New tools for precision measurement and quantum science
with narrow linewidth optical transitions
byMatthew A. Norcia
B.S., Stanford University, 2011

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
2017

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This thesis entitled:
New tools for precision measurement and quantum science with narrow linewidth opticaltransitions
written by Matthew A. Norcia
has been approved for the Department of Physics

Prof. James K. Thompson

Prof. Jun Ye

Date

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

iii
Norcia, Matthew A. (Ph.D., Physics)
New tools for precision measurement and quantum science with narrow linewidth optical transitions
Thesis directed by Prof. James K. Thompson

In my thesis work, I have explored a variety of ways in which narrow linewidth optical transitions can be utilized to extend the capabilities of precision measurement and quantum science.
These tools include non-destructive atom counting techniques that may enhance the performanceof optical lattice clocks, a novel form of cavity enhanced atomic spectroscopy appropriate for laser
frequency stabilization, a new form of laser cooling with reduced reliance on spontaneous emission,
several methods for enhancing the sensitivity of proposed atomic gravitational wave detectors, a
newly observed form of cavity-mediated spin-spin interactions, and a new class of laser based on
optical superradiance from narrow and ultra-narrow linewidth transitions. As part of the superradiant laser project, we have demonstrated the most precise active absolute frequency reference
realized to date, which still has great potential for improvement.
Many of these tools involve coupling atoms to an optical cavity, a field known as cavity
quantum electrodynamics, or cavity QED. In my thesis work, a key focus was the extension ofcavity QED techniques into the new regime of coupling large ensembles of atoms to a cavity using
narrow linewidth optical transitions (in contrast to the previously explored regimes of microwavefrequency transitions or broad-linewidth optical transitions). To this end, I constructed a newexperiment to couple an ensemble of atoms to an optical cavity using two optical transitions inatomic strontium, one with a narrow 7.5 kHz linewidth, and another with an ultra-narrow 1 mHz
linewidth. This new system has allowed us to access a unique new regime of cavity QED, and awealth of interesting and useful applications.

Dedication

To my parents.

v

Acknowledgements

Here I am, about to graduate the 22nd grade and looking for ways to stick around. For this,I have many people to thank.
First, I would like to thank my parents Pat and Tony. They have always been extremely
encouraging and tolerant of my projects, from taking up much of the street and yard with skateboard
ramps, to the five or so project cars that cycled through the driveway, to the ultimate entire
takeover of the back yard for the construction of an electric vehicle. I would also like to thank mybrother Michael (who has currently taken over my parents house as well as the yard for airplane
construction) for undertaking these technical adventures with me. These projects provided a great
background for my later transition into engineering and physics. My high-school auto-shop teacher
Mike Camicia had a lot to do with this as well.
In undergrad, I experienced just enough of physics research after my Freshman year to realizethat I was not mentally mature enough for it yet, so I focused my energies on engineering projects
with more immediate payoffs. Fortunately, an Engineering Physics degree had recently been introduced (thanks, Pat), so I could do this while leaving open the option of a physics career later on(once my frontal lobes had developed, as Tony would say).
That brings me to grad school where I was lucky enough to join James Thompson’s group.
The combination of physics, engineering and emphasis on simple visual explanations have madethis a far better choice than I could have known. James has been a terrific advisor, with excellent
physical intuition and a good sense of priorities both in science and in general.
During my time in the group, I have had the opportunity to work with some fantastic students,

vi
including Zilong Chen, Justin Bohnet, Josh Weiner, Kevin Cox, Graham Greve and Baochen Wu
on the Rubidium experiment, and Karl Mayer, Drew Morrill, Julia Cline, Matthew Winchester andpostdoc Juan Muniz on the Strontium experiment. Thank you all for the years of great discussions.
Finally, I would like to thank my friends and housemates here in Boulder who have made thelast five years here in Boulder a time that I will look back on with great fondness. Perhaps a new
high-point?

Contents

Chapter
1 Introduction
1.1

1.2

1

Overview of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.1.1

Appliction to Clocks: Non-Destructive Atom Counting . . . . . . . . . . . . .

5

1.1.2

Appliction to Laser Stabilization: Magnetically Induced Transparency . . . .

6

1.1.3

Superradiance from Narrow Linewidth Optical Transitions . . . . . . . . . . .

7

1.1.4

Cavity-Mediated Spin-Spin Interactions . . . . . . . . . . . . . . . . . . . . . 11

1.1.5

A Novel Cooling Mechanism

1.1.6

Enhanced Gravitational Wave Detection with Atoms . . . . . . . . . . . . . . 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 An Introduction to Atoms in Cavities and Narrow-Linewidth Transitions

15

2.1

Narrow-Linewidth Transitions in Strontium . . . . . . . . . . . . . . . . . . . . . . . 15

2.2

Atoms Interacting with a Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3

Cavity QED on Dipole-Forbidden Transitions . . . . . . . . . . . . . . . . . . . . . . 23

2.4

Cavity QED with Bloch Spheres and Pendula . . . . . . . . . . . . . . . . . . . . . . 26

3 Experimental Apparatus

31

3.1

Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2

Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3

Atomic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

viii
3.4

MOT Coils and Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5

Laser Systems and Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1

461 nm Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5.2

Initial Slowing and Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5.3

Repump Lasers at 707 and 679 nm . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5.4

689 nm System and Narrow-Line Cooling . . . . . . . . . . . . . . . . . . . . 46

3.5.5

698 nm Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5.6

Reference Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.6

The Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.7

The Optical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Strong Collective Coupling on 7.5 kHz Linewidth Transition and Nondestructive AtomCounting

60

4.1

Demonstration of Collective Strong Coupling . . . . . . . . . . . . . . . . . . . . . . 62

4.2

Nondestructive Atom Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Magnetically Induced Transparency

71

5.1

Basic Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2

Experimental Characterizations of MIT . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3

Mapping to EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4

Limits on Photon Retrieval Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Lasing on the 7.5 kHz Linewidth Transition: the Superradiant Crossover Regime

85

6.1

Pulsed Lasing in the Vacuum Rabi Splitting Limit . . . . . . . . . . . . . . . . . . . 87

6.2

Quasi Steady State Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3

Lasing Threshold Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

ix
6.4

Cavity Pulling, and Tuning Through the Superradiant Crossover Regime . . . . . . . 92

7 Pulsed Lasing on the milliHertz Linewidth Clock Transition

95

7.1

Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.2

Observation of Superradiant Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.3

Seeding Atomic Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.4

Simultaneous Lasing on Multiple Transitions . . . . . . . . . . . . . . . . . . . . . . 104

7.5

Simulations of Superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.5.1

Effects of Atomic Decoherence and Loss . . . . . . . . . . . . . . . . . . . . . 107

7.5.2

Effects of Inhomogeneous Coupling . . . . . . . . . . . . . . . . . . . . . . . . 108

8 Frequency Stability Characterization of Pulsed Lasing

111

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.2

Basic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.3

Stability Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.4

Cavity Pulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.5

Magnetic Field Insensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.6

Absolute Accuracy and Other Perturbations . . . . . . . . . . . . . . . . . . . . . . . 121
8.6.1

Lattice Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.6.2

Atomic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.6.3

Other Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

9 Cavity-Mediated Spin-Spin Interactions

130

9.1

Quantum Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9.2

Classical Description of OAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9.3

Experimental Characterization of OAT Term . . . . . . . . . . . . . . . . . . . . . . 138

9.4

The χJ 2 Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9.5

Experimental Characterization of Effects from the χJ 2 Term . . . . . . . . . . . . . 142

x
9.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

10 Sawtooth-Wave Adiabatic Passage (SWAP) Cooling151

10.1 How it Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.1.1 Adiabatic Passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.1.2 Basic Cooling Mechanism in One Dimension. . . . . . . . . . . . . . . . . . 154

10.1.3 Where Does the Entropy Go? . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.1.4 A Dressed-State Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.1.5 Regimes of Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.1.6 Simulated Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.2 Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.2.1 Forces in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.2.2 Cooling and Phase-Space Compression in 1D . . . . . . . . . . . . . . . . . . 167
10.2.3 Practical Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.3 Creating a Restoring Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.3.1 How it Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.3.2 Demonstration in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.3.3 Application to a Real Experiment . . . . . . . . . . . . . . . . . . . . . . . . 173
10.4 Extensions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
11 A new perspective on gravitational wave detection with cold atoms in space178

11.1 Optical Path Length Changes due to Gravitational Waves . . . . . . . . . . . . . . . 180
11.2 Detecting Changes in Optical Path Length. . . . . . . . . . . . . . . . . . . . . . . 182

11.3 Clock-like Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
11.4 Atom Interferometer Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.5 Enhanced sensitivity analogous to large-momentum-transfer . . . . . . . . . . . . . . 186
11.6 Evolution Times Beyond the Excited State Lifetime . . . . . . . . . . . . . . . . . . 190
11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

xi
11.8 Accounting for Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
12 Conclusion and future outlook

197

12.1 Spin squeezing in an optical lattice clock . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.2 Towards a steady-state superradiant laser . . . . . . . . . . . . . . . . . . . . . . . . 199

Bibliography203

TablesTable
2.1

Key parameters and scalings for cavity QED. . . . . . . . . . . . . . . . . . . . . . . 20

3.1

Overview of laser systems.

3.2

Cooling and trapping beam parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3

Typical lattice parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4

Key parameters of optical cavity.

7.1

Summary of key parameters and effects of inhomogeneous coupling for pulsed superradiance.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.1

Summary of quantum efficiencies in detection chain. . . . . . . . . . . . . . . . . . . 118

8.2

Summary of shifts and uncertainties in frequency comparison.

. . . . . . . . . . . . 123

Figures

Figure1.1(a) Strontium has two valence electrons, whose spins can either be aligned (spintriplet) or anti-aligned (spin singlet). Photons are very bad at flipping spins, so wecall transitions between these two types of states “forbidden,” meaning that theyoccur slowly if at all. (b) A basic overview of our experimental platform. Strontium
atoms are cooled to roughly 10 µK, and tightly confined in an optical lattice insideof an optical cavity. The wavelength of the lattice is carefully chosen so as not to
perturb atomic transition frequencies of interest — the “magic wavelength.” Theatoms interact with the cavity via one of several forbidden transitions.

2.1. . . . . . .

3A cavity QED system. Atoms exchange excitations with the cavity mode at a rate√
Ω = 2g N . Atoms can decay into free-space via single-particle spontaneous emission
at a rate γ, leading to loss of coherence. Photons in the cavity leak out through themirrors at rate κ. This channel of dissipation is easy to detect, and provides only
collective information about the state of the atomic ensemble.

2.2. . . . . . . . . . . . 17

One of the key manifestations of the strong coupling regime defined by Ω  κ, γ
is the vacuum Rabi splitting. As a weak probe tone coupled to the cavity is swept
over resonance with the cavity (here, δc = 0), a single transmission peak splits into
two. The splitting between these peaks is set by Ω, and the width of each peak is aweighted average of atomic and cavity decay rates.

. . . . . . . . . . . . . . . . . . 22

xiv
2.3

(a) The state of a two-level system with states |ei and |gi can be represented by a
Bloch vector that lives on a Bloch sphere. The figure shows a general state |ψi =sin(θ/2) |gi + cos(θ/2)eiφ |ei. The vertical projection of the Bloch vector tells us
the population inversion of the state (the relative likelyhood of finding the atom in
the ground or excited state), while the azimuthal angle φ tells us the relative phasebetween the |ei and |gi components. (b) An ensemble of many two-level systems
can be represented on a collective Bloch sphere by simply adding up all of theindividual Bloch vectors associated with the two-level systems to form a collective~ The inversion of the collective Bloch vector is labelled J~z , and the
Bloch vector J.
atomic coherence (projection onto the equatorial plane) is labelled J~+ .

2.4. . . . . . . 26

We can describe the dynamic of an ensemble of atoms interacting with the cavity
field using a Bloch sphere picture. The cavity field is represented by an axis C̆
about which the Bloch vector rotates at a rate proportional to g|C̆|. As pictured,
the rotation is leading to collective decay of the atoms from |ei to |gi. . . . . . . . . 27

2.5

With the substitutions given here and in the text, the dynamics of an ensemble ofatoms interacting with the mode of a resonant cavity can be mapped onto the motionof a damped pendulum under gravity. The angular velocity of the pendulum takes
the role of the cavity field while the decay of photons from the cavity is equivalent toviscous damping of the motion of the pendulum. The length of the pendulum scalesas (g 2 J)−1 , which leads to faster dynamics for many atoms or large coupling g.

. . 29xv
3.1

A simplified atom loading sequence. Atoms begin as a roughly 500 C collimatedbeam exiting the oven. Some of the atoms are slowed to around 50 m/s by a Zeeman
slower, then redirected by two 2D MOTs into the main experimental chamber. They
are then further cooled and spatially compressed by a 3D MOT, which we typicallyload for 500 ms. The Zeeman slower and both MOT stages also use 461 nm (blue)
light. The atoms are then transferred from the blue MOT to a narrow-line red MOTover roughly 50 ms. The magnetic field gradient (represented by the slope of the blackline) is reduced by a factor of roughly ten during this step. An asymmetric sawtooth
frequency sweep applied to the red MOT beams greatly improves the efficiency ofthis transfer (See Chapter 10). Atoms are further cooled, and loaded into the opticallattice with a typical temperature of around 10 µK.

3.2. . . . . . . . . . . . . . . . . . 32

Pictures of the experiment from July 2017, to give a sense of scale. Upper left: fourbore reference cavity, with associated optics. Upper right: main vacuum chamber
end of experiment. Lower left: the experiment spans two optics tables. Vacuum
system is in the foreground, with lasers and laser stabilization in background. Lowerright: enclosure houses all red and infrared lasers.

3.3. . . . . . . . . . . . . . . . . . . 34

The vacuum system consists of three main sections: the atom source, which contains an oven, Zeeman slower, and two 2D MOTs, the science chamber, which contains the optical cavity, and the pumping appendage. The system is pumped by a
75 liter/second ion pump and a titanium sublimation pump.

3.4. . . . . . . . . . . . . 35

Closeup of MOT and bias coils. MOT coils (large circular coils) are wound fromhollow square wire through which water flows. The vertical bias coils are wound
onto the MOT coils for support and cooling. The two sets of horizontal bias coils arewound around square frames. Additional freely wound coil in the foreground creates
an additional gradient along the cavity axis, but is not typically used. . . . . . . . . 37

xvi
3.5

Left: diagram of circuit used to control current through MOT coils. Right: actual
circuit used to control current through MOT coils. All high-power circuit elements
are mounted to a water-cooled plate. The MOSFET is the large white component
in the top of the image. The shunt resistor is the large black component below it,
and the diodes are below that. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6

Relevant energy levels of strontium. Level structure of 88 Sr, which lacks nuclear
spin and therefore hyperfine structure, are denoted with thick lines. When present,
additional states resulting from the hyperfine structure of 87 Sr are shown with thinlines. Zeeman sublevels are neglected. . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7

The 461 nm laser system consists of two master diode lasers and three slave diodes.
The frequencies of the two diode lasers are stabilized to a polarization-rotation spectroscopy setup, and by a beatnote lock between the two lasers. The three slave
diodes provide power for the Zeeman slower, 2D MOTs, and 3D MOTs respectively.

3.8

42

Closeup of AOsense source, with optics for Zeeman slower (far left) and 2D MOTs.
The thin tube with bellows visible in the lower left corner feeds atoms into the main
vacuum chamber.

3.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

689 nm laser system. An interference-filter ECDL from AOSense provides the masterlight for our 689 nm laser system. The laser is stabilized to our stable reference
cavity by a Pound-Drever-Hall lock on a sideband that provides a tunable offset.
Power for the MOT beams is increased using a tapered amplifier (TA). An AOM
both switches the cooling light on and off, and allows us to sweep its frequency in asawtooth ramp for efficient cooling. The 689 nm laser also provides optical pumping
for state preparation, among other functions. . . . . . . . . . . . . . . . . . . . . . . 47

xvii
3.10 Fiber-noise cancellation system for clock reference light. This architecture involves
two AOMs — one in the Ye lab (AOM1) and the other in the Thompson lab (AOM2).
Light that is deflected by AOM2 (giving it a frequency offset f ) is reflected off
the back mirror of the science cavity, and sent back to the Ye lab. This light isoverlapped with light that has not gone through the long fiber, and detected on afast photodiode. The resulting beatnote (near 2f in frequency) is used to lock thefrequency of a voltage controlled oscillator (VCO). The output of the VCO is divided
by two and used to drive AOM1 at a frequency near f . Any noise present in the fiber
leads to a perturbation of the frequency of the beatnote. Because we use opposite
diffraction orders on the two AOMs, this frequency offset is cancelled by AOM1.
Compared to simpler configurations that use a single AOM, this configuration has
the advantage of being immune to unwanted reflections before AOM2.

. . . . . . . 49

3.11 Transfer cavity from Stable Lasers Systems with four sets of mirrors sharing thesame spacer.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.12 Optical cavity assembly and vibration isolation. The cavity consists of two mirrors
separated by 4 cm. The axis if the cavity is horizontal. The mirrors are mounted
to piezoelectric actuators (PZTs), which allow for control of the cavity length. To
decouple the cavity from acoustic vibrations, the cavity spacer sits on viton balls on
top of a support block, which also sits on viton balls.

. . . . . . . . . . . . . . . . . 56

xviii3.13 Mirror assembly. The cavity mirrors are mounted to piezo actuator tubes (PZTs)
by an intermediate macor hat. This configuration allowed us to recess the mirrors
into the cavity spacer in order to avoid coating them with strontium, and provided
an easy way to align the mirrors. The two holes at either end of the cavity spacer
in which the mirrors sit were drilled at the same time, ensuring that they share acommon axis. All other alignments are referenced to those holes. All connections are
made by very thin beads of Torrseal epoxy. Electrical connections to the inner andouter surfaces of the PZT tubes are made by custom clips (not shown) that attach
through a slit in the macor hat (bottom, in image).
4.1. . . . . . . . . . . . . . . . . . 57(a) Experimental diagram. 88 Sr atoms are confined by 1D optical lattice in highfinesse cavity. π polarized probe light is coupled through cavity and is detected ona single-photon counting module. This probe interacts only with the 1 S0 to 3 P1 ,mJ = 0 transition. A magnetic field B shifts the mJ = ±1 Zeeman sub-levels by±15 MHz. (b) Relevant level structure of 88 Sr, showing dipole forbidden transitions
at 689 nm and 698 nm, and dipole allowed transition at 461 nm. . . . . . . . . . . . 61

4.2

Observation of the collective strong coupling regime on the forbidden 1 S0 to 3 P1transition. (a) We characterize the vacuum Rabi Splitting by recording transmitted
power as probe frequency is swept. Traces correspond to different cavity detunings,
δc . (b) Detected fluorescence versus vacuum Rabi splitting Ω. (c) Fitted linewidths
κ0± of the two resonances at ω+ (red) and ω− (blue) versus δc , with prediction basedon known atomic linewidth γ, measured Ω and cavity linewidth κ. . . . . . . . . . . 64

4.3(a) To probe the Vacuum Rabi Splitting, two probe tones at ωp± are simultaneously
swept across the two normal mode resonances at ω± while total transmitted power
is recorded. (b) The noise power spectra relative to photon shot noise for cavity
probe with a single tone (red trace) and two tones (black trace), demonstrate noise
cancellation of two-tone probing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

xix
4.4

(a) Black points show measurement noise relative to hypothetical projection noise inadjacent windows R, versus length of a measurement window. The lowest two points
represent R = −2.40.9
1.1 dB. Red line shows the the number of scattered photons per
atom ms in a single measurement window. (b) Fractional change in Ω2 = 4g 2 N
due to additional sweep  ≡ (Ω20 − Ω2s )/Ω20 , versus Mt , the number of photons in
the additional sweep. Ωs and Ω0 are the measured vacuum Rabi splittings with andwithout the additional scattering sweep. Solid red line is fit to data, with shaded
region representing 1 sigma error on the fit. Dashed line represents a prediction
based on the theoretically predicted free space scattering. The number of photons
used in two typical 100 µs windows is represented by the thickness of the left axis. . 69

5.1(a) Simplified experimental diagram. The system is probed with horizontally polarized probe light. The light can be coherently absorbed by the atoms (brown ovals)
and reemitted into the cavity at collective vacuum Rabi frequency Ω. The transmitted power is detected on a single photon counting module (SPCM). A magnetic
field is applied along the vertical direction. (b) The atomic energy level diagram of
the ground 1 S0 and excited states 3 P1 |mj i with quantization axis q̂ parallel to the~ This is the viewpoint adopted in this letter. Here, theapplied magnetic field B.
applied magnetic field creates a Zeeman splitting ∆ between the excited states |±1i.
Both of these transitions interact equally with the horizontally polarized light inside√
the cavity with collectively enhanced Rabi frequency Ω/ 2. The transmitted probe
light is measured as the probe’s detuning δp is swept. The |0i state is shown, but it
does not interact with the horizontally polarized cavity-field. . . . . . . . . . . . . . 75

xx
5.2

(a) The transmitted power through the cavity versus the probe detuning δp , withδc = 0. Each trace was taken for different applied magnetic fields, creating different
Zeeman splittings ∆ labeled on the vertical. The central red trace is taken for ∆ = 0
and displays a collective vacuum Rabi splitting Ω/2π = 5 MHz. When a magnetic
field is applied perpendicular to the probe polarization, inducing a Zeeman splitting∆, a new transmission feature appears in between the two original resonances of thevacuum Rabi splitting. (b) Linearized theory showing the power PT and phase ψ ofthe transmitted light, plotted here for Ω/2π = 5 MHz and ∆/2π = 1 MHz.

5.3

. . . . . 76(a) The measured linewidth of the central MIT transmission feature versus the induced Zeeman splitting between excited states. The traces are taken for three different collective vacuum Rabi frequencies Ω/2π = 4.6(5) (red), 10(1) (blue), and
16(1) (green) MHz, with values set by changing the total atom number N . The
upper dashed line is the empty cavity’s linewidth κ, and the lower dashed line isthe atomic transition’s linewidth γ. The minimum observed linewidth was 11 kHz.
The shaded regions are no-free parameter predictions from the linearized model introduced in the text, indicating the ±1 standard deviation uncertainty bands basedon independent measurements of Ω. (b) The measured peak transmitted power ofthe central MIT transmission feature for the same collective Rabi frequencies. Here,the transmitted power is normalized to the peak transmitted power when the cavity
is empty. Again the shaded regions indicate the ±1 standard deviation uncertainty
bands for the predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4

The pulling coefficient P versus Zeeman splitting ∆ for several collective vacuum
Rabi frequencies Ω/2π = 5(1) (red), 10(1) (blue), and 17(1) (green) MHz. The
prediction from the linearized theory is shown with ±1 standard deviation bands. . . 80

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5.5

In the ‘EIT picture’, the quantization axis is perpendicular the magnetic field andand parallel to the probe light’s polarization. (left) The cavity now couples theground and excited Zeeman state |mj⊥ = 0i. The magnetic field couples |0⊥ i toneighboring excited Zeeman states |mj⊥ = ±1i. (center) Equivalently, the magnetic
field couples to a symmetric combination of Zeeman states |Si while the antisymmetric combination |Di does not participate. (right) For comparison, a traditional
Raman EIT system is shown for comparison with the equivalent coupling strengths
labeled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.1

Energy level and experimental diagrams. The lasing transition is the 7.5 kHz wide
3 P , m = 0 to 1 S transition in 88 Sr. A magnetic field B is oriented perpendicular10J

to the cavity axis, and pump lasers at 688 and 689 nm are applied with directions
and polarizations as shown. Collectively enhanced emission into the TEM00 modeof the cavity is collected in an optical fiber and sent to one of various detectors.
6.2. . 86(a) Laser power emitted from the cavity during a collectively enhanced pulse, with
N = 36k atoms. The atoms are optically pumped into |ei at t = 0. (b) Numericalsimulation of (a) with no free parameters. (c) The power emitted from the cavity
during continuously repumped lasing with N = 60k atoms. Each atom emits around35 photons into the cavity before repump-induced heating causes laser operation tocease. For scale, 1 nW corresponds to 3500 intracavity photons. (d) Repumping
scheme. Atoms are incoherently repumped through 3 P1 , mJ = −1 to 3 S1 , where
they decay into the 3 P manifold. Atoms that fall into 3 P0 or 3 P2 are then repumped
by additional lasers via 3 S1 . For display purposes, 3 P0,2 and 1 P1 states are shownat arbitrary vertical positions.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xxii
6.3

Threshold behavior of laser. Laser output power in quasi steady state is plotted
0 = 0, (red points) and added
versus repumping rate w for no added broadening γ⊥
0 ' 2π × 3 MHz  γ, κ, black points). Atomic decay rate γ isbroadening, (2γ⊥displayed as vertical blue line, with shaded blue region representing uncertainty incalibration of w. For both conditions, the measured threshold repumping rate wt is
consistent with wt = γ.
6.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90(a) Pulling coefficient P versus the total homogeneous broadening of the lasing transition, along with prediction (blue). The vertical line at 160 kHz represents the cavity
decay rate κ. The inset displays the left five points on a linear scale. (b) Averaged
heterodyne power spectral density (PSD) between output light and 689 nm pumplaser, with recentering of each individual trial before averaging. Lorentzian (Gaussian) fits, shown as red (blue) lines indicate FWHM linewidths of 6.0(3) (4.7(3))
kHz.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

xxiii
7.1(a) The atoms can be represented by a collective Bloch vector on a Bloch sphere.
√
The cavity field C ∝ Mc , where Mc is the number of photons in the cavity, isrepresented by a vector that lies in the equatorial plane perpendicular to the atomic
coherence J⊥ (when the cavity is on resonance with the atomic transition). The
Bloch vector rotates about the axis that represents the cavity field. (b) The Blochvector behaves like a highly damped pendulum that starts inverted at the north poleof the Bloch sphere (all atoms in |ei). Quantum fluctuations disturb the system fromits unstable equilibrium position, causing the Bloch vector to swing down the Bloch
sphere, emitting peak radiation at the equator and ultimately relaxing to the southpole (all atoms in |gi) as inversion is lost. The radiated electric field (red trace) is
proportional to the perpendicular projection of the Bloch vector, J⊥ , which at its
peak is proportional to N . The radiated power (black trace) is proportional to thesquare of the radiated electric field, and at its peak is therefore proportional to N 2 .
This is one way to understand the origin of the collective enhancement in emission
rate. The black output power trace on the projection plane is actual data.

7.2. . . . . 96

Spontaneously generated superradiant pulses. Representative single time traces ofphoton output rate R for pulses at different atom number N ≈ 100 × 103 (green),
125×103 (blue), 150×103 (red), 200×103 (black). The equivalent average intracavity
photon number is calculated on the right as Mc = R/κ. . . . . . . . . . . . . . . . . 98

xxiv
7.3

(a) Peak photon output rate, Rpeak , versus initial total atom number. The blackline is a quadratic fit to the output power. We observe a horizontal offset in the fit,indicating a threshold atom number Nt (black vertical line in b,c,d). The thresholdresults from decoherence and atom loss and is a signature of lasing that would not
be present for single-atom emission. (b) FWHM pulse duration tw (blue) and delayof peak power td (red) versus initial total atom number. The blue line is a fit to
the predicted functional form for the pulse duration, with Nt determined from thefit to Rpeak . The dashed red line is a fit to the pulse delay, assuming threshold isdue to atomic homogeneous broadening without atom loss. The solid red line is a fit
to the pulse delay, assuming the threshold is set only by atom loss from the lattice.
(c) The ratio of emitted photons Mtot to the number of atoms in excess of threshold
Nx , plotted versus atom number. The dashed line is the average ratio, showing that
48% of the atoms in excess of threshold participate in the superradiant pulse, largely
independent of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.4

Seeded superradiant pulses. (a) By terminating adiabatic transfer with atoms in asuperposition of |ei and |gi, we seed coherence in the atomic ensemble. Here, theatomic Bloch vector is rotated to just above the equator (i.e. a small amount ofinitial inversion.) Seeding leads to the immediate onset of superradiant emission, in
contrast to the non-seeded pulses shown in Fig. 7.2 for which quantum noise seeds thecoherence. (b) Peak output power for seeded pulses exhibits N 2 scaling. In contrast
to results of Figure 2, seeded pulses exhibit a peak photon output rate consistent
with no threshold: Nt = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xxv
7.5

Lasing on multiple transitions at the same time leads to beating in output power(a) Atoms prepared in the mf = 9/2 and mf = 7/2 Zeeman sub-levels radiate
simultaneously into the cavity. (b) Interference between the electric fields radiatedfrom the two transitions leads to a modulation of the output power. An average of
20 time-traces is plotted, showing that the modulation has the same phase between
trials of the experiment. (c) We compute a Fourier transform of an averaged time
trace, showing a peak at the frequency splitting of adjacent Zeeman sub-levels (≈ 200
Hz). The peak at ≈ 400 Hz indicates that a smaller number of atoms have been
prepared in mf = 5/2. A Lorentzian fit to the average power spectrum (red line)
returns a FWHM linewidth of 11 Hz, due to the finite length of the pulse. The insetof (b) shows the center frequency of the Lorentzian fit versus applied magnetic field.
The blue line shows the predicted splitting between adjacent Zeeman sub-levels.

8.1. . 104

Superradiant pulses in the time and frequency domains for different configurationsof spin states. (a) We can populate a single nuclear spin state to observe lasing
on a single transition. In the time domain (center column) this leads to a simple
superradiant pulse. In the frequency domain (right column), we see a single peak.
(b) When two nuclear spin states are populated, we observe a beating of the twotransitions in the time domain, and two peaks in the frequency domain. (c) When wepopulate all ten nuclear spin states, we observe an interesting interference pattern inthe time domain from simultaneous lasing on the ten transitions. In the frequency
domain, we observe peaks associated with the different transitions, with heights
determined by the Clebsch Gordan coefficients of the different transitions.

. . . . . 113

xxvi
8.2

Stability of superradiant pulses expressed as a fractional Allan deviation σ(τ ) (seetext for definition). Two stretched (mf = ±9/2) transitions lase simultaneously.
The black points correspond to the average frequency between the two transitions,while the red points correspond to half the difference. Lines represent fits of the√small-τ points to σ(τ ) ∝ 1/ τ , the expected scaling for uncorrelated noise between
measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.3

Sensitivity of laser output frequency ω` to changes in the cavity detuning ∆c fordifferent initial atomic inversions. Red points correspond to an inversion empiricallytuned to minimize sensitivity to cavity detuning. Black and blue points correspondto slightly higher and lower inversions. The slope of the red points is 2 × 10−6 . . . . 119

8.4

A magnetic-field insensitive configuration. The frequencies of the mf = 9/2 andmf = −9/2 transitions (labelled fupper and flower here) respond in equal and opposite
manners to an applied magnetic field, making the average frequency of the two peaks
highly insensitive to magnetic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.5

Frequency difference between our laser and Ye lab optical lattice clock frequency at
different lattice laser detunings and power levels. Different colors of points correspond to different lattice depths. A simultaneous fit (see text) to all data returns
a nominal frequency offset consistent with zero, as represented by the horizontal
dashed line, with the uncertainty of the fit represented by the blue band. The vertical dashed line represents the fitted magic wavelength (relative to the expected value
based on our wavemeter) with fit uncertainty given by the green band.

8.6. . . . . . . 124

Measured shift to superradiant output frequency associated with density-dependent
collisions versus fraction of atoms prepared in 3 P0 at beginning of superradiant pulse.
Data taken with roughly 5 × 105 atoms in lattice at beginning of superradiance.

. . 126

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8.7

Calibration of DC Stark shifts. Measured frequency is plotted against voltage applied to electrode on large window of vacuum chamber. Frequency shifts due tobackground fields in the direction of the applied field from the electrode manifest
as an offset between the minimum of the parabola and its value at zero electrodevoltage. Shifts here are consistent with zero.

9.1. . . . . . . . . . . . . . . . . . . . . . 128(a) The cavity field mediates interactions between the effective spins corresponding
to the optical 1 S0 (g)- 3 P0 (e) clock transition. This leads to both a collectively
enhanced decay at rate Γ which leads to light leaking out of the cavity, and tointeresting unitary dynamics that lead to shifts in the frequency of the emitted lightω` . (b) The resonance of the optical cavity is detuned from the atomic transition bya frequency ∆. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9.2

Illustration of the effects of the two terms in H = ~χ[Jˆ2 − Jˆz2 ] (a) The χJˆz2 termcauses a twisting of the Bloch sphere, referred to as one-axis-twisting (OAT). A
quantum noise distribution (represented by the dashed circle) will be deformed bythis twisting, leading to a potentially useful entangled state. (b) The χJˆ2 term leads
to a so-called many-body gap protection effect, where dephasing of the atoms due
to inhomogeneous noise sources is supressed.

9.3. . . . . . . . . . . . . . . . . . . . . . 133

A Bloch sphere representation of the atomic ensemble radiating into a detuned cavity.
The phase of the cavity field C (φC ) is locked to the phase of the atomic coherence
J + (φ) up to a fixed offset φr = arctan(2∆/κ) + π/2. The Bloch vector precesses
about the cavity field axis, which is the source of both collectively enhanced decay
and of shifts in the output frequency ω` .

