noise
reduction

-15
-20

0

10000

20000

30000

Incident probe photons, Mi

Figure 6.12: Homogeneous squeezing results. (a) Projection noise scaling versus total atom number N , measured in the lattice (green points) including a theoretical prediction and in the dipole
trap (red points, including a fit to infer a coupling fraction ζ with 68% confidence interval bands).
The sequence is above, with microwave pulses (black) and cavity frequency measurements (green).
Tfall is only pertinent for the dipole trap configuration. (b) Spin noise reduction and a fit to contrast
loss (log10 (C 2 /CBG )) vs. incident photon number per window, Mi . Squeezing was optimized at
W = 11(1) dB near Mi = 104 , where spin noise is reduced but significant contrast still remains.
The measurement sequence is above.

We measure spin squeezing within the dipole trap using the Wineland criterion for phaseenhancement relative to the SQL, Eqns. 6.2 and 6.3. The conditional squeezing measurement sequence is shown in the inset of Fig. 6.12(b) and is the same as that of Sec. 6.1. After preparing

108
N = 630(30) × 103 atoms in |↑i, the dipole trap was introduced for Tfall = 13 ms, with gravity accelerating the atoms enough to average over approximately 13 cycles of the probe standing wave during the 40 µs measurement window. A π/2-pulse puts atoms in a superposition of
spin-states, and a measurement window N↑p and microwave π-pulse compensate inhomogeneouslight shifts as before. Consecutive pre- and final measurements of the cavity frequency are labeled N↓p and N↓f and converted to Jzp and Jzf . Trial-to-trial noise in the difference exhibits spinnoise reduction R = (∆(Jzf − Jzp )/∆Jz,QPN )2 = 13.9(6) dB below quantum projection noise near
Mi = 1.2 × 104 . The fractional coherence remaining after the pre-measurement is separately measured to be C = 0.70(5), yielding a directly observed phase resolution of W = 11(1) dB below the
SQL

6.3.1

Noise reduction limits from imperfectly time-averaging
A few limitations of this method are explored here and in the supplemental material of

Ref. [94]. Imperfect averaging over the probe standing waves leads to a small amount of classical dephasing and a significant impact on the obtainable spin noise reduction. For the 25 µK
ensemble, Tfall = 13 ms, and 40 µs measurements, we predict a noise floor 15 dB below QPN which
is a primary limitation for the results above.
To understand this effect, consider a thermal velocity distribution of atoms P (fi ), where
the ith atom moves along the cavity axis at velocity vi , moving from probe anti-node to anti-nodeat coupling oscillation frequency fi = 2vi /λp . The time dependent coupling can be written as
gi2 (t) = g02 sin2 (πfi t + φi ) where φi sets the coupling at t = 0.
We assume both pre- and final measurements last for times [0, Twin ] and [Tdiff , Tdiff + Twin ]respectively. We take the projection noise level to be that for a perfectly time-averaged scenario in
which each atom moves exactly an integer number of cycles of the standing wave: (∆ωQPN )2 =N g04 /16δc2 . The obtainable spin-noise reduction is
Z ∞R=

P (fi )−∞

4 sin2 (πfi Tdiff ) sin2 (πfi Twin )dfi .
2fi2 π 2 Twin

(6.9)

109
To gain some insight, consider an example when the atomic distribution P (fi ) is Gaussian withmean f0 and standard deviation ∆f . In the limits of our experiment, Tdiff , Twin  1/∆f and
f0  ∆f , so the terms sin2 (πfi Tdiff ) and sin2 (πfi Twin ) in Eq. 6.9 will oscillate rapidly with fi andcan be replaced in the integrand by the average, 1/2. The resulting spin-noise reduction is thenR = 1/(πNosc )2 where Nosc = f0 Twin is the number of cycles averaged by an atom at coupling
oscillation frequency f0 .
We estimate the maximum possible spin-noise reduction using Tdiff = Twin = 40 µs andthe directly measured distribution of coupling oscillation frequencies shown in Fig. 6.13 to obtain an experimentally measured probability distribution P (fi ). We find a limit from imperfect
averaging R ≈ −15 dB which would be a primary limit to the observed spin-noise reduction of
R = −13.9(6) dB. In the future, this effect could be reduced by allowing the atoms to fall for longer,
by using longer measurement windows Twin , or by employing velocity selection strategies as we
do in Ch. 7.
In addition to limiting spin noise reduction, imperfect cycle averaging of the probe standing
wave leads to classical dephasing as each atom receives a different average AC Stark shift. The
phase shift on a single atom during a single measurement window can be written
Z Twinψi =0

Mc gi2 (t)dtδc(6.10)

where Mc is the average intracavity photon number, taken to be constant for this calculation. The
variance in ψi over the atomic distribution in our simplified model is
2

Z ∞

Z 2π

∆ψ =

dφi
0

−∞

P (fi , φpi )ψi2 dfi −

dφi
0

2

Z ∞

Z 2π

P (fi , φi )ψi dfi(6.11)−∞

with a normalized probability distribution for velocity fi and initial phase φi denoted P (fi , φi ).
Assuming the phase φi of the coupling oscillations is uniformly distributed, the variance becomes∆ψ 2 =Z ∞P (fi )−∞12



g02 Mc2πfi δc2

sin2 (πfi Twin ) dfi ,(6.12)

2

reducing the contrast after a single pre-measurement by e−∆ψ /2 . This result can be interpreted
as arising due to the phase shift from the uncancelled non-integer fraction of an atom’s coupling

110
oscillation. The result will be slightly modified by the use of a spin-echo pulse during the premeasurement, but because the phase of each atom’s coupling oscillation changes for each window
based on it’s velocity, significant spin-echo cancellation of this dephasing is not expected. For oursystem we estimate that classical dephasing from imperfect time-averaging led to a contrast lossof less than 1 dB at the optimal squeezing.

6.3.2

Coupling oscillations
The ac signals encoded in probe light reflected from the cavity (Fig. 6.13) yield additional

information about the spin state of each velocity component of the atomic ensemble, but time
averaging has a minuscule effect on the added noise for our system parameters.
The continuous time trace of the total atomic coupling to the cavity is N (t) =

PN↑
i

2 .
gi2 (t)/grms

The squared Fourier transform of the signal Ñ (f )2 , with units of atoms/Hz, is closely related tothe atomic velocity distribution. Figure 6.13(a) shows Ñ (f )2 recorded over 2 ms and averaging
approximately 65 trials for both the lattice and dipole trap configurations. The data was taken aftert = 1 ms, 7.5 ms, and 15 ms of free fall following release into the dipole trap. Each power spectrum
is fit to a folded 1D Boltzmann distribution that accounts for the inability to distinguish between
upwards and downwards velocities. The fitted center frequency follows f0 (t) = g t/(λp /2), whereg = 9.8 m/s2 is the acceleration due to gravity. The widths of the distributions are consistent with
Boltzmann distributions giving final axial temperatures of 25 µK. For atoms held in the lattice,Ñ (f )2 shows only a narrow distribution at the lattice trap frequency, ωax = 200 kHz.
These oscillating signals yield information about the spin state of atoms moving at a particular velocity and therefore must cause some degree of additional quantum collapse or back-action.
A full treatment of this effect is difficult but the argument produced in Ref. [94] considers a classicalback-action model driven by quantum noise on the probing field. Photon shot noise (PSN) of the
probing beam causes an unknown, noisy contribution to the phase shift with rms fluctuation ∆ψthat is equivalent to the observed quantum back-action in the azimuthal angle. PSN in frequency
space is a power spectral density of photon number fluctuations in the cavity SM = 4Mc /κ (dot-

111
b)

[atoms/Hz]

7
6

Axial oscillations in lattice

5

Falling for 1 ms

4

Falling for 7.5 ms

3

Falling for 15 ms

2
1
0

0

200

400

600

800

v = 0 m/s

Sensitivity functions

a)

1000

Frequency [kHz]Thermal distributionv = 10 m/s
Photon shot noise

0

100

200

300

400

500

Coupling oscillation frequency, fi [kHz]Figure 6.13: (a) Power spectra showing measured coupling oscillations for ensembles in the dipole
trap with fall times of 1 ms (blue), 7.5 ms (brown) and 15 ms (green) with their respective fits. Axial
oscillations for atoms held in the lattice (red) are comparatively coherent. (b) Sensitivity to photon
shot noise (dashed line) for different velocity classes. A stationary atom only samples PSN at DC(red), but a moving atom couples with a transfer function (blue) sensitive at DC and frequency ficorresponding to its velocity, shown for an atom with velocity 10 cm/s. The atomic distribution ofoscillation frequencies is given by the Boltzmann distribution P (fi ) (black).

ted purple line in Fig. 6.13(b)) where Mc is the average number of photons in the cavity mode2[169]. Stationary atoms sample this PSN in a window centered at zero frequency with characteristic bandwidth 1/Twin (shown in red in Fig. 6.13(b)), while moving atoms sample the PSN witha transfer function sensitive at DC due to the time-averaged component of gi2 (t) and at fi due tocoupling oscillations (shown in blue).
We suppose a thermal velocity distribution where sub-ensembles of characteristic frequency
receive uncorrelated back-action noise. By parameterizing in terms of photon number, it is shown
in Ref. [94] that the physical mechanism is then similar to ordinary quantum back-action except thatthese sub-ensembles reduce the collective Bloch vector length. The upshot is that this effect only
1
provides an additional limit to angular resolution near W ∼ qN
(of order the Heisenberg limit,

where q is the total quantum efficiency of the experiment), several orders of magnitude below
current experiments.

2

Valid for frequencies much less than the cavity linewidth κ.

Chapter 7
An entanglement-enhanced matter-wave interferometer

Matter-wave interferometers were gently introduced in Sec. 1.6 and described with somedetail in Sec. 2.4. As sensors, they provide extraordinary sensitivity to external forces, making themvaluable for both applied and fundamental science. Applications include inertial sensing [63],mineral exploration, groundwater monitoring, measurements of gravity [38, 40], measurements ofthe fine structure constant [4, 5], tests of general relativity and the equivalence principle [6, 7, 170–173], and even the search for new proposed forces, quantum gravity, dark energy, and dark matter[34, 174, 175]. As with other quantum sensors, matter-wave interferometers using independent,
unentangled atoms are fundamentally limited by quantum collapse to an rms angular uncertainty√∆θSQL = 1/ N rad. This uncertainty restricts the precision, bandwidth, size, and accuracy ofquantum sensors. State-of-the-art interferometers operate at [176] or near this limit, but future
generations of sensors will benefit from surpassing the standard quantum limit via entanglement.
As we have seen, most previous experiments have generated spin-squeezing between theinternal degrees of freedom, with large amounts of entanglement between the spin and spatial
degrees of freedom. Pioneering work with collisional interactions in BECs [48–54] demonstrated
entangled spatial interferometry in trapped geometries [48, 54]. There have been several proposals
for transferring entanglement to external states suitable for free-fall interferometry [177–180]. Most
recently, twin-Fock states were generated with collisional interactions and the entanglement was
subsequently mapped onto momentum states [55].
Here, we demonstrate the first realizations of both cavity-enabled quantum non-demolition

113
generation and one-axis twisting generation of entanglement between the external atomic momentum states of different atoms. We further demonstrate the first injection of momentum-squeezed
states into a light-pulse matter-wave interferometer for a spectroscopic enhancement that surpasses the standard quantum limit as we successfully inject OAT-squeezed states into a MachZehnder interferometer for 1.7+0.5−0.5 dB of directly observed metrological enhancement. Matterwave interferometric control is achieved through intracavity Bragg or Raman transitions [42, 62].
We thus simultaneously combine two of the most striking features of quantum mechanics: the concept that a particle can appear to be in two places at once and entanglement between distinct particles. This work opens the path for a next generation of matter-wave interferometers that exploit
the many-body nature of the atomic sensor rather than simply operating N independent copiesof the experiment in parallel. A summary of spectroscopic enhancement results for the chapter isprovided in Table 7.1.
Configuration

NSpectroscopic enhancement W

QND momentum squeezing

1170(30)

0.46(11)

3.4+1.1−0.9 dB

OAT momentum squeezing

730(10)0.56(8)2.5+0.6−0.6 dB

OAT-squeezed interferometer

660(15)0.68(8)1.7+0.5−0.5 dB

Unentangled interferometer

660(15)

1.49(31)−1.7+0.7−0.8 dB

Table 7.1: Summary of spectroscopic enhancement results for this chapter.