. . . . . . . . . . . . . . . . . . . . . . . . 134

xxviii
9.4(a) When the cavity is on resonance with the atomic transition (∆ = 0) the cavity
field C is perpendicular to J + . This causes a rotation of the Bloch vector downwards,
and is the physical origin of superradiant emission. (b) When the cavity is tuned
far off resonance from the atomic transition (∆  κ), the cavity field C is nearlyaligned with J + . Now, the rotation of the Bloch vector about C is primarily sideways.
Because C follows the Bloch vector as it rotates, this leads to a frequency shift of
the Bloch vector precession, and of the emitted light.

9.5. . . . . . . . . . . . . . . . . 135

Shifts in frequency of emitted light ω` versus cavity detuning ∆, showing expected
dispersive behavior. Blue line is a fit with cavity linewidth held to its independently
measured value.

9.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Main: Pulling coefficient P (black points) and atomic inversion Jz (blue points)
versus state prep pulse duration. Blue line is a fit to the blue points with the
expected functional form of inversion versus state preparation angle, allowing for a
fixed fraction of atoms that remain in the ground state. Black line is predicted value
for pulling coefficient given the fit parameters of the blue line, up to an overall scale
factor. Inset: the same data can be viewed by plotting P against the value of Jz0 (0)inferred from the blue fit line, showing the expected linear scaling. . . . . . . . . . . 139

9.7

Illustration of the effects of the χJ 2 term. We consider two sub-ensembles, represented by the green and orange Bloch vectors. (a) When the two sub-ensembles havethe same transition frequency, the χJ 2 interactions simply lead to a precession about
the total Bloch vector J (blue arrow). (b) When the transition frequencies of thetwo sub-ensembles are split by δ, the two Bloch vectors rotate in opposite directions
along the equator when the χJ 2 interactions are absent. (c) When both δ and χ are
finite, and χN  δ, the two sub-ensembles follow small circles and never develop alarge opening angle. This is the origin of the many-body gap protection.

. . . . . . 141

xxix
9.8

(a) We introduce controlled inhomogeneity by simultaneously populating the mF =±9/2 states and applying a magnetic field to split the Zeeman sublevels by a frequency δ. (b) Output power spectra for lasing on the two transitions, with atoms
prepared near the south pole of the Bloch sphere.

In the presence of a large

detuning δ, the two branches correspond to light emitted independently on the
mF = ±9/2 transitions. When the cavity mode is detuned from atomic resonanceby ∆/2π = 150 kHz ' κ/2π (top row), we experimentally observe an asymmetry
in the two branches (left panel). This asymmetry is reversed for opposite cavity
detuning (middle row). A simulation with the full Hamiltonian Ĥ = ~χ[Jˆ2 − Jˆz2 ]qualitatively reproduces these features (center column), while a simulation that includes only Ĥ = ~χJˆz2 does not (right column). This indicates the important role ofthe term χJ 2 , which is associated with many-body gap protection. Power spectra at
fixed δ (bottom row) highlight this comparison,taken at a detuning δ indicated by
the horizontal green lines. (c) Cartoon projections of the Bloch vectors associatedwith the two spin ensembles onto the xy plane for pure OAT Hamiltonian (upper)and full Hamiltonian (lower) in reference frame of total Bloch vector. For pure OAT,
the two ensembles precess independently, leading to a large opening angle. For fulldynamics, the two Bloch vectors synchronize, preventing a large opening angle fromdeveloping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.9(a) We introduce controlled, reversible dephasing by illuminating the atoms withlight near resonance with the 1 S0 to 3 P1 transition. The beam is offset from centerwith the atomic cloud, so the atoms experience a spatially varying intensity, andtherefore a spatially varying light shift. (b) Applying the beam for roughly 1 ms
scrambles the phases of the atoms, reducing atomic coherence and superradiantemission. Applying the laser again, this time with the opposite detuning from the
1 S to 3 P transition unscrambles the atomic phases, restoring atomic coherence and01superradiance.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

xxx
9.10 We measure the change in the difference between optical phase measurements before
dephasing (ψ1 ) and after rephasing (ψ2 ), as the cavity is toggled between opposite
detunings ∆/2π = ±30 kHz: ∆ψ = (ψ2 − ψ1 )|30 kHz − (ψ2 − ψ1 )|−30 kHz . Red (black)points correspond to trials where the atoms were dephased (not dephased) during
Thold , corresponding to the bright (dark) normal modes. ∆ψ shows no dependenceon Thold for the dephased case, and a phase shift that scales linearly with Thold when
the atoms are not dephased, confirming a frequency shift between the bright anddark portions of the atomic coherence.

. . . . . . . . . . . . . . . . . . . . . . . . . 147

9.11 The connection between our observed phase shift and an energy gap can be understood explicitly by considering the effect of dephasing in a Dicke basis for states
prepared near Jz = −N/2, subject to a pure χJ 2 Hamiltonian. Atoms begin in asuperposition of maximally symmetric states with J = N/2 (far left states). Dephasing (red arrows) transfers the atomic ensemble to states of minimal coherence for agiven Jz . Because the difference in J between the states of maximal and minimal
coherence for a given Jz is Jz + N/2 for states with Jz ' −N/2, the energy difference
between these states is approximately χ(Jz +N/2). When coherence is restored in therephasing step (blue arrows), the different Jz states will have accumulated a phase
proportional to this energy shift. Because a linear phase shift between adjacent Jz
states corresponds to an azimuthal rotation of the Bloch vector, the measured phaseshifts of (b) provide a direct measurement of the energy splitting between manifolds
of different J, which is the origin of many-body gap protection.

. . . . . . . . . . . 148

9.12 We may vary the power in the dephasing beams to only partially dephase the atoms.
We can infer the residual coherence from the amplitude of the superradiant emission.
We observe that the pulling coefficient is uneffected by this process, maintaining thesame value as the fully coherent case (to within our experimental uncertainty). . . . 149

xxxi
10.1 Illustration of Landau-Zener crossing in a dressed state picture. Black lines represent
energy eigenstates in the presence of drive coupling Ω. An atom that begins in |gi
when the drive has a large negative detuning δd will be transferred to |ei as the drive
sweeps to large positive detuning (red arrows). . . . . . . . . . . . . . . . . . . . . . 152
10.2 Illustration of basic cooling mechanism. (a) Atoms moving in one dimension withvelocity v are illuminated by counter-propagating laser beams. (b) Atoms have asingle relevant atomic transition at frequency ωa . In our case, this transition is dipoleforbidden, so |ei has a relatvely long lifetime. (c) Representation of frequencies in lab
frame. Because of Doppler shifts resulting from atomic motion, the co-propagating(counter-propagating) laser beams are resonant with |gi to |ei transition when thelaser frequency ω = ωa − kv (ωa + kv). Because ω increases with time, the counterpropagating laser sweeps over the transition before the co-propagating laser. If theatom starts in |gi, the counter-propagating laser transfers the atom from |gi to |ei
as is sweeps over resonance, and the co-propagating laser transfers the atom back to|gi. This results in the transfer of two photon recoils of momenta to the atom in thecorrect direction to cool.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

10.3 Eigenstate diagram of coupled atom-light modes versus detuning of the lasers fromatomic resonance. Uncoupled eigenstates are labeled as |m, n, ai (defined in text).
Atoms start at point (1), then are adiabatically transferred to (2) by the laser sweep.
When the laser detuning jumps, they non-adiabatically project to (3), having given
up 2~k of momentum by transfering a photon from the left-going to right-goingmodes. Higher-order crossings, which lead to the exchange of multiple photons
become significant when kv ∼ Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.4 Simulated trajectories for atoms with different initial momenta. Produced by asimulation based on a Monte Carlo wave function trajectory method written by
Murray Holland and John Bartolotta. . . . . . . . . . . . . . . . . . . . . . . . . . . 161

xxxii10.5 Cooling into a moving reference frame. (a) We apply a frequency offset in the lab
reference frame between the two beams, which creates a moving reference frame inwhich the beams have equal frequency. (b) If laser frequency is swept upwards,(green points) the atoms settle into the moving reference frame, which has a velocity
indicated by the dashed black line. If frequency is swept downwards (orange points),
the atoms accelerate in the other direction, without cooling. If the frequency sweep
is symmetric (purple points), atoms experience a small acceleration due to radiationpressure imbalance.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

10.6 Acceleration versus polarization angle between two beams. Acceleration is enhanced
when the polarizations of the two beams are aligned. The blue line represents the
maximum acceleration for a saturated two-level atom without stimulated emission.
The dashed blue line represents the measured acceleration with a single beam. Black
points represent acceleration with both beams applied. The dashed black line represents the acceleration that two photon recoils per sweep of the laser frequency would
produce.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

10.7 Measured acceleration versus the sweep center frequency. As long as the lasers sweep
over both atomic resonance (blue region), the acceleration is roughly constant. The
sweep range here was 4 MHz.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

10.8 Force versus velocity of moving reference frame, as set by the relative detuning/Delta between counterpropagating cooling beams. Red points correspond to an
upwards frequency sweep, while black points correspond to a downwards frequency
sweep. Blue line represents half of the sum of the red and black points, while thegreen line represents half of the difference. We attribute the contributions to theacceleration represented by the blue line to transient effects not directly associatedwith our cooling mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

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

. . . . . . . . . . . 167

10.10Temperature versus Rabi frequency for a sweep range of 7 MHz and repetition rateof 15 kHz. Red line represents the results of our simulations, and black points areexperimental data. The red band indicates uncertainty of prediction due to statisticaluncertainty in simulation results, experimental calibration of Ω and variations in Rabi
frequency during the sweep. The vertical line at 130 kHz represents the approximate
point at which we expect adiabaticity to break down: Ω2 = α.

. . . . . . . . . . . . 169

10.11Mechanism for realizing a confining force with swept lasers. Counter-propagating
beams with opposite circular polarizations illuminate the atoms in the presence ofa magnetic field gradient, as in a standard MOT (left). The atoms are located near
the zero of the magnetic field. The MOT lasers are swept upwards in frequency
in a sawtooth manner, as before. In the simplest system, each laser connects theground state to a unique excited state, which shift in opposite directions in an appliedmagnetic field. The lasers sweep over the transition to the state that is shifted tolower energy first, causing acceleration in the direction of the beam that addresses
that state, and depleting the population in the ground state (right). Because theexcited state is relatively long lived, the atom is less likely to be in the ground state
when the other laser sweeps over its resonance. This leads to an imbalance in thenumber of photons absorbed from each beam, and causes a magnetic field dependent
force that can be used to spatially confine the atoms near a magnetic field zero.

. . 171

xxxiv
10.12Atomic acceleration in the configuration described in Fig. 10.11. The direction of
the force switches sign depending on the sign of the magnetic field applied along thedirection of the counter-propagating beams, and depending on the direction of thesweep (blue and red points). This is the necessary criteria for generating a restoring
force toward a zero in a spatially dependent magnetic field. A symmetric frequency
sweep generates a smaller force, which does not switch sign with the magnetic field(black points), likely due to an intensity imbalance between the two beams.

. . . . 173

10.13Fraction of atoms transferred from the broad-line MOT at 461 nm to the narrow-line
MOT at 689 nm versus center frequency of the sweep for 87 Sr and 88 Sr for different
sweep profiles. Red points represent an upwards-swept asymmetric sawtooth ramp,
which we refer to as the SWAP MOT configuration. Black points represent the morecommonly used symmetric triangle ramp. Blue points represent a downwards-swept
asymmetric sawtooth ramp, which captures almost no atoms.

. . . . . . . . . . . . 174

11.1 Optical lattice clock and atom interferometer detectors for gravitational waves. Each
type of proposal relies on two ensembles of atoms, one in each satellite. Laserstransmitted between the satellites encode phase shifts due to changing optical path
lengths caused by a gravitational wave. The key difference between clock and atominterferometer proposals is that in clocks, atomic recoils due to the momentum ofabsorbed/emitted photons are supressed by tightly confining the atoms, while itatom interferometers the atoms are free to recoil.

. . . . . . . . . . . . . . . . . . . 180

xxxv
11.2 A toy model for gravitational wave detection. (a) Two atoms (a and b) are addressed by a single laser resonant with a transition between the ground state |gi anda long-lived excited state |ei. A phase modulator mimics the effect of a gravitational
wave by modifying the optical path length between the two atoms. (b) For tightly
confined atoms (clocks), a simple Ramsey sequence can be used to detect changes
in optical path length. A first π/2 pulse imprints the laser phase on the two atoms
when the path length is shifted by L1 , and a second π/2 pulse acts when the path
length is shifted by L2 . Random variations in the laser phase (φ1 and φ2 ) are common to the two atoms, while phase shifts resulting from changes in path length arenot. The signal from the phase modulator manifests as phase shift Φs = k(L2 − L1 )between the excited state probability oscillations of the two atoms. (c) For unconfined atoms (atom interferometers), transitions between |gi and |ei are accompanied
by momentum kicks, which necessitate an additional laser π pulse in the middle ofthe sequence. The phase shift between excited state oscillations of the two atoms is
now Φs = k(L3 − 2L2 + L1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
11.3 LMT-like enhanement with confined atoms. (a) A toy model to demonstrate enhancement mechanism. Atoms are addressed by counter-propagating lasers, bothof which pass through the phase modulator that sits between the atoms. (b) Twoblocks of M pairs of π pulses (blue boxes) are inserted between the π/2 pulses ofthe standard Ramsey sequence. Each pair of π pulses contains a pulse originatingfrom each direction. The duration of each block is assumed here to be short relative
to the GW period, while a long evolution time (grey box) between the two blocks
may be comparable to the GW period. These additional pulses lead to a factor of
4M + 1 enhancement in the signal phase Φs compared to simple Ramsey sequence.

187

xxxvi11.4 Use of three-state system to allow atoms to stay in ground state for much of evolution
time. (a) Our procedure utilizes two ground states, |g1 i and |g2 i and a single longlived optically excited state |ei. |g1 i and |g2 i can independently be coupled to |eiusing laser pulses with different frequency and/or polarization (red and blue arrows).
(b) After preparing an atomic superposition of |g1 i and |ei using a π/2 pulse fromthe left (atoms begin in |g1 i), the portion of the atoms in |ei is transferred to |g2 iusing a π pulse from the right. The atoms are then in a superposition of |g1 i and|g2 i with phase dictated by the lasers and the path length L1 . The sequence ofpulses is reversed after a long evolution time, during which time the atoms are notsusceptible to excited state decay. The phase shift in the excited state probabilitiesof the two atoms Φs = 2k(L2 − L1 ) now encodes changes in the optical path lengththat occurred while the atoms were in the ground states.

. . . . . . . . . . . . . . . 191

12.1 A proposed architecture for spin squeezing in an optical lattice clock. Atoms aretightly confined in two dimensions by intersecting lattices supported by two crossed
optical cavities. One of these cavities is used to perform atom counting measurementson the 7.5 kHz linewidth 3 P1 transition. Clock rotations are performed by a 698 nm
laser applied in the plane defined by the two cavity modes, along which direction theatoms are tightly confined and Doppler-shift free.

. . . . . . . . . . . . . . . . . . . 199

12.2 A proposed architecture for a continuously loaded superradiant laser. Atoms emitted
from oven are slowed by a Zeeman slower, then captured in a 2D MOT formed on
the 461 nm transition. Some atoms that fall into 3 P2 are magnetically trapped andfall guided along the magnetic field zero. They are then depumped to the groundstate and cooled transversely with 689 nm light before falling into the mode of the
optical cavity. Here, they are loaded and cooled into a moving optical lattice, whichtransports them to the lasing region, where each atom emits up to one photon.

. . 202

Chapter 1

Introduction

The exquisite control of cold atoms, ions, and molecules in tabletop experiments has opened
new windows in our understanding of the universe at its most fundamental levels. This has occurred
through several primary avenues, notably precision measurement as well as quantum information
science and simulation.
Precision measurements have provided experimental values or bounds for fundamental anduniversal quantities with critical implications for our understanding of the most foundational aspects of physics. These quantities include the electron electric dipole moment [1, 2], the anomalous
magnetic moment of the electron and fine structure constant [3], and precision tests of key predictions of relativity [4, 5], to name a few. These studies complement and sometimes extend the results
obtained by more established, and far more expensive techniques such as high energy (accelerator)
physics.
The same kinds of tools can be applied in a completely different context, exploiting thequantum nature of the experimental platform provided by cold atoms, ions, and molecules. Theseexperiments provide direct experimental confirmation and great intuitive insight into some of the
puzzling aspects of quantum mechanics [6, 7]. The current cutting edge of this sub-field involves theuse of quantum systems for information processing and simulation tasks, and the related quest tounderstand the highly non-classical states that may arise in systems of large numbers of interacting
quantum particles. It is conceivable that the new tools for computation and simulation will upend
our current information and economic infrastructure [8], and would lead to a greater understanding

2of fascinating phenomena that arise in condensed matter systems [9].
This continuing bounty of insights and applications has been made possible by the everdeveloping tools used to prepare, manipulate, and detect useful states of quantum particles. Theseadvances include continuous improvements in laser cooling and trapping, as well as other methodsof creating low-entropy quantum systems such as evaporative cooling and quantum gas assembly[10, 11, 12, 13]. Advances in techniques for the coherent manipulation of the quantum states ofparticles using microwave and laser radiation have provided crucial tools for enabling precision
measurement and information processing, and advances in detection techniques have given experimentalists the capability to observe increasingly interesting states [14, 15, 16]. Finally, creating
controllable interactions between the quantum particles provides the potential to study complex
quantum systems and for the enhancement of precision measurements [17, 18, 19].
In this thesis, I present a collection of new tools for further advancing precision measurement
and quantum science, based on narrow-linewidth optical transitions. These tools include a nondestructive atom counting technique that may enhance the performance of optical lattice clocks
and other quantum sensors [20], a novel form of cavity-enhanced atomic spectroscopy appropriate
for laser frequency stabilization for portable sensing applications [21], a new form of laser coolingwith reduced reliance on spontaneous emission that may improve and extend the use of laser
cooling in new experimental systems [22], a method of enhancing the sensitivity of proposed atomic
gravitational wave detectors [23], cavity-mediated spin-spin interactions with implications both for
precision quantum sensors and for engineering interesting interactions between particles, and a newclass of laser (a “superradiant laser”) based on optical superradiance from narrow and ultra-narrow
linewidth transitions [24, 25]. As part of the superradiant laser project, we have demonstrated
the most precise active absolute frequency standard realized to date, providing an encouraging
indication that a superradiant laser may become a key experimental tool for precision measurementin the near future.
With the exception of the gravitational wave detection project (which was purely theoretical),this work was performed in a new experimental system that I began building in the fall of 2013. In3Figure 1.1: (a) Strontium has two valence electrons, whose spins can either be aligned (spin triplet)or anti-aligned (spin singlet). Photons are very bad at flipping spins, so we call transitions between
these two types of states “forbidden,” meaning that they occur slowly if at all. (b) A basic overview
of our experimental platform. Strontium atoms are cooled to roughly 10 µK, and tightly confinedin an optical lattice inside of an optical cavity. The wavelength of the lattice is carefully chosen so
as not to perturb atomic transition frequencies of interest — the “magic wavelength.” The atoms
interact with the cavity via one of several forbidden transitions.

this system, an ensemble of strontium atoms interact in a collective manner with an optical cavity
via narrow-linewidth optical transitions.
The important property of strontium for this work is the presence of two valence electrons.
The spins (intrinsic magnetic moments) of these two electrons can either be aligned in a configuration known as a “triplet” state, or anti-aligned in a configuration known as a “singlet” state (SeeFig. 1.1a). To go from a singlet state to a triplet state or vice-versa requires flipping the spin of oneof the electrons. Photons are very bad at doing this, so these transitions happen very slowly if atall, and are thus known as “forbidden transitions” or “dipole-forbidden transitions.” The work here
utilizes two forbidden transitions in atomic strontium: the somewhat forbidden 7.5 kHz linewidth
3P

1 to

1 S transition and the far more forbidden 1 mHz linewidth 3 P to 1 S clock transition. These
0
0
0

4
transition linewidths correspond to the inverse of the excited state lifetime, which is 20 µs for 3 P1
and 160 s for 3 P0 . Both of these transitions involve the absorption or emission of a red photon.
For scale, an “allowed” transition (which does not involve flipping a spin, or any other processes
that a photon cannot do) of this color would have a linewidth of order 10 MHz, or a decay rate one
thousand and ten billion times faster than the two forbidden transitions. These narrow-linewidthtransitions in strontium and similar elements have already proved very useful, and have enabled
the development of the most precise and accurate clocks to date [26, 27, 28, 29]. The tools that I
describe in this thesis have the potential to improve the performance of these clocks even further,and to extend the impact of forbidden optical transitions to new types of systems and sensors.
The key enabling technology for much of my work is the coupling of strontium atoms to an
optical cavity by these narrow linewidth transitions (See Fig. 1.1b). The optical cavity consistsof two highly reflective mirrors that light can bounce back and forth between many times. The
cavity provides distinct resonant optical modes, defined by their spatial profile and frequency, withwhich the atoms can interact through the exchange of photons. The strontium atoms are cooledusing laser light to a temperature of order 10−5 Kelvin, and suspended within the optical cavity
by an optical lattice — a standing wave of light that is resonant with the optical cavity. Atoms
are attracted to regions of high intensity of the lattice light, which pins the cold atoms in place
during the experiment, greatly reducing a variety of problems associated with atomic motion. The
wavelength of the lattice light is set near a “magic” value, for which it does not significantly perturb
the frequency of the narrow linewidth clock transition.
The study of atoms coupled to an optical cavity constitutes a field known as cavity quantum electrodynamics, or cavity QED. Historically, cavity QED with atoms has typically focusedon coupling the atoms to the cavity using either microwave transitions between Rydberg states[30], or strong allowed optical transitions [31]. The same kind of ideas have been extended tosuperconducting circuits in the microwave domain [32]. A key focus of my thesis work is the extension of cavity QED techniques into a new regime by instead coupling the atoms to the cavity
via narrow-linewidth optical atomic transitions. The slow interaction rates associated with these5narrow-linewidth transitions required substantial care in order for the atom-cavity interactions todominate over technical sources of decoherence.
Before beginning the construction of the strontium experiment, I contributed to several experiments that I will mention only briefly here. These experiments were performed on the preexisting
rubidium cavity QED apparatus with fellow graduate students Justin Bohnet, Josh Weiner and
Kevin Cox. The first was a spin-squeezing experiment where we demonstrated for the first time anorder of magnitude improvement in sensor performance (10 dB of directly observed spin squeezing) using entanglement generated from collective measurements of atoms inside an optical cavity[33]. In the second experiment, we demonstrated a technique in which we imparted controlled,
reversible disorder to an ensemble of atoms in order to suppress technical errors in coherent atomic
manipulations [34].
I summarize experiments performed in the strontium cavity QED system next, then discuss
two sets of additional tools that are enabled by narrow linewidth optical transitions but do not
involve an optical cavity — a new form of laser cooling that relies on narrow-linewidth transitions,and a method for enhancing the performance of proposed gravitational wave detectors.

1.1

Overview of Applications

1.1.1

Appliction to Clocks: Non-Destructive Atom Counting

The coupling of ensembles of atoms to a cavity via a narrow linewidth optical transition
holds promise for improving the performance of optical lattice clocks, which are currently themost precise sensors of time and frequency in existence [26, 27, 28, 29]. Like many clocks, opticallattice clocks rely on state-dependent atom number measurements to infer frequency. Typically,
these measurements are made through fluorescence detection, where atom number is inferred byscattering light off of an atomic transition. This technique has the advantage of simplicity and goodsignal-to-noise, but has the disadvantage that the scattered photons heat the atoms to the point
where a new ensemble must be prepared. During the time that new atoms are being prepared, the6frequency of the clock may wander, which can degrade its performance [35].
Another alternative approach to fluorescence detection is to couple the atoms to an opticalcavity, and count the atoms by measuring perturbations to the resonance frequency of the cavity.
In addition to excellent precision, this method has the advantage that it may be performed in a
non-destructive manner, eliminating the need to prepare a new ensemble of atoms, and reducing
problematic dead-time in the clock sequence. Further, by making collective measurements of atomic
population one can generate useful entanglement or “spin squeezing” in the ensemble, which canfurther improve the performance of the clock [36].
Proof-of-principle demonstrations of non-destructive atom counting and spin squeezing havebeen performed using dipole allowed transitions in alkali atoms coupled to optical cavities [14, 37,
33, 15, 17]. An outstanding goal is to apply these techniques to a state-of-the-art optical latticeclock based on an ultra-narrow optical transition in an alkaline-earth or alkaline-earth-like atomsuch as strontium or ytterbium. A promising path towards this goal is to couple the atoms to acavity via the 7.5 kHz linewidth dipole-forbidden transition to perform atom counting and spinsqueezing.
As a first application of the new strontium system, we demonstrated that this transition could
indeed be used to perform non-destructive atom counting at a level compatible with spin squeezing
in a system applicable to state-of-the-art optical clocks. We showed that we could count atoms with
precision compatible with quantum-limited clock operation, and inferred that the measurement was
sufficiently non-destructive to enable spin squeezing.

1.1.2

Appliction to Laser Stabilization: Magnetically Induced Transparency

We have also observed and characterized an interesting and potentially useful effect thatarises when an atom is coupled to the optical cavity by multiple narrow-linewidth transitions.
For most applications of cavity QED, each atom is viewed as a two-level system with a single
transition. By carefully preparing the initial state of the atoms interacting with a cavity mode ofwell-defined frequency and polarization, it is often possible to neglect additional atomic structure,

7which greatly simplifies the understanding of the system. In this work, we probe a cavity mode mode
whose polarization is othogonal to an applied magnetic field, and thus interacts with two atomic
transitions that share a ground state. This allows us to observe a feature that we call “magneticallyinduced transparency” (MIT) in analogy with a well-studied phenomena called electromagenticallyinduced transparency (EIT) [38, 39, 40]. This feature manifests as a spectrally narrow transmission
window through the cavity that appears when the applied magnetic field is sufficiently large. The
center frequency of this window has low sensitivity to changes in the cavity resonance frequency,
making it an appealing candidate for laser stabilization. Futher, the ability to tune the nature ofthis feature by changing the magnitude of the applied magnetic field could enable its use for storing
and retrieving photons in a manner similar to EIT [41, 42, 43, 44, 45].

1.1.3

Superradiance from Narrow Linewidth Optical Transitions

Much of my thesis focused on the study of superradiance from narrow-linewidth opticaltransitions, with the ultimate goal of creating a narrow linewidth superradiant laser. Superradiancerefers to a phenomena in which the rate that photons are emitted from an ensemble of atoms
greatly exceeds the sum of the rates at which photons would be emitted from the individual atoms.
Superradiance itself has been studied in detail as a curiousity for decades [46, 47, 48, 49]. Morerecently it has been proposed [50, 51] that one could take advantage of an ultra-narrow linewidth
clock transition like the one in strontium to realize a superradiant laser with linewidth of order 1mHz (surpassing the current records for laser stability [52]).
Like any laser, a superradiant laser would consist of a gain medium that sits within the modeof an optical buildup cavity. In contrast to a conventional laser, where the gain medium provides
broad-band amplification of power and the lasing frequency is set by the resonance frequency of the
optical cavity, a superradiant laser would operate in a regime where the gain medium is by far themost spectrally selective element in the system, and the optical cavity serves to enhance emission
in a comparitively broadband manner. The output frequency of the laser is then determined by thecenter frequency of the gain medium. Because cold atoms with narrow linewidth optical transitions

8
form a very narrow linewidth and stable gain medium, this could be a major advantage.
Today’s state-of-the-art stable lasers [53, 54, 52], which are used in state-of-the-art clocks[29, 27, 55, 56, 26] consist of a fairly standard laser whose frequency is stabilized [57] to theresonance of a highly engineered optical cavity, which consists of a spacer to which two mirrors
are stuck. The frequency of the laser is thus determined by the length of the optical cavity. Overthe last several decades, heroic efforts have been made to design cavities with very stable lengths[58]. Currently, the stability of optical cavities is limited by thermal brownian motion in thespacer, mirror substrates, and reflective coatings [59, 60]. To mitigate these effects, researchers
are currently developing cavities with crystaline spacers, substrates and reflective coatings that
operate at cryogenic temperatures [54, 52]. This is very hard, I am told. Reaching the regime
where these thermal fluctuations limit the performance of the laser also requires elaborate vibration
isolation techniques, which might make it difficult to use these lasers outside of a carefully controlledlaboratory environment.
A superradiant laser would offer an alternate, and perhaps complementary approach to creating a stable laser. Because the frequency of a superradiant laser is determined by the atomic
transition frequency rather than the cavity resonance frequency, a superradiant laser should belargely insensitive to these effects that continue to limit current stable lasers.
From another perspective, because the light emitted from a superradiant laser comes directlyfrom a narrow-linewidth atomic transition, one can view these devices as active optical frequency
standards. A superradiant laser would basically be the optical equivalent of a maser, which is anactive microwave frequency standard [61]. The fact that the superradiant laser operates in the
optical domain instead of the microwave domain is a major advantage.
The advantage of operating a frequency reference at high frequency is quite general. For agiven phase resolution and measurement time, the uncertainty in one’s ability to measure frequency,
expressed in absolute terms, is fixed. However, because a reference tone can be transferred todifferent frequency bands while maintaining its original fractional frequency error [62], the key
figure of merit for a frequency reference is its fractional uncertainty. This gives high-frequency

9standards a huge advantage over low frequency standards with similar absolute uncertainties.
Fixed phase resolution is a reasonable assumption to make here, as most frequency measurements are fundamentally limited by shot noise associated with measuring a finite number ofquantum particles (e.g. atoms or photons). This shot noise generally limits one’s ability to resolve√a phase to 1/ N radians, where N is the number of particles measured. Apart from phase resolution, many other technical limitations such a shifts from magnetic fields contribute frequency
noise whose magnitude is independent of the frequency at which the frequency standard operates.
These errors are also fractionally smaller for optical frequency references than for their microwave
counterparts.
The power of working at optical frequencies can be seen in the great improvement in theperformance of atomic clocks when the jump was made from the microwave to optical domain.
Because of their higher operating frequency, the fractional frequency stability and accuracy of theseclocks quickly overtook their microwave counterparts, and are now several orders of magnitude morefavorable. Because a superradiant laser would operate in the optical domain, one might reasonably
expect to see similar improvements over microwave masers to those that optical clocks achieved
relative to microwave clocks.
In this thesis, I will present a series of milestones on the path towards a technologically usefulsuperradiant laser. Before I joined the group, previous members of the Thompson lab — Justin
Bohnet, Zilong Chen, Joshua Weiner, and Kevin Cox — performed the first demonstrations andcharacterizations of a quasi steady-state superradiant laser [63, 64, 65, 66]. This experiment used
Raman transitions to create an effective long-lived optically excited state in an ensemble of cold
rubidium atoms. While not useful as a frequency reference (the frequency noise on the “dressing”
laser used to create the Raman system appears on the laser output), this system was an ideal
test-bed for studying many aspects of steady-state superradiance. Unlike rubidium, strontium
has several narrow-linewidth optically excited states, suitable for the operation of a true narrow
linewidth optical superradiant laser.
As a first step towards a superradiant laser, we demonstrated pulsed and quasi-steady state

10
lasing on the γ = 2π× 7.5 kHz linewidth 1 S0 to 3 P1 transition in 88 Sr [24]. The partially forbidden
nature of the 7.5 kHz transition puts the system in an interesting regime. The cavity linewidth(κ = 2π × 160 kHz) is much broader than the atomic transition linewidth, so excitations leave
the cavity primarily through the cavity mirrors rather than through spontaneous emission. The
frequency of coherent, collective interactions between the atoms and cavity is typically of orderΩ = 2π × 1 MHz, much larger than either decay rate. In pulsed operation, a laser operating withΩ  κ, γ exhibits interesting ringing behavior as photons are emitted into the optical cavity andreabsorbed.
We also demonstrated that it is possible to repump this laser, returning atoms that had
emitted a photon to the excited state to achieve quasi steady-state operation. In this steady-state
regime, the collectively enhanced emission rate is limited by the repumping rate. The repump
process also broadens the linewidth of the lasing transition, which we can use to continuously tune
the laser from the conventional “good cavity” regime of laser physics where the laser frequency is
set by the cavity to the “bad cavity,” or superradiant regime where the laser frequency is set by
the atoms.
The next step was a demonstration of lasing on the 1 mHz linewidth clock transition in
87 Sr [25].

In contrast to the laser based on the 7.5 kHz transition, this laser operates deep inthe superradiant regime, where cavity decay far exceeds both atomic dissipation and the rate ofcoherent interactions. We demonstrated collectively enhanced emission in which atoms emitted
photons into the cavity a factor of up to 10,000 times faster than would independently radiating
atoms.
After demonstrating and characterizing superradiance from the perspective of the radiatedpower, we turned to a characterization of the spectral properties of the emitted pulses of lightby comparison with a very stable laser from Jun Ye’s lab down the hall. We characterized key
elements of the performance of our superradiance as a frequency reference, including the short-timestability, absolute accuracy, and sensitivity to perturbations such as cavity frequency shifts andmagnetic fields. We have found the short-term stability to be within an order of magnitude of

11the current state-of-the-art optical lattice clocks (making ours the most precise active frequency
reference demonstrated to date), and the absolute frequency to be within roughly 1 Hz of its
predicted value. We have demonstrated a method to effectively eliminate sensitivity to magnetic
field fluctuations, and a reduction in sensitivity to changes in cavity frequency by more than fiveorders of magnitude. These observations are very encouraging for the prospect of creating evenmore precise superradiant frequency references.

1.1.4

Cavity-Mediated Spin-Spin Interactions

When we tune our cavity on resonance with a narrow linewidth optical transition, the primaryeffect is an increase in the rate of decay from the excited state (superradiance). If we instead
detune the cavity from atomic resonance, interesting Hamiltonian dynamics emerge, which we
can describe as cavity-mediated interactions between the effective spins formed by the upper andlower states of the lasing transition. In our case, these interactions manifest as both an energy
shift to the atomic transition that depends on both atomic inversion and cavity frequency, andas a self-energy term that leads to a so-called many-body gap protection. The energy shift leads
to the buildup of atom-atom correlations via an effect known as “One-Axis-Twisting” (OAT).
OAT has been demonstrated in driven systems as a way to create entangled spin-squeezed states[17, 67, 68, 69]. In our system, the atoms themselves create the drive that leads to twisting. Thisidea was independently proposed in ref. [70]. The second effect — many-body gap protection —leads to a suppression of decoherence in the presence of interactions mediated by the optical cavity.
A similar effect, mediated by collisions between atoms, has been observed and utilized in opticalmagnetometers [71], and cold-atom systems [72, 73, 74].
Experimentally, the superradiant light exiting the cavity gives us a precise realtime nondestructive probe of the atomic evolution [65]. By observing frequency shifts of the superradiantlight, we can probe the energy shifts of the atomic transition. We use this to characterize the OAT
dynamics at the mean-field level. We can also simultaneously populate two sets of atomic states,and observe simultaneous lasing on transitions between pairs of these states. This allows us to12controllably introduce a frequency splitting between the two transitions, which in turn allowed us
to observe signatures of many-body gap protection. Finally, we demonstrate a technique in which
we introduce controllable reversable dephasing to perform a direct spectroscopic measurement of
the many-body gap.
This work has important implications for understanding potential perturbations to the superradiant laser, for generating metrologically useful entangled states, and also as a demonstrationof a potentially appealing new type of system for studying many-body physics phenomena.

1.1.5

A Novel Cooling Mechanism

The following two projects do not rely on coupling atoms to an optical cavity, but do rely on
properties enabled by narrow-linewidth optical transitions. The first concerns several related newmechanisms for laser cooling and trapping.
The development of laser cooling has revolutionized atomic physics. Typically, laser coolinghas relied on broad-linewidth optical transitions. Because atoms decay very quickly from theexcited state of these transitions, we usually view the atoms as primarily living in the ground state,
and occupying the excited state only fleetingly before emitting a photon. Similar techniques tothose used on broad transitions have been extended experimentally to narrow linewidth transitions[75, 76, 77, 78, 79, 80], but the conceptual framework used to understand the cooling is very similar
to the broad linewidth case.
The use of narrow linewidth optical transitions opens up fundamentally new possibilities forlaser cooling beyond those previously explored. Because the excited state of such transitions islong-lived, we can now consider the use of mechanisms that rely on populating the excited state
for significant periods of time. In this work, we demonstrate two related but distinct mechanismsthat rely on transferring significant population to the excited state of a narrow linewidth opticaltransition.
The first mechanism, which we have named SWAP (sawtooth wave adiabatic passage) cooling,
relies on subsequent Landau-Zener transitions to and from an excited state driven by counterprop-

13
agating laser beams whose frequencies are swept over atomic resonance in a sawtooth waveform.
This method enables the generation of large forces even on very weak transitions, and the possibilityof cooling with reduced amounts of spontaneous emission. This may make it an appealing mechanism for the cooling of molecules, for which spontaneous emission causes major challenges. SWAP
cooling generates a velocity dependent force that leads to cooling and can increase phase-spacedensity in an atomic ensemble.
We also demonstrate a related mechanism that provides a confining force as well as cooling.
This mechanism utilizes a similar sawtooth ramp to SWAP cooling, but relies on shelving atoms in
one of several long-lived excited states to generate a differential scattering force that depends on
the local magnetic field. This allows for the formation of a magneto-opical trap (MOT). From apractical perspective, this second mechanism has proved to be very useful in our experiment, andforms the basis of our cooling in everyday operation. It’s use has enabled us to increase our atom
number, improve the robustness of our experiment, and greatly simplify our cooling and trapping
protocol.

1.1.6

Enhanced Gravitational Wave Detection with Atoms

The final project that I will describe here involves a new class of proposals for gravitational
wave detectors based on cold atoms with narrow linewidth optical transitions [81, 82, 83, 84].
These proposals rely on ensembles of atoms contained within two distant satellites. Laser pulses
are passed back and forth between the two satellites, interacting with the atoms on each end. A
passing gravitational wave modifies the phase of the light as it interacts with the far ensemble, andthis phase shift is encoded in atomic observables. The use of narrow linewidth optical transitions
enables protocols that appear to be impossible with more conventional approaches using broad
linewidth transitions or Raman transitions between ground states.
These new proposals fall into two categories: those described as “clocks,” where the atoms
are tightly confined by an optical potential, and those described as “atom interferometers” (AI),in which the atoms are not confined and as a result experience momentum kicks from absorbing

14
or emitting photons. My goal was to understand the fundamental differences between these two
types of proposals. In particular, the AI proposals rely on a technique known as “large momentum
transfer” (LMT) to enhance sensitivity, while clocks had no such analog.
In this work, we provided a framework in which the fundamental mechanism for sensitivityof the two types of proposal can be understood. We reached the conclusion that the two types of
proposals were fundamentally the same, and that the photon recoils and resulting spatial separationof the atoms’ wavefunctions in the AI proposals have no fundamental role in the gravitational wave
detection. We further showed that a technique analagous to LMT could be applied to a clock,though no momentum is actually transferred to the atoms. Finally, we proposed an enhancement
mechanism in which useful evolution time can be achieved with the atomic population entirely
in ground states, which could be separated by acoustic rather than optical frequencies. Theseenhancement mechanisms do not make sense if we think of the detectors that utilize confined atoms
simply as clocks whose sole purpose is to measure frequency. Rather, our perspective highlightsthat we should think of the atoms as phase memories that allow the phases of light pulses arriving
at different times to be compared.

1.2

Outline of Thesis
The rest of this thesis is organized as follows: In Chapter 2 I will outline language and theory

needed to understand systems where atoms are coupled to cavities. I will particularly focus on thoseelements that are different when using narrow-linewidth optical transitions. In Chapter 3, I will
describe the design, construction and basic operation of the experimental apparatus. In chapters 410, I cover a series of experiments performed using the new apparatus and narrow-linewidth opticaltransitions. Chapter 11 describes a new proposal for improved detection of gravitational waves using
cold atoms with narrow-linewidth optical transitions. Finally, in Chapter 12 I provide an outlook
to future extensions of this work and rough outlines for possible experimental implementations ofspin squeezing in an optical lattice clock, and for a steady-state superradiant laser.

Chapter 2

An Introduction to Atoms in Cavities and Narrow-Linewidth Transitions

Much of the work presented in this thesis centers around coupling large ensembles of atoms to
optical cavities via narrow-linewidth optical transitions. This chapter provides a basic introduction
to the physics of atoms interacting with cavities. I will introduce some of the language and parameters used to define atom-cavity systems, along with simple scaling arguments where applicable.
This will enable a discussion of different regimes in which one might operate, and the advantages
and challenges of each. I will particularly focus on these tradeoffs in the context of coupling atoms
to cavities via narrow-linewidth dipole-forbidden optical transitions. Finally, I will present a simple
and intuitive approach to describing and visualizing the atom-cavity system, taking advantage of
the Bloch sphere formalism and an analogy to a classical pendulum.

2.1

Narrow-Linewidth Transitions in Strontium
The primary distinguishing feature of strontium, as compared to the more commonly usedalkali atoms, is the presence of two valence electrons. This property leads to rich internal structure.
Importantly, the spin of the two valence electrons can either be aligned in a symmetric manner(spin S = 1), known as a triplet configuration, or in an anti-symmetric or singlet configuration(S = 0). Any transition between a singlet and a triplet state is in principle forbidden, as it would
involve flipping the spin of one electron relative to the other.
In reality, the states we write down are not actually triplet or singlet states. Rather, when
we write down a state like 3 P1 , where the 3 stands for triplet and which decays to the nominally16
singlet 1 S0 state at a rate of γ = 2π × 7.5 kHz, we are actually referring to a state that is mostly
triplet-like, but has some singlet character mixed in. In the case of strontium (and other alkaline
earth atoms), the finite lifetime of 3 P1 is due to the presence of spin-orbit coupling, which mixes insome 1 P1 character into the state that we call 3 P1 . In 87 Sr, which has nuclear spin I = 9/2, I · S
coupling mixes some 3 P1 character into 3 P0 , which allows it to decay very weakly, resulting in atransition linewidth of roughly 1 mHz.
Great detail and calculations of these effects can be found in [85, 86]. For the rest of thiswork however, the key takeaway is that atomic strontium has two particularly useful forbidden
transitions, with very weak dipole matrix elements. The 7.5 kHz linewidth transition is present
in all isotopes of strontium, while the 1 mHz linewidth transition is present only in the fermionicisotope, 87 Sr, as this is the only isotope with I 6= 0.

2.2

Atoms Interacting with a Cavity
Two of the simplest systems in quantum mechanics are the two-level system and the harmonic

oscillator. The interaction between an atom and a cavity provides a clean experimental realizationof the coupling of these two canonical systems, with the atom serving as the two-level system (withground and excited states |gi and |ei) and the cavity providing the harmonic oscillator. Despite (ormore likely because of) the simplicity of this system, the field of cavity quantum electrodynamics(cavity QED) has yielded a wealth of fundamental understanding of quantum mechanics, as wellas great potential for technologically useful systems.
The typical goal of cavity QED is to realize a situation where an atom (or ensemble of atoms)
interacts primarily with a single mode of electromagnetic radiation. The cavity supports discrete
resonant modes, each with a well-defined frequency and spatial profile. Each of these modes isanalogous to a harmonic oscillator, with a ladder of evenly spaced energy levels representing integer
numbers of photons occupying the mode. When an atom is located within one of these modes whose
resonance frequency is near that of the atomic transition, then the interaction between the atomand this mode may dominate over coupling to other modes of the cavity or the environment.

17

Figure √2.1: A cavity QED system. Atoms exchange excitations with the cavity mode at a rate
Ω = 2g N . Atoms can decay into free-space via single-particle spontaneous emission at a rate γ,
leading to loss of coherence. Photons in the cavity leak out through the mirrors at rate κ. This
channel of dissipation is easy to detect, and provides only collective information about the state of
the atomic ensemble.

Starting with the case of a single atom in the cavity, the atom-cavity system can be describedby the so-called Jaynes-Cummings Hamiltonian:Ĥ = ~ωa σ̂ + σ̂ − + ~ωĉ† ĉ + ~g(σ̂ − ĉ† + σ̂ + ĉ)The first term corresponds to the energy of the atom — ωa is the atomic transition frequency andσ̂ + = |ei hg| (σ̂ − = |gi he|) is the atomic raising (lowering) operator. The second term represents
the energy of the cavity field — ω is the cavity resonance frequency, and ĉ† (ĉ) are the photon
creation (annihilation) operators. Finally, the third term represents the interaction between theatom and the cavity. The atom can go from |ei to |gi by emitting a photon, or from |gi to |ei byabsorbing one. The frequency at which this happens is given by g, the single particle coupling.
We can rewrite this Hamiltonian in a rotating frame at the atomic transition frequency ωa(basically just subtracting out the atom’s excited state energy, and assuming that the transition
from |ei to |gi is accompanied by the addition of a photon to the cavity) as:Ĥ = ~δc ĉ† ĉ + ~g(σ̂ − ĉ† + σ̂ + ĉ)
where δc = ω − ωa is the cavity detuning .

18In addition to the coherent exchange of excitations between the atom and cavity, we must
consider the coupling of the system to the environment. The atom can decay by emitting a photon
into a mode other than the cavity mode at a rate γ, the single-atom spontaneous decay rate from|ei to |gi in free space. Photons in the cavity decay at a rate κ, the cavity linewidth. Note that
κ represents a power decay rate, not an amplitude decay rate. This is illustrated in Fig. 2.1.
Additional sources of broadening such as Doppler shifts of the atomic transition cause dephasingof the collective excitations stored in the atoms. These additional sources of broadening must be
carefully controlled so as not to dominate the dynamics.
The single atom coupling g is a very important parameter in cavity QED, so it is worth
understanding how it scales. Following the definitions in Dan Steck’s quantum optics notes [87](which I will do as much as possible in this thesis), g is given by:r~~g = −ε̂ · dω20 ~Vε̂ represents the polarization of the cavity field, d~ is the dipole matrix element of the atomic
transition, and V is the volume of the cavity mode (the cross-sectional area at the location of the~E~ associated with the atomic
atoms times the cavity length). Physically, ~g represents the energy d·dipole interacting with the field associated with a single photon in the cavity. A summary of thescalings of basic and derived parameters is given in Table 2.1.
From an experimental design perspective, for a fixed frequency ω, g depends only on atomic~ and the volume of the cavity mode. A short cavity with a tightly focused mode will
properties (d),thus have a large g, while a long cavity with a spatially broad mode will have a small value of g.
Note that the quality of the mirror coatings do not effect the value of g because we have assumed
that there is a single photon in the cavity.
If N atoms are placed inside the cavity mode and excited by the cavity field, the Hamiltonian
becomes19

Ĥ = ~δc ĉ† ĉ + ~g(Jˆ− ĉ† + Jˆ+ ĉ)
where Jˆ± = 12±i=1 σ̂i are the collective raising and lowering operators for the atoms. If we focus

PN

on the case where the number of atomic excitations is much less than the total number of atoms,
we can approximate this Hamiltonian [88, 89] as√Ĥ = ~δc ĉ† ĉ + ~g N (âĉ† + â† ĉ)

(2.1)

√
√
where we replace Jˆ± with creation and annihilation operators â = J+ / N and â† = J− / N .
The takeaway from this exercise is that when there are many spins inside the cavity, the rate√
at which a photon is exchanged between the cavity and the atoms is enhanced from 2g to 2g N ,which I will follow convention and label Ω, the collective vacuum Rabi frequency. While thisderivation assumed a low-excitation limit, the basic scaling is true in general so long as dissipation
rates are sufficiently low.

20
Quantity/Label
Atomic Transition Frequency ωaCavity Field Frequency ωCavity Length L
Loss
Cavity Finesse F
Wavelength λMode Waist w
Cavity Mode Volume V
Dipole Matrix Element d
Atomic Linewidth γ
Cavity Linewidth κ
Coupling g

Cooperativity Parameter C

Formula/Definition

Key Scalingω ' ωapower loss in round trip
2π/Loss
2πc/ω
1/e radius of field
πw2 L
ωa3 d23π0 ~c3

c/2LFqd 20ω~3 V(rms value of averaged
over standing wave)
4g 2 /κγ(averaged over standingwave)

d2 ωa3
Loss/Lpd ω/Vλ2 /(w2 Loss)

Table 2.1: Key parameters and scalings for cavity QED.

Generally, the goal of cavity QED is to create a situation where interactions between the atoms
and the cavity dominate in some way over other effects. These situations are broadly referred toas the strong coupling regime. There are several useful definitions of the strong coupling regime,
which are each relevant in different situations and for different tasks. The simplest criteria is that√the rate of coherent interactions — 2g in the single-atom case and 2g N in the multi-atom case —should exceed the dissipative rates κ and γ (as well as other sources of inhomogeneous broadening).
These criteria define the resolved vacuum Rabi splitting regime.
A second definition of the strong coupling regime involves the cooperativity parameter ofcavity QED, C = 4g 2 /κγ. In the limit that κ  γ, the cooperativity parameter quantifies therelative probability that an atom placed in the excited state in the cavity mode will decay byemitting a photon into the cavity mode, versus by emitting a photon into free space. For many
applications of cavity QED, C  1 is a key figure of merit. If many atoms are used, C becomes
collectively enhanced to a value of N C, which can greatly exceed unity.
Historically, several approaches have been used to attain strong coupling. I will briefly review

21these, to provide context for the new regime explored in this thesis.
The first of these approaches, exemplified by the work of the Haroche group, involves coupling
single Rydberg atoms to a superconducting microwave cavity. In these experiments, Rubidium
atoms prepared in circular Rydberg states traverse the mode of a microwave resonator coatedin superconducting niobium. Because the cavity is superconducting, losses are very low (small
κ). Further, because transitions between circular Rydberg states have very large dipole moments,
they couple very strongly to the microwave field (large g). Finally, because the transitions occur at
microwave frequencies, the available phase-space for spontaneous emission is relatively small, whichleads to a long lifetime (small γ). These qualities, combined with efficient state-selective detection
techniques, made this an ideal system for early explorations of the strong coupling regime, leading
to a wealth of pioneering work [90, 49, 30] and a Nobel prize.
An alternate approach, initially pursued by the groups of Jeff Kimble and Michael Feld,
among others, is to couple single Alkali atoms to optical resonators by dipole-allowed optical transitions. Certain physical scalings lead to persistent challenges with this approach, but the allure ofcoupling atoms to optical fields (whose thermal occupation at room temperature is low [91]) has
justified great efforts in this domain and some impressive successes [92, 31].
In order to overcome the rapid spontaneous emission rate associated with optical transitions(which scales as the cube of frequency), one must confine the cavity mode to very small volumes
to increase the local electric field seen by the atoms, and thus the rate of coherent interactions.
Even if one assumes that the mode volume scales with the wavelength of the radiation as 1/ω 3(which practically gets really hard to do), the ratio g/γ scales unfavorably from the perspective of
optical transitions as 1/ω. Counterintuitively, it is the relatively small dipole matrix element ofthese optical transitions that makes them viable at all for cavity QED applications — g scales with~ while γ scales with d~2 , so achieving g/γ  1 is actually easier with a weak matrix element d,~d,though this does make it harder to achieve g  κ. This is a hint that transitions with even smaller
dipole matrix elements may be a favorable way to go.
In order to maximize g (to achieve g  γ), these systems end up employing very short22
optical cavities (sub-millimeter lengths) with very tightly focused waists in order to attain a small√
mode volume V . Unfortunately, while g ∝ 1/ L, κ ∝ 1/L. This means that the short cavity
length L makes it challenging to achieve the condition g  κ, which requires very high mirror
reflectivity. Breakthroughs in dielectric mirror coatings enabled the realization of cavities withTransmitted Powerfinesse F exceeding 105 , and the attainment of single-atom strong collective coupling.

Probe Frequency
Figure 2.2: One of the key manifestations of the strong coupling regime defined by Ω  κ, γ is thevacuum Rabi splitting. As a weak probe tone coupled to the cavity is swept over resonance withthe cavity (here, δc = 0), a single transmission peak splits into two. The splitting between thesepeaks is set by Ω, and the width of each peak is a weighted average of atomic and cavity decay
rates.

The key signature of the single-atom resolved vacuum Rabi splitting regime (g  κ, γ) is
the single atom vacuum Rabi splitting (pictured in Fig. 2.2), an effect where a single transmission
peak of the optical cavity splits into two peaks separated by 2g (when δc = 0), corresponding tohybrid atom-cavity modes [93]. The equivalent feature for the collective regime is the collective√
vacuum Rabi splitting, in which the peaks are separated by Ω = 2g N . The eigenfrequenciesof our Hamiltonian for δc = 0 are ±Ω/2. These determine the center frequency of the two peaks.
The eigenmodes that correspond to these eigenfrequencies are half atom-like and half photon-like incharacter, and thus decay at a rate given by the average of the atom and cavity decays κ0 = (κ+γ)/2.

23This decay rate determines the width of each of the two peaks.
If the two peaks are resolved, it means that their separation, which is set by the rate ofcoherent interactions, exceeds their width, which is set by decay rates from the system. A very
nice review of the origin and characteristics of the vacuum Rabi splitting can be found in ref. [88].
For many applications, it is not necessary (or desirable) to operate with only a single atom.
For these applications, collective enhancement may be used to ease technical requirements. The
strong collective coupling regime has been applied to the generation of entanglement through effective unitary dynamics [17] and measurement [14, 15, 33], the creation of single photon sources[94] and studies of optomechanical effects with cold atoms [95, 96], among others.

2.3

Cavity QED on Dipole-Forbidden Transitions
Much of my thesis work centers around extending the collective coupling approach to dipole-

forbidden optical transitions. This has been previously explored in the context of cavity-enhanced
non-linear spectroscopy in the group of Jan Thomsen, where an ensemble of roughly milliKelvintemperature strontium atoms are coupled to a moderate-finesse optical cavity via a dipole-forbidden
7.5 kHz linewidth optical transition to obtain cavity-enhanced spectroscopic features useful for laser
stabilization [97, 98]. In these experiments, the relatively high temperature of the atoms led toinhomogeneous Doppler broadening that exceeded the rate of coherent interactions between atoms
and cavity and prevented the observation of some of the canonical features of cavity QED, such as
the vacuum Rabi splitting.
In the work I present in this thesis, we both further cool the atoms to temperatures of a few
microKelvin and tightly confine the atoms to less than a wavelength using an optical standing wave
lattice. This enables the observation of canonical cavity QED features on the 7.5 kHz linewidthstrontium transition, as well as the exploration of a far more extreme regime of cavity QED using
the even more forbidden 1 mHz linewidth optical clock transition.
In this section, I will discuss the differences between using narrow-linewidth optical transitions
and the more conventional regimes explored previously. I will focus on what gets harder or easier24with narrow transitions, and what advantages they may provide over broad-linewidth transitions.
Let us consider the first criteria for strong coupling presented earlier, the resolved vacuum
Rabi splitting: Ω  γ, κ. A forbidden transition has a long-lived excited state because the dipole
matrix element d associated with the transition is small. This leads to a reduction in g, and√therefore Ω = 2g N compared to a dipole-allowed transition. Fortunately for the goal of achieving
this definition of the strong coupling regime, the atomic linewidth γ ∝ d2 , so Ω  γ is actuallyeasier to achieve for a dipole-forbidden transition.
However, if we account for sources of atomic broadening besides photon emission into freespace, things get much harder when using narrow linewidth transitions. Effects like Doppler shifts(∆ωD = 2πv/λ, where v is the atomic velocity and λ is the wavelength of the radiation) do notscale with the strength of the atomic transition. Ensuring that these effects do not overwhelm
coherent interactions, which are now much slower, requires far more care and lower temperatures
than for dipole-allowed transitions. Specifically, assuming a fixed wavelength, in order to fix thesame ratio of g/∆ωD between a narrow linewidth transition and a broad linewidth transition, onemust scale the temperature by the ratio of the transition linewidths.
Further, achieving the condition that Ω  κ is also more challenging. For the 7.5 kHz
linewidth strontium transition, this is still relatively easy to achieve with modern mirrors. However,accessing this regime on the 1 mHz linewidth transition would be experimentally challenging.
For our other definition of strong coupling, N C  1, the scalings work out somewhat differently. Because g ∝ d and γ ∝ d2 , the dipole matrix element actually cancels from C. Thus,
for this definition of the strong coupling regime, the fact that we use dipole-forbidden transitions
is fundamentally unimportant. However, it is sometimes important that the rate N Cγ, the rateof collectively enhanced emission per atom in an overdamped regime, exceeds the rate of inhomogeneous dephasing. This leads to a more stringent requirement that the temperature must scaleas the square of the transition linewidth in order to maintain the same ratio of N Cγ/∆ωD , which
would be very hard to achieve using unconfined atoms and narrow linewidth transitions.
There is one more consideration in a cavity QED system where the transition linewidth is25
critical: the ratio of γ to κ. The regime where κ  γ is often referred to as the bad-cavity regime,
while γ  κ defines the good-cavity regime. In the bad-cavity regime, excitations are more likely toleave the atom-cavity system as photons emitted from the cavity mirrors, while in the good-cavity
regime they are more likely to leave as spontaneously emitted photons. In many applications ofcavity QED, photons transmitted through the cavity mirrors are desirable, as they provide only
collective information about the atomic ensemble and are easily collected. Photons scattered into
free-space are typically detrimental, as they lead to dephasing of the atomic ensemble and are
difficult to utilize.
This makes the bad-cavity regime desirable for many applications, and therefore makes theuse of dipole-forbidden transitions very advantageous. In cavity QED experiments that involve
dipole-allowed optical transitions (which typically have linewidths of a few MHz), one must work
hard to access the bad-cavity regime. In order to achieve a high value of C, one would like to use ahigh-finesse cavity. The down-side is that this implies a long photon storage time and corresponding
narrow linewidth, unless one makes the cavity very short and difficult to work with. In contrast,
when working with a narrow-linewidth optical transition, one can work in the bad-cavity regime
with a cavity of convenient length (allowing ample optical access to atoms trapped within the cavity
and a relatively small free spectral range) and reasonably high finesse.
In certain applications one can attain some of the advantages of being in the bad-cavity regime
by detuning the optical cavity resonance from atomic resonance or using Raman transitions. Whilethese techniques have been applied with great success in a variety of systems [63, 17, 14, 15, 99], theiruse does lead to certain limitations not present with narrow-linewidth transitions. For example, byworking in the bad-cavity regime with the cavity on resonance with a narrow-linewidth transition,
we were able to demonstrate a noise-insensitive probing scheme that would not be possible with adetuned cavity [20].

26

(a)

(b)

Figure 2.3: (a) The state of a two-level system with states |ei and |gi can be represented by a
Bloch vector that lives on a Bloch sphere. The figure shows a general state |ψi = sin(θ/2) |gi +cos(θ/2)eiφ |ei. The vertical projection of the Bloch vector tells us the population inversion of
the state (the relative likelyhood of finding the atom in the ground or excited state), while theazimuthal angle φ tells us the relative phase between the |ei and |gi components. (b) An ensembleof many two-level systems can be represented on a collective Bloch sphere by simply adding up
all of the individual Bloch vectors associated with the two-level systems to form a collective Bloch~ The inversion of the collective Bloch vector is labelled J~z , and the atomic coherence
vector J.
(projection onto the equatorial plane) is labelled J~+ .

2.4

Cavity QED with Bloch Spheres and Pendula
This section introduces a very powerful and intuitive way of thinking about cavity QED using

Bloch spheres. This model is totally classical, yet captures almost all important aspects of cavity
QED for the purposes of the work in this thesis. Those that it does not can often be added in byhand as necessary. I will also introduce an even more clearly classical picture, where the behaviourof the atoms and cavity are mapped onto that of a pendulum experiencing gravity.
The goal is to provide a visual formalism for describing the dynamics of an atom interactingwith a cavity. Detailed mathematical formalism can be found, for example, in Daniel Steck’s quantum optics notes [87]. We find that this simple visual picture provides not only great explanatory
power, but rapid predictive power that is key to running the experiment.
I will start with a description of how we represent the different parts of the physical system.
A single atom will be treated as a two-level system, which can be represented by a vector on a

27
Bloch sphere of radius 1/2. For pure quantum states, the tip of the vector lies on the surface ofthe sphere. If the atom is in the excited state |ei, the vector points to the north pole of the sphere.
If the atom is in the ground state |gi, the vector points to the south pole. An atom in the state|ψi = sin(θ/2) |gi + cos(θ/2)eiφ |ei is depicted in Fig. 2.3a.

Figure 2.4: We can describe the dynamic of an ensemble of atoms interacting with the cavity fieldusing a Bloch sphere picture. The cavity field is represented by an axis C̆ about which the Blochvector rotates at a rate proportional to g|C̆|. As pictured, the rotation is leading to collective decayof the atoms from |ei to |gi.

For systems with more than one atom, we simply add the Bloch vectors of the individual~ which lives on a collective Bloch sphere. In
atoms together to form a collective Bloch vector J,
order to account for the fact that the atoms are located at different points along the cavity axis,
and therefore experience a different optical phase from the cavity field, we define the phase of eachatom’s Bloch vector relative to the phase of the optical field of the resonant mode. So long as theatoms do not move around, and coherent rotations are only performed using the cavity mode (as is
the case for pretty much everything in this thesis), we can safely ignore this detail. We describe J~
in terms of it’s vertical projection Jz , which is −N/2 at the south pole, 0 at the equator and N/2
at the north pole, and its projection onto the equatorial plane J + = Jx + ıJy , which is a complex
number whose real and imaginary parts are Jx = Jsinθcosφ and Jy = Jsinθsinφ respectively. These
quantities are labelled in Fig. 2.3b.
We describe the cavity field C (not the cooperativity parameter introduced earlier, but this28
is a somewhat common notation) as a vector that originates from the center of the Bloch sphere.
For a field whose frequency is on resonance with the atomic transition (which is the relevant casefor the primary applications of this model in this thesis), this vector lies in the equatorial plane of
the Bloch sphere. The Bloch vector rotates about this cavity field vector at a frequency 2g|C|, with
C expressed in units of the amplitude associated with a single photon. This is shown in Fig. 2.4.
In general, C can be any arbitrary cavity field, with any general source. Here though, I
will specialize to the case where the cavity field is radiated by the atoms themselves (hence theassumption that it is on resonance with the atomic transition). We wish to describe the effects ofthis self-radiated field back on the atoms. Following [64], we may write down differential equations(known as optical Bloch equations) that describe the interactions with the cavity field C, as well
as dissipation. We assume that both the atomic coherence and cavity field oscillate at the same asof yet unknown frequency ωγ , and define J˘+ = J + eıωγ t and C̆ = Ceıωγ t . I will keep the˘s aroundfor the remainder of this chapter, but will treat them as implied after that. We may then write
equations of motion for J˘+ and C̆:˙J˘+ = −[γ⊥ + ı(ωa − ωγ )]J˘+ + ı2g C̆Jz

(2.2)

∗
C̆)
J˙z = −γ(Jz + N/2)/2 + ig(J˘+ C̆ ∗ − J˘−

(2.3)

˙
C̆ = −[κ/2 + ı(ωc − ωγ )]C̆ − ıg J˘+(2.4)

Stepping through the terms in these equations, γ⊥ defines the decay rate of J + due to excitedstate decay or other processes. In the absence dephasing processes other than excited state decay,γ⊥ = γ/2. Note that these equations only capture processes that would lead to an exponential
decay of J + , such as spontaneous emission or collisions that lead to large, instantaneous phaseshifts. Other processes that lead to a Gaussian decay of J⊥ such as Doppler decoherence requirea more sophisticated treatment. ωa is the frequency of the bare atomic transition, and 2g is the
single-atom vacuum Rabi frequency. γ represents the excited state decay rate, κ is the power decayrate of the cavity, and ωc is the bare cavity resonance frequency.

29
We can also understand the nature of the collective enhancement of the vacuum Rabi frequency in this picture. Each atom’s Bloch vector rotates at a rate proportional to |C̆|. If we startwith N atoms near the excited state, they will emit photons into the cavity, causing the Bloch√vector to swing down to the south pole. At this point, |C̆| ∝ N , as each atom has emitted one
photon into the cavity. The rate at which the Bloch vector is rotating is thus enhanced by a factor√of N , which is one manifestation of the collective enhancement. In general, whenever the atoms
are interacting with a self-generated field, the rate of the interactions is enhanced because each
atom sees the field radiated by all of the other atoms.
We can gain further intuition though a simple mapping: For a cavity tuned to resonancewith the atomic transition and no damping of the atomic coherence (though cavity damping is still
captured), the equations of motion for the Bloch vector are equivalent to those for a pendulum
under gravity. This picture can be very useful for describing cavity QED phenomena.

Figure 2.5: With the substitutions given here and in the text, the dynamics of an ensemble ofatoms interacting with the mode of a resonant cavity can be mapped onto the motion of a damped
pendulum under gravity. The angular velocity of the pendulum takes the role of the cavity field
while the decay of photons from the cavity is equivalent to viscous damping of the motion of thependulum. The length of the pendulum scales as (g 2 J)−1 , which leads to faster dynamics for many
atoms or large coupling g.

Because we are neglecting damping of the atomic coherence (setting γ = γ⊥ = 0), thelength J of the Bloch vector is fixed. Further, because we assume that the cavity is on resonance30(ωa = ωc = ωγ ), we can assume that J˘+ is real. We can then describe the position of the Blochvector by its polar angle θ alone. Rewriting the above equations of motion in terms of θ, we get:θ̇ = i2g C̆

(2.5)θ̈ = 2g 2 Jsinθ − κθ̇/2(2.6)

We can compare these to the equation of motion for the angular velocity θ̇ of a pendulum:θ̈ = ggrav sinθ/L − β θ̇

(2.7)

Here, ggrav is the gravitational acceleration, L is the length of the pendulum, and β is the velocity
damping rate, due for example to the pendulum being immersed in a viscous fluid.
The cavity field C̆ thus corresponds to the angular velocity of the pendulum (up to a factorof 2g). The strength of gravity and length of the pendulum are constrained by ggrav /L = 2g 2 J, anddamping of the cavity field corresponds to viscous drag on the pendulum. Strong coupling (highg) or large atom number (large J) correspond to a short pendulum in a strong gravitational field.
I will return to this pendulum analogy several times throughout this thesis, but for now will
give one example of its utility. An obvious quantity of interest for a pendulum is its resonancefrequency. Does this correspond to anything meaningful in cavity QED? The resonance frequency√√
pof a pendulum is ggrav /L, which maps to g 2J = g N . If we excite the pendulum, it will swing√
back and forth with a frequency g N . The angular velocity (and thus the cavity field) of thependulum also oscillates at this frequency. If we look at the field inside the cavity, it will thus be√amplitude modulated as C = cos(g N t)eiωc t . The Fourier spectrum of this field consists of two√
peaks, centered around the cavity resonance ωc , and split by 2g N . This spectrum corresponds tothe famous collective vacuum Rabi splitting of cavity QED.
This pendulum mapping is also very useful for providing intuition into different regimes ofsuperradiance, and I will return to it in later sections of this thesis.