7.1

Setup and state preparation
The conceptual setup of the experiment is shown in Fig. 7.1. Strong collective coupling to thecavity N C  1 is the key requirement for both approaches to generate entanglement, where C is the
single particle cooperativity parameter [29–31]. We achieve strong collective coupling N C ≈ 500
by operating inside a high cavity finesse F = 1.3 × 105 with small mode waist w0 = 72 µm. A
cavity-based interferometer with N C  1 was demonstrated in [181], though primarily to provide
spatial mode filtering and power build-up. Around 700 atoms participate in the interferometer,

114
falling freely under gravity but simultaneously traversing two paths through space while alsoentangled with the other atoms. We utilize the magnetically-insensitive 87 Rb clock states, |↓i ≡|F = 1, mF = 0i and |↑i ≡ |F = 2, mF = 0i, separated by the hyperfine transition frequency ωHF ≈2π × 6.835 GHz (Fig. 7.3).

Squeezing
light

Raman
tones
Homodyne readout

Entangled anddelocalized atoms

Figure 7.1: System overview. Ultracold atoms undergo guided free fall in a vertical high-finesse
cavity. The atomic wavepackets are split (up to 12 µm at Tevol = 1 ms) and recombined by driving two-photon Raman transitions to provide quantized momentum kicks to the atoms. (insets)
Intracavity atomic probe light generates entanglement between the atoms via either one-axis twisting dynamics or quantum non-demolition measurements made by detecting the reflected atomic
probe field with a homodyne detector.

The two-mirror cavity is vertically-oriented along Ẑ (Fig. 7.1(a)). A 1 G bias magnetic fieldalso sets the quantization axis along Ẑ. The cavity has a power decay rate κ = 2π × 56(3) kHz. To
ensure significant intracavity power for the Raman tones, the length of the cavity was adjusted tobe near ωHF (mirror separation L = 2.2 cm, and free spectral range ωFSR = 2π × 6.7879 GHz). Every
700 ms, rubidium atoms are loaded into a red-detuned 813 nm standing-wave intracavity latticeand laser-cooled to a radial temperature of 0.7(3) µK (Ch. 5). The lattice is adiabatically reduced
to allow the atoms to accelerate under gravity for a duration Tfall , guided tightly along the cavity
axis by a hollow (Laguerre-Gauss LG01 -like) blue-detuned optical dipole guide. An overview ofthe lasers and cavity modes involved in this chapter is shown in Fig. 7.2. Technical details for these115
subsystems are now provided before the matter-wave story resumes in Sec. 7.2.
HG01

HG10

Cavitymodes

163 GHz6.79 GHz

Laser
modes

HomodyneLO

Path lengthstab. tone

Atomicprobe tone
Red lattice
813 nmAtomic probe laser
780.25 nm

6.835 GHz
AP lockto CP

PDH lockto cavity
RL lock
to CP

Raman laser
780.1 nm

Cavity probe
779.9 nmBlue donut guide
760 nmFrequency

Figure 7.2: Overview of cavity modes and laser frequencies for matter-wave interferometry.
The bare cavity is detuned δc to the blue of the |↑i → |ei transition, and the presence of atoms inN↑ shifts the dressed cavity resonance by ωc0 − ωc . A cavity probe sideband is locked to a TEM00
mode of the cavity, and the Raman laser and atomic probe are locked relative to the cavity probe.

Blue dipole guide

The blue dipole guide laser is a 760 nm interference filter ECDL locked

to a reference cavity for improved long-term stability. The laser is modulated by a fiber EOM withmodulation index β ≈ 1.3 at the cavity free spectral range ωFSR . By exciting adjacent longitudinalmodes of the cavity with opposite spatial parity with respect to the center of the cavity, one creates
an axially-uniform blue dipole guide near the center of the cavity (Sec. 4.5) [94]. The donut-mode
LG01 profile is created by a phase plate and inserted on transverse modes as described in Sec. 4.6.
The cavity is locked to the blue dipole guide laser via PDH.

Laser cooling Each trial, a 2D MOT loads a 3D MOT with 108 atoms near the cavity center
over approximately 0.5 s. The MOT coils are turned off, and around 2 × 105 atoms are cooled
via polarization gradient cooling to 15 µK and loaded into the red-detuned 813 nm lattice. The
red lattice depth is ramped down to ∼ 100~ωr and an extra stage of Λ-enhanced gray molasses
cools on the D2 transition. After 5 ms, the temperature of the ensemble is reduced to 6 µK. Twodimensional degenerate Raman sideband cooling (RSBC) further cools the radial temperature to
0.5-1 µK (Sec. 5.5) [131]. To improve the coupling of the atoms to the cavity (and thus increase the116phase space density of the atoms) we apply multiple cooling cycles. After three RSBC cycles, we
slowly turn off the remaining red lattice and the RSBC lattice at the same time so that the atoms
start to free fall with rms radial extent rrms = 3.0(5) µm  w0 .

Raman tones High fidelity Raman and Bragg pulse sequences require agile control of lowphase noise microwaves. The frequency source used for driving Raman pulses is described inSec. 3.1.9. Briefly, a low-noise 6.800 GHz source is mixed with 35 MHz DDS tones at a single sideband modulator. This provides phase, frequency, and amplitude control for arbitrary rotations
on the Bloch sphere, for selecting different momentum-changing transitions, velocimetry, etc. The
DDS tones are continuously ramped in frequency at a rate b = 2gk ≈ 2π×25.1 kHz/ms to match thetime variation of the two-photon Doppler shift as the atoms fall under gravity. The modulator output near 6.835 GHz is divided by two using a low-noise divider (PN: Analog Devices HMC862A)
and applied to a fiber-coupled EOM to generate the desired Raman tones as the ±1st order sidebands. We estimate that the noise contributed by this frequency source is at least 30 dB below the
SQL for 1000 atoms.
The laser that drives the Raman transitions is detuned ∆ = 2π × 85 GHz blue of the |↑i ↔
|ei ≡ |52 P3/2 , F = 3i transition (Fig. 7.3(a, b)). This laser is phase-locked to the cavity probe andcentered between two adjacent longitudinal TEM00 cavity modes. The two Raman tones are symmetrically detuned from the cavity resonances by (ωHF − ωFSR )/2 = 2π × 23 MHz. With 2.5 mWof total σ + -polarized light incident on the cavity, the EOM modulation index allows a two-photon
Rabi frequency of up to Ω = 2π × 15 kHz.

Cavity and atomic probes As in the previous chapter, a separate cavity probe laser is used
to stabilize the frequencies of the Raman lasers and the atomic probe relative to the cavity. The
cavity probe is locked to a TEM00 mode 160 GHz to the blue of the atomic transition frequency |↑i
to |ei at wavelength 779.915 nm. The locking of the cavity probe to the cavity is done via a PoundDrever-Hall lock at very low phase modulation index. To reduce the amount of power entering

117
Raman tones

out

c)

read

Ram
a

n

b)

QND

a)

Frequency Figure 7.3: Energy diagrams for the interferometry experiment. (a) Basis states for the interferometer are the clock states. Raman, OAT, and QND tones use σ + -polarized light. Final state readout is performed on the stretched transition after optically pumping atoms to |F = 2, mF = 2i. (b)Simplified energy-level diagram. The empty-cavity (green) is detuned by δc from ωa . The Raman
tones drive a |↑i ↔ |↓i transition with two-photon detuning δ in a falling reference frame. (c) The
Raman tones (blue) are derived from a laser locked between two adjacent TEM00 modes (gray) andmodulated at ωR ∼ ωHF /2.

the cavity to only 200 pW while still operating above the technical noise floor of the photodiode,we lock to one of the weak sidebands. This lock is always engaged. We then phase lock the otherlasers to the cavity probe, so some of the cavity probe light is passed through a fiber EOM driven
strongly at 13.6 GHz to generate very high order sidebands.
The atomic probe laser is phase-locked 13.6 × 12 = 160 GHz to the red of the cavity probe.
The atomic probe laser is maintained 80 MHz blue of the cavity resonance frequency. The reflected(carrier) tone provides a reference for a homodyne LO path length stabilization lock. To generatesqueezing or perform atomic population measurements, the atomic probe carrier tone is modulated by a fiber EOM at 80 MHz, allowing the atomic probe’s red sideband to interrogate the dressedcavity frequency shift. The laser could be actively locked to the dressed resonance as in [20], or the
linear part of the dispersive could be used to estimate small frequency shifts, but for this work, we
sweep the atomic probe through resonance at 1.5 MHz/ms. This simplifies the experiment but it
results in a 6 dB loss of quantum efficiency compared to true homodyne detection on resonance.
Although π-polarization would be preferable for the atomic probe, the light traverses the same
path as the Raman tones, which need to be σ + -polarized for sufficient transition strength.
The cavity and atomic probe lasers are DBRs with free-running linewidths of hundreds of118kHz. We use active feedback of an external cavity for narrowing the linewidths to less than 1 kHz(Sec. 4.3). This was instrumental for improving the technical noise floor for estimating populations
and removing undesired Bragg scattering.

7.2

Manipulating matter-waves in a high-finesse cavity
We manipulate matter-wave wavepackets using velocity-sensitive two-photon transitions

with wavelength λ = 780 nm. Within the cavity, Raman tones are propagating in both directions.
A co-propagating pair of tones induces a Doppler-free clock transition if they are separated by
ωHF , and these transitions are unimportant for the context of this dissertation. But the combined
absorption and stimulated emission of counter-propagating pairs of photons imparts 2~k momentum kicks oriented along the cavity axis, where k = 2π/λ and ~ is the reduced Planck constant.
As the atoms fall within the cavity, the relative Doppler shift for light propagating upwards
versus downwards chirps linearly in time. We compensate this effect by linearly ramping the
instantaneous frequency of the sidebands as 2ωR = ωHF + δ − b(t − tvs ) with b = 2kg = 2π ×25.11 kHz/ms. Here the local acceleration due to gravity is g = 9.8 m s−2 , δ is the two-photon
detuning in the falling frame of reference, and tvs is the time at which we will apply the first πpulse for velocity selection described below (Fig. 7.4(b)). The velocity-insensitive Doppler-freetransition is driven at δ = 0 but we avoid operating near this detuning by letting the atoms fall fora significant duration.

Coherent Raman transitions and velocity selection In Fig. 7.4(c), we show the initial axial
velocity spectrum of the atoms as mapped out by inducing velocity-dependent spin flips. Atoms
are prepared in |↓i and dropped for a duration Tfall , and the two-photon detuning δ is scanned
to resonantly excite a narrow velocity class to |↑i for measurement. We use this same process toselect atoms within a narrow range of initial velocities for coherent manipulation of matter-waves
in the remainder of this work. In the velocity selection sequence, atoms fall for Tfall = 15 ms
and are optically-pumped to |↑i, and the two-photon detuning is set to δvs = 2π × −400 kHz to

119
a)

b)
800 atoms,
Δp=0.1ħk

400
200
0

-600

-500

-400

-300

Two-photon detuning

-200

[kHz]

-100

0

EnergyResonant atoms[atoms/kHz, arb.]

Resonant atoms[atoms/kHz]

600

-30

0

30

Two-photon detuning

60

90

0 2 4 6 8 10

[kHz]

Momentum []Figure 7.4: Velocity-selective Raman transitions. (a) Atoms are prepared in |↓i and dropped
for Tfall = 7.5 ms (orange) or 15 ms (blue). The axial velocity distributions are mapped out by
scanning the detuning δ. The 5~k FWHMs are too broad for interferometry. During velocity selection, a group of about 800 atoms with rms momentum spread ∆p = 0.1~k (red) are kept fromthe latter distribution while the rest are removed with transverse radiation pressure. (b) After velocity selection, Raman transitions can be used to place atoms into a superposition of |0~k, ↓i and|4~k, ↓i. For the second transition, δ = −δvs + 2b(t − tvs ) − 8ωr , where the photon recoil frequency
is ωr = 2π × 3.77 kHz. Scanning δ demonstrates discrete momentum distributions.

transfer a group of atoms to |↓i from the center of the axial velocity distribution [182]. Atoms in|↑i are removed by a transverse radiation pressure force. The velocity-selected atoms are returnedto |↑i with a Raman π pulse and the selection process is repeated, resulting in approximately N0 =800 − 1200 atoms in |↓i with rms momentum spread ∆p = 0.1~k set by choice of the two-photon
Rabi frequency Ωab = 2π × 1.4 kHz. By using two stages, unwanted Rabi formula sinc-lobe Fouriercomponents are largely suppressed. Without this technique, roughly 20% of atoms are outside the∆p velocity group.
In Fig. 7.4(d) we demonstrate the quantized nature of the momentum kicks imparted by
the intracavity Raman transitions. After velocity selection, a π/2 pulse is followed by a second
Raman π pulse to place the atoms into a superposition of |0~k, ↓i and |4~k, ↓i in the falling frameof reference. We observe this as two distinct peaks separated in the subsequent velocity spectrum.
Though not leveraged here, future interferometers might evolve in such superpositions so as tominimize systematic errors and dephasing due to environmental couplings to the spin degree of

120
freedom.

Bragg transitions and large momentum transfer

Complementary to hyperfine spin-statechanging Raman transitions, we also demonstrate intracavity Bragg transitions in this high finesse
and high cooperativity cavity. Although these transitions still use two photons, the Bragg coupling
connects states |n~ki ↔ |(n + 2)~ki with no change in the spin degree of freedom as shown in
Fig. 7.5(a). The Bragg transitions are driven by two laser tones derived from the same laser withdifference frequency ωB = δvs − b (t − tvs ). To achieve 2π × 10 kHz Rabi frequencies, the center
frequency of the driving laser is shifted 10 MHz such that the two tones responsible for the Bragg
transitions are approximately 13 MHz from cavity resonance.
In Fig. 7.5(a), the ensemble is coherently split by a Bragg π/2 pulse, followed by successive
π pulses to transfer momentum to one of the wavepacket components for a momentum differenceof up to 10 ~k. An interferometer fringe for a Mach-Zehnder interferometer with 4~k maximum
momentum separation is shown in Fig. 7.5(b). The number of atoms in |0~k, ↓i is measured while
scanning the azimuthal phase of the final Bragg π/2 pulse from 0 to 2π. The 4~k Bragg interferometer has an evolution time of Tevol = 50 µs. Access to Bragg transitions opens the door to both large
momentum transfer operations for greater sensitivity and to improved coherence times in future
work.

7.3

Squeezed momentum states
Quantum entanglement between the atoms allows the atoms to conspire together to re-duce their total quantum noise relative to their total signal during the phase estimation process[13, 14]. Strong collective atom-cavity coupling has been used to either mediate pseudo spin-spin
interactions [25, 56, 59, 150] or to realize high resolution quantum non-demolition measurements[17, 20–24, 46, 47, 151]. Whether entanglement could really be extended over a distance was atonce hotly debated [1], but we can now exploit the phenomenon for metrological purposes.
In this section, we directly prepare squeezed momentum states through two separate cavity-

121
b)

-30

0

30

60

90

Two-photon detuning

Atoms in

EnergyResonant atoms[atoms/kHz, arb.][arb.]a)120 150

0 2 4 6 8 10

[kHz]

Momentum [

]

0

0

π

2π

3π

4π

Readout phase [rad.]Figure 7.5: Bragg-pulse superpositions and interferometry. (a) Two-photon Bragg transitions candrive coherent changes in momentum. Here we use a Bragg π/2 pulse followed by consecutive
π pulses to transfer momentum into superpositions of |0~k, ↓i and |2n~k, ↓i. The π pulses are
kept on resonance with successive transitions and thus require that the detuning δ be stepped in
8ωr increments between transitions. (b) An interferometer fringe for a Bragg-pulse Mach-Zehnderinterferometer with 4~k maximum momentum separation and Tevol = 50 µs evolution time.

based interactions: quantum non-demolition (QND) measurements [16, 20, 21, 29] and one-axis
twisting (OAT) [14, 30, 56], demonstrating the first generation of squeezed momentum states, with
+0.6
metrological gain 3.4+1.1−0.9 dB and 2.5−0.6 dB below the standard quantum limit respectively. These

techniques represent both of the most successful methods for generating large amounts of atomic
spin-squeezing to date. In both cases, quantum noise is reduced in one spin-momentum projection
at the expense of increased quantum noise along the orthogonal projection. The resulting states
are well-defined, well-resolved momentum modes useful for matter-wave interferometry. It will
be useful to establish terminology for the rest of the chapter, including a useful formulation of thespectroscopic enhancement W , before describing these results in more detail.

The Wineland criterion in terms of angular resolution. Contrary to the previous chapter,the spectroscopic enhancement of our interferometer will be evaluated directly in terms of angularresolution. This formulation is equivalent to the Wineland criterion [13] presented earlier, but it ismore closely connected to the sequences we run.
In order to emphasize the coupling between spin and momentum, we describe the basis

122states of our interferometer as |aii = |2~k, ↑ii and |bii = |0~k, ↓ii for the ith atom. The collective
state is depicted using a Bloch sphere with Bloch vector J~ = hJˆx x̂ + Jˆy ŷ + Jˆz ẑi of length J ≡ J~ ≤N0 /2 bounded by the number of velocity-selected atoms N0 . The collective population projection
P 0
PN0
operators are N̂↑ = N
i |aii i ha| and N̂↓ =i |bii i hb|, to define pseudospin projection operatorsJˆz ≡ 12 N̂↑ − N̂↓ , etc.
For the experiments that follow, an initial Raman π/2 pulse nominally prepares all atoms in
10√
an unentangled coherent spin state |ψi = ΠN(|aii + |bii ) described by the Bloch vector J~ = J x̂.
i2

The quantum noise that will appear in a measurement manifests in the non-zero variance of thespin projection operators (∆Jz )2 = hJˆz2 i − hJˆz i2 6= 0, etc. and is visualized on the Bloch sphereas a quasi probability distribution of the orientation of the Bloch vector from trial to trial. It is theratio of rms spin projection noise amplitude to the length of the Bloch vector that sets the standard
quantum limit in the quantum phase estimation of the polar and azimuthal angles.
The Wineland parameter is again defined by[∆(Jzf )]2 Ci
W =,2∆Jz,SQL
Cf2(7.1)

where the initial contrast Ci ≡ 2Jc /N0 is evaluated in the absence of squeezing interactions, andthe final contrast Cf ≡ 2Js /N0 is evaluated with squeezing, just as in Ch. 6. This expression is√rearranged using Jz,SQL = N0 /2 to arrive at
h 
i2
N −N

2
∆ N↑0 Cf ↓
∆θ
W =
≡
(7.2)
1/N0 Ci∆θSQL
Put into words, the Wineland criterion compares the angular resolution of the final measurement√√
to the standard quantum limit ∆θSQL = 1/ N0 Ci ≡ 1/ N obtained with the number of atoms
remaining after the initial contrast is evaluated. Physically, it is the reduction in the angular noise
variance of the phase estimation relative to the standard quantum limit one would have for a purestate with a Bloch vector length Jc = N/2 equal to that of the actually prepared partially decohered(mixed) state absent the squeezing operation.

Preparing QND-squeezed momentum states Population readout is achieved through col-123
lective or QND measurements of the free falling atomic samples that ideally give information about
the fraction of the atoms in spin-momentum states |ai and |bi without revealing single-particle information [29, 94]. We tune a TEM00 cavity mode with resonance frequency ωc to the blue of the|↑i → |ei transition ωa by δc = ωc − ωa . The |↓i → |ei transition is far detuned from ωa . The
QND Hamiltonian was introduced in Eq. 2.3. After adiabatically eliminating the excited state |ei[65] and ignoring mean-field light shifts that will be spin-echoed away, the effective Hamiltonian
describing the atom-cavity QND interaction can be expressed as [30]
ĤQND = δc + χQND N̂↑ â† â

(7.3)

where the cavity field is described by creation and annihilation operators â† and â. Physically,
atoms in |↑i create a round trip phase shift of the intracavity light that causes the cavity resonanceto shift. The two momentum states therefore interact differently with the optical cavity because
they carry distinct spin labels. For example, at the detuning we use for QND squeezing, δc =2π ×175 MHz, the dressed cavity shifts by χQND = 2π ×336(2) Hz per atom in |2~k, ↑i (see Sec. 7.5.3
and Fig. 7.6(a)). At the detuning used for OAT-squeezing, δc = 2π × 350 MHz, the dressed cavity
shifts by χQND = 2π × 183(1) Hz per atom in |↑i.
The population of atoms N↑ in the momentum state with spin label |↑i is inferred from themeasurement of this cavity frequency shift. The frequency shift is estimated by detecting the Q
quadrature of probe light reflected from the cavity input mirror as the laser frequency is swept
across resonance during a 150 µs measurement (Sec. 3.1.8). The population N↓ of atoms in the
momentum state with spin label |↓i is measured with the same technique after transferring theatoms to |↑i using a Raman π pulse. The Raman π pulse serves the additional functions of reoverlapping the wavepackets and cancelling the average light shift of the probe.
Collective QND measurements are not only used for readout, but also in creating conditional
spin-squeezing. The spin-momentum projection in the population basis is measured once with the
pre-measurement outcome Jzp = 12 (N↑ − N↓ ) |pre . The same projection is then measured a second
time with the final measurement outcome labeled Jzf = 12 (N↑ − N↓ ) |fin . Each final population

a)

124

b)

QND

-100

0

100

200

300

400

Frequency offset frombare cavity resonance [kHz]QND spectroscopicenhancement

432Standard
quantum limit

19
8
7
6
5
4
3

2

3

4

5

6

7 8 9

10³

2

3

Interacting photons

Figure 7.6: QND momentum squeezing. (a) Atomic probe frequency setup. Collective population measurements are made by sweeping the probe laser frequency over cavity resonance anddetecting the Q quadrature of the reflected field. (b) QND measurements are used to pre-measure
the quantum noise in the spin projection Jz and subtract it from a final measurement as in [20].
Increasing the number of probe photons Mi overcomes photon shot noise for a more precise premeasurement, but at too high of a photon number, free space scattering causes shortening of the
Bloch vector and spontaneous Raman scattering to other states. Squeezing is characterized by thespectroscopic enhancement W which reaches an optimum below the standard quantum limit at
Mi = 600 photons.

measurement is made after first optically pumping atoms in |↑i to |F = 2, mF = 2i to achieve lower
readout noise (estimated at more than 15 dB below the projection noise level) by probing on the
optical cycling transition to |F = 3, mF = 3i.
If sufficiently precise, the pre-measurement localizes the state to below the initial coherent
spin state level, producing a squeezed state. It can be considered a measurement of the quantum fluctuation of the orientation of the state on a given trial, and the measurement outcome canthen be used to partially subtract the quantum fluctuation from the final measurement outcome
by considering the difference Jzd = Jzf − Jzp . The quantum fluctuation is common to the two
measurements, but any rotation of the state (i.e. the signal) that occurs in between the two measurements appears only in the final measurement outcome so that one can estimate the angulardisplacement as sin(θ) ≡ Jzd /Js . The length of the Bloch vector Js after the pre-measurement
has prepared a squeezed state is measured in a separate set of experiments in which a π/2 pulse
about azimuthal angle φ is inserted between the pre- and final measurements. The length of the
Bloch vector is estimated from the fringe amplitude of Jzf versus φ as it is varied between 0 to 2π.