Chapter 3

Experimental Apparatus

In this chapter, I describe the basic experimental hardware and processes that allows us tooperate our experiment. I will focus on those elements that are general to most or all experimentsthat we perform, and will leave atomic manipulation and detection techniques specific to particularexperiments to their corresponding chapters.
From a broad perspective, the key requirement of the apparatus is to achieve a situation where
the rate of collective interactions between the atoms and cavity, mediated by the mHz linewidthtransition, exceeds all atomic decoherence rates in the system. This is the requirement to obtainsuperradiance, and dictated most design choices in the experiment.
For superradiance experiments operating in the deep bad-cavity regime (where the cavity
is much more lossy than the atoms), the rate of collective interactions is given by N Cγ, where
N is the atom number, C is the cavity cooperativity parameter, and γ is the decay rate of thetransition. We want this rate to be large, which for a given atomic transition requires many atoms
and a high-cooperativity cavity. In practice, much effort has gone into optimizing the number ofatoms in the cavity.
Achieving a low enough decoherence rate involves removing any factors that would causea scrambling of the relative phase between the superposition states of the different atoms. Suchfactors include collisions with background gas, which is eliminated through the use of an ultra-highvacuum environment, Doppler shifts, which are eliminated by cooling and tightly confining theatoms, and interactions with electromagnetic fields, which are managed by carefully controlling32
the optical and magnetic fields experienced by the atoms. These requirements provide the key
constraints for the experimental design and implementation.

3.1

Experimental Overview
As is typical of cold-atom experiments, this experiment runs in a cyclical manner. In each cy-cle, a new ensemble of atoms is prepared, manipulated, measured, then discarded. This is repeatedwith a cycle time of roughly one second. The details of the manipulation and measurement portionsof the sequence depend on the exact experiment being performed, but the initial preparation step
is for the most part common.
Load Blue MOT (~500ms)
Oven(770 K)
Zeeman Slower(461 nm)2D MOTs(461 nm)Hot
Window

3D MOTs(461 nm,~10-3K)

Recapture Into
Red MOT (~50ms)

Load Into
Lattice(~100ms)

B
Turn OffBlue MOT
B
Turn Down
B Field, Turn
On Red MOT
B (Ramped)

Adjust Bias
Fields,
MaintainRampedMOT(10-5 K)

Figure 3.1: A simplified atom loading sequence. Atoms begin as a roughly 500 C collimated beam
exiting the oven. Some of the atoms are slowed to around 50 m/s by a Zeeman slower, thenredirected by two 2D MOTs into the main experimental chamber. They are then further cooled
and spatially compressed by a 3D MOT, which we typically load for 500 ms. The Zeeman slower
and both MOT stages also use 461 nm (blue) light. The atoms are then transferred from the blue
MOT to a narrow-line red MOT over roughly 50 ms. The magnetic field gradient (represented by
the slope of the black line) is reduced by a factor of roughly ten during this step. An asymmetricsawtooth frequency sweep applied to the red MOT beams greatly improves the efficiency of thistransfer (See Chapter 10). Atoms are further cooled, and loaded into the optical lattice with atypical temperature of around 10 µK.

The process of preparing an ensemble of confined cold atoms involves several steps that aresummarized in Fig. 3.1. The atoms originate as a collimated beam emanating from a hot oven.
They are then slowed and cooled from this initial high temperature using dissipative laser cooling33
techniques. This portion of the cycle uses the broad-linewidth “blue” transition at 461 nm, andbrings the atoms down to a few milliKelvin. At this point, further cooling is performed using the“red” transition at 689 nm, which has a much narrower linewidth and is compatible with coolingto much lower temperatures. For this step, we utilize a new form of cooling and trapping, which
we refer to as Sawtooth-Wave Adiabatic Passage (SWAP) cooling. This technique has significant
advantages over conventional laser cooling techniques, and is described in detail in Chapter 10.
Once the atoms are appropriately cooled using SWAP cooling, they are loaded into theconservative potential of the optical lattice. This optical lattice is supported by the same cavity
that the atoms interact with via narrow-linewidth transitions, ensuring spatial overlap between theatoms and cavity mode. The lattice both holds the atoms within the cavity mode for the durationof the experiment and tightly confines them along the cavity axis, eliminating Doppler shifts inthat important direction. The wavelength of the lattice is set to the so-called “magic” wavelength,
which minimizes perturbations of the milliHertz linewidth clock transition [100, 101].
After this point in the experimental sequence, the procedure differs depending on the specific experiment being performed. I will cover most of the specifics of these procedures in thecorresponding experimental sections (Chapter 4-10). One important distinction between different
experiments is the choice of isotope. We use one of two isotopes: 88 Sr (the easy isotope), and 87 Sr(the hard isotope).

88 Sr has roughly ten times the natural abundance of 87 Sr, making it mucheasier to achieve high atom numbers. The most significant difference between the isotopes howeveris the nuclear spin — 88 Sr has none, while 87 Sr has a nuclear spin I = 9/2. The presence of nuclear
spin in 87 Sr causes the 3 P0 to 1 S0 transition to become weakly allowed [85]. The large nuclear spinof 87 Sr also leads to a great proliferation in the number of atomic states. 3 P0 and 1 S0 now eachhave ten Zeeman sublevels, and other states have additional hyperfine structure. While we aresometimes able to exploit this multitude of states, they generally lead to additional complications
when working with 87 Sr.

34

Figure 3.2: Pictures of the experiment from July 2017, to give a sense of scale. Upper left: four-borereference cavity, with associated optics. Upper right: main vacuum chamber end of experiment.
Lower left: the experiment spans two optics tables. Vacuum system is in the foreground, with lasers
and laser stabilization in background. Lower right: enclosure houses all red and infrared lasers.

3.2

Vacuum System
The atoms must be prepared, cooled, and manipulated within an ultra-high vacuum (UHV)

environment in order to avoid loss and perturbations associated with background-gas collisions.
Our vacuum system consists of three main sections: the atomic source (provided by AOSense,
Inc.), the main science chamber, and the pump appendage. A model of the vacuum system ispictured in Fig. 3.3
The function of the atomic source is to provide a beam of pre-cooled strontium atoms to
the main chamber, and will be described in more detail in Sections 3.3 and 3.5.2. From a vacuum
perspective, the source does not present a significant load on the rest of the chamber. In fact,

35

Figure 3.3: The vacuum system consists of three main sections: the atom source, which contains anoven, Zeeman slower, and two 2D MOTs, the science chamber, which contains the optical cavity,
and the pumping appendage. The system is pumped by a 75 liter/second ion pump and a titanium
sublimation pump.

the pressure measured within the atom source chamber is lower than that of the main sciencechamber, even with the oven at full temperature. This is because strontium is highly reactive, andthe coating that forms in the atomic source chamber serves as an excellent pump. The atom sourceis connected to the main chamber by an all-metal valve (VAT 48124-CE01-0001) that allows us tovent the main chamber without compromising the vacuum inside the atomic source chamber andoxidizing the strontium in the oven.
The main chamber is a Kimball physics 8 inch spherical octagon (MCF800-ExtOct-G2C8A16).
This chamber houses the optical cavity and provides ample optical access. All windows have abroadband anti-reflective coating provided by TAKOS with a specification of less than 1 percent
reflection for wavelengths between 461 and 1064 nm.
The pump appendage is connected to the bottom 8 inch conflat flange of the main chamber.
The pump appendage contains an angled mirror to redirect the vertical MOT beams through thecenter of the cavity spacer. Pumping is performed by a 75 l/s ion pump (gamma vacuum TiTan

36
ion pump, 75S). We have also installed a titanium sublimation pump, but did not notice a majorchange in vacuum levels from running it. Typical pressures measured by the main ion pump arein the mid 10−10 Torr range, sufficient for our needs. This level of vacuum is probably limitedby the relatively large amount of macor, epoxy, viton and Kapton coated wire associated with thecavity and its support structure, and by the relatively low bake temperature that these components
require.

3.3

Atomic Source
We used an atomic source provided by AOSense, Inc. This source consists of an oven thatprovides a collimated beam of hot strontium atoms, a permanent-magnet Zeeman slower, two 2D
MOT’s with in-vacuum permanent magnets and optics, a heated saphire window upon which thebeam of strontium atoms impinges, and necessary vacuum pumps for UHV operation. During
typical operation, the pressure inside the source chamber is 3 × 10−11 Torr, lower than the pressurein our main science chamber. Operating details are given in Section 3.5.2.

3.4

MOT Coils and Driver
Because of the relatively large size of the main science chamber and the broad linewidth of

the 461 nm cooling transition (32 MHz), the MOT coils ended up being quite serious. The goal was
to be able to create magnetic field gradients of up to 50 G/cm without the coils getting hot, as this
could heat the chamber and the cavity spacer, causing drifts in beam alignment and in the cavity
resonance frequency. The coils must also be capable of switching off within a few milliseconds.
I wound the MOT coils out of 3/16” hollow square wire potted with Araldite 2011 epoxy. The
hollow core of the wire allows chilled water to flow through the coils. Each of the two MOT coils
has an inner diameter of 8.25”, an outer diameter of 10.5”, and a length of 1.75”. The centers of thecoils are spaced by 6”. These dimensions were chosen to maintain a relatively homogeneous field
gradient (by staying near an anti-Helmholtz configuration), while fitting over the 8 inch flanges ofour chamber. Each coil has 60 turns, comprised of three sets of 20 turns that are wired electrically37
in series by jumper bars, but whose coolant passages are wired in parallel. Running the coolant
through three parallel paths increases the total coolant flow by a factor of nine relative to a single
path. In current operating conditions, the coils are wired together in series. We flow 80 Amps
through the coils to generate a gradient in the vertical direction of 30 G/cm, and half that in theother two directions. Heating of the coils is negligible under these conditions.
Three sets of bias coils in near-Helmholtz geometry provide uniform bias fields in threeorthogonal directions. Each coil that produces a horizontal field has 51 (yes, 51) loops of 14AWG
copper wire. A closeup of the coils and support structures is shown in Fig. 3.4. The vertical bias
coils (50 turns each) are wound around the MOT coils for structural support and cooling. The MOT
coils are held in place by Garolite (cotton fabric impregnated with phenolic resin) supports. This
material is chosen because it is nonmagnetic and non-conductive, which means it will not support
eddy currents when the coils are switched on and off. The bias coils are wound on aluminum
supports, which are connected in the corners by Delrin plastic blocks so they do not form complete
conductive loops (also to eliminate eddy currents).

Figure 3.4: Closeup of MOT and bias coils. MOT coils (large circular coils) are wound from hollow
square wire through which water flows. The vertical bias coils are wound onto the MOT coils
for support and cooling. The two sets of horizontal bias coils are wound around square frames.
Additional freely wound coil in the foreground creates an additional gradient along the cavity axis,
but is not typically used.

38

Figure 3.5: Left: diagram of circuit used to control current through MOT coils. Right: actual
circuit used to control current through MOT coils. All high-power circuit elements are mounted
to a water-cooled plate. The MOSFET is the large white component in the top of the image. The
shunt resistor is the large black component below it, and the diodes are below that.

The circuitry that controls and switches the current through the MOT coils also requires
some degree of care in its design, and is shown in Fig. 3.5. The amount of current flowing throughthe coils is controlled by a very large MOSFET (microsemi APTM50UM09FAG, rated for 500 V,
497 A, 5000W dissipation), and sensed by a 0.5 mΩ shunt resistor (Ohmite TGHGCR0005FE, 1%,
60ppm/C) in series with the coils. A JILA coil driver with PID controller provides a control voltage
to the gate of the MOSFET in order to drive the current through the shunt resistor to a desired
level. The control electronics are mounted to a water-cooled plate. Copper bars and plates areused to connect the various components.
When the coils are fully energized, a substantial amount of energy is stored in the magnetic
field. When the coils are turned off quickly, we must provide a path for the current to go, andsafely dissipate this energy. Otherwise, the voltage across the MOSFET would rise until something
fails dangerously. This is the job of the flyback circuit. Currently, the flyback circuit consists of a39
1 Ohm resistor (TAP800K1R0E) and two diodes (UFL230FA60), all in series. Upon initial turnoff,
the voltage across this resistor jumps up to Imax ∗ 1Ohm, where Imax is the maximum current
flowing through the MOT coils. The current through the coils then decays exponentially with atime constant of L/R until the voltage drop becomes dominated by the diodes, at which point
it decreases linearly to zero. In their current configuration, the coils settle from their operating
current to zero in 10 ms, and to 10 percent of operating current in 6 ms. If desired, this could bemade faster by using a larger flyback resistor at the expense of creating a higher voltage spike inthe flyback circuit. Quickly switching the coils also leads to mechanical shaking of the table as wellas pickup on the cavity piezos, so we choose not to be more aggressive with the switching speeds.
As a word of caution, a previous version of the flyback circuit proved unable to handle
the peak power dissipation that we needed, and after six months of operation failed in an open
configuration. The current from the MOT coils then had nowhere to go and exploded the large
switching transistor which was rated for 600 V and 300 A, presumably higher than a human. Thishighlights how dangerous these circuits can be if not properly designed, even when hooked up to a
low-voltage power supply.

3.5

Laser Systems and Laser Cooling
In contrast to Alkali atoms, the more complicated level structure of Alkaline Earth atoms

requires many more laser wavelengths to utilize. The laser systems used in our experiment aresummarized in Table 3.1, along with their common functions. Details on the various systems
follow.
An energy level diagram with relevant states and transitions is shown in Fig. 3.6. Dipoleforbidden transitions, which are of particular interest for this thesis, are shown as dashed lines.
Spin singlet states are shown on the left, and spin triplet states are shown on the right. Transitions
shown between states of the same total spin are dipole allowed, and have decay rates of at least
several MHz, while transitions between singlet and triplet are dipole forbidden and decay much
more slowly.

40
Transition

Laser systems
Wavelength (nm) Functions

1 S to 1 P
0
1

461

1 S to 3 P
0
1

689

1 S to 3 P
0
0

698

3 P to 3 S
0
1

679

3 P to 3 S
1
1

688

3 P to 3 S
2
1

707

NA

813

Zeeman slower,
2D MOTs, 3D
MOT,
fluorescencenarrow-line
MOT,
opticalpumping, heterodyne
frequency
reference, cavity
probe
seeding superradiance,
heterodyne
frequency
reference
depopulate 3 P0
repumpforsteady-statelasingdepopulate 3 P2

magic wavelength
lattice

Stabilization
Methodpolarization
spectroscopy inhollow-cathodelamp
reference cavity

reference cavity,
fiber from Ye lab

reference cavity
or free-runningreference cavity

FM spectroscopy
in hollow-cathodelamp
science and reference cavities

Table 3.1: Overview of laser systems.

3.5.1

461 nm Laser System

The current generation of the 461 nm laser system used for initial cooling is shown in Fig. 3.7.
It provides power for the Zeeman slower, 2D MOTs and 3D MOT. A master laser (Toptica DL-pro)
is frequency stabilized to a polarization-rotation spectroscopy signal detailed in [102] and modeled
after the system described in [103].
It is desirable to have far more 461 nm power available than our single diode laser can
supply. To achieve this, we use three injection locked “slave” diodes and a second master diode41

Triplet States

m
,1

m

69

8n

68

9n

m,

Hz

7.5
k

Hz

461 nm707 nm
688 nm679 nmSinglet States

Figure 3.6: Relevant energy levels of strontium. Level structure of 88 Sr, which lacks nuclear spin
and therefore hyperfine structure, are denoted with thick lines. When present, additional states
resulting from the hyperfine structure of 87 Sr are shown with thin lines. Zeeman sublevels are
neglected.

laser (another Toptica DL-pro). One of these diodes provides power for the Zeeman slower, thesecond to the 2D MOTs, and the third (seeded by the second master laser) provides power to the
3D MOT. There are two types of injection locked diodes in the current setup, which operate ina very different manner. The diodes used for the Zeeman slower and 2D MOTs do not have anantireflection (AR) coating on their output facet, while the diode used for the 3D MOTs does. All
three are supplied (somewhat unwillingly) by the Nichia corporation (part numbers NDBA116T(AR coated) and NDB4216E (non AR coated)).
The diodes without the AR coating have a typical lasing threshold of roughly 30 mA. We

42Main Switchyard

Fiber Input/Output
To Spectroscopy(+50 MHz)

Master 1 ECDL(+50 MHz)AOM -270 MHz

To Beatnote Lock(-210 MHz)

Laser/Slave Diode
Beamsplitter Cube

QWP

Opto Isolator
To Slave 1
Zeeman Slower(-490 MHz)

Acousto-Optic
Modulator (AOM)

To Slave 2
2D MOTs (+50 MHz)

Slave 2 (2D MOT)
From Main
Switchyard

Master Laser 2

To Beatnote Lock(+50 MHz)

Master 2 ECDL(+~50 MHz)To Slave 3(-30 MHz)To 2D MOTs
AOM -80 MHz

Non-AR CoatedSlave DiodeAOM -80 MHz

Slave 1 (Zeeman Slower)
From Main
Switchyard

Slave 3 (3D MOTs)
From Master
Laser 2

To Slower
Non-AR CoatedSlave Diode

AOM(-80 MHz, Chopped at 100kHz)

90/10 BS
Pump
Probe

AR Coated
Slave DiodeTo 3D MOTs
(1x3 Fiber Splitter)

QWP
Hollow Cathode
Lamp

Differential Photodiode(Polarimeter)
HWP

Figure 3.7: The 461 nm laser system consists of two master diode lasers and three slave diodes. The
frequencies of the two diode lasers are stabilized to a polarization-rotation spectroscopy setup, andby a beatnote lock between the two lasers. The three slave diodes provide power for the Zeeman
slower, 2D MOTs, and 3D MOTs respectively.

typically operate them around 140 mA. In the absence of seed light from the master laser, thediodes emit with an ill-defined frequency. When roughly 0.5 mW of seed light is applied the output
power of the diode remains roughly the same, but the frequency of the emitted light matches thatof the master laser if the current applied to the slave diode is within a certain range. These diodes
output roughly 100 mW, giving them an effective gain of around 200.
The diode with the AR coating has a lasing threshold far above our operating current. In
the absence of seed light, the diode emits negligible amounts of light. When seed light is applied,the diode acts to amplify it by a factor of around 5. A similar configuration is described in [103].

43
The non-AR coated diodes thus have a much higher effective gain than the AR coated diodes,
making them a far more efficient way of amplifying optical power. They are also much cheaper($3300 instead if $7600 at the time of purchase). The downside of the non-AR coated diodes isthat they are much less stable to operate, and I would not recommend them for use at 461 nm.
The range of current, temperature and input power over which the diodes will injection-lock isquite narrow and also highly hysteretic. Drifts or transient changes in any of these parameters willcause the diode to either drop out of lock entirely or to operate in a partially injection-locked state,
where only a portion of its power is at the correct frequency. Further, these properties vary greatly
with individual diodes. We have one diode that will stay injection locked for days at a time. Other
diodes, under seemingly identical operating conditions, drop at an unacceptably short timescale ofa few hours or less. We have installed these diodes in Thorlabs laser diode mounts (TCLDM9), andalso in a custom mount designed for thermal and mechanical stability. The behavior was largely
unmodified with the new mount, and seems to be more a property of the diode than the mount.
Other groups have reportedly had both better and worse success with these diodes. It is unclear
whether this is due to variation between individual diodes or some element of the experimental
design.
These frustrations lead us to replace the most critical of these diodes (the 3D MOT diode)with an AR coated version. Because of the lower gain of the AR coated diode, this required asecond master laser devoted entirely to providing seed light for this diode. This second master
laser is stabilized to the first master laser by a beat-note lock. This has the added advantage ofallowing us to jump the frequency of the MOT beams quickly and precisely over at least 10’s of MHz
during the experimental sequence, which has proved useful for fluorescence imaging. Because theAR coated diode is operated below threshold, it cannot drop out of lock, and we can be confident
that all emitted power is actually at the correct frequency.

3.5.2

Initial Slowing and Cooling44

Figure 3.8: Closeup of AOsense source, with optics for Zeeman slower (far left) and 2D MOTs. The
thin tube with bellows visible in the lower left corner feeds atoms into the main vacuum chamber.

45
Role
Zeeman Slower
2D MOTsBlue MOT
Red MOT
3 P Repump
2
3 P Repump0

Key Beam Parameters
WavelengthPower461 nm
44 mW
461 nm
15 mW each461 nm
13mW each
689 nm3 mW each
707 nm2 mW
679 nm2 mW

Beam Diameter
8 mm
5x25 mm
10 mm
10 mm
5 mm
5 mm

Table 3.2: Cooling and trapping beam parameters.

The atoms begin at an initial temperature of roughly 500 C inside the AOSense oven, thenexit the oven through a nozzle consisting of an array of narrow capillaries that collimate the atoms
into a narrow beam. A portion of the atomic beam is slowed in the Zeeman slower to roughly
50 m/s, and is redirected into our main chamber by two 2D MOTs, whose axes are at small angles
from the thermal beam. AOSense provides in-vacuum magnetics and optics for these. We provide
light for the Zeeman slower and 2D MOTs from external optics, as shown in Fig. 3.8. Both Zeeman
slower and 2D MOTs require circularly polarized light. We use powers of 44 mW in the Zeeman
slower, and 15 mW in each of the 2D MOTs. All slowing light up to this point is near resonancewith the 1 S0 to 1 P1 dipole allowed transition near 461 nm, which has a linewidth of roughly 32 MHz.
Key beam parameters for lasers involved in slowing and cooling are summarized in Table 3.2.
At this point, the atoms have entered the main chamber, where we capture and cool themusing a standard 3D MOT formed again on the 461 nm transition. The MOT beams here have adiameter of roughly 1 cm, limited by the optical access afforded by the cavity spacer (See Fig. 3.12
for geometry). The beams are retro-reflected after passing through the vacuum chamber, with thetypical power in a single pass of around 13 mW. A 1X3 fiber splitter from Evanescent Optics isused to efficiently distribute the power.

3.5.3

Repump Lasers at 707 and 679 nm

The 1 S0 to 1 P1 transition is not perfectly cycling. There is a branching ratio of roughly
2 × 10−5 to the 1 D2 state [86], which eventually decays to 3 P2 . To avoid this source of atom loss,

46
we apply a repump laser to the 3 P2 to 3 S1 transition near 707 nm. This pumps atoms into the 3 P1
and 3 P0 states. 3 P1 decays quickly (20 µs) to the ground state, while 3 P0 is metastable. To avoid
accumulation of population in 3 P0 , we apply a second repump laser to the 3 P0 to 3 S1 transition
near 679 nm. Both lasers are homemade ECDLs using AR coated diodes from Sacher Laser.
The 707 nm repump laser is stabilized to an error signal derived from the same hollow cathodelamp as the 461 nm error signal. While this transition connects two excited states, there is sufficient
population of 3 P2 in the hollow cathode lamp to stabilize a laser, presumably because of electron
collisions. We have detailed the frequency stabilization technique in [102]. Unfortunately, the usefulgeometry of hollow cathode lamp for this application has become very difficult to find.
The 679 nm laser can either be left free-running, or can be locked via a frequency sideband
derived from a fiber modulator to our stable reference cavity.
For 88 Sr, which has no nuclear spin and therefore no hyperfine structure, a single tone at eachof 707 nm and 679 nm is sufficient to provide efficient depopulation of 3 P2 and 3 P0 . In 87 Sr, thepresence of nuclear spin leads to hyperfine structure in 3 P2 , requiring multiple tones to efficiently
depopulate this state. We generate these tones by sending the 707 nm repump light through a fiber
modulator with high power-handling capabilities (AdvR WPM-K0707-P78P78ALO). We apply atone generated by a DDS board (AD9914) to this modulator to create sidebands to address thevarious 3 P2 to 3 S1 transitions. One tone is applied at a time, with eight pre-programed tones
that we jump between at a rate of 100 kHz by toggling the DDS profile pins with an Arduino.
We empirically find that tones at 225, 500, 550, 575, 800, 1150, and 1300 MHz lead to efficient
depumping, with the carrier tone at roughly 40 MHz red relative to the 3 P2 to 3 S1 transition in
88 Sr because of various AOM offsets.

3.5.4

689 nm System and Narrow-Line Cooling

After initial slowing and cooling using the 461 nm transition, we use the narrow linewidth(γ = 2π × 7.5 kHz) transition at 689 nm to provide further cooling and to load the atoms into our
optical lattice. We use a new form of MOT that we have developed to do this, as described in

47
689 nmIF ECDLTo MOT
Switchyard

TA

EOM
To Transfer
CavityEOM

Fiber Input/
OutputScience
Cavity Probe

Laser/
TA
Beamsplitter
Cube

689 nm MOT Switchyard
AOM - 80 MHz

From TAEOMAOM -~80 MHz
To Red MOT
Fiber Splitter

To Optical
Pumping

Opto Isolator
Acousto-Optic
Modulator (AOM)Electro-Optic
Modulator (EOM)

Figure 3.9: 689 nm laser system. An interference-filter ECDL from AOSense provides the masterlight for our 689 nm laser system. The laser is stabilized to our stable reference cavity by aPound-Drever-Hall lock on a sideband that provides a tunable offset. Power for the MOT beams isincreased using a tapered amplifier (TA). An AOM both switches the cooling light on and off, andallows us to sweep its frequency in a sawtooth ramp for efficient cooling. The 689 nm laser also
provides optical pumping for state preparation, among other functions.

Chapter 10. This MOT uses standard polarizations and magentic field configurations to operate,
but the frequency of the MOT beams is swept in an asymmetric sawtooth waveform during thetime from when the blue MOT is turned off until the atoms are loaded and cooled into the lattice.
The frequency sweep goes to slightly above the transition between 1 S0 and 3 P1 F = 11/2, andcovers a frequency range of 7 MHz every 50 µs. Using this technique, we avoid the need for astirring laser on the 1 S0 and 3 P1 F = 9/2 transition that is typically required.
The 689 nm laser system is shown if Fig. 3.9. An AOSense interference filter ECDL (IFECDL) provides initial power to seed a tapered amplifier (Eagleyard EYP-TPA-0690, in JILAmade mount). The laser is stablized to our reference cavity (see 3.5.6) with a variable offset
provided by an EOM sideband. The power and frequency of the MOT beams are controlled by
an AOM that shifts the light roughly 80 MHz to the red. Additional light is picked off both
before and after the tapered amplifier to provide optical pumping, cavity probe tones, or for otherexperiment-dependent functions.

48
The red MOT beams are also distributed via a fiber splitter from Evanescent Optics. The red
MOT beams pass through separate polarization optics from the blue beams, and are overlappedwith and separated from the blue beams by dichroic beam-splitter plates before and after the atoms.

3.5.5

698 nm Laser System

The role of our laser system at 698 nm is two-fold. It is used to transfer population between
1 S and 3 P and to establish coherence on this transition, and also as a frequency reference against
00which to measure the frequency of superradiant emission. For population transfer, we use our own
moderately stable laser (the local laser). For a frequency reference, we rely on very stable lightfrom the Ye lab’s stable laser system (the Ye lab laser).
Our local laser is another IF ECDL from AOSense. It is stabilized to our stable reference
cavity via a standard PDH lock, with a tunable fiber EOM providing a frequency offset (see 3.5.6).
A beatnote between our local laser and the Ye lab laser indicates a short-term linewidth of 200 Hz(Gaussian FWHM) when we float the optical table on which the laser and cavity live. The relative
linewidth is 2 kHz when the table is not floating. The 200 Hz FWHM represents an upper boundon the laser linewidth, as it includes noise from roughly five meters of unprotected optical fiber.
A small portion of our 698 nm light is coupled to the science cavity via a second fiber-modulator.
This allows us to apply agile frequency sweeps to a laser tone that addresses the atoms.
The light from the Ye lab comes from a laser stabilized to a 40 cm long ULE reference
cavity [104], which goes by the name “Mike Martin.” We pick off a portion of this light to seed an
injection-lock located in the Ye lab, and send the output to our lab through a 75 meter long fiberthat runs through the ceiling. This injection lock (which uses a non-AR coated HL6738MG diodefrom Thorlabs) is very robust compared to the blue ones, and outputs 67 mW after an isolator. We
use a typical seed power of 400 µW, a home-built diode mount, and a JILA current and temperature
controller.
To eliminate noise associated with vibrations in the fiber that connects the two labs (which
contribute roughly 20 kHz of frequency deviation if left uncancelled, as inferred from the FWHM of

49

Figure 3.10: Fiber-noise cancellation system for clock reference light. This architecture involves
two AOMs — one in the Ye lab (AOM1) and the other in the Thompson lab (AOM2). Light thatis deflected by AOM2 (giving it a frequency offset f ) is reflected off the back mirror of the sciencecavity, and sent back to the Ye lab. This light is overlapped with light that has not gone through thelong fiber, and detected on a fast photodiode. The resulting beatnote (near 2f in frequency) is used
to lock the frequency of a voltage controlled oscillator (VCO). The output of the VCO is divided
by two and used to drive AOM1 at a frequency near f . Any noise present in the fiber leads to a
perturbation of the frequency of the beatnote. Because we use opposite diffraction orders on the
two AOMs, this frequency offset is cancelled by AOM1. Compared to simpler configurations thatuse a single AOM, this configuration has the advantage of being immune to unwanted reflections
before AOM2.

an observed beatnote with our local laser), we retro-reflect a portion of the light back down the fiberand actively cancel the noise due to vibrations using the same architecture as demonstrated in [105].
In order to provide a stable reference for superradiant light emitted from one end of the cavity, we
use the mirror on the opposite side of the optical cavity as a reference surface. This configuration isshown in Fig. 3.10, and provides cancellation of noise associated with both vibrations in the opticalfiber and movement of the cavity. Doppler shifts from cavity motion contribute around 20 Hz ofnoise if left uncancelled.
There is a 2 m optical fiber that transfers reference light to the far side of the optical cavity.
Acoustic noise in this fiber, as well as roughly 2 m of free-space path are imposed on the heterodyne
beatnote. While it is possible that these parts of the path do contribute noise on our reference
laser, we have found no significant increase in noise when we direct a fan over the optical table.

50
3.5.6

Reference Cavity

Figure 3.11: Transfer cavity from Stable Lasers Systems with four sets of mirrors sharing the same
spacer.

All lasers that require precise frequency stabilization (those at 689 nm, 698 nm, 688 nm, andsometimes 679 nm), as well as the science cavity via the lattice laser, are stabilized via standard
Pound-Drever-Hall (PDH) locks to a stable reference cavity from Stable Lasers Systems (SLS).
SLS also provided a vacuum and isolation enclosure for the cavity. The reference cavity has highFinesse (roughly 25000) near both 700 nm and 813 nm. We measure a drift rate of 7 kHz/day,sufficiently low that we typically do not need to account for its drift during a typical dataset. The
reference cavity spacer (a cylinder of 100 mm diameter and length, made from ultra low expansion
glass) is temperature stabilized near its point of zero coefficient of thermal expansion (21C) by
heaters surrounding the vacuum can. Unfortunately for us, the zero-crossing ended up being atroom temperature for our spacer, meaning that we must bias a few degrees away from the optimal
point to actively stabilize the temperature with heaters alone. Nonetheless, the measured drift
rate is adequate for our needs. The spacer has four sets of mirrors mounted to it, which gives us
four reference cavities to which we can stabilize lasers without overlapping beam-paths. We havefound this to be advantageous to avoid issues of cross-talk from multiple lasers addressing the same

51
cavity.
The resonances of the reference cavities are at arbitrary frequencies, so we need a way toprovide offsets to the laser locks of up to half a free spectral range of the reference cavity (FSR =3GHz). To do this, we first send each laser through a fiber-coupled EOM (from EOSpace). We apply
a tone to these EOMs to generate “offset sidebands” on the laser. We lock these offset sidebands
to the reference cavity, and can tune the frequencies of the lasers by changing their frequencies.
We set the modulation index to empirically maximize the size of the first order sidebands. Afterthe fiber-coupled EOM, we send each laser through a free-space EOM. This free-space EOM ismodulated near 20 MHz to provide PDH sidebands to lock the offset sideband to the cavity. The
PDH sidebands on the different lasers are at slightly different frequencies (separated by roughly
1 MHz) in order to avoid coupling between the different locks.

3.6

The Optical Lattice
The optical lattice serves three key functions. First, it provides a way to trap many atoms

within the cavity mode volume, and keep them there for the length of the experimental sequence.
Additionally, the lattice localizes the atoms to a fraction of the optical wavelength along the axisof the cavity, eliminating the effects of Doppler shifts. Finally, the lattice laser provides a way tostabilize the length of the cavity. The lattice light forms a standing wave within the optical cavity,with regions of high intensity and low intensity separated by 813/2 nm along the cavity axis. The
atoms are attracted to regions of high intensity through optical dipole forces, which tightly confines
them in the direction of the cavity axis. In the directions perpendicular to the cavity axis, the cavity
mode has a Gaussian profile with a characteristic size of around 80 µm. This leads to a much looser
confinement of the atoms in the radial directions.
The energy shift associated with the optical dipole potential (also known as an AC Stark
shift) arises from the lattice laser introducing off-resonant coupling between the state of interest(whichever state is populated, often the ground state) and other states. If there is a single transition
involved, the shift of the lower-energy state is given by ∆E/~ = Ω2 /4∆, where ∆ here is the52
detuning of the laser frequency ωL from the atomic transition frequency ωa , ∆ = ωL − ωa , and Ω
is the Rabi frequency associated with the light. If the laser is higher in frequency (blue detuned)than the atomic transition, the energy of the lower state is increased, while if the laser is lower
in frequency (red detuned), the energy of the lower state is decreased, leading to an attractive
potential.
Real atoms have more than one atomic transition, so to find the total energy shift associatedwith a state we must sum up the contributions from all relevant states. For our purposes, we wanta potential that is attractive, and has the same magnitude for the 1 S0 and 3 P0 states. This ensuresthat the atomic transition frequency between these two states is unperturbed by the optical lattice.
This is exactly the same requirement for strontium optical lattice clocks, and has thus been thesubject of extensive research [101, 100]. It turns out that light near 813.4274(4) nm[106] fulfills
these conditions, leading it to be referred to as a magic wavelength. Deviations from the magic
wavelength lead to shifts of the clock transition of 0.07 Hz/(GHz µK). For typical experiments, weTypical Lattice Parameterstransmitted lattice power (mW)
5.3
lattice depth (µK)
100longitudinal frequency (kHz)170
radial frequency (Hz)60
Lamb-Dicke parameter
0.17
atom temperature (µK)
10
rms radial extent (µm)
20
Table 3.3: Typical lattice parameters.

inject several mW of light into the optical cavity (which is described in the next section) to createa lattice with a depth of order 100 µK. The lattice light is supplied by a home-built 813 nm diodelaser, which uses an AR coated diode from Eagleyard.
Along the cavity axis, the atoms are tightly confined in the so-called Lamb-Dicke regime,
where the effects of atomic motion are highly suppressed. This regime is of great importance tothe operation of optical clocks, and so is discussed in great detail in early optical clock theses (for

53
example ref. [106]). Since this is pretty well established by now, I will provide a summary thathighlights a few of the key implications rather than reproduce the full detail of the derivations here.
In the Lamb-Dicke regime, we can describe the atomic motion as a quantum harmonic oscillator along the direction of propagation of the light with which the atom is interacting. The Lambp
Dicke regime is quantified by the Lamb-Dicke parameter η = kx0  1, where x0 = ~/2mωm isthe extent of the ground-state atomic wavefunction along k and ωm is the harmonic oscillator frequency in that direction. Equivalently, the Lamb Dicke parameter can be defined as η 2 = ωR /ωm ,where ωR is the recoil-frequency ~k 2 /2m. The recoil frequency corresponds to the average change
in kinetic energy of the atom when it absorbs or emits a photon. We can thus understand theLamb-Dicke regime in two ways: the atom is localized to much less than the wavelength of thelight, or the energy spacing of the harmonic oscillator is much greater than the recoil energy.
There are two limits in which the Lamb-Dicke regime is important. The first is when an
atom absorbs or emits a photon quickly, for example through spontaneous emission or due to astrong drive. We are interested in the probability that the atom makes a jump from one harmonic
oscillator level to another (assuming that the photon was emitted or absorbed in the direction ofconfinement, for simplicity). In this limit, we must calculate the matrix element associated withthe transition from one harmonic oscillator state to another, coupled by the phase factor associatedwith the photon:
P (n → n0 ) ∝hn0 eıkx |ni

i2

=

h

i2
†
n0 eıη(â+â ) |ni

where â, â† are the raising and lowering operators for the atomic motion. In the Lamb-Dicke regime(η  1), we can Taylor expand this to see that the probability of changing harmonic oscillator states
P (n → n0 ) ∝ η 2 for n0 = n ± 1, and is even smaller for larger changes in motional quantum number.
Changes in the atom’s axial quantum number are greatly suppressed when absorbing or emitting
a photon in the Lamb-Dicke regime.
In fact, the effects of the Lamb-Dicke regime are even more pronounced when the Rabi
frequency Ωd associated with the light that drives the transition between internal states is much

54
lower than the harmonic oscillator frequency of the atom, as is typically the case for optical clocks.
In this regime, not only does the photon lack the spatial variation to efficiently connect different
harmonic oscillator states (the previous effect described), but it may be off resonance from all n → n0transitions but one, as each transition that is accompanied by a change in the harmonic oscillator
level is separated by ωm . This regime is referred to as the resolved sideband limit. For example,
if the drive is tuned on resonance with the n → n transition, then the probability of exciting to
n0 = n±1 is reduced by an additional ωa2 /Ω2d compared to the case of instantaneous transitions. We
typically operate in this limit, meaning that transitions between different harmonic oscillator statesin the axial direction are quite negligible. However, the finite value of the Lamb-Dicke parameter
does lead to a slight reduction in the atom-cavity coupling g.
While we usually wish to operate in the Lamb-Dicke regime, for some applications, we wish to
maintain radial confinement while allowing the atoms to move freely along the cavity axis. We canaccomplish this by driving longitudinal modes of the cavity with opposite parity about the center ofthe cavity, such that the maxima of one mode are aligned to the minima of the other. In practice,
we do this by creating frequency-modulation sidebands on the lattice laser using a fiber-coupledelectro-optic modulator (EOM). If these frequencies are equal to the free spectral range (FSR) of
the cavity, then we primarily drive one mode with the carrier tone, and two modes with opposite
parity with the sidebands. By adjusting the sideband amplitude, we can flatten out the potential
experienced by the atoms. To achieve our desired lattice depth we need more power than the EOM
can handle, so we use a tapered amplifier (TA) after the EOM to increase the total available power.
We have measured atomic lifetimes in the lattice of up to 1.1 seconds when all atoms are in
the ground state, though this value depends in a sensitive manner on the exact operating conditionsof the lattice laser. Because of the relatively narrow linewidth of the optical cavity even at 813 nm,
small frequency excursions of the lattice laser can lead to significant heating. Small tweaks to thelaser current seem to dramatically alter the frequency noise properties of the laser. Because weare most sensitive to intensity fluctuations at even multiples of the axial trap frequency (typicallyseveral hundred kHz), supressing these fluctuations with electronic feedback is difficult.

55
One annoying detail of the far-detuned optical lattice is that because the wavelength of thelattice differs from the wavelength associated with the forbidden transitions that we wish to couple
to the cavity, the coupling between the atoms and the cavity is inhomogeneous. The atoms aretrapped at antinodes of the optical lattice, which are spaced by 813/2 = 406.5 nm, while themagnitude of the electric field associated with the cavity modes at 689 and 698 nm oscillates witha spacing of 344.5 and 349 nm, respectively. The location of the atoms and the cavity mode
antinodes thus get out of phase with each other on a roughly 1 µm length-scale, and so over theroughly 1 mm extent of the atomic cloud can be treated as random. To account for this effect, we
can treat the atoms as though they are randomly distributed along the cavity axis, and assign a
modified coupling rate to atoms at different locations x: g = g0 cos[kx], where k is the wavenumberof the transition of interest (though we typically average over a full cycle anyway so it does notreally matter what k refers to), and 2g0 is the vacuum Rabi frequency for an atom located at an
antinode of the cavity mode associated with the coupling. Depending on the experiment being
performed, this inhomogeneous coupling will lead to either a simple renormalization of quantities
of interest, or to qualitatively different dynamics. I will treat these cases as they arise.

3.7

The Optical Cavity
The heart of the experiment is the optical cavity. The cavity serves two main functions.

First, it supports the magic wavelength optical lattice, and second it provides a mode with which
the atoms interact via either of the narrow linewidth transitions at 689 nm and 698 nm. The cavity
was designed to achieve a moderately high cavity cooperativity parameter (C ' 0.4) at 689 nm and698 nm. This value was chosen with the goal of maximizing the collective cooperativity parameter
N C within other technical constraints. The cavity has a lower (but still substantial) finesse at
813 nm to enable a deep lattice with moderate laser power. The cavity parameters are summarizedin table 3.4.
We could have increased C (which would have been nice) by either using a smaller mode waist
or higher finesse cavity. The smaller waist would be accompanied by a higher atomic density and56

Figure 3.12: Optical cavity assembly and vibration isolation. The cavity consists of two mirrors
separated by 4 cm. The axis if the cavity is horizontal. The mirrors are mounted to piezoelectric
actuators (PZTs), which allow for control of the cavity length. To decouple the cavity from acoustic
vibrations, the cavity spacer sits on viton balls on top of a support block, which also sits on viton
balls.

collision rate, which we thought would be detrimental. Using a higher finesse would have decreased
the cavity linewidth (for a fixed length). In doing so, we would have sacrificed the favorable ratio
κ/γ  1 on the 7.5 kHz linewidth transition, though likely benefitted from the higher product N C
for work on the milliHertz linewidth transition.
We chose a relatively long cavity length of 4 cm. Several factors dictated this choice. First,
we wanted to ensure ample optical access for MOT and other beams. We also chose to recess thecavity mirrors into the spacer to avoid a direct line-of-sight from the atomic source, as we were
worried about coating the cavity mirrors with strontium. These constraints encouraged a relatively
long cavity.
Additionally, we wanted to keep the free-spectral range (FSR, the frequency spacing between
modes of the cavity, equal to c/2L) relatively small. For superradiance experiments using themilliHertz transition, a mode of the cavity near 698 nm must be made resonant with the clocktransition. At the same time, we want to keep the lattice as close to the magic wavelength as

57
cavity spacer
Torrseal epoxy
Macor “hat”

cavity mirror

PZT tube

6ROLG:RUNV6WXGHQW(GLWLRQ)RU$FDGHPLF8VH2QO\Figure 3.13: Mirror assembly. The cavity mirrors are mounted to piezo actuator tubes (PZTs) by anintermediate macor hat. This configuration allowed us to recess the mirrors into the cavity spacer
in order to avoid coating them with strontium, and provided an easy way to align the mirrors. The
two holes at either end of the cavity spacer in which the mirrors sit were drilled at the same time,ensuring that they share a common axis. All other alignments are referenced to those holes. Allconnections are made by very thin beads of Torrseal epoxy. Electrical connections to the inner andouter surfaces of the PZT tubes are made by custom clips (not shown) that attach through a slitin the macor hat (bottom, in image).

possible. In the worst-case scenario, the lattice laser is forced to be one half of a FSR from themagic wavelength. Reducing the FSR thus minimizes the expected frequency offset of the lattice
relative to the magic wavelength.
To compensate for this magic wavelength offset, one could simultaneously drive cavity modes
at 0, ±2 FSRs. For the correct relative power between these modes, one could create an effective
magic wavelength lattice. This leads to the practical constraint that the FSR should be less thanhalf of the bandwidth of available phase-modulators, which would be used to generate the additional
tones. This factor ultimately constrained our cavity length to its chosen value.

The cavity mirrors were supplied by Advanced Thin Films, with transmission specificationsof 95 parts per million (ppm) at 689 nm, 106 ppm at 698 nm and 1339 ppm at 813 nm. The radius

58C
2g0 /2π (Hz)L (cm)
finesse F
κ/2π(kHz)mode waist w(µm)
mirror transmission (ppm)
mirror losses (ppm)
FSR (MHz)

Cavity Parameters
698 nm689 nm
0.4433(4)0.4230(4)4.04 (spec)
11,000 (spec)
4.03517(1)25,570(20)24,700 (100)145.3(1)150.3(4)(measured)(measured)
73.7
73.3
106 (spec)
95 (spec)
30 (estimated)
3714.750(1)813 nm
NA
NA
2346
1600
(spec)
79.6
1339 (spec)

Table 3.4: Key parameters of optical cavity.

of curvature of the mirrors is 5 cm. We would expect to achieve a finesse of 33,000 at 689 nm, and29,500 at 698 nm. We measured cavity linewidths of 150.3(4) kHz and 145.3(1) kHz at 689 nm and698 nm respectively, implying finesses of 24,700(100) and 25,570(20). These measurements imply
losses of around 30 ppm per mirror.
The length of the optical cavity can be tuned by roughly two FSRs using piezo actuators(PZTs) on each of the two mirrors. The PZTs are cut from tubes of PZT-5H material supplied by
Piezomechanic GMBH, and are driven with a voltage of 0-120 V. The cavity mirrors nest inside of
the PZT tubes and are recessed into the cavity spacer. This assembly is shown in Fig. 3.13, anddescribed in detail in the figure caption.
To enable additional slow tuning of the cavity length, we have installed three automotive
tail-light bulbs that warm the cavity through the large viewport. These are controlled by a tunable
power supply, and can tune the cavity length by several FSR, enabling better alignment of thelattice to the magic wavelength. The settling time associated with this tuning is very slow (several
hours), so we have not been using them recently.
The ability to tune and stabilize the resonance frequency of the science cavity is critical forour experiments. A convenient way of doing this is to use the optical lattice.
The lattice laser frequency is locked to the resonance frequency of the science cavity using astandard PDH lock. This lock is performed with fairly high bandwidth (roughly 100 kHz unity gain

59
frequency). Some power from the lattice laser is split off and sent through a fiber-coupled EOM tothe stable reference cavity. As described earlier, this EOM generates offset sidebands at frequenciesof up to a few GHz. Additionally, we use a free-space EOM to apply sidebands at roughly 10 MHz,
which are used to generate PDH error signals on the carrier and each of the offset sidebands. We
stabilize the frequency of the science cavity to the reference cavity by feeding back on the sciencecavity PZTs based on a PDH error signal generated from one of the offset sidebands. By tuning thefrequency of the offset sideband, we can then control length of the science cavity in a well-definedmanner.
The bandwidth with which the science cavity frequency is stabilized relative to the reference
cavity is limited by the PZTs to which the cavity mirrors are mounted. Before installing the sciencecavity, I measured the resonance response of the cavity mirror/PZT system by forming a Michelsoninterferometer that included one of the cavity mirrors. The first mechanical resonance appeared at
10 kHz, which limits feedback bandwidth to the PZTs to roughly 200 Hz before oscillations become
apparent at 10 kHz, if a simple PI controller is used. This dramatically limits the degree to which
we can suppress harmonics of 60 Hz that appear on the output of our high voltage source (unless we
decided to do something fancy [107]). These harmonics lead to cavity frequency excursions of order
5 kHz. These have not been a problem for us so far, but could be improved for future applications.

Chapter 4

Strong Collective Coupling on 7.5 kHz Linewidth Transition and
Nondestructive Atom Counting

The first scientific project on the new system was the first demonstration of strong collective
coupling between an optical cavity and atoms, via a forbidden optical transition [20]. Typically,strong transitions are used in cavity QED experiments to couple the atoms to the cavity field. In
the optical domain, this means using dipole-allowed transitions, while in the microwave domain
it means using Rydberg atoms with large dipole moments. Here, we instead demonstrate strong
collective coupling using the dipole forbidden 1 S0 to 3 P1 transition in 88 Sr. This transition has awavelength of 689 nm, and a decay linewidth of 7.5 kHz, roughly a factor of 1000 slower than adipole allowed transition of the same wavelength.
Demonstrating strong collective coupling in our system is of particular interest, as strontium
atoms (and other Alkaline Earth-like atoms with similar structure) have proved very useful for
quantum metrology [108]. Example technologies include optical lattice clocks [55, 27, 29] and ultranarrow lasers [109, 110, 111, 112], along with their associated broad range of potential applications[113, 114, 115, 116, 117, 118].
Strong coupling on dipole-forbidden transitions in a system like ours enables state-selective,
non-destructive counting [119] of these useful species of atoms. Such counting has been used togenerate highly spin-squeezed states that surpass the standard quantum limit on phase estimation[33, 37, 120, 121, 122, 123, 124]. If applied to optical lattice clocks, this could provide a rareopportunity to use entanglement to enhance the performance of a range of precision measurements.

61

69
cloc 8 nm
k tra
nsit
ion

461 nmcavity mode
689 nmb.

fluorescencea.

Probe Pol.

Figure 4.1: (a) Experimental diagram. 88 Sr atoms are confined by 1D optical lattice in high-finesse
cavity. π polarized probe light is coupled through cavity and is detected on a single-photon counting
module. This probe interacts only with the 1 S0 to 3 P1 , mJ = 0 transition. A magnetic field B
shifts the mJ = ±1 Zeeman sub-levels by ±15 MHz. (b) Relevant level structure of 88 Sr, showing
dipole forbidden transitions at 689 nm and 698 nm, and dipole allowed transition at 461 nm.

More simply, but quite importantly, non-destructive readout methods [125] can reduce the highly
deleterious effects of local oscillator noise aliasing [126, 35] that limit current state of the art opticallattice clocks.
In this work, we utilize the vacuum Rabi splitting (a key signature of strong collective coupling) to make non-destructive differential atom number measurements at the equivalent of subprojection noise sensitivity. We verify that the number of photon recoils imparted per atom toreach this sensitivity is much less than one (0.01). This is a key criteria for non-destructiveness, as
it characterizes the degree to which atoms are heated by the measurement regardless of trap depth,
as well as the measurement’s applicability for the generation of spin-squeezed states[88].
Cavity-enhanced nonlinear spectroscopy [127] of the same transition has been performed in a
MOT in [98, 97], but inhomogeneous doppler broadening prevented the observation of a collective
vacuum Rabi splitting. Non-fluorescence based atom-counting in 87 Sr [125, 126] and 171 Yb [128]have been performed using the dipole-allowed transition 1 S0 to 1 P1 and no optical cavity. Atomic

62
projection noise level sensitivity was achieved, but with the scattering of ∼ 100 photons per atom
[125, 126] and with uncharacterized scattering [128]. While [125, 126] demonstrated that the atoms
were retained in the lattice trap, the need to re-cool the atoms leads to additional stochastic
losses [129]. Finally, at values of ' 0.5 recoils per atom one would be unable to generate entangledstates for fundamental enhancements in measurement precision beyond the standard quantum limit[130, 88].

4.1

Demonstration of Collective Strong Coupling
As noted in Chapter 2, there are certain aspects of strong collective coupling that become

hard when using a dipole-forbidden transition, and others that do not. For the work we present
here, two conditions of the strong collective coupling regime must be met. The first is that the√collective cavity cooperativity parameter N C = Ω2 /κγ  1. As a reminder, Ω = 2g N is thecollective vacuum Rabi frequency, κ is the cavity linewidth and γ is the atomic transition linewidth.
This condition is required in order to perform atom-counting measurements compatible with spin
~ and γ ∝ |d|~ 2 , where d~ is the dipole matrix element of the transition
squeezing. Because g ∝ |d|which is much smaller for dipole-forbidden transitions than for dipole-allowed transitions, C isactually independent of the strength of the transition, and attaining N C  1 is not harder whenusing a dipole-forbidden transition.
A second requirement for the observations that we present here, and a second definition of
the strong collective coupling regime is that the collective vacuum Rabi frequency must exceed
relevant decoherence rates in the system: Ω  κ, γ. This requirement allows for the observationof a resolved vacuum Rabi splitting. Fundamentally, Ω  γ is actually easier to achieve on a
~ and γ ∝ |d|~ 2 . Achieving Ω  κ is harder, and requires the use of
weak transition, as g ∝ |d|a relatively high-finesse optical cavity. The collective vacuum Rabi frequency must exceed othernon-fundamental sources of broadening as well, notably Doppler decoherence. This requirement ismuch harder to meet for a narrow linewidth transition, and has prevented the observation of thisregime in previous works [98, 97].

63
To confirm that we had achieved this more stringent definition of the strong collective coupling
regime, we show that we can observe a highly-resolved splitting of the normal modes of the coupledatom-cavity system, known as a collective vacuum Rabi splitting [131, 132]. This constitutes adirect verification of the hierarchy Ω  κ, γ.
To do this, we begin by loading atoms into the lattice as described in the experimental
apparatus section. Because we used 88 Sr for this experiment, which has a unique ground state,
no further state preparation was required. We probe the atoms with a weak tone coupled to thecavity that is polarized along the direction of an applied magnetic field, as shown in Fig. 4.1a, andmeasure transmitted power on a single photon counting module (SPCM).
For this work, the probe laser frequency was stabilized to a transfer cavity whose length isactively stabilized by saturated absorption FM spectroscopy on the 1 S0 to 3 P1 , mJ = 0 transition.
A phase-modulation sideband of the 813 nm lattice laser was locked to the stabilized transfer cavity,
providing a tunable offset frequency. The science cavity was frequency stabilized to the lattice laser.
At 689 nm, the cavity has a finesse of F = 2.4(2) × 104 . The atom-cavity coupling is√enhanced by using up to N = 1.25 × 105 atoms, such that Ω = N 2g = 2π × 6.3 MHz, whereg = 2π × 9.2(1) kHz [133, 134]. The quoted N and g are the effective atom numbers and couplings
used to account for inhomogeneous coupling of atoms to the standing wave probe mode, and for thefinite extent of the atomic ensemble [121, 122]. The effective coupling is related to the maximumppossible coupling g0 by g = 3/4g0 . The effective atom number is related to the total number ofatoms in the lattice Ntot by N = 2Ntot /3. Using these definitions consistently reproduces the sizeof the vacuum Rabi splitting and the magnitude of atomic projection noise as though all atoms
were homogeneously coupled. A quick note on cavity linewidths: at the time that I performed thiswork, I had measured the cavity linewidth to be 160(16) kHz. A later, more precise measurement
put the linewidth at 150.3(4) kHz. In this section, I will use the previous measurement, as thatis what appears in the published version. It is important to note here that the key quantitative
claims made in this work depend on g, which is independent of κ, and whose value we can calculate
based on well-known geometric factors up to our knowledge of the transition linewidth γ.

64
Inhomogeneous doppler broadening is highly suppressed along the cavity axis by confining
the atoms in the 1D optical lattice to much less than the 689 nm probe wavelength (Lamb-Dicke parameter of 0.16). We estimate that inhomogeneous transition broadening from the lattice is at most
44 kHz rms. This broadening results from the fact that 813 nm is not a magic wavelength for the
1 S to 3 P transition, which leads to polarization-dependent shifts [135]. Since the inhomogeneous
0
1

broadening is much less than Ω, its impact on the dressed modes is negligible [136].

Transmitted Power
5
3
1
-1
-3
-5
-4

Fluorescence(arb. units)

b.

-2

1
0.5
0

0

2

Probe Frequency δp /2π (MHz)0 1 2 3 4 5Ω/2π (MHz)4c. 160Linewidth(kHz)

Cavity Detuning, δc /2π
(MHz)

a.

120
80
40
0

-6 -2 2 6
δc /2π (MHz)Figure 4.2: Observation of the collective strong coupling regime on the forbidden 1 S0 to 3 P1 transition. (a) We characterize the vacuum Rabi Splitting by recording transmitted power as probe
frequency is swept. Traces correspond to different cavity detunings, δc . (b) Detected fluorescenceversus vacuum Rabi splitting Ω. (c) Fitted linewidths κ0± of the two resonances at ω+ (red) andω− (blue) versus δc , with prediction based on known atomic linewidth γ, measured Ω and cavity
linewidth κ.

Figure 4.2a shows a vacuum Rabi splitting obtained by tuning the empty cavity resonanceωc near the atomic transition frequency ωa , loading atoms into the lattice, and then recording thetransmitted power as a cavity probe is linearly swept in frequency. The two transmission peaks
correspond to the new normal modes of the dressed atom-cavity system at frequencies ω+ and65
ω− . The different traces correspond to different detunings δc = ωc − ωa , and exhibit the expectedpavoided-crossing behavior ω± = (δc ± δc2 + Ω2 )/2 (see ref. [88] for a nice review of the features√of a vacuum Rabi splitting). In Fig. 4.2b, we confirm the N scaling of the size of the normal
mode splitting with atom number by measuring the frequency splitting of the two modes at zerodetuning, and comparing to the subsequently measured free-space fluorescence signal when 461 nm
light is applied from the sides of the cavity.
The fitted widths κ0± of the upper and lower transmission peaks are plotted in Fig. 4.2b,along with a theoretical prediction based on the known atomic decay rate γ and the measured
cavity linewidth κ [131]. The width of each feature is a weighted average of the atomic and barecavity linewidths [88]. The more cavity-like the normal mode is, the closer its width is to κ, while amore atom-like mode has width closer to γ. The fact that the splitting between the normal modes
is always much larger than the width of either mode indicates that we have indeed achieved thehierarchy Ω  κ, γ.

4.2

Nondestructive Atom Counting
We next demonstrated that the strong coupling achieved here could in principle be used toachieve spin squeezing. By measuring the size of the vacuum Rabi splitting, we can infer the atom
number N with great precision. If we can do so in a manner that does not reveal single-atom
information (or equivalently without scattering photons into free space), then this can be used
to create an entangled state. Similar techniques have been used in systems with dipole-allowedtransitions to generate large amounts of spin squeezing [33, 15, 14].
We characterize the non-destructiveness of our differential population measurement by theaverage number of photon recoils per atom ms imparted while achieving a measurement imprecisionequal to the projection noise level of the atoms, mPs N [36, 137].
At the time that we performed these measurements, we did not have the ability to drive theatoms into a superposition state on the clock transition. Even now, we can only do so by taking
advantage of a cavity resonance near the clock transition wavelength, which is incompatible with

66
the measurement scheme we demonstrate here. As a result, the atoms are not in a superpositionof states, so there is no atomic projection noise. However, to give a point of comparison, we define
the projection noise level as though our N atoms were actually 2N atoms, each in a superpositionof the ground state and the excited 3 P0 clock state, represented in Fig. 4.1b. The rms quantumpprojection noise in the ground state population would then be ∆NP N = N/2. We consider the
rms noise ∆N = ∆(Nf − Np ) in the difference of two consecutive measurements Np and Nf of the
population in the ground state 1 S0 . We define the spin noise reduction as R = (∆N/∆NP N )2 . Two
figures of merit for the measurement are the minimum R attainable, and mPs N .
Two consecutive measurements of the vacuum Rabi splitting Ω, with measurement outcomeslabeled Ωp and Ωf , are used to infer the populations of the ground state Np and Nf . For δc =0, the rms fluctuations ∆NP N lead to rms fluctuations in the measured vacuum Rabi splitting√∆ΩP N = g/( 2) = 2π × 6.48(6) kHz, regardless of atom number N , provided Ω  κ, γ [88].
The spin noise reduction can then be written in terms of measured and known quantities as R =(∆(Ωf − Ωp )/∆ΩP N )2 . The vacuum Rabi splitting frequency Ω = ω+ − ω− is determined bymeasuring the transmitted power when probe light is scanned over cavity resonance.
To gain common mode rejection of frequency noise between the cavity and probe laser, we
use two frequency components or tones to simultaneously probe the two dressed cavity modes,
as shown in Fig. 4.3a [122]. We create the two probe frequencies ωp± by amplitude modulatinga fixed probe laser whose center frequency is resonant with the empty cavity. As we ramp themodulation frequency ωm over 1 MHz in 40 ms, the lower and higher frequency AM sidebands ωp±simultaneously sweep across the resonances at ω± .
When the probe tones are on the side of the resonance feature of the dressed modes, changes
in the number of atoms in 1 S0 cause first order changes in the total transmitted power. In thesmall noise limit, relative frequency noise between the probe laser and the empty cavity causes
no net change in the total transmitted power. To demonstrate the noise suppression of this twotone technique, Fig. 4.3b shows the measured power spectral density of cavity-probe frequency
noise, using a single sideband to probe the bare cavity resonance versus using the two tones ωp±

67
simultaneously. The single tone probing exhibits broad band noise and resonances, likely due tovoltage noise on the science cavity’s piezo driver, that are strongly suppressed using the two-tone
technique.
To demonstrate the probing method with atoms, we perform sweeps as described above andfit the total transmitted power to a Lorentzian. We then extract the portion of the time datathat corresponds to probing on side of resonance, defined as the portion between .5 and .75 ×κ/2.
To convert changes in transmitted power into changes in frequency, we fit a linear slope S oftransmitted power versus the known probe frequency. The time data is broken into bins of lengthτ , and the average transmitted power, Pi is computed within each time bin i. Differences aretaken between adjacent bins, and the desired standard deviation ∆Ω = ∆P/S = ∆(Pi+1 − Pi )/S
is computed over non-overlapping pairs of bins.

Transmitted
Power (pW)a.

Relative
Noise Power (dB)

b.

20
15
10
5
0
-2
20
15
10
5
0

0

ω- ω

ωp + ω +

p-

-1

0

1

Single-Tone Probe Frequency (MHz)

2

4

6

8

Frequency (kHz)

10

2

12

Figure 4.3: (a) To probe the Vacuum Rabi Splitting, two probe tones at ωp± are simultaneously
swept across the two normal mode resonances at ω± while total transmitted power is recorded. (b)The noise power spectra relative to photon shot noise for cavity probe with a single tone (red trace)and two tones (black trace), demonstrate noise cancellation of two-tone probing.

Fig. 4.4a shows measurement noise relative to hypothetical projection noise R as a functionof the chosen window length τ at N = 9 × 104 . Similar results were obtained for atom numbers
between N = 1.5 × 104 and 1.25 × 105 . At a given atom number, the incident probe power is held

68
fixed at a value that does not saturate the detector or the atomic ensemble. Therefore, increasingτ is equivalent to collecting more probe photons to average down photon shot noise. At large τ ,R rises due to uncanceled low frequency probe noise. For window lengths between 50 and 175 µs,
corresponding to detected photon numbers of 750 and 2650 per window, we achieve measurement
noise R < 1. The lowest value measured is R = 0.58(13), without any detector or shot noise
subtractions applied.
The goal of being in the bad-cavity, strongly-coupled regime is to achieve a high ratio ofdetected photons to free-space scattered photons, which lead to atomic state collapse, heating, andpotentially atom loss. We characterize these effects by measuring how passing photons through thecavity leads to changes in Ω, as shown in Fig. 4.4b. Changes in Ω allow us to infer changes in the√
product g N . These changes could either be due to atom loss or heating in directions perpendicular
to the cavity axis. We make our observations at much higher transmitted probe photon numbers
Mt than those used for observing R < 1 by inserting a third sweep between two probe sweeps used
to measure a change in Ω. From the fit to the change in Ω2 vs Mt , we extrapolate to the typicalnumber of transmitted photons Mt ≈ 5000 in a single window that achieves R ≤ 1. At N = 5 × 104
atoms, we find 0.01 photons scattered per atom/measurement window. This corresponds to atemperature increase for two windows of 8 nK, or in the context of generating entanglement, a lossof atomic coherence of only 1% or 0.09 dB of signal loss due to the first measurement.
We theoretically expect one free-space scattered photon per 11(1) photons transmitted froma single end of the cavity, a value set simply by the ratio κ/2γ. The resulting change in Ω2 predictedfrom radial recoil heating due to scattering is shown as a dashed line in Fig. 4.4b, and is consistent
within the error of the fit to the data.
Because we operate with κ  γ there is little fundamental loss for generating atomic squeezing
or for non-destructive readout from probing with δc = 0, and applying the noise-immune two-tone
technique [88]. This would not be true for the current cavity geometry and finesse on a fully allowed
dipole transition as the magnitudes of the atomic damping and cavity damping would be reversedκ  γ. This is a key advantage of working with a dipole-forbidden transition.

2.0
1.5
1.0
0.5
0.0

0

16
12
8
4
0
50 100 150 200 250
Window Length (µs)

103 msRelative Noise Ra.

69

b. 2 10
Change ² in Ω2

×

−2

1.5
1
.5
0
-.5
0.0

0.4

0.8

1.2
1.6
6
Mt in Additional Sweep (10 photons)
Figure 4.4: (a) Black points show measurement noise relative to hypothetical projection noise inadjacent windows R, versus length of a measurement window. The lowest two points represent
R = −2.40.9
1.1 dB. Red line shows the the number of scattered photons per atom ms in a single
measurement window. (b) Fractional change in Ω2 = 4g 2 N due to additional sweep  ≡ (Ω20 −Ω2s )/Ω20 , versus Mt , the number of photons in the additional sweep. Ωs and Ω0 are the measured
vacuum Rabi splittings with and without the additional scattering sweep. Solid red line is fit todata, with shaded region representing 1 sigma error on the fit. Dashed line represents a prediction
based on the theoretically predicted free space scattering. The number of photons used in two
typical 100 µs windows is represented by the thickness of the left axis.

For certain applications, including spin squeezing, another relevant metric for the nondestructiveness of a measurement is the degree to which it imparts inhomogeneous AC Stark or
light shifts of the ground state 1 S0 . While we cannot measure this directly, we can estimate it byconsidering the AC Stark shifts associated with the two probe tones. For typical probing parameters, the time integrated phase shift contributed by a single probe tone during a measurement
window would be of order π radians for maximally coupled atoms. With our two-tone probing
scheme, the phase shifts cancel due to their opposite detunings from the excited state 3 P1 [88]. If
we assume a power balance at the 10% level, we could expect a phase shift of order 0.1 radians for
maximally coupled atoms, such that the loss of atomic contrast would be < 0.5% or 0.04 dB loss of

70
signal. This cancellation would potentially eliminate the need for a spin-echo sequence to maintaincontrast in an experimental sequence. Cavity-optomechanical effects that can limit the observed
spin noise reduction would be partially canceled as well [33].
From mPs N , we can estimate the fundamental limit of our probing scheme [88]. If technicalsources of noise were addressed, one could achieve 22 dB of spin squeezing at 2 × 105 total atoms.
This fundamental limit on spectroscopic enhancement could reach 30 dB by improving the net
effective quantum efficiency for detecting the probe light from the current 3.5% to a value of
25%. For context, Kevin Graham and Josh obtained a quantum efficiency of 37% on the rubidium
experiment [15].