125
The initial length of the Bloch vector Jc needed for estimating the spectroscopic enhancement isestimated in the same manner, but without the pre-measurement applied.
Fig. 7.6(b) shows the spectroscopic enhancement W versus the strength of the QND interaction as parameterized by Mi , the average number of incident photons that enter the cavity during
each population pre-measurement window. At low Mi , the probe’s vacuum noise limits the spectroscopic enhancement, while at high Mi , the spectroscopic enhancement is limited by free spacescattering of the probe light that leads to a reduction in Js and transitions to other ground statesthat decorrelate the pre- and final measurements. Near Mi = 600, N = 1170(30) atoms, andδc = 2π × 175 MHz, we achieve W = 0.46(11) or 3.4+1.1−0.9 dB of directly observed squeezing in the
momentum-spin basis.

Preparing OAT-squeezed momentum states.

We also realize momentum-spin entangle-

ment via cavity-mediated interactions (Sec. 2.3) [30, 31, 56]. The one-axis twisting (OAT) Hamiltonian [14]ĤOAT = χOAT Jˆz2

(7.4)

is generated by applying a fixed frequency drive tone offset from the average dressed cavity resonance by δp & κ/2. Briefly, the populations in each momentum-spin state tune the cavity closer
to or further from resonance with the fixed frequency drive tone, allowing more or less light into
the cavity to produce an AC Stark shift on the states that depends on the population of atoms in|↑i. After adiabatic elimination of the dressed cavity mode [65] and neglecting terms that will be
spin-echoed away, one finds that to first approximation, ĉ† ĉ ∝ N̂↑ . As a result, the spin-dependent
portion of the QND Hamiltonian is transformed after a spin-echo pulse into Eq. 7.4, a pure spinspin Hamiltonian proportional to N̂↑2 + N̂↓2 = 4Jˆz2 with the cavity mode acting as a mediator of
the interaction. The unitary interactions drive shearing of the atomic quantum noise distribution
with a resulting squeezed state minimum noise projection oriented at a small angle α0 from ẑ
(Fig. 7.10(b, inset)).

126
To suppress free-space scattering, it is ideal to operate at δp = κ/2, but we work at larger
detunings to both reduce the QND interactions that can compete with the desired one-axis twisting
interaction [57] and to allow operation in a linearized regime even in the presence of shot-to-shot
total atom number fluctuations. In general, the optimum probe detuning increases with higher
N C. We empirically find an optimum detuning of δp = 2.7 × κ/2 with χOAT ≈ 2π × 10 Hz.
After the OAT interaction is applied, the state is rotated using the Raman beams to couple
the momentum-spin states so that the minimum noise projection is along ẑ. The momentum-spin
populations are destructively read out as described previously with measurement outcome labeled
Jzf . The Bloch vector lengths Js (Jc ) with (without) OAT squeezing are measured in the samemanner as was done for the QND measurements by measuring the fringe amplitude of Jzf versus
the azimuthal phase of a π/2 rotation prior to the final readout. We achieve a directly observed
spectroscopic enhancement from OAT of W = 0.56(8) or 2.5+0.6−0.6 dB. The optimal configuration
was realized with Mi ≈ 700 photons, δc = 2π × 350 MHz, and N = 730(10) atoms.

Tomography of momentum-squeezed states

Having prepared coherent states and QND-

and OAT-squeezed momentum states, we can visualize the quasi-probability distribution by measuring various projections along Jz . After preparing each state along x̂, we insert a pulse of varied
duration ψ to rotate the state about the Bloch vector. Statistics are collected for about ten durations of ψ and an inverse Radon transform reconstructs the roughly-Gaussian states [20, 154]. In
Fig. 7.7, the quasi-probability information is used in a height-and-color map to visualize the quantum states.

7.4

An entangled matter-wave interferometer
We now turn to injecting the prepared entangled states into a matter-wave interferometer.

An overview of the protocol is given in Fig. 7.8: generate momentum squeezing in the phasequadrature, run a low-noise interferometer, perform sub-SQL readout. The multitude of relevant
sequences are shown schematically in Sec. 7.5.1. We use a slightly modified Mach-Zehnder Raman

127

CSS

QND

OAT

Figure 7.7: Quasi-probability distributions are reconstructed and plotted in the Jy − Jz plane fora coherent spin state (CSS), a QND-squeezed state, and an OAT-squeezed state.

interferometer with (π/2 - π - π/2) pulse sequence to coherently separate, undo the separation, andinterfere the atomic wavepackets. In analogy to an optical Mach-Zehnder interferometer, the π/2pulses play the role of 50/50 beam splitters at the entrance and exit of the interferometer and theπ pulse plays the role of the mirrors that redirect the beams from the input beam splitter to the
output beam splitter.
After preparing a squeezed state with OAT, a Raman beam splitter rotation orients the squeezing along ŷ. The spin projection Jy will change if a small signal phase φ is applied. The orienting of
the squeezing is accomplished via a (π/2 + α0 ) pulse aligned to the atomic Bloch vector along x̂. A
relative phase accumulates between the wavepackets during a free evolution time Tevol , a Raman
π “mirror” pulse is applied, followed by another free evolution time Tevol . Finally, a readout π/2pulse transfers the signal φ and the squeezing into a displacement in the momentum-spin population basis ẑ with a measurement outcome Jzf . The Bloch vector lengths Js and Jc are measured inseparate experiments with and without OAT applied by scanning the azimuthal phase of the final
π/2 pulse of the interferometer and measuring the fringe amplitude as before (see Fig. 7.10(a)).
We achieve a directly observed spectroscopic enhancement as measured by the Winelandparameter of 1.7+0.5−0.5 dB beyond the standard quantum limit with N = 660(15) atoms as shown

128
Momentum squeezing
via QND or OAT
Generate
entanglement

Matter-wave interferometer

BeamsplitterMirrorpulse

Homodyne
readout

Readout
beam
splitterFigure 7.8: Squeezed interferometer space-time diagram and Bloch spheres for the generationand injection of the entanglement into a Mach-Zehnder matter-wave interferometer. Squeezing is
first generated in the population basis, and then a Raman beam splitter pulse orients the squeezing
for enhanced interferometer phase sensitivity. Wavepackets traversing the two paths (red and blue)
accrue a relative phase φ over time Tevol , the mirror pulse serves to reoverlap the wavepackets, andthe readout beam splitter pulse creates interference that is read out as a population difference with
sub-standard quantum limit sensitivity. Representative noise distributions are depicted on the
Bloch sphere for various points in the interferometer.

in Fig. 7.9(b). We also note that the actual phase variance of the interferometer is enhanced by
3.4+0.9−1.2 dB compared to with no one axis twisting, consistent with expectations from Jc .
Phase sensitivity beyond the SQL was limited to evolution times Tevol < 0.7 ms (Fig. 7.10(b)).
Evidence from an interferometer run with purely microwave rotations (i.e. with no momentum
states involved) suggests this loss of observable squeezing was primarily due to magnetic field
fluctuations in the lab, just like the earliest free-fall interferometer [42]. The matter-wave interferometer is sensitive to vibration noise, but the measured accelerations are not sufficient to explain
the loss of directly observed entanglement (Sec. 7.5.5). Single-particle decoherence is also inadequate as an explanation because Jc decreased by less than 5% over these evolution times. However,when increasing the bias magnetic field to ∼ 4 G, evolution times were limited to less than 0.3 ms.

129
a)
OAT

OAT

Readout

Timeb)Spectroscopicenhancement

10²

10¹

Standard quantum limit

100

-2

10

-1

10

0

10

Rotated angle α [rad]Figure 7.9: Interferometer sensitivity beyond the standard quantum limit. (a) The squeezedinterferometer sequence, including entanglement generation (purple), the interferometer (blue),and state readout (green). Each Raman transition is labeled with magnitude (within) and axis ofrotation (below). (b) The spectroscopic enhancement W is compared for three configurations: a
Mach-Zehnder interferometer with OAT (red circles, sequence above), an unentangled interferometer without OAT (black squares), and OAT-squeezed states without the interferometer (bluecircles). The duration of a π/2 + α rotation is scanned to minimize the projected spin noise at α0 .
An ellipse is fit with 68% confidence bands to the OAT-squeezed interferometer data, giving a minimum variance of W = 0.68(8) or 1.7+0.5−0.5 dB. The interferometer here has Tevol = 0.112 ms.

Despite working on the first-order magnetically-insensitive clock states, the second order Zeeman
effect leads to fluctuations in the azimuthal phase accrued during the interferometer evolution
times. With Fig. 7.10(b), we further observe that if the squeezed spin projection is left in the population basis Jz during the interferometer, then the squeezing persists for several milliseconds.
From this, we conclude that the entangled state persists for longer than we can directly confirm
because the interferometer is detecting an undesired magnetic field signal that masks the quantum
noise reduction as Tevol increases.

130
a)

b)
4
3

Spectroscopicenhancement[atoms]

600

0

-600
0

π

2π

3π

2

1.0

4π

Readout phase [rad]Standard quantum limit

8
6
4

0.1

Evolution time

1.0

[ms]

Figure 7.10: (a) Interferometer contrast fringes with Tevol = 0.112 ms shown for no squeezing Jc(black) and with squeezing Js (red). The ratio of Bloch vector lengths, which accounts for decoherence due to OAT photons, is Js /Jc = 88%. (b) Phase sensitivity is maintained below the SQL for
the squeezed interferometer (red circles, left Bloch sphere) up to Tevol = 0.7 ms. By contrast, if thesqueezed spin projection is oriented along the population basis (blue circles, right Bloch sphere),
spectroscopic enhancement was seen to persist beyond Tevol = 1 ms because this orientation is insensitive to phases accrued during the evolution time.

As an accelerometer or gravimeter, more work would need to be done to characterize systematic effects including the quadratic Zeeman effect, radial motion, and gradients of the intracavity
2fields. The phase of a Mach-Zehnder interferometer scales as φ = 2k · aTevol
for an accelerationa. The parameters of our Raman interferometer system allow for measuring accelerations such as√
gravity at the ∆a/a ∼ 10−4 level (100 µg/ Hz), in the same realm as other compact interferometers [181]. The maximum wavepacket separation is 2~kTevol /mRb = Tevol × 0.0117 mm/ms. For
the longest evolution times of Tevol = 1 ms, the maximum separation is 12 µm, an order of magnitude larger than the wavepacket size ∆x ≈ ~2 ∆p = 0.6 µm. Compared to a unitary-squeezed state∆θSQL ∆φSQL , the area of the OAT-squeezed state in the interferometer ∆θ∆φ is increased by about
6.5, similar to the QND-squeezed states described earlier.

7.5

Supplementary details

7.5.1

Experimental sequences
A schematic timing diagram for each sequence relevant to this chapter is provided in Fig. 7.11.

To measure N , OAT or QND is temporarily disabled by using Mi = 0, but the window durations

131
are left intact. The azimuthal phase of the π/2 readout pulse is scanned through [0, 2π], and thepeak-to-peak amplitude of the fringe is converted to N↑ using the known detuning δc and grms . For
estimates of angular resolution ∆θ, Mi is re-enabled. The size of the interferometer fringe N0 Cf
is measured by again scanning the readout phase. Measurements of ∆θ consist of 100-200 points
with readout phase tuned to the side of the fringe.

7.5.2

Optics layout
The complex array of frequencies and polarizations requires careful consideration of the lay-out on the optical table. ABCD matrices are first used to mode-match beams into lasers [26]. Highquantum efficiency (low optical loss) is required for the path of the reflected cavity field to thedetector. Various filters are used for overlapping beams at different wavelengths, and the angle ofincidence can affect transmission coefficients significantly. Finally, the spatial constraints of optic
mounts and clamps cannot be neglected. A suitable 2D geometry was drawn in Solidworks andconverted for print in Fig. 7.12.