4.3

Conclusions
While we were able to demonstrate an atom-counting method that is fundamentally compat-

ible with spin squeezing, significant challenges remain to actually use it to create spin squeezing.
The primary challenge is to create the necessary superposition on the optical clock transition. In
addition to a very stable laser, this requires us to address the atoms in a Doppler-shift-free manner.
We can currently do so only along the cavity axis, where the atoms are tightly confined. Thisrequires us to shine the clock light through the cavity mirrors, which only works well when thecavity is on resonance with the clock transition at 698 nm. Because the atom-counting relies on
placing a cavity mode very near resonance with the 1 S0 to 3 P1 transition at 689 nm, we would haveto get very lucky for this to work out with a single cavity.
A way around this would be to use a two-dimensional lattice to provide a direction otherthan the cavity axis along which the atoms could be addressed by the clock laser in a Doppler freemanner. This idea is discussed further in Chapter 12.

Chapter 5

Magnetically Induced Transparency

In this work, we explore a new cavity-enhanced spectroscopy technique for stabilizing a laser
to narrow linewidth optical transitions. We call this technique “magnetically induced transparency”(MIT) in analogy to the much-studied phenomena of “electromagnetically induced transparency”(EIT) [38, 39, 40]. MIT allows us to create a narrow spectroscopic feature from an ensemble ofatoms, whose frequency is set primarily by atomic properties, making it suitable for the stabilizationof optical frequencies. EIT can also be used to generate a narrow-linewidth feature, but its frequency
is set by that of an applied laser, making it unsuitable for optical frequency stabilization. It is theuse of narrow-linewidth, dipole forbidden transitions that makes such applications possible with
MIT. While this work focuses primarily on the utility of MIT for frequency metrology, it may also be
suitable for the photon-storage type applications that have made EIT so famous [41, 42, 43, 44, 45].
I will provide a brief discussion of these applications at the end of this chapter.
As with much of the work in this thesis, this project is motivated by the quest to create
lasers with ever-higher frequency stability, as well as novel kinds of absolute frequency references.
Currently, the frequency stability of state-of-the-art lasers is limited by thermal fluctuations in thereference cavity mirror coatings, substrates, and spacer [59]. Further, the frequency of a laser is not
fixed in absolute terms — the reference cavity frequency is arbitrary and can drift in an unbounded
manner. One approach to getting around these limitations is to create systems whose frequency isreferenced to narrow-linewidth optical atomic transitions. These techniques can take several forms:
active frequency references (superradiant lasers [63, 24, 25, 51]), which will be discussed in detail72
in later chapters, and passive frequency references.
Depending on one’s desired tradeoffs between accuracy, complexity, and bandwidth, passive
optical frequency references may take the form of a fully fledged optical clock, operated in apulsed mode, or a continuously operated interferometer using an atomic beam [138]. These freespace beam solutions are appealing for thier simplicity, but face challenges resulting from motional
effects, limited optical depth, and the difficulty of collecting photons emitted in random directions.
To mitigate these effects while keeping the system simpler than a state-of-the-art clock, cavity
enhanced spectroscopy techniques have been proposed and developed for narrow-linewidth opticaltransitions [127, 98, 139].
In these techniques, an ensemble of cold atoms is placed inside an optical cavity, such thatlight incident on the cavity will bounce back and forth many times through the atoms, increasing
the effective optical depth. Because the light is incident from both sides, certain errors resulting
from atomic motion are suppressed. Further, because the cavity imposes a resonance condition on
the system, the spectroscopic features that emerge are the result both of atomic absorption and of
dispersive phase shifts caused by the atoms. MIT represents a new variant on these cavity-enhanced
spectroscopy techniques.

5.1

Basic Mechanism
Very roughly (a detailed description follows), our technique works by placing atoms inside

the mode of an optical cavity that is nearly resonant with a set of transitions between a uniqueground state and several Zeeman sublevels of an excited state. A probe laser is shone through thecavity with a polarization that couples the ground state to two of the excited states that experience
opposite frequency shifts in the presence of a magnetic field. If the cavity is placed on resonancewith the zero-field frequency of the transitions (which are the same), very little light will get throughthe cavity, as the atoms efficiently absorb and scatter the light into other directions. If a magnetic
field is applied, it opens up a window between the two transitions, allowing light to circulate in
the cavity without being absorbed, and creating a transparent window between the two atomic

73
absorption features.
If this experiment were performed in free-space (no cavity), the width of this window would
increase without bound with the magnetic field, making it a poor spectroscopic feature. If a large
enough field were applied such that the probe were transmitted through the ensemble, the frequency
width of the transmission window would be too wide to serve as a useful frequency reference.
The key role of the cavity is to impose a strict requirement on the round-trip phase shift of thecirculating light. For light with a frequency exactly between the two atomic transitions, the phaseshifts contributed by the two transitions perfectly cancels, allowing the cavity resonance condition
to be fulfilled and light to be transmitted. For probe frequencies not exactly centered between
the two atomic transitions, the phase shift from the closer transition exceeds that of the farthertransition and the two no longer cancel, breaking the cavity resonance condition. For appropriate
system parameters, this leads the cavity resonance condition to be dominated by the atoms, rather
than the cavity length. This can make the transmission feature narrower with atoms than without,and can make the frequency of the transmission feature largely insensitive to changes in the cavity
length.
As in the previous chapter, we create a strongly coupled atom-cavity system by loading up
to N = 1.3 × 106 88 Sr atoms (we got much better at this since the last chapter) into our usual 1D
optical lattice supported by the high-finesse optical cavity. The peak trap depth is 100(10) µK and
the atoms are laser-cooled to 10(1) µK (see Fig. 5.1a. We tune a TEM00 resonance of the cavity
at frequency ωc to be near resonance with the dipole-forbidden γ = 2π × 7.5 kHz linewidth 1 S0
to 3 P1 transition at frequency ω0 or wavelength 689 nm (see Fig. 5.1b). The cavity decay rate at
this wavelength is κ = 2π × 150.3(4) kHz. In 88 Sr, the absence of nuclear spin means that the 1 S0ground state is unique, while the 3 P1 excited state has three Zeeman sublevels (see Fig. 5.1b.)
In the limit of zero applied magnetic field, our system responds to an applied probe as thougheach atom were a simple two-level system. The ensemble can collectively absorb light from and then√collectively reemit light into the cavity mode at the collective vacuum Rabi frequency Ω = N 2g.
Here 2g/2π = 15 kHz is the rms value of the single-atom vacuum Rabi frequency, which accounts for74
averaging over the standing-wave cavity mode. This exchange creates two new transmission modes(the usual vacuum Rabi splitting) that are shifted away from the empty cavity’s transmission peakby ±Ω/2, as shown by the central red trace in Fig. 5.2a.
Because two orthogonal components of the probe light can couple to two distinct Zeeman
sublevels, the coupled atom-cavity system has three normal modes of excitation, not two (seeFig. 5.1b and c.) In addition to the two modes at ±Ω/2 that lead to the transmission peaks, which
we will refer to as the ‘bright’ modes, there is a third ‘dark’ mode whose frequency is equal to thatof the atomic transition. The dark mode is composed of an equal superposition of the two atomic
excited states and a photonic component that vanishes as the magnetic field approaches zero.
By applying a magnetic field B, we can mix photonic character into the dark mode, inducing transmission (also refered to as transparency) for probe light near ω0 . The probe light ishorizontally polarized and is perpendicular to the vertically oriented magnetic field. Each trace
in Fig. 5.2a corresponds to a different applied magnetic field with strength parameterized by the
induced Zeeman frequency splitting ∆/2π = (2.1 MHz/G)B.
To describe the system, we extend the linearized input-output equations of [88] to include an
additional atomic transition written in a rotating frame at the average atomic transition frequency
ω0 as:11ȧ = − (γ + ı∆) a − ı √ Ω c
2
2 2
1
1
ḃ = − (γ − ı∆) b − ı √ Ω c
2
2 2
√
1
1
ċ = − (κ + ı2δc ) c − ı √ Ω (a + b) + κ1 ci eıδp t .
2
2 2

(5.1)
(5.2)
(5.3)

Here, δc = ωc − ω0 is the detuning of the cavity resonance frequency ωc from atomic resonance, Ω isthe observed collective vacuum Rabi splitting when ∆ = 0, γ is the decay rate of the excited atomic
states, κ is the cavity power decay rate, and κ1 is the coupling of the input cavity mirror that isdriven by an externally incident probe field with complex amplitude ci and at a probe frequency
ωp and detuning from atomic resonance δp = ωp − ω0 . The complex variables a = hâi, b = hb̂i,

75

(a)

SPCM

(b)

Figure 5.1: (a) Simplified experimental diagram. The system is probed with horizontally polarizedprobe light. The light can be coherently absorbed by the atoms (brown ovals) and reemitted into
the cavity at collective vacuum Rabi frequency Ω. The transmitted power is detected on a single
photon counting module (SPCM). A magnetic field is applied along the vertical direction. (b) The
atomic energy level diagram of the ground 1 S0 and excited states 3 P1 |mj i with quantization axis
~ This is the viewpoint adopted in this letter. Here, theq̂ parallel to the applied magnetic field B.
applied magnetic field creates a Zeeman splitting ∆ between the excited states |±1i. Both of thesetransitions interact equally with√ the horizontally polarized light inside the cavity with collectively
enhanced Rabi frequency Ω/ 2. The transmitted probe light is measured as the probe’s detuning δp
is swept. The |0i state is shown, but it does not interact with the horizontally polarized cavity-field.

c = hĉi are expectation values of bosonic lowering operators describing the cavity c, and collective
excitations of the two atomic transitions a and b. The required Holstein-Primakoff approximation
assumes weak excitation such that the number of atoms in the excited states Ma = |a|2 , Mb =|b|2  N is a small fraction of the total atom number N . The average number of photons in thecavity is given by Mc = |c|2 , and the complex field transmitted through the cavity is ct =

√

κ2 c

with κ2 the coupling of the output mirror. The transmitted probe power relative to incident probe
power is PT = |ct /ci |2 and the relative phase is ψ = arg (ct /ci ).

76

Figure 5.2: (a) The transmitted power through the cavity versus the probe detuning δp , with δc = 0.
Each trace was taken for different applied magnetic fields, creating different Zeeman splittings ∆labeled on the vertical. The central red trace is taken for ∆ = 0 and displays a collective vacuum
Rabi splitting Ω/2π = 5 MHz. When a magnetic field is applied perpendicular to the probe
polarization, inducing a Zeeman splitting ∆, a new transmission feature appears in between thetwo original resonances of the vacuum Rabi splitting. (b) Linearized theory showing the power PT
and phase ψ of the transmitted light, plotted here for Ω/2π = 5 MHz and ∆/2π = 1 MHz.

Figure 5.2b shows the calculated steady-state transmitted power and phase for a single Zeeman splitting. The phase response changes rapidly near zero probe detuning, which results in anarrow MIT resonance compared to the broad vacuum Rabi splitting or bright modes for which
the phase changes more slowly.
In order to describe the linewidth of the dark state resonance, we introduce a mixing angle θ¯ 2 /(Ω2 + ∆¯ 2 ). Here, the effective detuning is ∆¯ 2 = ∆2 + γ 2 . The character ofdefined by sin2 θ = ∆77
the dark state excitation is given by the ratio of the probability that the excitation is photonic-like
Pc = Mc /(Mc + Ma + Mb ) = sin2 θ versus atomic-like Pab = (Ma + Mb )/(Mc + Ma + Mb ) = cos2 θ.
The dark state excitation can decay into free space at rate Rab or by emission through the cavity¯ 2 , where
mirrors Rc , with the ratio of the rates given simply by Rab /Rc = γ/(κ tan2 θ) = N C(γ/∆)
the single particle cooperativity parameter is C = 4g 2 /κγ.
The linewidth of the dark state resonance can be written as:κ0 = (γ cos2 θ + κ sin2 θ)/b.

(5.4)

The term in parentheses is a weighted average of atom and cavity linewidths that reflects the¯ 2.
character of the mode. The correction factor is b = d cos2 θ + sin2 θ, where d = (∆2 − γ 2 )/∆When ∆  γ, both b and d approach unity. At small detunings ∆ ∼ γ, the responses of the dark
and bright modes to the applied drive become comparable, causing a modification of the correction
factor. In the regime experimentally explored here (b ≈ 1), κ0 is simply the full width at half
maximum linewidth of the power transmission feature. To define a linewidth valid in general, we
define the linewidth via κ0 = 2 (dψ/dδp )−1 |δc =δp =0 . For Ω  ∆  γ, the mixing angle is small
and the linewidth approaches the atomic linewidth κ0 ≈ γ, which can be much narrower than thecavity linewidth κ.

5.2

Experimental Characterizations of MIT
We measure the linewidth for the central dark resonance by linearly sweeping the probe

laser’s frequency over the cavity resonance and recording a time-trace of power transmitted on asingle photon counting module. A Lorentzian is fit to the central feature to extract the full width at
half maximum. This measurement is taken for a range of different Zeeman splittings and vacuum
Rabi splittings by varying the applied dc magnetic field and atom number respectively. Figure
5.3a shows collected data plotted against the theoretical prediction at several Rabi frequencies. Forvery small ∆ the feature becomes increasingly narrow, approaching the atomic transition linewidth

78
γ = 2π × 7.5 kHz. For very large ∆ the feature linewidth approaches the cavity decay rate κ.

(a)

(b)

Figure 5.3: (a) The measured linewidth of the central MIT transmission feature versus the induced
Zeeman splitting between excited states. The traces are taken for three different collective vacuum
Rabi frequencies Ω/2π = 4.6(5) (red), 10(1) (blue), and 16(1) (green) MHz, with values set bychanging the total atom number N . The upper dashed line is the empty cavity’s linewidth κ, andthe lower dashed line is the atomic transition’s linewidth γ. The minimum observed linewidth was
11 kHz. The shaded regions are no-free parameter predictions from the linearized model introducedin the text, indicating the ±1 standard deviation uncertainty bands based on independent measurements of Ω. (b) The measured peak transmitted power of the central MIT transmission feature
for the same collective Rabi frequencies. Here, the transmitted power is normalized to the peaktransmitted power when the cavity is empty. Again the shaded regions indicate the ±1 standard
deviation uncertainty bands for the predictions.

Figure 5.3b shows the peak transmitted power at the MIT feature’s resonance for the samedata show in Fig. 5.3a. The linearized theory predicts that the peak transmitted power is given by:

Pmax =

4κ1 κ2122γ
κ
1 + κ tan
2θ

(5.5)

Note that the term in the denominator above is just the ratio of excitation decay rates Rab /Rc . For
large detunings ∆  Ω, γ, the peak transmission goes to that of an empty cavity Pmax → Pempty =79
4κ1 κ2 /κ2 .
In the regime experimentally explored here, a change in the cavity resonance frequency ωc by∆ωc leads to a change in the dark state resonance frequency ωD by a much smaller amount ∆ωD .
The pulling coefficient P = ∆ωD /∆ωc expresses this ratio. A small pulling coefficient P  1 isdesirable for a frequency reference as it will be less sensitive to thermal fluctuations and technicalnoise on the reference cavity. We can extract a pulling coefficient applicable to all parameter
regimes from the linearized theory by considering how much the probe and cavity detunings would
have to change to create equal changes in the quadrature amplitude of the transmitted field (suchas one might measure using homodyne detection). This general pulling coefficient can be expressed
as:

P =

sin2 θ
.
b(5.6)

In the typical regime of operation (b ≈ 1), this is simply the cavity-like fraction of the dark
excitation.
In Fig. 5.4, we show the measured pulling coefficient versus splitting ∆ for several values ofΩ, along with the predicted pulling coefficients from the linearized theory. The pulling coefficientswere measured by sweeping the probe laser frequency across the dark resonance and fitting thecenter frequency ωD with a Lorentzian fit model. This is then repeated while toggling ωc between
two values separated by 100 kHz, and the pulling coefficient is determined from the change in ωDversus ωc . Our lowest measured pulling coefficients are below P = 0.05.
In principle, ∆ can be reduced further to reach a smaller pulling coefficient, at the expenseof transmitted power. The theoretical pulling coefficient reaches a minimum value of P = 1/(1 +√N/(8Mc )) at ∆ = 3γ, where Mc = (γ/(2g))2 is the so-called critical photon number. The criticalphoton number is proportional to the cavity mode volume and atomic linewidth, but does notdepend on the mirror reflectivity. As a result, small pulling coefficients are reached by working
with small cavity volumes and very narrow linewidth transitions. For spectroscopic applications,

80
one would want to balance the desire for a low pulling coefficient against the need to collecttransmitted photons without inducing heating in the atomic ensemble due to free-space scattering.
The optimal parameter regime will depend on the specific requirements of the system.

1.0

Pulling Coefficient P

0.8
0.6
0.4
0.2
0.00

10

20

30

40

Zeeman Splitting ∆/2π [MHz]50

Figure 5.4: The pulling coefficient P versus Zeeman splitting ∆ for several collective vacuum Rabi
frequencies Ω/2π = 5(1) (red), 10(1) (blue), and 17(1) (green) MHz. The prediction from thelinearized theory is shown with ±1 standard deviation bands.

While the majority of this work has been done with the atoms trapped in the Lamb-Dicke
regime (i.e. confined to much less than the wavelength of the probe light) with respect to the cavity
axis, we have also performed scans of the cavity transmission spectrum in which the atoms wereunconfined along the cavity axis. In this configuration, the rms Doppler shift along the cavity axis
is roughly 45 kHz. Despite this inhomogeneous broadening, we observe a center feature linewidthof 18.5 kHz, which we believe is limited by technical noise on the cavity frequency that arises when
we turn down the lattice depth to release the atoms. We expect the linewidth of the dark feature
to be insensitive to inhomogeneous broadening so long as ∆ is much larger than the inhomogeneous
broadening [136]. This insensitivity to Doppler broadening may make such techniques suitable tocontinuously operating atomic beam experiments, where confining the atoms to the Lamb-Dicke
regime would be challenging.

81

5.3

Mapping to EIT
In this section, we make explicit the connection between magnetically induced transparency(MIT) and a more traditional Raman electromagnetically induced transparency (EIT) system.
Previously, we adopted the perspective where the quantization axis q̂ is taken to be along~ We will call this the ‘MIT picture’. In this picture,
the direction of the applied magnetic field B.
the horizontally polarized probe light can be decomposed into circularly polarized σ ± components√that collectively couple to the |mj = ±1i states respectively with Rabi frequency Ω/ 2. Since thequantization axis is along the magnetic field, these |±1i states have a Zeeman splitting given by∆. The |0i state does not interact with the horizontally polarized light. When no magnetic field isapplied, the dark state mode is entirely atom–like, and no probe light is transmitted through thecavity. However, when the magnetic field is applied, it mixes a photon–like component into thedark state, resulting in transmitted light.
In this picture, we can think of the transmission as resulting from two simultaneous effects.
First, the absorption profiles of the |mj = ±1i states are pushed off resonance with the dark state.
At the same time, the dark state is nearly pinned to the center of the two transition frequencies
because of the opposite dispersive shifts from the two states |mj = ±1i.
However, we may also view the system from an alternative perspective where the quantization
axis is perpendicular to both the cavity axis and the axis of the magnetic field (Fig. 5.5). We willcall this the ‘EIT picture’. For clarity, the excited Zeeman states written in this new basis aredenoted by |mj⊥ i. The probe light is π polarized and collectively couples the ground state to thesingle excited state |0⊥ i with Rabi frequency Ω.
The three states |mj⊥ i are degenerate in this EIT picture, but are coupled to their neighboring√states by the transverse magnetic field with a Rabi frequency ∆/ 2. Because of the symmetry√of the couplings, the magnetic field couples |0i to the symmetric state |Si = (|+1⊥ i + |−1⊥ i) / 2
with Rabi frequency ∆. This symmetric state |Si is not coupled to the cavity since this statephysically corresponds to an optical dipole aligned along the cavity axis. The difference state82√|Di = (|+1⊥ i − |−1⊥ i) / 2 has no coupling to other states, but for completeness we note that thedifference state would correspond to a verical dipole moment that would couple to the verticallypolarized cavity mode.
The ground state, |0⊥ i, and |Si form a traditional three–level EIT system, where the control
optical field has been replaced by a control dc magnetic field. In contrast to a traditional EIT system
however, the state |Si has the same linewidth as |0⊥ i. While there is a coherent cancellation of
excitation to |0⊥ i as in EIT, the state |Si does become populated in our system, and may radiate
into free space. This prevents the observation of spectral features narrower than |0⊥ i, as areobserved in EIT, and motivates the use of narrow optical transitions.

Figure 5.5: In the ‘EIT picture’, the quantization axis is perpendicular the magnetic field and andparallel to the probe light’s polarization. (left) The cavity now couples the ground and excited
Zeeman state |mj⊥ = 0i. The magnetic field couples |0⊥ i to neighboring excited Zeeman states|mj⊥ = ±1i. (center) Equivalently, the magnetic field couples to a symmetric combination of
Zeeman states |Si while the antisymmetric combination |Di does not participate. (right) For
comparison, a traditional Raman EIT system is shown for comparison with the equivalent couplingstrengths labeled.

5.4

Limits on Photon Retrieval Efficiency
EIT systems have been used to store light. The hypothetical photon capture and retrieval

efficiency is PR , defined such that PR = 1 corresponds to perfect photon storage or retrieval
efficiency. Compared to more traditional EIT schemes in which the two ground states are long

83
lived, here the photon storage state |Si decays at the same rate γ as the intermediate state |0⊥ i.
To estimate the theoretical maximum efficiency PR , we adopt the EIT quantization axis. We
assume that a photon is perfectly stored as a collective excitation in state |Si at zero magnetic
field (i.e. ∆ = 0.) The goal is to convert the excitation into a photon in the cavity as quickly as
possible, returning all atoms to the ground state.
To accomplish this conversion, at t = 0 one switches on a large magnetic field ∆  Ω fora duration that creates a π-pulse. This swaps the collective excitation from the state |Si into thestate |0⊥ i. Physically, this corresponds to using a transverse magnetic field to rotate the axis of thecollective optical dipole by 90◦ from being oriented along the cavity (i.e. cannot radiate into thecavity) to being oriented along the quantization axis (i.e. can radiate into the cavity.) The magnetic
field is then returned to zero. We will assume that the magnetic field can be made arbitrarily large
and can be switched on and off arbitrarily fast such that atomic decay at rate γ during this step
can be neglected.
Having swapped the excitation into |0⊥ i, the excitation will now automatically Rabi flop intoa cavity photon and back at frequency Ω. We assume here that Ω  κ, as is the case in our system.
The time to accomplish this second π pulse, converting the excitation into a cavity photon, is justtπ = π/Ω. At tπ one immediately switches the magnetic field back on to a large value ∆  Ω
coupling |0⊥ i to |Si such that the photon in the cavity is no longer re-absorbed by the atoms. The
photon will then exponentially leak out of the cavity at power decay rate κ.
During this second π-pulse, the excitation spends on average 50% of the time as a collective
atomic excitation that decays at rate γ. Accounting for the atomic decay, one finds a retrieval
efficiency PR = 1 − 12 tπ γ = 1 − (π/2)γ/Ω, the result given in the main text. Numerical simulation of
the photon retrieval process indicates that the above expression is accurate to 20% even when Ω = κ.
From time reversal symmetry, photon storage will have the same efficiency. For the current systemparameters, the theoretical limit is PR ≈ 99.8%. It is likely that at this level, other experimental
imperfections (finite magnetic field switching speeds, finite magnitude maximum magnetic fields,
and other) would limit storage and retrieval efficiencies to less than this fundamental limit.

84
We note that here the photon loss scales as 1 − PR ∝p
mc /N where mc is the critical photon

number of cavity QED [134]. This is to be compared to more traditional EIT schemes in which one
assumes κ  Ω and κγ/Ω2  1, in which case 1 − PR ∝ nc /N [140]. Here, nc is the critical atom
number nc = 1/C of cavity QED. The critical atom number does not depend on the strength of theatomic transition, while here, the critical photon number is improved (i.e. made smaller) by using
narrower linewidth optical transitions since mc ∝ γ. Also note that the critical photon number
does not depend on the cavity finesse, so long as the finesse is high enough such that Ω  κ.

5.5Conclusion
To summarize, we have demonstrated a technique to realize a narrow spectroscopic feature

based on collective interaction between an ensemble of atoms and a high finesse optical cavity.
The center frequency of the feature can be made highly insensitive to changes in cavity resonancefrequency. In analogy to EIT, this technique may also be applicable to tasks relevant for information
processing.
The observed linewidth of the feature approaches the natural linewidth of the 7.5 kHz opticaltransition and can be insensitive to inhomogeneous broadening of the atomic transition frequency.
In the future, it might be possible to extend this technique to even narrower optical transitions forenhanced spectroscopic sensitivity in atoms such as calcium and magnesium.

Chapter 6

Lasing on the 7.5 kHz Linewidth Transition: the Superradiant Crossover
RegimeOur first foray into superradiance on narrow-linewidth optical transitions was a study ofsuperradiance on the 7.5 kHz linewidth 3 P1 to 1 S0 transition in 88 Sr. The hierarchy of decay rates
κ/γ ' 20  1 places this laser in the bad-cavity regime with respect to decay rates — excitationsin the system are much more likely to be lost as photons transmitted through the cavity mirrors
than as photons emitted into free space by the atoms. However, for typical atom numbers used in√this experiment, the collective vacuum Rabi frequency Ω = 2g N was several MHz, much greaterthan either κ or γ. This can place the system in a regime where coherent dynamics dominate overdissipation — the resolved vacuum Rabi splitting regime.
Pulsed lasing in this regime (Ω  κ, γ) exhibits interesting ringing behavior associated withthese coherent dynamics as the atoms exchange photons with the cavity mode several times before
they are lost to transmission through the mirrors.
We can also apply repumping lasers to achieve quasi steady state operation. This modifies
the hierarchy of rates, and thus the qualitative dynamics of the system. This occurs because
the repumping process (which involves spontaneous emission) both causes damping of the atomic
coherence and takes the atomic system out of a fully symmetric state (reduces the length of the
Bloch vector J), which reduces the rate of coherent interactions. A more general expression for the√rate of coherent interactions between the atoms and cavities is Ω0 = 2g 2J, which can be much lessthan Ω. By introducing variable rates of repumping, we can continuously tune between the resolved

86
vacuum Rabi splitting regime Ω0  κ, γ to the more traditional bad-cavity regime κ  Ω0 , γ.
For context, the bad-cavity regime of laser physics has been explored in the optical domain ingas lasers [141], with homogeneous and inhomogeneous transition linewidths of hundreds of MHz,and in a 4-level system in Cesium with a Doppler-broadened gain bandwidth of 9 MHz [142]. In
the microwave domain, masers operate deep in the bad-cavity regime [143]. Rare-earth doped solidstate lasers utilize transitions with long-lived excited states, but with inhomogeneous broadening of
hundreds of GHz [144]. Raman dressing has been used to create long-lived virtual states with low
inhomogeneous broadening, which have been used to demonstrate key properties of optical lasing
very deep into the bad-cavity regime [145, 63, 65, 66], and deep into the good-cavity regime [146].
This work is unique in that we explore lasing on a true optical transition in a regime where boththe homogeneous and inhomogeous linewidths of the gain medium can be made small compared tothe cavity linewidth, and in which the rate of coherent interactions between atoms and cavity can

Lasing Transition

be tuned to be either large or small compared to the cavity decay rate.

B

688 nmpump pol.
689 nmpump pol.

Figure 6.1: Energy level and experimental diagrams. The lasing transition is the 7.5 kHz wide
3 P , m = 0 to 1 S transition in 88 Sr. A magnetic field B is oriented perpendicular to the cavity
10J
axis, and pump lasers at 688 and 689 nm are applied with directions and polarizations as shown.
Collectively enhanced emission into the TEM00 mode of the cavity is collected in an optical fiberand sent to one of various detectors.

87
For this work, we used up to N ∼ 100k 88 Sr atoms. The atoms were cooled to 9 µK
and confined as usual in our 813 nm optical lattice (Fig. 6.1). The lasing occurs on the |ei ≡3P , m = 0
1
J

to |gi ≡ 1 S0 transition at 689 nm, which has a natural decay linewidth γ = 2π ×7.5

kHz. A single atom trapped at an antinode of the lasing mode exchanges excitations with the
TEM00 mode at a frequency 2g0 = 2π × 21.2 kHz. Accounting for inhomogeneous coupling
between the atoms and and cavity mode, this frequency is collectively enhanced to as much as√Ω = g0 2N ∼ 2π × (1 to 5) MHz.

Pulsed Lasing in the Vacuum Rabi Splitting Limit

707, 679 nm
repumps

1

nm

MOT

-1

461

Lasing Transition

689 nm repump

688 nm repump

6.1

Figure 6.2: (a) Laser power emitted from the cavity during a collectively enhanced pulse, with
N = 36k atoms. The atoms are optically pumped into |ei at t = 0. (b) Numerical simulation of(a) with no free parameters. (c) The power emitted from the cavity during continuously repumped
lasing with N = 60k atoms. Each atom emits around 35 photons into the cavity before repumpinduced heating causes laser operation to cease. For scale, 1 nW corresponds to 3500 intracavity
photons. (d) Repumping scheme. Atoms are incoherently repumped through 3 P1 , mJ = −1 to
3 S , where they decay into the 3 P manifold. Atoms that fall into 3 P or 3 P are then repumped by1
02
additional lasers via 3 S1 . For display purposes, 3 P0,2 and 1 P1 states are shown at arbitrary vertical
positions.

88First, we characterized pulsed lasing in the resolved vacuum Rabi splitting regime. Whenmany atoms are placed in state |ei, the collectively enhanced coupling to the cavity causes theatoms to quickly decay to |gi by emitting a pulse of light into the cavity. We detect this light on
an avalanche photodiode, with a single trace shown in Fig. 6.2a. In this regime, the collectively
enhanced emission rate exceeds the cavity linewidth, which in turn is much greater than the atomic
decay rate; i.e, Ω0  κ  γ. The light pulse is partially re-absorbed by the atoms, and re-emitted
into the cavity several times before escaping the system via transmission through a cavity mirror,
resulting in oscillations in output power from the cavity, as shown in Fig. 6.2a. The timescale ofthese oscillations is of order 2π/Ω, but because of inhomogeneous coupling to the cavity, the atoms
do not remain in a fully symmetric state. This modifies the period of oscillations and results inincomplete reabsorption of the initially emitted pulse.
By measuring a vacuum Rabi splitting following the pulse, we infer an atom number of
N = 36, 000 for the trial shown in Fig. 6.2a, up to fluctuations between trials of around 20 percent.
We simulate the pulse dynamics for this atom number by integrating a set of optical Bloch equations(see Chapter 2.4 and ref. [64]) that also account for the inhomogeneous coupling to the cavity mode.
The results are shown in Fig. 6.2b, with good qualitative agreement to observed pulses.
We can get some intuition into the ringing behavior from the pendulum model presented in
Chapter 2.4. Neglecting the fact that the atoms are inhomogeneously coupled to the cavity, thesystem is equivalent to an under-damped pendulum that starts out fully inverted. The pendulum
swings downward, turning potential energy into kinetic energy (signifying atomic inversion being
turned into excitation of the cavity field). Because the system is under-damped, the pendulum will
oscillate about its resting position several times before settling.

6.2

Quasi Steady State Operation
We next apply continuous repumping from |gi back to |ei to operate the laser in a steady-state manner. The repumping process both maintains population inversion and causes decay of
the transverse coherence of the atomic ensemble, which reduces the collectively enhanced emission

89rate. When a steady state operating condition exists, the total rate of emission from 3 P1 matches
the single-particle rate w at which atoms are optically pumped out of |gi. The repumping process
also homogeneously broadens the ground state to a width w. When operating with w  κ, we
access the bad cavity or superradiant regime, where the cavity decay rate exceeds the collectively
enhanced emission rate, other damping rates, and the bandwidth of the gain medium.
Fig. 6.2c, shows the laser output power for a representative run of the experiment, with
repumping applied to an ensemble of N ' 60k atoms initially in |ei. We can sustain lasing for up
to 1.5 ms, with higher power operation possible for durations of around 500 µs, as shown. In such a
trial, each atom emits up to 35 photons into the cavity before lasing ceases. This seems consistent
with heating we would expect from photon recoils during the repump process.
In addition to operating in the bad-cavity regime defined by κ  γ, our laser operates in an
unusual regime with respect to inhomogeneous broadening. In contrast to gas and solid state lasersthat operate on similarly long-lived transitions, our laser has very low inhomogeneous broadening.
We should compare the spread in frequencies associated with this broadening to the collectively
enhanced emission rate, which in the steady state case is set by w [110, 112]. When inhomogeneous
broadening is comparable to or less than the collectively enhanced emission rate, the majority ofatoms in the ensemble can phase-synchronize and contribute to collectively enhanced emission.
Because we operate in the Lamb-Dicke regime, Doppler shifts are negligible in our system.
However, the lattice contributes a polarization-dependent shift of the lasing transition [135]. The
degree to which this shift is inhomogeneous depends on the temperature of the atoms, which
increases during laser operation. At the beginning of laser operation, we estimate inhomogeneous
broadening to be less than 30 kHz FWHM for our typical lattice polarization. In principle, thisbroadening can be tuned to zero by changing the polarization of the lattice. For our range of
operating parameters, the majority of atoms have transition frequencies within w of the averagetransition frequency, allowing them to simultaneously contribute to lasing.
The repumping mechanism is shown in Fig. 2d, with beam directions and polarizations shownin Fig. 6.1a. A magnetic field is applied perpendicular to the cavity axis, splitting out the mJ = ±1

90states by ±7 MHz. The 689 nm repump light is applied near resonance with the 1 S0 to 3 P1 , mJ = −1
transition. Additional π-polarized 688 nm repump light resonant with the dipole-allowed 3 P1 to
3 S transition is applied from the same direction.
1

Because of dipole selection rules, this 688 nm

pump couples the 3 P1 mJ = ±1 states to 3 S1 , while the mJ = 0 state is unaffected. Single-particle
spontaneous emission takes atoms from the 3 S1 states to the 3 P states, and additional repump lasers
at 707 nm and 679 nm are applied to depopulate the metastable 3 P2 and 3 P0 states. Because
atoms reach 3 P1 by single-particle spontaneous decay, we do not expect any coherence between thepump lasers and the emitted light.
It is likely that lasing terminates when heating from free-space scattering caused by the
repump process reduces the product N C both by causing atom loss and by reducing the couplingof remaining atoms to the cavity. For each lasing photon emitted into the cavity, we estimate
roughly 13 photon recoils are imparted during the repump process. For 35 lasing photons peratom, this would contribute heating comparable to the lattice depth.

Laser Output Power (nW)1.5γ/2π =

7.5 kHz
1.0
0.5
0.0

0

5
10
15
20
Repumping Rate w/2π (kHz)25

Figure 6.3: Threshold behavior of laser. Laser output power in quasi steady state is plotted
0 = 0, (red points) and added broadening,
versus repumping rate w for no added broadening γ⊥
0(2γ⊥ ' 2π × 3 MHz  γ, κ, black points). Atomic decay rate γ is displayed as vertical blueline, with shaded blue region representing uncertainty in calibration of w. For both conditions, themeasured threshold repumping rate wt is consistent with wt = γ.

/2π =
160 kHz

1.0

1.0
0.5
0.4
0.3
0.2
0.1
0.0

0.4
0.2
0.0

Normalized PSD

0.8
0.6

101

91

b.

P

Pulling Coefficient Pa.

0

50 100 150 200(w + γ +2γ0 )/2π (kHz)

102
103
0
(w + γ +2γ )/2π (kHz)

104

0.75
0.5
0.25
0.0
-50

0
Freq. Offset (kHz)50

Figure 6.4:(a) Pulling coefficient P versus the total homogeneous broadening of the lasingtransition, along with prediction (blue). The vertical line at 160 kHz represents the cavity decayrate κ. The inset displays the left five points on a linear scale. (b) Averaged heterodyne power
spectral density (PSD) between output light and 689 nm pump laser, with recentering of eachindividual trial before averaging. Lorentzian (Gaussian) fits, shown as red (blue) lines indicate
FWHM linewidths of 6.0(3) (4.7(3)) kHz.

6.3

Lasing Threshold Behavior
The output power in our laser exhibits a characteristic threshold behavior versus w, as shown

0
in Fig. 6.3. In the regime that the total atomic transverse decoherence rate γ⊥ = γ/2 + w/2 + γ⊥

satisfies γ⊥  N Cγ, the threshold repumping rate for lasing is simply wt = γ. This is expected
to be true even in the presence of homogeneous or inhomogeneous broadening that exceeds the
0 accounts for other mechanisms
natural decay rate of the lasing transition [51]. In the above, γ⊥that lead to a decay of atomic coherence. Intuitively, this threshold requirement states that inorder to establish population inversion, atoms must be removed from |gi at a rate faster than they
can undergo single particle decay from |ei to |gi.
We measure output power from the laser in a window between 20 and 60 µs after the beginningof laser operation. We tune w by changing the intensity of the 689 nm repump laser and measure
wt = 2π × 8(3) kHz, with the error dominated by the calibration of w. This is consistent with thepredicted threshold wt =γ = 2π × 7.5 kHz.

92
By inducing Rayleigh scattering on the dipole-allowed |gi to 1 P1 transition at 461 nm using
0 ' 2π × 3 MHz  κ, γ. Despite
our MOT beams, we add additional homogeneous broadening 2γ⊥

this high scattering rate, we measure the same threshold value of wt = 2π × 8(3) kHz, again
consistent with the prediction. By rotating the lattice polarization, we can introduce inhomogeneous
broadening that can be varied from less than 30 kHz to as much as 140 kHz FWHM. Between thesetwo conditions, we measure wt to differ by less than 3 kHz. This confirms that the dominant factor
in determining threshold is indeed the bare atomic decay rate, not the effective transition linewidth.

6.4

Cavity Pulling, and Tuning Through the Superradiant Crossover Regime
An important technological appeal of bad-cavity lasers is their reduced sensitivity to fluctu-

ations in cavity resonance frequency. We define the pulling coefficient P = ∆f` /∆fc , where ∆f`
is the change in the lasing frequency created by a change ∆fc in the cavity resonance frequency.
We expect the pulling coefficient to be given by P = 2γ⊥ /(2γ⊥ + κ) when the laser is operatingin steady-state [141, 63, 66]. A pulling coefficient P  1/2 is a key signature that the laser isoperating in the bad-cavity, or superradiant regime with coherence primarily stored in the atoms,
not in the light field.
To measure the pulling coefficient, we overlap the laser light emitted from the cavity witha heterodyne beam from our 689 nm laser. We changed the cavity frequency between trials andcomputed power spectra from the heterodyne data. From Gaussian fits to these power spectra, we
extract the peak frequency of the emitted light. We obtain the pulling coefficient from a linear fit tof` versus fc , with fc scanned by 1 MHz. We repeat this with different values of w. When we turndown the repump beams to just above threshold, w = 2π × 14 kHz, we measure a pulling coefficientof P = 0.09(2). This indicates that we have reached the bad-cavity regime, where the spectral
properties of the emitted light are dominated by the atomic gain medium. When we increase therepump rate such that the effective atomic linewidth becomes comparable to the cavity decay rate,
we reach a crossover regime where coherence is shared between atoms and light field, and measure
a pulling coefficient consistent with P = 1/2.

93
The right-most point in Fig. 6.4a was obtained by turning on the 461 nm MOT beams to
0 ' 2π × 3 MHz. With thisfurther increase γ⊥ by Rayleigh scattering from |gi at a rate 2γ⊥

scattering rate, we measure a pulling coefficient of P = 0.97(3). In this highly broadened regime,
scattering from the 461 nm MOT beams removes coherence between 1 S0 and 3 P1 without directly
affecting the population inversion. By removing the phase coherence from the atoms, we access the
good-cavity regime, where the spectral properties of the emitted light are set by the optical cavity.
We bound the linewidth of the emitted light by computing an average power spectrum fromheterodyne data taken over many trials (Fig. 6.4). From trial to trial, the center frequency ofthe power spectrum fluctuates due to low frequency noise on the heterodyne laser and opticalcavity (at the time of this work, we did not have a very stable reference cavity, so our laser moved
around quite a bit). We align each power spectrum by shifting its frequency axis by the center
frequency obtained from a Gaussian fit, which reduces sensitivity to low-frequency noise. We thenfit a Lorentzian and Gaussian to points from all recentered spectra simultaneously. From 150 µs
long subsets of the time data, we measure a Lorentzian (Gaussian) FWHM linewidth of 6.0(3)(4.7(3)) kHz. We expect that the fundamental quantum linewidth for these operating conditions
would be around 1 Hz [141], far below our current ability to resolve. Our measured linewidth isFourier-limited for shorter subsets, and limited by acoustic noise on the heterodyne laser for longer
subsets. However, our measured linewidth is slightly narrower than the natural linewidth of thelasing transition (7.5 kHz), and far narrower than the linewidth imposed by repumping (∼ 100kHz), exhibiting the frequency narrowing characteristic of synchronization in a laser.
This work served as a first step in demonstrating that the results achieved in Raman systems could indeed be applied to a system with true narrow-linewidth optical transitions. It also
demonstrated a basic mechanism for achieving quasi steady state operation in strontium. Beyond
its current role as a scientific curiosity, a laser based on the 7.5 kHz linewidth strontium transition
could perhaps be considered as a moderate performance but relatively simple frequency reference.
If one wished to take this path, it would be advantageous to work with a broader linewidth opticalcavity in order to further reduce the effects of cavity frequency noise. To achieve steady-state94
operation, one would need to provide a constant flux of new atoms into the cavity mode in orderto replace those lost to heating. Because of the relatively broad linewidth of the transition, it isconceivable that one could operate such a laser using as a gain medium a beam of pre-cooled strontium atoms that pass through the cavity mode without axial confinement. This would represent a
substantial simplification relative to a laser operating on the milliHertz linewidth transitions.

Chapter 7

Pulsed Lasing on the milliHertz Linewidth Clock Transition

The core element of my thesis is the demonstration and characterization of superradiance fromthe milliHertz linewidth strontium clock transition. A superradiant laser based on this transition
provides a promising path to a active optical frequency reference with performance far beyond thecurrent state of the art [51].
The first major milestone towards this goal, described in this chapter and in [25], was todemonstrate the collective enhancement of radiation from the milliHertz linewidth transition byobserving superradiant pulses. In this work, I was able to demonstrate a factor of up to 10,000
enhancement in the rate at which atoms radiate into the cavity, compared to the expected rateof emission for independently radiating atoms. The temporal form of the pulses, as well as key
scalings are well-described by simple theory presented later in this section. Also in this section, I
describe a method for seeding coherence in the atoms that has proved very useful in the study ofour superradiant source. While a much more thorough characterization of the spectral properties ofthe superradiant pulses appears in the next chapter, I include here a preliminary characterizationof the relative coherence between two simultaneously lasing milliHertz linewidth transitions.

7.1

Experimental System
For this work, we switched over from 88 Sr, for which the 3 P0 is effectively infinitely long

lived for our purposes, to 87 Sr. In 87 Sr, the presence of nuclear spin causes 3 P0 to decay in about
160 seconds [85]. We used an ensemble of up to N = 2.5 × 105 87 Sr atoms cooled and tightly

96

Figure 7.1: (a) The√ atoms can be represented by a collective Bloch vector on a Bloch sphere. The
cavity field C ∝ Mc , where Mc is the number of photons in the cavity, is represented by a vector
that lies in the equatorial plane perpendicular to the atomic coherence J⊥ (when the cavity is onresonance with the atomic transition). The Bloch vector rotates about the axis that represents thecavity field. (b) The Bloch vector behaves like a highly damped pendulum that starts inverted at
the north pole of the Bloch sphere (all atoms in |ei). Quantum fluctuations disturb the systemfrom its unstable equilibrium position, causing the Bloch vector to swing down the Bloch sphere,
emitting peak radiation at the equator and ultimately relaxing to the south pole (all atoms in|gi) as inversion is lost. The radiated electric field (red trace) is proportional to the perpendicular
projection of the Bloch vector, J⊥ , which at its peak is proportional to N . The radiated power(black trace) is proportional to the square of the radiated electric field, and at its peak is therefore
proportional to N 2 . This is one way to understand the origin of the collective enhancement inemission rate. The black output power trace on the projection plane is actual data.

trapped within our usual magic wavelength lattice (near 813.4274 nm, which unlike in previous
sections, is finally actually a magic wavelength). Our goal was to get the atoms to collectively emit
light from the mHz linewidth transition.
To describe this process, I will apply our usual Bloch sphere description to this system: As areminder, the state of the atomic ensemble can be represented by a collective Bloch vector, whichis the vector sum of the Bloch vectors of the N individual atoms. An atom in the excited state, |ei(3 P0 ), has a Bloch vector pointing up, while an atom in the ground state, |gi (1 S0 ), has a Blochvector pointing down. An atom in an equal superposition of |ei and |gi corresponds to a Blochvector on the equator of its Bloch sphere, with phase φ determined by the phase of its superposition|gi + eiφ |ei. A collective Bloch vector on the equator of the Bloch sphere corresponds to each atom
in a superposition of |ei and |gi with the appropriate phases to collectively radiate into the cavity

97
mode (See Fig. 11.1a).
As described in Chapter 2.4, the atoms radiate an electric field C into the cavity at a rate
proportional to J⊥ , the magnitude of the projection of the Bloch vector onto the equatorial planeof the Bloch sphere. The collective enhancement of emission results from the fact that the powerradiated is proportional to the square of the electric field. Because the electric field is proportional
to the atom number N , the radiated power scales with N 2 [147].
For this work, we place the cavity on resonance with the atomic transition. In this configuration, the cavity field C ∝ Mc , where Mc is the average number of photons in the cavity, isrepresented by a vector that in the equatorial plane of the Bloch sphere that is perpendicular toJ⊥ . The electric field radiated into the cavity acts on the atoms by causing rotation of the Blochvector about C. This is the mechanism of stimulated emission in the superradiant regime. For
an ensemble whose Bloch vector lies in the northern hemisphere of the Bloch sphere, this leads topositive feedback for emission. The atoms will radiate into the cavity, and the radiated electricfield will then cause the Bloch vector to tip further from the north pole, increasing J⊥ and thus
causing the atoms to radiate more strongly. The result is a pulse of light that builds up gradually,reaches a peak in power as the Bloch vector passes the equator, and falls to zero as the atoms reachthe ground state (Fig. 11.1b).
The dynamics of the Bloch vector in this regime (neglecting the effects of inhomogeneous
coupling of the atoms to the cavity) are well-described by the motion of an initially inverted, highly
damped pendulum. In contrast to the pulsed emission on the 7.5 kHz transition, the high levelof cavity damping here relative to the rate of coherent interactions means that there is no ringing
in the pulsed emission, which would be analogous to the pendulum swinging about its equilibrium
position.

7.2

Observation of Superradiant Pulses
To observe superradiant pulses, we prepare atoms in |ei, the nuclear |F = 9/2, mf = 9/2i

sub-level of 3 P0 . We first optically pump the atoms to |gi, the |F = 9/2, mf = 9/2i sub-level of98

Figure 7.2: Spontaneously generated superradiant pulses. Representative single time traces ofphoton output rate R for pulses at different atom number N ≈ 100 × 103 (green), 125 × 103 (blue),
150 × 103 (red), 200 × 103 (black). The equivalent average intracavity photon number is calculatedon the right as Mc = R/κ.

1 S , then adiabatically transfer up to 75% of the atoms to |ei using a frequency-swept 698 nm
0transfer beam applied through the cavity.
This transfer beam is derived from our 698 nm diode laser, which is frequency-stabilized toour transfer cavity. At the time of this experiment we were using a home-built transfer cavity,
which was not stable enough to allow for high-fidelity optical rotations without sweeping. Further,
we are constrained to drive the optical rotations along the cavity axis, as this is the only direction inwhich Doppler shifts are sufficiently suppressed to efficiently drive the narrow-linewidth transition.
This means that we are forced to drive the transition with a standing wave with spatially varying
Rabi frequency. This would further limit our ability to do a high-fidelity optical rotation (although
this actually works quite well for other modes of operation, as will be described in the following
section). Our solution was to rely on Landau-Zener transitions (adiabatic transfer) from the groundto excited state. We accomplish this by coupling a sideband from the 698 nm laser to a TEM00
mode of the cavity. This mode is placed on resonance with the atomic clock transition. The laser isthen swept in frequency over the atomic and cavity resonances. For this experiment, we typicallyswept the laser by 200 kHz in 20 ms. The Rabi frequency experienced by an atom at an antinode99of the standing wave was typically around 5 kHz. Because the transfer beam is linearly polarizedalong the magnetic field, the atoms maintain their spin polarization mf when transferred to 3 P0 .
This adiabatic transfer process was sufficient to transfer most of the atoms to the desired
state with minimal heating. Further, those that were left behind were presumably those that werelocated near nodes of the 698 nm cavity mode, and so would contribute little to the superradiantpulse anyway. To prepare the atoms with full inversion (no atoms in |gi) and to ensure that the laser
pulses are initiated by quantum noise rather than residual atomic coherence associated with the
adiabatic transfer process, we next briefly apply lasers to the dipole-allowed 1 S0 to 1 P1 transition
to heat any atoms remaining in the ground state out of the lattice.
When all atoms are initially prepared in |ei, we observe collectively enhanced decay on theclock transition. To detect the superradiant laser pulses, the output of the cavity is coupled to asingle-mode fiber and detected on a single-photon counting module (SPCM) whose TTL output is
low-pass filtered to provide a signal proportional to the photon emission rate.
Both quantitative and qualitative features of the collectively enhanced emission are dramatically different from that of independently radiating atoms. Not only does the collective enhancement
lead to an emission rate into the cavity mode of up to 104 times greater than that of independently
emitting atoms, but the functional form of the decay versus time is distinctly non-exponential.
Figure 7.2 shows the photon output rate R for four representative pulses recorded with different
initial atom numbers. Because the rate of collectively enhanced emission per atom scales with N ,for higher atom numbers the pulses appear sooner, have shorter duration, and have a higher peak
power than for lower atom numbers.
We quantitatively characterize our superradiant pulses in terms of the scaling of key observables with atom number N in Fig. 7.3. Figure 7.3a shows the characteristic N 2 scaling of thepeak output power Rpeak versus atom number that one expects for superradiance. In the presenceof decoherence or atom loss, the atom number must exceed a threshold Nt for superradiance tooccur. Nt is set by the requirement that the collectively enhanced decay rate exceeds the atomic
1
decoherence rate. Above this threshold, we predict Rpeak = 4ξNx2 Cγ, where Nx = N − Nt is the

100

(a)

(b)

(c)

Figure 7.3: (a) Peak photon output rate, Rpeak , versus initial total atom number. The black line isa quadratic fit to the output power. We observe a horizontal offset in the fit, indicating a thresholdatom number Nt (black vertical line in b,c,d). The threshold results from decoherence and atom
loss and is a signature of lasing that would not be present for single-atom emission. (b) FWHM
pulse duration tw (blue) and delay of peak power td (red) versus initial total atom number. The
blue line is a fit to the predicted functional form for the pulse duration, with Nt determined fromthe fit to Rpeak . The dashed red line is a fit to the pulse delay, assuming threshold is due to atomic
homogeneous broadening without atom loss. The solid red line is a fit to the pulse delay, assuming
the threshold is set only by atom loss from the lattice. (c) The ratio of emitted photons Mtot to thenumber of atoms in excess of threshold Nx , plotted versus atom number. The dashed line is theaverage ratio, showing that 48% of the atoms in excess of threshold participate in the superradiantpulse, largely independent of N .

total number of atoms in the lattice in excess of the theshold atom number. The inhomogeneous
coupling of the atoms to the cavity mode is accounted for by the numerical factor ξ ≈ 2.95 (Details
on how we account for decoherence and inhomogeneous coupling follow). From a fit of this form, we
extract a threshold of Nt = 3.3 × 104 atoms. From the fitted Nt and known C and γ, the measured
peak photon emission rate Rpeak is 0.7(4) times the above predicted rate.
The time duration of the superradiant pulse provides a measure of the collectively enhanced
decay rate. The measured FWHM time duration versus atom number N is shown in Figure 7.3b

101(blue points). We predict that the FWHM duration tw of the pulse is given by tw ≈ 7.05/(Nx Cγ),
such that the enhanced decay rate scales linearly with the excess atom number Nx . We fit thisfunctional form (blue line) to the data, with the threshold held fixed to Nt = 3.3 × 104 atoms fromabove. From this fit we find that the measured FWHM is 1.4(7) times the predicted FWHM given
the known C and γ.
The measured delay time td of the peak in output power versus atom number N is shownin Figure 7.3b (red points). In the presence of homogeneous broadening of the atomic transition,+γe )but with no atom loss, we expect the delay time to be given by td ≈ 2(lnNN(dashed red fit line
x Cγwith Nt fixed and C fitted), where γe ≈ 0.577 is the Euler-Mascheroni constant. In the presence of
atom loss from the lattice at rate γ` , we observe a delay time in numerical simulations of the form+γe )td ≈ αγ` + 2(lnNNCγ
where α is a constant, a functional form which seems to better describe thedata (solid red fit line with Nt fixed and C and α fit.)
We define the number of atoms that participated in a superradiant pulse in terms of theintegrated number of photons Mtot emitted from the cavity mode, i.e. one photon equals one participating atom. Figure 7.3c shows the number of emitted photons per atom in excess of threshold,
Mtot /Nx . We observe that above threshold, Mtot /Nx = 0.48(15), independent of atom number.
Because the peak photon output rate Rpeak and pulse duration tw scale as Nx2 and 1/Nx , respectively, we do not expect this quantity to depend on atom number. Inhomogeneous coupling to
the cavity mode would predict Mtot /Nx = 0.7 partially accounting for the observed participation.
Atomic collisions, which lead to an atom-number dependent contribution to Nt may account for
the additional reduction of participation. We may contrast this level of participation to the casewhere atoms emit independently: if there were no stimulation, only 0.1% of atoms would emit a
photon into the cavity mode during our measurement time.
The preceding measurements all rely on a calibration of the number of atoms in the opticallattice. As is often the case, this is somewhat tricky. After allowing a fixed time T (typically
300 ms) to record the superradiant pulses, we perform atom counting using a resonant 461 nm
fluorescence beam and a CCD camera. We calibrate the fluorescence signal by measuring a vacuum

102
Rabi splitting on the 1 S0 to 3 P1 , F = 9/2 transition. To measure total atom number, we apply
pump lasers to the 3 P0 and 3 P2 to 3 S1 transitions to drive all atoms to the ground state. We
then infer the number of atoms at the beginning of the 300 ms superradiance time window from ameasured background loss rate from the lattice of γ0 = 2.0(4)/s and a measured additional atomnumber dependent collisional loss rate of γN = 10(5) × 10−6 /(atom s) for excited state atoms.
To account for the state-dependence of the collisions, we make a lowest order correction for thedecrease in excited state population after the superradiant pulse, which leads to a 5% correction
at 175,000 atoms, and a <1% correction below 100,000 atoms.

7.3

Seeding Atomic CoherenceInstead of preparing the atomic ensemble in the excited state with no initial coherence as

before, we can terminate the adiabatic transfer process early and prepare the atoms in a superposition of |gi and |ei. Because the transfer is performed with light that is resonant with the cavity
mode at 698 nm, the phase factors written in to the atoms are exactly right to allow those atoms
to radiate in phase into the cavity mode. This seeds the collectively enhanced emission. Unlikein the non-seeded pulses of figure 7.2, we detect an immediate output of light from the cavity.
Figure 7.4a shows a typical output trace resulting from terminating the adiabatic transfer with the
Bloch vector just above the equator of the Bloch sphere.
Seeding the coherence in this manner also eliminates the threshold behavior observed in
Figure 7.3. Figure 7.4b shows the peak output power Rpeak versus N for pulses seeded with the
Bloch vector just above the equator of the Bloch sphere. This data is well described by a quadratic
fit with no offset, i.e. Nt = 0.
In contrast to the non-seeded case described earlier, the positive feedback mechanism is
not required for seeded pulses. If prepared with the Bloch vector at the equator, the atoms will
immediately radiate with a power proportional to N 2 . Sources of dephasing will lead the pulse todie out sooner than it otherwise would, but will not effect the initial radiated power.
We view this technique for establishing collectively enhanced emission with no threshold

a.

Rpeak

(106 photons/s)

b.

4

3

3

2

2

1

1

0

0

4
3
2
1
0
0

10

20

30

Time (ms)40

50 100 150 200 250

Intracavity Photons Mc

4

0
4
3
2
1
0

Mc (photons)(106 photons/s)

Photon Output Rate R

103

Total Atoms (103 )Figure 7.4: Seeded superradiant pulses. (a) By terminating adiabatic transfer with atoms in asuperposition of |ei and |gi, we seed coherence in the atomic ensemble. Here, the atomic Blochvector is rotated to just above the equator (i.e. a small amount of initial inversion.) Seeding leads
to the immediate onset of superradiant emission, in contrast to the non-seeded pulses shown in
Fig. 7.2 for which quantum noise seeds the coherence. (b) Peak output power for seeded pulses
exhibits N 2 scaling. In contrast to results of Figure 2, seeded pulses exhibit a peak photon output
rate consistent with no threshold: Nt = 0.

or delay time to be a key tool for the development of superradiant sources. In this work, being
able to use these signatures of collectively enhanced emission to incrementally tune the system
to meet threshold was essential. In subsequent work, we typically operate with seeded pulses, as
they are more robust, less sensitive to fluctuations in atom number and cavity detuning, and offer
the additional control of being able to target a desired population inversion. More fundamentally,
seeding allows collectively enhanced emission to be achieved in systems that are incapable of meeting
threshold, but that may still be of metrological value. If superradiant pulses are used to stabilize

104
the frequency of another laser, seeding could be used to reduce dead-time that contributes to Dicknoise aliasing [35].

7.4

Simultaneous Lasing on Multiple Transitions

Figure 7.5: Lasing on multiple transitions at the same time leads to beating in output power (a)Atoms prepared in the mf = 9/2 and mf = 7/2 Zeeman sub-levels radiate simultaneously into
the cavity. (b) Interference between the electric fields radiated from the two transitions leads toa modulation of the output power. An average of 20 time-traces is plotted, showing that themodulation has the same phase between trials of the experiment. (c) We compute a Fouriertransform of an averaged time trace, showing a peak at the frequency splitting of adjacent Zeeman
sub-levels (≈ 200 Hz). The peak at ≈ 400 Hz indicates that a smaller number of atoms havebeen prepared in mf = 5/2. A Lorentzian fit to the average power spectrum (red line) returns a
FWHM linewidth of 11 Hz, due to the finite length of the pulse. The inset of (b) shows the center
frequency of the Lorentzian fit versus applied magnetic field. The blue line shows the predicted
splitting between adjacent Zeeman sub-levels.

For all preceding data, we applied optical pumping to populate primarily the 1 S0 , mf =9/2 sub-level before adiabatic transfer, resulting in a single relevant lasing transition. We candeliberately reduce the efficiency of the optical pumping to populate both the mf = 9/2 andmf = 7/2 ground states, and then adiabatically transfer the atoms into superpositions of groundand excited states with different mf projections, represented in Figure 7.5a. This creates two
separate sub-ensembles of atoms that interact with the same cavity mode but have slightly different

105transition frequencies. We observe a modulation in the output power at the magnetic-field induced
frequency difference between the two transitions. This modulation is the result of interference
between the fields radiated by the atoms lasing on the two transitions. Because more atoms
are prepared in the mf = 9/2 sublevel, the total field radiated never goes through zero and thecontrast of the modulation is not full. Later on (as described in the next chapters), we became
more sophisticated with this kind of technique, but this represents our first attempt at spectral
characterization of the superradiant pulse.
Figure 7.5b shows the average of 20 time traces recorded under these conditions, illustratingthat the phase of the modulation is the same between trials, a result of seeding coherence into thetwo transitions. To verify that the observed amplitude modulation is the result of beating between
adjacent Zeeman transitions, we compute a Fourier transform of the emitted power (Figure 7.5c)
and fit the peak in the power spectrum that corresponds to the output power modulation. The
frequency of this peak is plotted against our applied magnetic field in the inset of Figure 7.5b.
The slope and offset of beat frequency are consistent with the expected Zeeman splitting between
the mf = 9/2 and mf = 7/2 transitions. The smaller peak near 400 Hz indicates that a smaller
number of atoms have been left in the mf = 5/2 state.
A Lorentzian fit to the peak in the average power spectrum returns a FWHM linewidth of
11 Hz, primarily reflecting the finite length of the pulse. Because many photons are detected in atrial of the experiment, we can fit the center of the Lorentzian peak with deviation much smaller
than its width.
Treating this as a differential frequency measurement of two lasers, we compute a fractional
Allan deviation of 2.6 × 10−15 at ≈ 2 seconds, the repetition rate of our experiment. Because manysources of frequency errors that are common-mode to the two lasing transitions are not capturedby this measurement, this number does not indicate the ultimate performance of the system as
a frequency reference. It does, however, reflect a bound on its quantum-limited instability. In
the next chapter, we perform a more complete characterization of the spectral properties of our
superradiant pulses by comparison to a very stable reference laser.

106

7.5

Simulations of Superradiance
To provide theoretical predictions for superradiance in the presence of inhomogeneous cou-pling to the cavity mode and atomic decoherence and loss, we integrate a set of simplified optical
Bloch equations:2J˙z = −CγJ⊥− γ` J z ,(7.1)

J˙⊥ = (CγJz − γ⊥ − γ` )J⊥ .

(7.2)

Here, Jz and J⊥ are the atomic inversion and the magnitude of the coherence, γ` accounts
for atom loss, and γ⊥ accounts for other homogeneous atomic decoherence. In reality, both γ` andγ⊥ depend on N , and on the instantaneous distribution of population in ground and excited states.
For simplicity, and to arrive at the analytic expressions given below, we assume γ` and γ⊥ to beconstants. In arriving at these equations, we have assumed that the cavity mode is on resonance withthe atomic transition, and that the cavity mode occupation quickly equilibrates to the instantaneous
value of J⊥ . As discussed in ref. [47], the dynamics of superradiance consist of an initial time period,
during which quantum fluctuations are important in providing initial coherence, and a later time
period during which the classically radiated field greatly exceeds quantum fluctuations and the
dynamics proceed classically, governed by the above equations. In the later time period, theseequations can easily be solved for the case of homogeneous coupling and γ⊥ , γ` = 0, with solutionsof the form:
N Cγ(t − td )
,
2


N
N Cγ(t − td )Jz (t) = − tanh.
22NJ⊥ (t) = sech2

To simulate the quantum fluctuations in the cavity field that lead to the onset of superradiance, we include a drive to J⊥ that represents random vacuum fluctuations that are replaced at
the cavity amplitude decay rate κ/2. This allows us to simulate the initial conditions that lead tothe time delay td . We verify that in the limit of large atom number our simulation reproduces the107
expected peak power, as well as the pulse duration tw and time delay td of the peak output power
derived in ref. [47].

7.5.1

Effects of Atomic Decoherence and Loss

Atomic decoherence, or a decay of J⊥ at a rate γ⊥ , has the effect of setting a threshold atom
number Nt below which a superradiant pulse will not occur without seeding of coherence. This
threshold can be derived from Equation 7.2 by setting Jz = N/2 to determine the minimum valueof N for which fluctuations in J⊥ will grow. For atoms homogeneously coupled with cooperativity
C, Nt = 2γ⊥ /(Cγ). For atom numbers above this threshold value, N atoms radiating in thepresence of decoherence produce a pulse identical to the pulse that the number of excess atoms,
Nx = N − Nt , would produce in the absence of decoherence. The solutions for J⊥ and Jz forhomogeneous coupling in the presence of decoherence γ⊥ areNx
Nx Cγ(t − td )
sech
,
2
2


Nx
Nx Cγ(t − td )Jz (t) = −tanh+ Nt /2.
2
2J⊥ (t) =To treat the effects of atom loss, we rely on numerical simulation. In the presence of atom
loss at a rate γ` , we observe a threshold-like behavior at an atom number of Nt` ≈ 13.6γ` /(Cγ),
assuming homogeneous coupling. For atom numbers well above threshold, peak output power andpulse duration are well-described by assuming an ensemble of N − Nt` atoms radiating in theabsence of loss. Near threshold, however, the peak power radiated is higher and the pulse durationis shorter than that predicted by this analogy. The delay time of the pulse exhibits a more striking
difference: at fixed atom number, the delay time of the pulse peak increases linearly with loss rate
up to the threshold loss rate.
From our measured atom loss rate, assuming uniform inhomogeneous coupling (defined below), we expect a contribution to the threshold atom number of 3.0(6) × 104 . The measured valueis 3.3(8) × 104 , which may indicate that the threshold behavior is due primarily to atom loss.

108
The rates of atom loss and decoherence due to atomic collisions scale in proportion to thenumber of atoms, N . This leads to a threshold atom number whose value depends on N . This raises
the threshold atom number compared to its value in the absence of N -dependent decoherence andleads to a constant fractional reduction in Nx . For our measured atom-number dependent loss rate,
we expect the threshold atom number to increase by 0.13(7) for each atom added. This effectively
decreases Nx by 13% compared to its value if the same threshold were the result of atom number
independent processes only. Collisions that do not lead to atom loss, but whose rate would also
scale with N could lead to additional decrease in Nx .