7.5.3

Dispersive atom-cavity coupling in a multi-level system
The dispersive shift of the σ + -polarized cavity mode’s frequency for a given number of atoms

N↑ involves interactions with multiple excited state hyperfine levels F 0 [29]. The problem is complicated by the time-dependent atom-cavity coupling, but a hand-waving argument for the calculation is included here. We will define the states |0γ i and |1γ i to denote 0 and 1 photons in the


1 P
0
0
p
cavity, and the notation |F 0γ i ≡ |0γ ii |↑ · · · Fi · · · ↑i to indicate one collective atomicN↑excitation to a particular state F 0 . The ground state |↑ 0γ i has all atoms in |↑i and no cavity excitations. In the weak-probing (single excitation) limit, |↑ 0γ i couples to excited states |↑ 1γ i (onecavity excitation) and to |F 0 0γ i (one atomic excitation) for all excited states F 0 . The atom-cavity

132
1. Velocity selection

2. Raman superposition
update δ

3. Bragg superposition

scan δ

total n-1 pulses
update δ

update δscan δ

4. QND spin-noise

5. OAT spin-noise

6.

for momentum squeezing

7.

for momentum squeezing

8. Tomography

9. Squeezed interferometer (and10.

)

for squeezed interferometer

11. Number-squeezed interferometer

Figure 7.11: Sequence timing diagrams. BA: blow away |↑i atoms. OP: optically-pump |↑i atoms
for stretched-state readout. For rotation axes marked φ, the azimuthal phase is scanned out toextract a fringe.

133
Cavity795 Bragg laser(future only)

Homodyne
detector

760 nmBlue dipole guide
λ/4

λ/2

Ondax
Noiseblock ASE

Semrock
LD01-785

Thorlabs
FESH0800

λ/4
λ/4

PBS

λ/2

λ/2

Semrock
FF01-786-22
BSX11

Atomic probe,
Raman tones

PBS

Cavity
probe
λ/2

99:1
(R:T)

EOM

λ/4

λ/2

λ/2

λ/2

λ/2

HomoyneLO

ity
Cav PDHe
prob

Reflected cavity field

λ/2

λ/2

Red lattice
PDH

Red
lattice

6 inches

Figure 7.12: Optics layout for coupling the multitude of beams into the bottom port of the cavity.
Raman tones, QND and OAT light follow the same atomic probe path (red). The 795 Bragg laser
path is currently unused. Not only are the design wavelengths of the filters important (PN: Ondax
Noiseblock ASE Suppression 780.25, Semrock LD01-785 and FF01-786-22, Thorlabs FESH0800), but
their angles of incidence are critical as well.

interactions are described in a rotating frame of the |↑i → |F 0 = 3i transition frequency bypN↑ g3 â† σ̂3− + âσ̂3+ + δc â† âp+ N↑ g2 â† σ̂2− + âσ̂2+ + δ2 |20 0γ i h20 0γ |p+ N↑ g1 â† σ̂1− + âσ̂1+ + δ1 |10 0γ i h10 0γ |

Ĥ1 =

(7.5)

Here the cavity photon creation and annihilation operators are â† and â, and the atomic raisingP
and lowering operators are σ̂3+ = (|0γ i i |↑ · · · 3i · · · ↑i) h↑ 0γ |, etc. The detunings δf are the hyperfine energy splittings from F 0 = f to the F 0 = 3 state. The coupling constants gF =√αF grmsincorporate the time-averaged single-atom coupling constant grms and branching ratios αF fromF 0 to |↑i. In our interferometer system, grms = 2π × 0.341(2) MHz.
We invoke time-averaging and collective intracavity indistinguishability to switch to sym-

134
metrized basis states with a single excitation. The dispersive shift ωc0 − ωc can be found by solving
for the cavity-like normal mode of the effective Hamiltonian,


Ω3
Ω2
Ω1
2
2 
−δc 2


 Ω3

0
0
0
 2
,
Hˆeff = ~  Ω2
0 δ2 0 
 2



Ω1
0
0 δ1
2

p
1
where ΩF ≡ 2gF N↑ . For probing atoms in |↑i ≡ |F = 2, mF = 0i in particular, α3,2,1 = 25 , 14 , 60
and δ2,1 = 2π × {267, 424} MHz. For counting atoms in |F = 2, mF = 2i, α3,2,1 = {1, 0, 0}.
 2

g
g22
g12
To first order, the cavity shift is well-approximated by ωc0 − ωc = N↑ δ3c + δc +δ+
δc +δ1 ,2
but we use the full Hamiltonian for all results presented. In this section, we have assumed N↓ = 0
to neglect the contribution from the far-detuned F = 1 manifold, but the shift from N↓ atoms is
also accounted for in an analogous way.

7.5.4

Mitigating unwanted atomic transitions and light shifts
The cavity probe, blue dipole guide, the Raman tones, and even the atomic probe have thecapacity to drive unwanted atomic transitions during the interferometer experiment. Laser frequency noise and residual potentials have the capacity to drive unwanted rotations and undesired
Bragg scattering. The cavity probe standing wave was simply kept low, and the blue 760 nm laser
was injected on multiple free spectral ranges to keep an axially smooth potential.
At one point, we noticed that sweeping the atomic probe LO frequency with the probe tone
turned off was still producing AC Stark shifts the same magnitude as the non-demolition measurements. A small amount of light from the LO was reflecting off homodyne optics and entering thecavity resonantly. Inserting an optical isolator on the probing path mitigates these light shifts by atleast 30 dB. This should serve as an example of the subtle difficulties of working with such a highcavity finesse.
As for the Raman tones, the Raman laser is a DBR laser with a free-running linewidth of approximately ∆ν = 500 kHz. The laser is locked between two TEM00 modes separated by 6.788 GHz,135
so the sidebands split by hyperfine frequency 6.835 GHz are each 23 MHz from resonance when2ωR = ωHF . By injecting the Raman tones non-resonantly and with opposite detunings, we greatly
suppress laser frequency noise from being converted into phase and amplitude noise inside thecavity.
Nevertheless, we observed that frequency to amplitude noise conversion around the frequency of the velocity selection drove undesired Bragg transitions, leading to a significant loss of
population from the desired two-level basis. To suppress this, we implemented optical feedback
to narrow the laser to a Lorentzian linewidth of less than 1 kHz (Sec 4.3), after which we found
the fraction of total atoms lost out of the desired two-level manifold is less than 3(3)% for all the
Raman pulses involved in the interferometer sequence combined.
When an off-resonant tone excites a cavity mode, it builds up a field at the original frequency
ωp , but it also creates an electric field on resonance with the cavity at ωc that decays quickly. For
an instantaneous turn-on, we solve the damped oscillator equation
x00 (t) + κx0 (t) + ωc2 x(t) = E0 eiωp t(7.6)

subject to the boundary conditions x(0) = x0 (0) = 0. The turn-off case is similar but the bound
ary conditions are derived from the steady-state drive, x(0) = E0 / ωc2 + iκωp − ωp2 and x0 (0) =
iE0 ωp / ωc2 + iκωp − ωp2 . After simplifying with κ  ωp , ωc , the intracavity field isEc (t) = E0ωc cos (ωc t) + iωp sin (ωc t) − κteiωp t2+e.
ωc2 + iκωp − ωp2
ωc3 + iωc (κ + iωp ) ωp
|
{z
}
|
{z
}
driving field(7.7)

cavity-resonant field

The familiar picture of total intracavity power build-up with timescale κ (field build-up withtimescale κ/2) is valid, but the cavity-resonant field is instantly established with equal amplitude
and decays in a few µs. Atoms are subject to this short-lived field unless the laser is adiabaticallyintroduced.
We also observe residual off-resonance transitions to other momentum states if the turn on
and off of the Raman beams is too rapid. The temporal intensity profile of the Raman rotations isnominally a square pulse with the fiber EOM controlled by an rf switch. When using a switch with

136
5 ns rise-time, we observed transitions to other momentum states shown in Fig. 7.13. The fractionof atoms lost was reduced below 2(2)% by using an rf switch with 3 µs rise-time to gate the Raman
tones.

Atoms [arb.]

Fast switch
Slow switch
Unwanted
Bragg transition

0-60

Unwanted
Bragg transition-30

0

Two photon detuning

30

60

[kHz]

Figure 7.13: Unwanted Bragg transitions induced by rapid turn-on of the Raman tones. Witha fast TTL (red), higher-order Fourier components drive ±2~k transitions. With a 3 µs rise-timeswitch (blue), the loss is ameliorated. Opaque lines are spectroscopic data after smoothing.

7.5.5

Mirror vibrations
In a typical matter-wave interferometer, the position of the retro-reflecting mirror sets an in-ertial frame for the measurement. In contrast, both cavity mirrors here set the inertial frame. We
treat our cavity as a rigid body stabilized in length by piezos, but mirror vibrations neverthelessshow up as phase noise in the interferometer (see Refs. [66, 183]). In the limit of zero-duration
42
ωTevolpulses, the transfer function for a Mach-Zehnder interferometer |T (ω)|2 = 64k
sin
con4
2
ω
verts accelerations to an integrated phase noise,2σΦ(τ ) =Z ∞|T (ω)|2 Sa (ω) dω.

(7.8)

0

A more complete approach that includes finite pulse durations is detailed in Sec. 2.4.
A commercial vibration analysis system (PN: Herzan VA-2) was used to measure the spectral
density Sa (ω) of acceleration noise at a location on the optical table close to the portion that supports the vacuum chamber. The optical table is not floating due to (possibly superstitious) fears
that an extreme shock could topple the cavity spacer. The cavity spacer rests on Viton balls for

137
additional passive vibration isolation, so the table measurements provide only an estimation of
the noise seen by the Raman tones.
This vibrometer provides three-axis acceleration data for bandwidths between 1 Hz and 1 kHz.
The noise floor of the probe limited the upper frequency of our integration region to roughly 2.5
kHz. For a sequence with Tevol = 0.3 ms, we estimate the phase noise caused by vibrations is 20 dB
lower than the phase resolution set by the SQL of 1000 atoms. Repeating this measurement using
the voltage power spectral density of the path length stabilization lock output led to similar conclusions: mirror vibrations can be ignored for our current experimental parameters, but they may
contribute significantly for longer interferometer sequences. For a sequence with Tevol = 0.3 ms,
we estimate the phase noise caused by vibrations is 20 dB lower than the phase resolution set by
the SQL of 1000 atoms. The acceleration noise power spectral density (corrected for the frequency
response of the probe), a representative transfer function, and the results of numerical integration
are shown in Fig. 7.14.
a)Rb optics table
Sr optics table (floating)
Grad student hands

10-5

b)

c)

10-7

10-9

10-11
100

10¹

10¹

10²

Frequency [Hz]

10³

10²

10³

Frequency [Hz]Integrated noise [rad]Acceleration spectral
density

100

0.03
0.02
0.01
0.00
100

10¹

10²

10³

Upper integration bound fu [Hz]Figure 7.14: (a) Acceleration power spectral density with the probe on the Rb experiment optical
table, on a strontium experiment optical table floating on vibration isolators, and held in a gradstudent’s hands. The data has been corrected to account for the probe’s response curve. (b) A
simple two-window transfer function for Tevol = 5 ms windows showing insensitivity to vibrations
at multiples of 1/Tevol . (c) The integrated contribution of vibration noise σΦ for Tevol = 0.3 ms (blue)
compared to the standard quantum limit for 1000 atoms (red).

Chapter 8
Conclusion and outlook

8.1

Summary of accomplishments
While the concept of entanglement has been theorized for nearly a century, we can now

create truly macroscopic amounts of entanglement using atom-cavity systems. Just a few years
after the first demonstrations of squeezing in cavity QED systems, atomic spin-squeezing has surpassed optical squeezing in terms of the enhancement afforded to phase sensitivity. Together with
Ref. [21], the experiments of Ch. 6 constitute the largest amounts of spin-squeezing to date. Quantum non-demolition measurements have achieved nearly two orders of magnitude improvement
in variance for estimating a quantum phase over the classically-imposed standard quantum limit.
It is now of utmost interest to extend these capabilities to state-of-the-art sensors including atomic
clocks, magnetometers, and atom interferometers.
Along the way, our lab has explored ideas that we hope will benefit others. The Raman
SWAP cooling technique in Sec. 5.6 has real potential to lower temperatures with reduced heatingfrom spontaneous emission. We have demonstrated that adiabatic transfers may be employed forcooling atoms without single-photon narrow transitions, achieving final temperatures well below
the Doppler cooling limit. SWAP cooling is straightforward to implement, amenable to the presence of a large magnetic field, robust against small changes in atomic transition frequency, andpotentially useful for cooling molecules. The site-dependent loading technique of Sec. 4.4 may
provide unobscured insight into cavity QED physics. Narrowing DBRs with external optical feedback, transforming optical lattices into axially-smooth potentials, and inserting LG01 modes into

139
cavities could prove useful in a variety of contexts as well. The phase-tracking proposal of Sec. 6.0.1
represents in some sense a holy grail of quantum metrology, allowing continuous observation ofan evolving quantum phase with sub-SQL resolution.
We have demonstrated two additional extensions for bringing spin-squeezing into new classesof quantum devices. Feeding back on a quantum state, as we do in the deterministic squeezing
experiment of Sec. 6.2, could one day facilitate Heisenberg-limited sensors and other quantum information protocols. The homogeneous coupling of a free-falling ensemble in Sec. 6.3 opens thepath to entangled matter-wave interferometers and other devices involving particle delocalization.
Here the collective, time-averaged measurements of the atoms ensure that every atom participates
equally in the measurement process, uninhibited by a standing wave lattice.
Finally, after years of planning and hard work, we have demonstrated the first entanglementenhanced free-fall matter-wave interferometer. In Ch. 7, momentum-squeezed states are directly
prepared with quantum non-demolition and one-axis twisting interactions. The states are injected
into a Mach-Zehnder interferometer, where each atom traverses multiple trajectories, accumulating a differential phase while simultaneously entangled to other atoms. We directly observea −1.7+0.5−0.5 dB enhancement in the interferometer phase resolution beyond the standard quantum limit. The sub-SQL enhancement remains for sequences with evolution times as long as
Tevol = 0.7 ms. At longer times, the interferometric phase is dominated by magnetic field noisein the lab, but there is evidence that the squeezing persists for longer than 1 ms.

8.2

Looking forward
Cold-atom quantum sensors are at the precipice of exploring a wealth of new physics [184].

In addition to the possibility of using entangled states, cavity QED systems provide a range ofbenefits for both applied and fundamental physics. Performing state readout via a cavity measurement can allow for reduced technical noise, higher bandwidth, cleaner optical modes, andpower buildup for Raman transitions [181]. Higher order transverse modes, atom-chip technologies [185, 186], trapping [183] or tailored potentials [187] might be combined with the cavity mea-

140
surement techniques presented here to create new varieties of matter-wave Sagnac interferometers
and other inertial sensors.
The prospects for improving the achievable amounts of squeezing are somewhat well-known.
√
Operating with higher finesse and atom number exploits the N C scaling of the squeezing. Utilizing optical cycling transitions for entanglement generation in rubidium, strontium, and ytterbium
improves the fundamental scaling of the Wineland parameter to W ∝ 1/N C [20, 25, 29, 94, 151]. In
general, rotation-added noise, scattering, and spin-flips remain important obstacles for squeezed
sensors. Learning to read out squeezed states with magnification and SATIN techniques [59, 60]
will likely reduce the importance of low-noise detection to the benefit of future sensors. Carefulconsideration of optical and electronic decisions will allow higher quantum efficiency, reducing
the role of back-action in non-unitary squeezing [20, 153].
State-of-the-art optical atomic clocks can achieve sub-10−18 fractional uncertainties [43, 145–
147, 188] and are poised to re-define the SI second [189]. The unprecedented accuracy and stabilityof optical clocks pave the way for tests of physics beyond the Standard Model, but current sensors are at least partially constrained by quantum projection noise. Multiple groups are nearing
demonstrations of entanglement-enhanced optical clocks [25, 190]. Atomic magnetometers, too,stand to benefit from entanglement, with several proof-of-principle devices already demonstrated[18, 19, 165]. Optically pumped magnetometers (OPMs) are now sensitive enough to probe weaksignals from hearts and brains, and they have a significant size advantage over superconducting
systems [191]. In general, entanglement stands to benefit not just the sensitivity, bandwidth, andaccuracy of quantum sensors, but potentially the dynamic range, size, weight, power, and resistance to environmental noise [192].
The entangled matter-wave interferometer of Ch. 7 is a first step toward future generation
devices based on particle delocalization. In the future for our system, the magnetic field noise
can be reduced by identifying sources or using magnetic shielding. The need for velocity selection limits our final number of atoms, so higher atom density in momentum space through improved axial cooling or the use of a Bose-Einstein condensate could lead to significant improve-141
ments [49, 156, 185, 193]. Looking further ahead, the Raman and Bragg techniques demonstrated in
Ch. 7 would enable the most delicate portion of the interferometer to be operated fully with the two
portions of the superposition possessing the same spin label. To further improve interferometer
sensitivity, the entanglement can be combined with large momentum transfer sequences, one could
inject the squeezed state into a lattice interferometer to hold the atoms longer [183], or prepare theentanglement in the cavity and allow the atoms to undergo free fall and subsequent fluorescencemeasurement readout [166]. Matter-wave interferometers have begun to place bounds on a wideclass of dark energy theories [34], explore general relativity [6, 7, 171, 173], and perform the mostprecise tests of a physical theory to date [4, 5]. These devices range from miniature, chip-based experiments to atomic fountains and are operated in drop towers, parabolic flights, sounding rockets,and outer space [194]. Momentum-entangled interferometry serves as a promising probe for the
weak equivalence principle [195], Bell inequalities [196], and fundamental decoherence models[197]. In coming years, large-scale experiments including MIGA and MAGIS-100 will search forgravitational waves [35, 36] and dark matter [32, 33]. The seemingly endless advances in atominterferometry all stand to benefit from entanglement.
Finally, entanglement and manipulation of matter-waves are harbingers of future quantum
many-body simulations using degenerate gases in cavities both with and without spin-orbit coupling [198–200]. Such work will be able to explore beyond mean-field physics by directly modifying and probing the quantum fluctuations leading to modifications of the physics, or by accessinga regime where the quantum measurement process itself might induce phase transitions [201].

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Appendix A
Acronyms

156
SymbolDescription

AI
AMO
AOM
APD
AR
BEC
DAQ
DBR
DDS
ECDL
EOM
FSR
FWHM
HG
JILA
LF
LG
LO
MOT
NEG
OAT
PBS
PDH
PFD
PGC
PID
PN
PSN
PZT
QED
QPN
REF
REP
RSBC
SAI
SQL
SWAP
TA
TEC
UHV
VCO
VGA
VIS
VS

Atom interferometer
Atomic, molecular, and optical physics
Acousto-optic modulator, diffract and shift frequency of light
Avalanche photodiodeAnti-reflective [coating]Bose-Einstein condensate
Data acquisition [system]
Distributed Bragg reflector laser
Direct digital synthesizer, creates analog signals (frequency, amplitude, and phase control)
Extended-cavity diode laser
Electro-optic (phase) modulator, modulates phase to add frequency sidebands
Free spectral range; spacing of longitudinal cavity modes & inverse round-trip time for light
Full-width at half maximum
Hermite-Gaussian, rectilinear-basis modes
Formerly the Joint Institute for Laboratory Astrophysics, where this work was carried out
Loop filter, a servo-controller for stabilizing a system with PID feedback
Laguerre-Gaussian, radially symmetric modes
Local-oscillator, a reference frequency
Magneto-optical trap, or the laser responsible near the 87 Rb 2 → F 0 transitionNon-evaporable getter, adsorbs molecules to maintain vacuum
One-axis twisting
Polarizing beam splitter cube
Pound-Drever-Hall laser frequency locking technique
Phase frequency detector, used in feedback loops
Polarization-gradient cooling
Proportional-integral-derivative control for feedback systems
Part number
Photon shot noise
Piezoelectric transducer (lead zirconate titanate), changes length with applied electric field
Quantum electrodynamics
Quantum projection noise, from the wavefunction collapse of unentangled atoms
Reference, as in reference signal or reference laser locked to 85 Rb 3 → 40 transition
Repumper laser, near the 87 Rb 1 → F 0 transition for repumping atoms out of F = 1.
(Degenerate) Raman sideband cooling
Squeezed atom interferometerStandard quantum limit (for estimating a phase with N unentangled atoms)
Sawtooth-wave adiabatic passage [cooling] [139]Tapered amplifier, amplifies laser power
Thermoelectric cooler, also known as a Peltier device
Ultra-high vacuum
Voltage-controlled oscillator, changes frequency based on voltage input
Variable-gain amplifier/attenuator
Velocity-insensitive [transition], as in Doppler-free
Velocity-selection or velocity-selective [transition]

Table A.1: Frequently used acronyms.

Appendix B
Raman SWAP cooling: models and simulations

As discussed in Sec. 5.6, Raman SWAP cooling is optimized by balancing the need to sweep
slowly to preserve adiabaticity and the need to sweep fast enough to avoid scattering. This appendix uses the optical Bloch equations for a simplified three-level system to better understandthe dynamics and limitations of adiabatic passage. Section B.0.1 includes the adiabatic eliminationof the intermediate state |ii. Section B.0.2 accounts for the internal momentum states of the atom todistinguish successful adiabatic passage from a scattering event that transferred an atom into thecorrect final state. Section B.0.3 presents a probabilistic model supported by numerical simulationsof the preceding sections to supplement claims from Sec. 5.6.
The model considered in this appendix is the same as in Fig. 5.4(b) except that minor mathematical conveniences have been made. This three-level system has stable ground states |ai and|bi and an optically excited state |ii that decays to each state with equal probability at total rate
Γ. The energy of these states are ~ωa , ~ωb , and ~ωi . A laser at frequency ω1 (t) couples |ai ↔ |ii
with single-photon Rabi frequency Ω1 , and a laser at frequency ω2 (t) couples |bi ↔ |ii with Rabi
frequency Ω2 . The average detuning ∆ of these lasers from the excited state is large compared tothe two-photon detuning δ(t), ∆  δ(t), allowing for adiabatic elimination.

158
B.0.1

Optical Bloch equations and adiabatic elimination
This three-level system is described by the density matrix ρii ρia ρib 





ρ ≡ ρai ρaa ρab 





ρbi ρba ρbb

and the Hamiltonian for the system can be written H = HA + V1 + V2 where V1 couples |ai ↔ |ii
and V2 couples |bi ↔ |ii. The interactions between an atom and a field are provided by
Z t
V1 = −d · E1 cos

0

0



ω1 (t )dt
0

 Rt

 Rt

Rt
Rt
~Ω1 
0
0
0
0
0
0
0
0
|ii ha| e−i 0 ω1 (t ) dt + ei 0 ω1 (t ) dt + |ai hi| e−i 0 ω1 (t ) dt + ei 0 ω1 (t ) dt
2

Rt
Rt
~Ω1 
0
0
0
0
≈
|ii ha| ei 0 ω1 (t ) dt + |ai hi| e−i 0 ω1 (t ) dt
2

Rt
Rt
~Ω2 
0
0
0
0
V2 ≈
|ii hb| ei 0 ω2 (t ) dt + |bi hi| e−i 0 ω2 (t ) dt
2
=

and

after making the rotating wave approximation, ~Ω1 ≡ − ha | d · E1 | ii and ~Ω2 ≡ − hb | d · E2 | ii.
Thus the Hamiltonian in matrix form is

~ωi



Rt
0
0
~Ω
H=
 2 1 ei 0 ω1 (t ) dtR~Ω2 i 0t ω2 (t0 ) dt0
2 eR~Ω1 −i 0t ω1 (t0 ) dt0
2 e



R
~Ω2 −i 0t ω2 (t0 ) dt0

2 e

~ωa

0

0

~ωb








where the swept laser frequencies are
δ(t)
2
δ(t)
ω2 (t) = ωi − ωb + ∆(t) −2
ω1 (t) = ωi − ωa + ∆(t) +such that ∆(t) is the large average detuning of the lasers from their respective transitions. In our
experiment and simulations, the lasers are swept linearly in time, δ(t) = αt −∆swp
2 . From the Li-

ouville equation, ρ̇ = ~i [ρ, H] − γρ, with the dissipation term γ representing population relaxation,

159
the equations of motion for the coherences are
Rt
Rt


0
0
0
0
Γ
iΩ1 e−i 0 ω1 (t ) dtiΩ2 e−i 0 ω2 (t ) dt
ρ̇ia = − i(ωi − ωa ) +ρia +(ρii − ρaa ) −
ρba ,
2
2
2
R
R
t
t


0
0
0
0
Γ
iΩ2 e−i 0 ω2 (t ) dt
iΩ1 e−i 0 ω1 (t ) dt
ρ̇ib = − i(ωi − ωb ) +ρib +(ρii − ρbb ) −
ρab ,
2
2
2
Rt

0

Rt

0

0

(B.1)
(B.2)

0

iΩ1 e−i 0 ω1 (t ) dtiΩ2 ei 0 ω2 (t ) dt
ρ̇ba = −i (ωb − ωa ) ρba +
ρbi −
ρia ,
2
2

(B.3)

and for the populations, because the branching ratios from the intermediate state are assumed to
be balanced,
ρ̇ii =

 iΩ  R t

Rt
RtiΩ1  i R t ω1 (t0 ) dt0
0
0
0
0
0
0
2
e 0
ρia − e−i 0 ω1 (t ) dt ρai +ei 0 ω2 (t ) dt ρib − e−i 0 ω2 (t ) dt ρbi − Γρii ,
2
2
(B.4)

 Γ
Rt
iΩ1  −i R t ω1 (t0 ) dt000
e 0
ρai − ei 0 ω1 (t ) dt ρia + ρii ,
2
2
 Γ
Rt
iΩ2  −i R t ω2 (t0 ) dt000ρ̇bb =e 0
ρbi − ei 0 ω2 (t ) dt ρib + ρii .
2
2

(B.5)

ρ̇aa =(B.6)

A transformation to “slow” variables is used to enter the rotating (natural) frame,
Rt

0

0

Rt

0

0

ρia = ρ̃ia e−i 0 ω1 (t ) dt ,
ρib = ρ̃ib e−i 0 ω2 (t ) dt ,−iρba = ρ̃ba e

R
t

0
0
0
0 ω1 (t )−ω2 (t ) dt



Substituting these into Eqns. B.1-B.3, the coherences in this frame become ˙ρ̃ia = i ∆(t) + δ(t) − Γ ρ̃ia + iΩ1 (ρii − ρaa ) − iΩ2 ρ̃ba ,
2
2
2
2
 


δ(t)
Γ
iΩ2
iΩ1
ρ̃˙ ib = i ∆(t) −−ρ̃ +(ρii − ρbb ) −
ρ̃ ,
2
2 ib
2
2 ab
iΩ1
iΩ2
ρ̃˙ ba =ρ̃bi −
ρ̃ia + iδ(t)ρ̃ba .
2
2
The populations of Eqns. B.4-B.6 can now be rewritteniΩ1iΩ2(ρ̃ia − ρ̃ai ) +(ρ̃ib − ρ̃bi ) − Γρii ,
2
2
iΩ1
Γ
ρ̇aa =(ρ̃ai − ρ̃ia ) + ρii ,
2
2
iΩ2
Γ
ρ̇bb =(ρ̃bi − ρ̃ib ) + ρii .
22ρ̇ii =

(B.7)
(B.8)
(B.9)

160
The equations of motion ar reduced to an effective two-level system by adiabatically eliminating ρia and ρib . This is justified when the population of the intermediate state is small, which isvalid when ∆  Γ, δ. The time derivatives ρ̃˙ ia , ρ̃˙ ii , and ρ̃˙ ib are set to zero, and henceiΩ1iΩ2(ρii − ρaa ) −ρ̃ba ,
Γ + 2i∆(t) − iδ(t)Γ + 2i∆(t) − iδ(t)Ω2Ω1ρ̃ib =(ρii − ρbb ) −ρ̃ .
Γ + 2i∆(t) + iδ(t)Γ + 2i∆(t) + iδ(t) abρ̃ia =

These coherences are substituted into Eqns. B.7-B.9, and after transforming back into the original
coordinates, the final set of Bloch equations areΓρiiΓΩ21 (ρii − ρaa )
Ω1 Ω2
ρ̃ba
ρ̃ab
ρ̇aa =+ 2−+,2
Γ + (2∆(t) + δ(t))22
Γ − 2i∆(t) − iδ(t) Γ + 2i∆(t) + iδ(t)ΓρiiΓΩ22 (ρii − ρbb )
Ω1 Ω2
ρ̃ba
ρ̃ab
ρ̇bb =+ 2−+,2
Γ + (2∆(t) − δ(t))22
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) + iδ(t)


ρ̃
Ω21
Ω22
ρ̃˙ ba = iδ(t)ρ̃ba − ba+2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) − iδ(t)


Ω1 Ω2
ρbb − ρii
ρaa − ρii−+2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) − iδ(t)
with
ρii ≈

Ω21 ρaa + Ω22 ρbb + Ω1 Ω2 ρ̃ba + Ω1 Ω2 ρ̃ab.
Γ2 + 4∆(t)2 + Ω21 + Ω22(B.10)

Equation B.10 has been greatly simplified by the assumption that δ(t)  ∆(t), Ω1 , Ω2 .

B.0.2

Including internal momentum states
To account for the momentum states of the atom, we use a second label as in |a, 0~ki. The

basis of states is truncated to only include 0~k, 1~k, 2~k, . . . , (m − 1)~k momentum states, and thedensity matrix becomes 2m × 2m. Terms that are non-physical, involving a momentum transfer
with no change of state, can also be negated:

161





0
0
. . . ρab00 ρab01 ρab02 . . .ρaa00




 0

ρaa11
0
ρab10 ρab11 ρab12




 .
..
..
..
..
.. 
 ..
.
.
.
.
.



ρ=


ρ
0
0
. . .
 ba00 ρba01 ρba02 . . . ρbb00





ρ

0
ρbb11
0
 ba10 ρba11 ρba12



 .
..
..
..
..
.. 
..
.
.
.
.
.
Ideal adiabatic passage from |a, 0~ki through |i, 1~ki to |b, 2~ki is represented by adapting the
Bloch equations from the previous section:Γρii00ΓΩ21 (ρii11 − ρaa00 )Ω1 Ω2
ρ̃ba20ρ̃ab02ρ̇aa00 =
+ 2
−
+
,
2
Γ + (2∆(t) + δ(t))22
Γ − 2i∆(t) − iδ(t) Γ + 2i∆(t) + iδ(t)Γρii22ΓΩ22 (ρii11 − ρbb22 )Ω1 Ω2
ρ̃ba20ρ̃ab02
ρ̇bb22 =
+ 2
−
+
,
2
Γ + (2∆(t) − δ(t))22
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) + iδ(t)


Ω21
Ω22
˙ρ̃ba20 = ρ̃ba20 2iδ(t) −−2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) − iδ(t)


Ω1 Ω2
ρbb22 − ρii11
ρaa00 − ρii11−+2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) − iδ(t)withρii11 ≈

Ω21 ρaa00 + Ω22 ρbb22 + Ω1 Ω2 ρ̃ba20 + Ω1 Ω2 ρ̃ab02.
Γ2 + 4∆(t)2 + Ω21 + Ω22

Other momentum states are coupled in due to the scattering. For example, an atom in |b, 2~ki may
absorb a photon, losing ~k of momenta to enter |i, 1~ki. This state may in turn decay incoherently
to |a, 1~ki or |b, 1~ki.
A Runge-Kutta fourth-order numerical integration method is used to simulate these equations (Fig. B.1). Because the momentum states are now distinguishable within the density matrix,
we can separate atoms that undergo adiabatic passage without scattering from other possibilities.
We also construct expectation values for the momentum change. We use a 10 × 10 density matrix
and initialize the entire population in |a, 1~ki (ρaa11 = 1). When determining final populations,

162
1.0

Population

0.8
0.6
0.4
0.2
0.0

0.0

0.25

0.50

0.75

1.0

Time

Figure B.1: (a) Simulation of populations including momentum labels during adiabatic transfer.
The parameters used match those of Fig. 5.4(b). The expected change in momentum at the end of
the adiabatic transfer is 1.71~k.

oscillations in the simulation can be significant, so we use the mean results over the last 1/30th oftime steps. To find the expected change of momentum, we compute"#X∆p(Γ, ∆, Ω, ∆swp )AT =n ~k (ρbbnn + ρaann ) − 1~k.
nFinally, this model has ignored effects such as laser noise and atomic dephasing which can set
substantial limitations on transfer fidelity [202, 203]. In Fig. 5.7(a), a separate two-level simulation
is used which allowed for entering this dephasing by hand to better model our system.

B.0.3

A probabilistic model for Raman SWAP coolingSuppose adiabatic transfer from |ai to |bi adds 2~k momentum to an atom, and transferfrom |bi to |ai removes 2~k momentum. During a scattering process, absorption from |ai → |ii
adds one photon’s worth of momentum and |bi → |ii removes one photon’s worth momentum
but decay from |ii into either state causes on average no net momentum change. Then to firstorder, the possible momentum changes are detailed in Table B.1. The first-order expected change
in momentum, from summing the tabulated momentum changes multiplied by the probability ofthat trajectory, ish∆pAT i = 1 + x − 2x1+Q × ~k

163with x ≡ e−Rsc

∆swp
α

2

2

Ω
, Rsc ≈ ΓΩ
, and Q ≡ π2 Γ∆. The simulations of the previous section serve to
4∆2
swp

validate this model exceptionally well.
Row

Before T /2

At T /2

After T /2

Probability

∆p

1
2
3
4
5
6
7
8
9
10
11
12

Scatter to |bi
Scatter to |bi
Scatter to |bi
Scatter to |bi
Scatter to |ai
Scatter to |ai
Scatter to |ai
Scatter to |ai
No scatter
No scatter
No scatter
No scatter

Adiabatic transfer to |ai
Adiabatic transfer to |ai
No adiabatic transfer
No adiabatic transfer
Adiabatic transfer to |bi
Adiabatic transfer to |bi
No adiabatic transfer
No adiabatic transfer
Adiabatic transfer to |bi
Adiabatic transfer to |bi
No adiabatic transfer
No adiabatic transfer

Scatter
No scatter
Scatter
No scatter
Scatter
No scatter
Scatter
No scatter
Scatter
No scatter
Scatter
No scatter

1
2 SAS

0 ~k
−1 ~k
0 ~k
1 ~k
2 ~k
3 ~k
2 ~k
1 ~k
1 ~k
2 ~k
1 ~k
0 ~k

1
2 SA(1 − S)1
2 S(1 − A)S1
2 S(1 − A)(1 − S)
1
2 SAS
1
2 SA(1 − S)1
2 S(1 − A)S1
2 S(1 − A)(1 − S)

(1 − S)AS(1 − S)A(1 − S)
(1 − S)(1 − A)S(1 − S)(1 − A)(1 − S)

Table B.1: First-order possibilities for atom state trajectories. An atom starts in |ai and may scatter
before time T /2. At time T /2, adiabatic transfer may occur. From T/2 to T ,the atom may scatterΩ2

again. The probability to undergo adiabatic transfer is A ≡ 1 − exp − π2 αab , and the probability


2
∆swp
to scatter is S ≡ 1 − exp −Rsc 2αwith scattering rate Rsc ≈ ΓΩ.
4∆2

The momentum transfer during a full SWAP cooling sweep comes from cooling during thefirst half of the sweep, and cooling or heating during the second half which depends on whether
an atom successfully transferred from |ai to |bi. The expected momentum transfer during a fullcooling sweep is then h∆pSWAP i = h∆pAT i (1 + (Pb − Pa )). The fraction of atoms cooled vs. heatedin the second half of the sweep is represented by (Pb − Pa ), where Ps is the probability to be instate |si at the conclusion of the first half of the sweep. We find the expected momentum transferfrom one SWAP cycle,
h∆pSWAP i = h∆pAT i 1 + x(1 − 2xQ ) .

(B.11)

To evaluate the benefits of SWAP cooling over Doppler cooling, we would like to know how
much momentum could be removed before an atom (or molecule) scatters a photon. Although wehave assumed equal branching ratios back to ground states until now, suppose that every scat-

164
tering event causes an atom to be lost. We also assume an atom scatters a photon from optical
repumping if it finishes the cooling sweep in |bi. The expected change in momentum per sweep
comes entirely from the trajectories described by Table B.1 rows 10 and 12:∆p0AT = (1 − S)A(1 − S) × 2~k = x(1 − xQ ) × 2~k
For consistency, we need to normalize to the number of atoms remaining in the system:⇒h∆pAT i =∆p0AT
(1 − S)A(1 − S) + (1 − S)(1 − A)(1 − S)

= (1 − xQ ) × 2~k⇒h∆pSWAP i = h∆pAT i × [1 + ((1 − S)A(1 − S) − (1 − S)(1 − A)(1 − S))]= 1 + x − xQ − 3x1+Q + 2x1+2Q × 2~kNext we find the average momentum transferred per atom at a time when 1/e atoms remain,
still under the assumption that any scattering event will cause the atom to be lost. The two ways
an atom does not scatter a photon (during the sweep or from optical repumping) come again fromrows 10 and 12:
P (no scatter) = [(1 − S)A(1 − S)]2 + [(1 − S)(1 − A)(1 − S)]2 .
Let the probability that an atom survives n sweeps be 1/e; then the momentum change for an atom
at the point when it has probability 1/e not to be lost ish∆pSWAP i− ln (P (no scatter))(1 − xQ )(x(2xQ − 1) − 1)=
× 2~k.
ln (x2 + 2xQ+2 (xQ − 1))n × h∆pSWAP i =(B.12)

This function is plotted in Fig. B.2(b) for Q = 25, near where our experiments operate, and maximized with respect to sweep rate in Fig. 5.7(b).
The optimal sweep rate for transferring momentum in the model where scattering is unimportant is found by taking the derivative of Eq. B.11 with respect to α and equating it to zero:
αopt ≈π Ω41.
2
2 4∆ ln [2 (1 + Q)](B.13)

165a)

Sweep rate αopt for maximal ∆p
per scattered photonb)

Momentum transfer before
photon is scattered
15

10

10

5
1

1

10

100

0
0.1

1.0

Quality parameter Q

Figure B.2: (a) The momentum transferred during a single simulated SWAP cooling sweep optimized with respect to the sweep rate α. When the quality parameter Q is prohibitively small, theideal strategy is to sweep very slowly in order to coherently scatter once. When Q is large, efficientadiabatic sweeps are possible. (b) The maximum momentum transferable before a photon scatters
due to SWAP cooling or optical pumping beams.

However, the optimal sweep rate for transferring momentum before atoms are lost to recoiled photons (Eq. B.12) could only be found numerically (Fig. B.2(a)). The scaling at significant Q remains
the same as in Eq. B.13 – the optimal sweep rate can be increased as roughly Ω4 /∆2 although Q
also changes as Ω2 .

Appendix C
Statistics of inhomogeneous coupling

In conditional squeezing, the results of a pre-measurement are used to cancel the quantum
noise found by a final measurement. Rearranging individual atomic couplings between the two
measurements sets a limit on the degree to which correlations can be utilized. This rearrangement may happen due to atomic motion, or it may be engineered, e.g. if the pre-measurement isperformed in a lattice and atoms are released into free space for the final measurement. The implications of these imperfect correlations are explored in the supplemental material of Ref. [94] but
the main results are reproduced in this section.
To understand the limits of inhomogeneous coupling, we consider a population measurement where each atom i couples to the cavity with a factor ηi (t) = gi2 (t)/δc 1 . In this expression,gi2 (t) is the time-dependent Jaynes-Cummings coupling parameter and δc is the cavity detuningfrom the atomic transition. We will denote the coupling as ηm,i omitting the time-dependence bytaking the average value during the m ∈ [p (pre-), f (final)] measurement.
The operator ω̂mi measures the time-averaged cavity frequency shift from the ith atom during
measurement m,0ω̂mi = σ̂z,i+ γ ηmi .

(C.1)

0 = (1 − γ)( |↑ i h↑ | − |↓ i h↓ | ) is the Pauli spin operator σ̂thHere, σ̂z,iiiiiz,i for the i atom rescaled by(1 − γ). The constant 0 ≤ γ ≤ 1/2 is used to account for measurements that are sensitive to a1√
Simple form valid for the dispersive coupling regime, δc  2g0 N

167
linear combination of N↑ and N↓ given by (N↑ − (1 − 2γ)N↓ ). For example, a number of squeezing
experiments have used a cavity detuned halfway between the |↑i → |ei and |↓i → |ei transitions[17, 21, 46]. Here, γ = 0 and the dispersive cavity shift is only sensitive to population differences
between |↑i and |↓i. For the experiments in this dissertation, the cavity resonance frequency is nearthe |↑i → |ei transition so there is minimal sensitivity to N↓ (γ = 0.5).
We wish to cancel the quantum projection noise in the final measurement of all N atoms
PPNthω̂f = Ni=1 ω̂f i using a pre-measurement ω̂p =i=1 ω̂pi . However, if the i atom’s couplings ηf i
and ηpi change, projection noise cannot be exactly canceled. In order to optimize the cancellationof quantum noise, we construct a weighted difference ω̂diff ≡ ω̂f − W ω̂p with a single weight factor
W for the whole ensemble. This procedure is similar to Bayesian estimation of phase resolution[204]. The variance in ω̂diff,i for a single atom is(∆ωdiff,i )2 ≡ h(ω̂f i − W ω̂pi )2 i − h(ω̂f i − W ω̂pi )i2 ,(C.2)

where the average over many independent experimental trials is denoted with h...i to evaluate the0 as well as uncorrelated fluctuations inquantum fluctuations of the spin projection operator σ̂z,ithe couplings ηmi . For the relevant case of atoms in an equal superposition of |↑i + |↓i, we have
0 i = 0. The contributions to (∆ω2hσ̂z,idiff,i ) may then be understood as





02
2
(∆ωdiff,i )2 = hσ̂z,ii hηf2i + W 2 ηpi− 2W ηf i ηpi i + γ 2 (∆ηf i )2 + W 2 (∆ηpi )2 − 2W cov(ηf i , ηpi )
{z
}
|
{z
} |
quantumclassical(C.3)
where cov(X, Y ) = hXY i−hXihY i is the covariance quantifying the correlation of the fluctuationsof X and Y . If X and Y are uncorrelated, then cov(X, Y ) = 0. If X and Y are perfectly correlated,then cov(X, Y ) = ∆X∆Y . There are two contributions to Eq. C.3. The first term arises from uncancelled quantum projection noise and will be nonzero if ηf i 6= W ηpi , illustrating the fundamental
problem of inhomogeneous coupling to the collective measurement.
The second term in Eq. C.3 results from trial-to-trial noise in the couplings ηf i and ηpi and canbe thought of as noise in the scale factor relating an observed cavity frequency shift to an estimated168
population of atoms in |↑i or |↓i. When γ = 0, the cavity frequency shifts are proportional toN↑ − N↓ which is on average zero for the superposition considered here, and any noise in the scale
factor contributes no additional noise. However, if γ = 1/2, then the cavity frequency shifts areproportional to N↑ = N/2. In this case, classical scale factor noise can dominate the quantum noise.
As an example, classical noise is important when the atoms are trapped at random positionsof an intracavity lattice for the pre-measurement and then released to into free space for the final
measurement [166]. An atom in the optical lattice draws its coupling ηpi from a uniform distribution from trial to trial. Because we cannot know every atom’s couplings on each trial, we are left
to measuring the average couplings hηf i i and hηpi i via the cavity shift. This leads to classical noisein the pre-measurement that is of the same order as quantum projection noise.
The total noise variance (∆ωdiff )2 is found by summing the presumably uncorrelated noise
contribution from each atom2(∆ωdiff ) =NX(∆ωdiff,i )2 ⇒ N (∆ωdiff,i )2 .

(C.4)

i=1

which follows when particle labels for every atom are interchangeable, or when the trial average
h...i can equivalently be viewed as an average over all of the atoms in the ensemble on a single
trial. To calculate the quantum projection noise (QPN) limit for different coupling configurations,
we sum the quantum term of Eq. C.3 with W = 0 for all atoms,(∆ωQPN )2 = N hσ̂z02 ihηf2 i.

(C.5)

Because we are assuming the expectation values are the same for all atoms, the indices i have been
02 i ≡ hσ̂ 02 i.
dropped, hηmi ηm0 i i ≡ hηm ηm0 i, hηmi i ≡ hηm i, and hσ̂z,iz

For Sec. 6.3, we want the projection noise fluctuations in the effective dipole trap whenfractionally (1 − ζ) of the atoms remained trapped in the residual lattice. We assume that theatoms are identically prepared in a pure state with an equal superposition of √12 (|↑i + |↓i) so thathσ̂z02 i = 1/(1 − γ)2 = 1/4. For the ζ atoms in the dipole trap, hηf2 i = g04 /(4δc2 ) where g0 is the
Jaynes-Cummings coupling parameter at an anti-node of the cavity. For the 1 − ζ atoms in the169
residual lattice, hηf2 i = 3g04 /(8δc2 ). Adding the projection noise variance of all of the atoms leads tothe observed decrease of the projection noise2 versus ζ,
N g04 1
3(∆ωQPN ) (ζ) =ζ + (1 − ζ) .
4δc2 48
2(C.6)

The observable spin-noise reduction relative to the quantum noise in the final measurement
is given by R = (∆ωdiff )2 /(∆ωQPN )2 . R is optimized with respect to the weight factor W by [94]Ropt = 1 +
|{z}
QPN

|



γ
1−γ

2

2

hηf ηp i +cov(ηf , ηp )(∆ηf )2−.
2
hηf2 iγ2222hηf ihηp i + 1−γ hηf i(∆ηp )
{z
}
|
{z
}
classical noise

γ
1−γ

2

(C.7)

optimum cancellation

This is the best noise reduction possible when the pre- and final measurements have different
coupling strengths. The first term is projection noise and the second term is classical noise in the
final measurement. The third term represents the optimum cancellation of the final measurement’s
noise provided by the optimally weighted pre-measurement.
A few cases deserve special comment. For a pre-measurement of atoms in an incommensurate standing wave followed by a uniform measurement (e.g. perfectly time-averaged coupling
due to atomic motion), the maximal spin-noise reduction is R = 1/2 for γ = 1/2 or R = 1/3 for
γ = 0. Squeezing is limited to 4.8 dB. In situations where only population differences are measured (γ = 0), Ropt simplifies to the ratio of the second order moments of the coupling strengths,
Ropt = 1 − hηf ηp i2 /(hηf2 ihηp2 i). Without weighting the final measurement with W , these limits areeven more stringent, but the optimal weight factor W for canceling quantum and coupling noise
is not necessarily the optimum for canceling the other technical noise sources such as photon shotnoise or laser frequency noise. In our experiments, optimum squeezing was always observed with
W = 1.

2√
Assuming the dispersive limit δc  2grms N , here, but not in Sec. 6.3.