7.5.2

Effects of Inhomogeneous Coupling

In our system, the atoms are trapped at anti-nodes of the 813 nm lattice, which are notaligned with the anti-nodes of the lasing cavity mode at 698 nm. The coupling of an individual
atom to the lasing mode thus depends on its location along the cavity axis.
We numerically simulate the effects of this inhomogeneous coupling by dividing the atoms
into many classes, each with its own location xi and value of coupling to the lasing mode gi =g cos(kxi ), where k is the k-vector associated with the 698 nm cavity mode. We may reasonably
assume that before atoms are transferred to 3 P0 , they are effectively uniformly distributed withrespect to the phase of the standing-wave 698 nm cavity mode. For simplicity, we analyze thecase where the atoms maintain this uniform distribution after being transferred to 3 P0 , meaningthat the values of xi are sampled from a uniform distribution. In reality, the adiabatic transfer isprobably more effective for atoms located at the anti-nodes of the 698 nm cavity mode, as theseexperience a higher Rabi frequency from the transfer beam. The true distribution of couplings
for different transfer parameters will thus lie somewhere between the uniform distribution and thehomogeneously coupled case.
Below, we summarize the results of the numerical simulations for key parameters underthese two conditions: homogeneous coupling and uniformly inhomogeneous coupling. Homogeneous
coupling refers to the case where all N atoms are at anti-nodes of the lasing mode and are coupled

109
with peak coupling g. Uniform inhomogeneous coupling refers to the case where N atoms are
distributed uniformly along the cavity mode. In all expressions, C stands for the peak cooperativity
2

C = 4gκγ .

Homogeneous Coupling

Uniform Inhomogeneous Coupling

peak output rate Rpeak

1
2
4 (N − Nt ) Cγ

1
≈ 11.8(N − Nt )2 Cγ

delay time td(ln N +γe )≈ (N−Nt )Cγ2(ln N +γe )≈ (N−Nt )Cγ

duration tw≈ 3.5/(N Cγ)≈ 7.05/(N Cγ)

threshold atom number Nt

2γ⊥ /Cγ

4γ⊥ /Cγ

participation Mtot /N(N − Nt )/N≈ 0.7(N − Nt )/N

Table 7.1: Summary of key parameters and effects of inhomogeneous coupling for pulsed superradiance.

The effect of uniform inhomogeneous coupling on td , tw , and Nt is to reduce C by a factor of2, corresponding to taking the spatial average of C over the ensemble of atoms. Its effects on thepeak photon output rate Rpeak and the fraction of atoms emitted into the cavity mode Mtot /N ,however, are more complex. Differing Rabi frequencies within the ensemble cause the atoms todephase with respect to the polar angle θ. This shortens the length of the collective Bloch vector
at the time of peak emission and strands poorly coupled atoms in |ei when the pulse terminates,
leaving 30% of atoms in the excited state.
The finite temperature of the atoms within the lattice leads to a reduction of C throughseveral mechanisms. The finite temperature of the atoms leads to a non-zero radial extent of theensemble, which causes the atoms to sample regions with lower coupling to the cavity mode. At
10 µK, this leads to a 10% reduction in the spatially averaged C. The finite Lamb-Dicke parameterη = 0.16 also leads to a reduction C for the motional carrier transition observed here. We estimate

110
this effect to lead to a 7.5% reduction in C. To compare measured values of Rpeak and tw to
predictions in the main text, we take into account these two temperature-related effects, as well as
the 13% reduction in Nx from the measured atom number dependent loss rate.

Chapter 8

Frequency Stability Characterization of Pulsed Lasing

8.1

Introduction
After demonstrating that we could achieve pulsed superradiant emission from the mHz

linewidth transition and characterizing the pulses in the amplitude domain, our next step was
to characterize the spectral properties of the pulses and their sensitivity to various perturbations.
To do this, we needed a frequency reference to compare against. Fortunately for us, our lab is down
the hall from the Ye lab, which has developed one of the most stable laser systems in the world fortheir optical lattice clocks. Because their clocks also use the strontium clock transition, their laser
is also at exactly the frequency we need to compare to our superradiant laser. Ye lab students andpostdocs John Robinson, Ed Marti, Aki Goban and Ross Hutson were particularly instrumental inenabling these comparisons.
Through these comparisons, we were able to characterize the short-term frequency stability√of our laser light, and found the fractional instability to scale with a coefficient below 10−15 / Hz (astandard deviation of around 300 mHz between subsequent laser pulses). This makes our superradiant laser the most precise active atomic frequency standard realized to date. We also performeda preliminary characterization of the frequency accuracy of our source by comparing to an opticallattice clock operating in the Ye lab. We have found agreement between the clock and our superradiant light at the Hz (fractionally 10−15 ) level, on par with the stability floor of a modern hydrogenmaser (e.g. Microsemi MHM 2010) achieved for averaging times of order one day.
In order to understand the sensitivity of our laser’s frequency to environmental perturbations,112
we have primarily focused on effects that are unique to our superradiant laser system (as opposedto optical lattice clocks in general). The most important of these effects is cavity pulling — ashift in the output frequency of the laser associated with the optical cavity’s detuning from atomic
resonance. We have found this puling to exhibit an interesting dependence on atomic inversion,
which is explored in more detail in Chapter 9. For typical operating conditions, we have foundthat for a given shift in the cavity frequency, our laser frequency shifts by five orders of magnitudeless. This is an encouraging confirmation of an important promise of superradiant lasers: they
have the potential to be many orders of magnitude less sensitive than conventional lasers to both
thermal and technical cavity length noise sources that limit the performance of today’s most stablelasers. We also demonstrate a mode of operation that provides real-time insensitivity to shifts due
to magnetic fields, which may also be of great use for future superradiant lasers.

8.2

Basic Techniques
The basic idea of this experiment is to generate seeded pulses of superradiant light, andcompare their frequency to that of the stable reference laser. We do this by using a heterodyne beatnote, a standard technique that involves overlapping the light from our weak superradiant sourcewith the much stronger reference light on a photodiode. In addition to a DC signal corresponding
to the total optical power incident on the photodiode, the detector produces an AC signal at thedifference frequency between the two lasers. The frequency of this AC signal is our measured
quantity of interest.
We can describe the superradiant light incident on the photodiode as a time-varying electricfield E` = E0` cos[ω` t], where ω` is the frequency of the superradiant laser light. The reference
light has an associated electric field of Er = E0r cos[ωr t], where we assume E0r  E0` . The
photodetector is sensitive to optical power, which is proportional to the square of the total electricfield P ∝ (Er + E` )2 ' (E0r cos[ωr t])2 + 2E0r E0` cos[ωr t]cos[ω` t] = (E0r cos[ωr t])2 + E0r E0` (cos[(ωr +ω` )t] + cos[(ωr − ω` )t]). The detector is too slow to respond to signals at frequencies near ωr or ω` ,and so provides a time-average over these signals. The first term in the final expression for P then113
corresponds to the DC voltage from the photodetector.
The term ∝ E0r E0` cos[(ωr − ω` )t]) is our signal of interest, and is within the bandwidth ofthe photodetector because the reference laser is near in frequency to our superradiant light. Notethat the large reference field E0r boosts the size of the signal. By using a sufficiently powerfulreference beam, we can make the heterodyne beat large compared to the electronic noise of thedetector, even though our superradiant laser is very weak. In our case, the signal frequency ωr − ω`is near 60 MHz, a convenient frequency band for low-noise detection. We further mix the signal
down using an IQ demodulator to obtain a signal at a few 10 kHz, which we directly sample into
our data acquisition system. Details of the reference laser system can be found in Chapter 3, andin Mike Martin’s thesis [148].

a)

b)

c)

Figure 8.1: Superradiant pulses in the time and frequency domains for different configurations ofspin states. (a) We can populate a single nuclear spin state to observe lasing on a single transition.
In the time domain (center column) this leads to a simple superradiant pulse. In the frequency
domain (right column), we see a single peak. (b) When two nuclear spin states are populated, we
observe a beating of the two transitions in the time domain, and two peaks in the frequency domain.
(c) When we populate all ten nuclear spin states, we observe an interesting interference patternin the time domain from simultaneous lasing on the ten transitions. In the frequency domain, we
observe peaks associated with the different transitions, with heights determined by the Clebsch
Gordan coefficients of the different transitions.

114
Fig. 8.1 shows typical signals in the time and frequency domain for a variety of experimental
configurations. The first row shows a configuration familiar from the previous chapter, with all
atoms prepared with a single nuclear spin projection mf . On a given run of the experiment, thesuperradiant light comes out in a pulse lasting of order 100 ms or shorter, shown in the power
domain in the center panels. We perform a Fourier transform of the heterodyne signal to view thepulse in the frequency domain. For the single transition of the first row, we see a single spectralfeature, with width of order 10 Hz, corresponding to the finite length of the pulse. What we aremost interested in here is the center frequency of this feature, which we can obtain by fitting a
Lorentzian to the peak in the Fourier domain. The choice of fit function is somewhat arbitrary, but
we find that a Lorentzian has a reasonable form at higher offset frequencies than does a Gaussian,
and was chosen for simplicity.
Instead of populating a single nuclear spin state, we may populate both stretched (maximal|mf |) states simultaneously, and oberve lasing on both transitions at the same time. This is shownin the second row of Fig. 8.1. In the power domain, we see a beating at the frequency of themagnetic-field induced splitting between the two transitions. In the frequency domain, we see two
distinct peaks corresponding to light emitted from the two transitions. We can independently fit
the center frequencies of these two peaks. Because the sensitivity to magnetic fields is equal andopposite for the two transitions, we can use this configuration to obtain a magnetic-field insensitive
signal (characterized in more detail in section 8.5).
Finally, we can populate all 10 nuclear spin states, and observe a fun interference effectbetween the fields radiated by all 10 transitions (bottom row). In the amplitude domain, thismanifests as a series of short bursts of power as ten transitions come into phase with one-another,
separated by dark periods when their radiated fields cancel. In the frequency domain, we see
peaks associated with each transition, with heights that reflect the relative strengths of the Clebsh
Gordan coefficients of the different transitions. This configuration may prove useful for schemesthat combine dark evolution periods with bright periods during which phase may be read out.
In each of these panels, the data shown represents a single run of the experiment, giving an

115
indication of the high signal to noise obtained from a single pulse. In contrast to a passive clock,where one would have to run the experiment at least once for every point on the frequency axis,
we get all of this information from a single pulse. Further, in a passive clock one must find thetransition by setting the clock laser frequency to within the width of the feature to be probed. Here,so long as the reference laser is within the detection bandwidth of the heterodyne system (which
can easily be GHz), the signal can be measured with uncompromised precision. This could be a
substantial advantage for active frequency references in terms of bandwidth and dynamic range.

8.3

Stability Characterization
We characterize the frequency stability of our superradiant light in the two-stretched-state

configuration. As a figure of merit, we compute the Allan deviation for different time windowspτ . The fractional Allan deviation defined as σ(τ ) = f10 h(fn+1 − fn )2 i/2, where fn is the averagefrequency in the nth window of length τ from the beginning of the measurement record and f0 is theabsolute frequency of the superradiant light, is a standard metric for quantifying the performanceof a frequency reference [149]. The average is performed over all adjacent pairs — each fn is a partof two pairs.
While each pulse has a width in the Fourier domain of roughly 10 Hz, we can extract thecenter frequency with much greater precision due to the high signal to noise. This comes from thefact that we collect many photons per pulse.
The Allan deviation of the frequencies of our superradiant pulses is shown in Fig. 8.2, after
subtracting a linear drift associated with the reference laser. We compute the Allan deviation for
two quantities: the average frequency of the two peaks corresponding to the two transitions (black
points), and half the difference between the two peaks (red points). The black points correspondto the stability of the system if used as a frequency reference. We observe an initial behavior of√σ = 1.04(4) × 10−15 / τ for the data shown, which quantifies the short-term stability of our laser.
√
Our most favorable measurement of this value was σ = 6.7(1) × 10−16 / τ , obtained for a separate
shorter set of data. This corresponds to a standard deviation of the frequencies of individual

116√
trials of 300 mHz. The recently-obtained record for this quantity is 6 × 10−17 / τ , achieved bysimultaneously interrogating two optical lattice clocks based on ytterbium at NIST [26]. Our value

Fractional Allan Deviation σ(τ)

is encouragingly close, given the relative maturity of superradiant lasers and optical lattice clocks.

10 -15

10 -1610 0

10 1

Averaging Time τ (s)10 2

Figure 8.2: Stability of superradiant pulses expressed as a fractional Allan deviation σ(τ ) (seetext for definition). Two stretched (mf = ±9/2) transitions lase simultaneously. The black points
correspond to the average frequency between the two transitions, while the red points correspond√to half the difference. Lines represent fits of the small-τ points to σ(τ ) ∝ 1/ τ , the expectedscaling for uncorrelated noise between measurements.

The red points (corresponding to half the measured difference in frequency between the
two transitions) provide a useful characterization of the fundamental limit on the precision of ourmeasurement. The difference frequency is insensitive to may forms of perturbation that plague
the average frequency, for example to noise on the reference laser, cavity pulling effects, and shifts
associated with atomic density. The difference frequency is sensitive to magnetic field noise, but
this appears to be small on short time-scales. The red points, which exhibit a short-term stability√of 4.5(2) × 10−16 / τ for the data shown provide an estimate for the quantum-limited stability of
our laser.

117
The quantum limit on our ability to estimate the frequency of the laser is given by ∆f '√1/(T M ), where T is the duration of the pulse and M is the number of photons detected. This is
the limit imposed by photon shot noise, which can in principle be improved by improving detection√
efficiency. A more fundamental quantum limit for this system is ∆f ' 1/(T N ) where N is thenumber of atoms, which is the limit imposed by atom shot noise. We confirm that we are photon
shot noise limited by placing an OD 0.6 ND filter after the output of the cavity (reference power
is unmodified). We measure a reduction in detected power to 0.25 of its original value, and anincrease in the short-term instability (first points in Allan deviation plots) of the difference points
of by a factor of 1.9(2), consistent with the expected photon shot noise scaling. For the averagefrequencies, the short-term instability increases by a factor of 1.4(2). This confirms that the averagefrequency stability is partially limited by effects other than photon shot noise, though of course wealready knew this because the sum frequencies are noisier than the difference frequencies.
A major contributor to the added noise on the average frequencies versus difference frequencies is noise on the reference laser. This appears worse than one might think in our measurements,
because the duration of our superradiant pulses is short relative to the experimental cycle time (aneffect known as Dick noise aliasing [35]). At short averaging times τ , this should contribute a noise√
contribution of around 5 × 10−16 / τ . For our most favorable dataset, with fractional stability√
σ = 6.7(1) × 10−16 / τ , this contribution explains the difference between sum and difference frequency standard deviations. For our longer dataset shown in Fig.8.2, there appear to be additional
contributions, perhaps due to imperfect perfromance of the fiber-noise cancellation system or tofluctuating atom number.
For longer averaging windows (large τ ), we observe that the measured Allan deviations flatten
out. The reference laser has a noise floor set by thermal perturbations to the length of the reference
cavity at around 1 × 10−16 , which should set the floor for the black points. We measure a floor thatis slightly higher than this, most likely due to long-term atom number drifts. The Allan deviation
for the difference frequencies (red points) should be insensitive to noise on the reference cavity.
These points also hit a floor at σ(τ ) ' 2 × 10−16 , perhaps due to slowly varying magnetic fields in

118
the lab.
Contribution to Detection Efficiency
cavity losses

0.7

other cavity mirror

0.5

heterodyne overlap beamsplitter

0.82

other optics

0.95

fiber-coupling to detector

0.7

detector quantum efficiency

0.84

detector electronic noise

0.9

heterodyne detection

0.5

Table 8.1: Summary of quantum efficiencies in detection chain.

To determine how close we are to being atom shot-noise limited, we must estimate the total
efficiency of our detection, η. Contributions to η for typical operating conditions are summarized inTable 8.1. From this, we estimate a total quantum efficiency of η = 0.06. If we were to work for aheroic total efficiency of η = 0.37, as demonstrated in [15] (which would involve asymmetric cavity
mirrors and homodyne detection), we would expect to improve our quantum-limited precision bya factor of 2.5. This, however, is not a priority at the moment.

8.4

Cavity Pulling
A detailed description of the mechanism behind cavity pulling is provided in the next chapter.

For our purposes here, the important consequence is that for small perturbations of the cavity
detuning ∆c from atomic resonance, the frequency of the output of the laser ω` is “pulled” by an

(Hz)

119

Figure 8.3: Sensitivity of laser output frequency ω` to changes in the cavity detuning ∆c fordifferent initial atomic inversions. Red points correspond to an inversion empirically tuned tominimize sensitivity to cavity detuning. Black and blue points correspond to slightly higher andlower inversions. The slope of the red points is 2 × 10−6 .

amount ∆ω` = P ∆c . We refer to P as the pulling coefficient. The value of P depends linearly on
atomic inversion, with a sign that switches between states prepared above and below the equatorof the Bloch sphere. Because inversion changes during a superradiant pulse, when we fit the center
frequency of the Fourier peak, we are sensitive to a value of P that is averaged over inversion. By
preparing states for which the average inversion during the superradiant pulse is near zero, we canmake P very small.
Unlike in the previous chapter, where we used adiabatic transfer to prepare the superradiantstate, here we prepare the superradiant states by performing a coherent rotation using light resonantwith the clock transition and coupled through the optical cavity. By varying the duration of thecoherent rotation, we can prepare states with differing initial inversion. In Fig. 8.3 we show thesensitivity of the frequency of ω` to changes in cavity detuning ∆ for different initial inversions. The
red points correspond to a case where the initial inversion is empirically tuned to give a low pulling
coefficient (measured P = 2(3) × 10−6 ). Compared to the red points, the black and blue points

120
correspond to preparation pulse areas 13% higher and 7% lower respectively, and have pulling
coefficients roughly an order of magnitude higher (P = −2.5(5) × 10−5 and P = 3.0(3) × 10−5 forthe black and blue points, respectively). While these values do differ, for any of these configurations
the output frequency of the laser is highly insensitive to the cavity frequency. This is a key promiseof superradiant lasers.

Magnetic Field Insensitivity

fUpper (Hz)

8.5

50
0
-50

-50

0

fLower (Hz)50

Figure 8.4: A magnetic-field insensitive configuration. The frequencies of the mf = 9/2 andmf = −9/2 transitions (labelled fupper and flower here) respond in equal and opposite manners
to an applied magnetic field, making the average frequency of the two peaks highly insensitive tomagnetic fields.

By operating with population in both stretched states, and taking the average frequency of
the upper and lower frequency peaks, we can obtain a high degree of insensitivity to magnetic field
fluctuations, which would become relevant at long timescales in our system. Fig. 8.4 shows thecenter frequency of the upper frequency peak plotted against that of the lower frequency peak as avarying magnetic field is applied on different trials. A linear fit to fU pper versus fLower yields a valueof -1.000(5). The average frequency between the two peaks is thus highly insensitive to magnetic

121
field fluctuations. At the bias field of 1 G used here, the second order Zeeman shift is 0.233 Hz/G.
By measuring the splitting between the two frequency components at the 5 Hz level (which would
be easy), one could subtract out this second order correction at the mHz level (provided sufficiently
precise knowledge of the second order Zeeman shift coefficient). Note that in contrast to standard
optical lattice clocks, where the two stretched transitions are alternately probed on subsequent
runs of the experiment, this method provides real-time insensitivity to fluctuating magnetic fields.
This provides insensitivity to magnetic fields that are fluctuating on timescales short compared tothe cycle time of the experiment, which can actually become quite long in clocks, especially as theduration of the spectroscopic sequence is extended.

8.6

Absolute Accuracy and Other Perturbations
We performed an initial measurement of the absolute accuracy of our superradiant light bymaking frequency measurements at the same time as measurements were being made on an opticallattice clock down the hall (Thanks go to Aki Goban, Ed Marti and Ross Hutson for making thispossible). The goal of this project was not so much to make a claim about the final accuracy of oursystem, as the current iteration of the experiment was not designed to have great accuracy. Rather,
the goal was to determine if there are any unexpectedly large shifts to our laser frequency thatwould have to be controlled at an unreasonable level in order to achieve a desired stability. With
that in mind, we sought to characterize key perturbations at roughly the 1 Hz level, acknowledging
that further improvements could be made in the future.
The clock (described in [150]) had systematic errors calibrated at the 0.3 Hz level at the timeof our measurement. These errors were dominated by uncertainty in the wavelength of the lattice,as it was referenced only to a wavemeter with 100 MHz accuracy at the time of our measurement.
For this comparison, we take into account known sources of error for both the clock andthe superradiant laser, then compare the resulting corrected frequency. This comparison was not
performed in a blind manner, but as this is an immature experiment and we are not measuring aquantity of universal importance, I think that this is acceptable for our rough comparison.

122
Table 8.2 summarizes shifts and uncertainties in our comparison, each of which is describedin more detail below.

8.6.1

Lattice Shifts

By far the largest perturbation, if we do not account for it properly, is the shift due to theoptical lattice. Because we are constrained to operate with the cavity on resonance with the clocktransition at 698 nm and the lattice must be on resonance with another mode near the magic
wavelength at 813.4272, we do not have complete freedom in setting the lattice wavelength — wemust work with the closest mode available, which could be up to half a FSR away. A deviationfrom the magic wavelength ∆latt (in frequency), in a trap of depth U contributes a shift to theclock transition frequency of -0.0155 Hz/(GHz Erec ) × U ∆latt [151], where Erec = ~k 2 /2m is thephoton recoil energy, which is used in the clock community as the unit for lattice depth. We use a
trap depth of up to around 1000 Erec (around 185 µK) deep (which is very deep compared to mostclocks).
We can fit away this shift by measuring the frequency of our laser at several wavelengths andpower levels. The wavelength of the lattice can be changed either by simply changing the frequencyof the lattice laser by 1 FSR (3.7 GHz) and locking to a different longitudinal mode of the cavity, orby changing the length of the cavity such that the clock transition is on resonance with a different
longitudinal mode of the cavity. This shifts the frequency of the mode to which the lattice locks
by FSR×698/813. The cavity PZTs have enough tuning range to place two different longitudinalmodes on resonance with the clock transition, and the lattice can lock to either of two modes foreach of these, one above and one below resonance. This gives us access to four lattice detunings.
In principle, we could detune the lattice laser further from resonance to access more detunings, but
we do not find this to be necessary.
For each of these lattice detunings, we measure the frequency of our laser at three different
lattice depths. We then perform a fit of the form: f = a(flatt − fmagic ) × (P − P0 ) + f0 to thesetwelve points. Here, flatt and P are the measured lattice frequencies (using a wavemeter) and123

Corrections and Uncertainties
EffectValueorShift (Hz)

Uncertainty(Hz)

Measured Offset

0.1

1

Cavity Drift, Magic Wavelength

00.7

Collisions

1

1

Hyperpolarizability

0.3

0.3

Clock Lattice0

0.3

2nd order Zeeman

-0.04

0.01

DC Stark Shift0+0, -0.1

Differential Blackbody Radiation

0

0.06

Table 8.2: Summary of shifts and uncertainties in frequency comparison.

124transmitted power for the various points, and a, fmagic , P0 , and f0 are fit parameters. The value
a represents the sensitivity of the clock transition to the magic wavelength. We find a value of-0.0147(3) Hz/(GHz Erec ), close to the previously measured value.
The results of this fit are shown in Fig. 8.5. The vertical dashed line represents the fit valueof fmagic . Our measured value of f0 is 0.1 ± 1 Hz (prior to the application of corrections described
below). To check the robustness of this fit, we have tried fixing the lattice power offset P0 = 0, andincluding a shift that depends quadratically on the measured PZT voltage for each point. These

Frequency Difference (Hz)

did not shift the measured frequency by more than the reported error on the fit.

60
40
20
0
20
40
60
3

2

1

0

Lattice detuning (GHz)12

Figure 8.5: Frequency difference between our laser and Ye lab optical lattice clock frequency at
different lattice laser detunings and power levels. Different colors of points correspond to different
lattice depths. A simultaneous fit (see text) to all data returns a nominal frequency offset consistent
with zero, as represented by the horizontal dashed line, with the uncertainty of the fit represented
by the blue band. The vertical dashed line represents the fitted magic wavelength (relative to the
expected value based on our wavemeter) with fit uncertainty given by the green band.

125
8.6.2

Atomic Collisions

The next largest source of frequency shifts that we characterized were those associated withcollisions between strontium atoms resulting from the relatively high atomic density in our lattice.
To get a sense of the scale of these shifts for our typical operating conditions, we spread out theatoms in order to modify the atomic density without modifying the atom number (which affects
the pulling coefficient, and could confuse the measurement). To spread out the atoms, we appliedsidebands to our lattice laser at ±1 FSR for 30 ms after initial lattice loading (see Chapter 3).
Cavity modes that differ by 1 FSR have opposite symmetry about the center of the cavity — thenodes of one mode correspond to the antinodes of the other. By setting the proper power ratio
between the lattice carrier and sidebands, we can flatten out the potential from the optical latticein a region near the center of the cavity. The atoms are still confined radially, but can expand
axially. Once the atoms have expanded to about four times their original extent, we turn off the
sidebands to restore the lattice, and then apply cooling for 20 ms. In doing so, we cause less than
a 10% change in atom number, and observe no measurable change in temperature between when
the atoms have expanded and when they have not. The density in the configuration in which theatoms were not spread out was roughly 5 × 1012 /cm3 , though we did not make a precise calibrationof this number as our goal was to get a rough estimate of the size of collisional shifts for our typicaloperating parameters.
By toggling between either spreading the atoms or not spreading the atoms, we can modify
the density of the atoms by a factor of roughly 4. We observe shifts to the frequency of the emitted
light associated with this toggling. Previous groups have observed a strong inversion dependence
to the collisional frequency shift [148], so we measured these density shifts versus the initial state
preparation angle (Fig. 8.6). Importantly, our data was taken with two spin projections populated,
as in our typical operating condition. In a typical optical lattice clock, a single spin state is used,which prevent so-called s-wave collisions from occurring in fermionic atoms, as these collisions only
occur between symmetric motional states of atoms. In our case, s-wave collisions are allowed and126
may be dominant for us. We do not observe a substantial inversion dependence in the density shiftof our laser.
During the absolute frequency measurement, we used a substantially lower atom number
compared to the density measurements. This was so that we could maintain the same atom number
when turning down lattice power. We thus apply a conservative shift and error of 1±1 Hz for ourcomparison. If a more precise absolute frequency measurement were desired, it would probably
be preferable to work in the low-density configuration for which these shifts are a factor of four
smaller.

3.5

Density Shift (Hz)

3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0

0.1

0.2

0.3

0.4

0.5

Fraction of Atoms in Excited State

0.6

Figure 8.6: Measured shift to superradiant output frequency associated with density-dependent
collisions versus fraction of atoms prepared in 3 P0 at beginning of superradiant pulse. Data takenwith roughly 5 × 105 atoms in lattice at beginning of superradiance.

8.6.3

Other Shifts

Other effects contribute fairly negligible contributions to our measured absolute frequency
and uncertainty at the level of our measurements. These include hyperpolarizability, 2nd order127
Zeeman shifts, DC Stark shifts and Blackbody Radiation shifts. These are summarized below.
To estimate the magnitude of the hyperpolarizability shift — a shift that scales with thesquare of the lattice intensity — we use the coefficient 0.3(3)µHz/Erec 2 from Travis Nicholson’s
thesis [152]. For the deepest lattice depths that we use, this would give a shift of 0.3(3) Hz.
To calculate the second order Zeeman shift, which is a symmetric shift to the mf = ±9/2transitions that depends on the square of the magnetic field, we simply need to know the value of
our field, which we extract from the spacing between the upper and lower transition peaks. We use
the coefficient -0.233 Hz/G2 , also from Travis Nicholson’s thesis, to infer a shift of -0.04(1) Hz.
DC Stark shifts describe shifts to the transition frequency due to static electric fields. It has
been reported that dielectric surfaces under vacuum can accumulate surprising amounts of charge,
and lead to electric fields that can cause significant perturbation to the frequency of atomic clocks[153]. This effect could be especially concerning for us, due to the ceramic cavity spacer and mirrorsthat are in close proximity to the atoms. Further, the piezo tubes that control the position of thecavity mirrors are driven with up to 120 V (although the geometry of the piezos should lead to ahigh degree of cancellation of the associated electric fields at the location of the atoms).
The magnitude of the DC stark shift scales with the square of the electric field at the locationof the atoms. This leads to a convenient way of measuring it’s value by applying an additional field~ 2 = |E~ bg + E~ applied |2 where E~ bg is the preexistingwith external electrodes. The shift ∆fstark ∝ |E|~ applied is the applied field. As the strength of the applied field is scanned, one
background field and E
observes a parabolic frequency shift of the clock transition (Fig. 8.7). By comparing the frequency~ applied = 0, one can measure
measured at the maximum of the parabola to frequency measured at E
~ bg that lies along E~ applied = 0.
the shift due to the component of E
~ applied , we simply placed an electrode (a screw embedded in
To generate an applied field Ea Teflon block) on the large upper window of our vacuum chamber. We applied a voltage to thiselectrode relative to the vacuum chamber (which is grounded), and varied this voltage over roughly±500 V. A sample scan is shown in Fig. 8.7. We repeated this measurement with different electrode~ applied , and observed frequency
positions on the large top window in order to vary the direction of E

128

0.5

Frequency Shift (Hz)

0.0
0.5
1.0
1.5
2.0
800

600

400

200

0

200

Electrode Voltage (V)

400

600

800

Figure 8.7: Calibration of DC Stark shifts. Measured frequency is plotted against voltage applied
to electrode on large window of vacuum chamber. Frequency shifts due to background fields in the
direction of the applied field from the electrode manifest as an offset between the minimum of theparabola and its value at zero electrode voltage. Shifts here are consistent with zero.

offsets always well below 0.1 Hz. We thus assign an uncertainty associated with DC Stark shifts+0, -0.1 Hz (since the sign of the shift is well-defined).
Finally, we consider the shifts to the clock transition frequency associated with blackbody
radiation, which have merited great effort in the clock community (See refs [29, 28, 55, 154, 155],for example). Because we are performing a comparison against another strontium system, we do
not care about the absolute value of these shifts, but rather the difference in these shifts between
our system and the clock. The slope of the shift versus temperature is roughly 0.03 Hz/µK. We
measure the temperature of the two vacuum chambers to be well within 1 K of each other. However,
because the lattice clock has recessed windows that are close to the atoms and to the MOT coils,
and which are not accessible for temperature probes, we assign a more conservative temperature
uncertainty of 2 K, corresponding to a differential error associated with blackbody shifts of 0.06 Hz.

129
Accounting for these corrections and uncertainties, we arrive at a measured frequency difference between our superradiant light and the value measured in the Ye lab optical lattice clock of
1.2 ± 1.6 Hz.

Chapter 9

Cavity-Mediated Spin-Spin Interactions

In the previous chapters, all measurements were performed with the optical cavity tuned
right on resonance with the atomic transition. In that regime, the collective interactions leadprimarily to an enhanced decay rate of the atoms into the cavity, which we can naturally describe
as a dissipative process.
In this chapter, I present a detailed exploration of the effects of detuning the cavity fromatomic resonance. This work was performed in collaboration with Ana Maria Rey’s theory group.
As one might expect, detuning the cavity from atomic resonance reduces the rate of collectively
enhanced decay from the cavity, representing a decrease in the strength of dissipative interactions.
More interestingly, interactions mediated by the detuned cavity also lead to inversion-dependent
frequency shifts of the atomic transition and radiated light as well as spin-locking effects, which
can be viewed as Hamiltonian or unitary dynamics.
The inversion-dependent frequency shifts lead to an effect known as One-axis-twisting (OAT),
which can generate useful spin squeezing [156], Schrodinger cat states [157], quantum phase magnification [158], and new measures of entanglement [159]. The spin-locking effects can protect
collective dynamics against single-particle sources of dephasing that limit the atomic coherence
times needed for high precision measurements and for preserving interesting quantum correlations[160].
This work has implications for two fairly distinct types of scientific pursuits: the developmentof precision metrology tools, and quantum simulation. For the development of a narrow-linewidth

131superradiant laser, it is clearly very important to understand how changes in the frequency of the
optical cavity perturb the frequency of the emitted light. We refer to this effect as cavity-pulling,
and the results presented here can be viewed as a detailed exploration of cavity pulling effects in
a superradiant laser. Further, there are good reasons why one might want to operate an opticallattice clock inside of an optical cavity, such as enabling non-destructive atom counting. This work
provides tools for understanding the nature and magnitude of freuquency shifts to the clock that
may be introduced in this configuration.
From the perspective of quantum simulation, this work provides a new way of generatinginteractions between atoms. Previous implementations of atomic coherent interactions result fromdirect atomic collisions [161] direct electric and magnetic dipole interactions [162, 163, 164, 165, 166,
167], phonon-mediated couplings in trapped ions [168, 169, 170], and photon-mediated coupling ina driven optical cavity [67]. We add to this list spin-exchange interactions between ultra-long-lived
optical dipoles mediated by photons in an undriven optical cavity, which are decoupled from atomic
motion, and are fully tunable by controlling the cavity parameters.

9.1

Quantum Description

Figure 9.1: (a) The cavity field mediates interactions between the effective spins corresponding tothe optical 1 S0 (g)- 3 P0 (e) clock transition. This leads to both a collectively enhanced decay at rate
Γ which leads to light leaking out of the cavity, and to interesting unitary dynamics that lead toshifts in the frequency of the emitted light ω` . (b) The resonance of the optical cavity is detuned
from the atomic transition by a frequency ∆.

132
The interactions present in our system can be described as a collective spin-exchange interaction acting on the effective spins formed by the 1 S0 (g)- 3 P0 (e) clock transition — the ensembleof atoms emits a photon into the cavity, which is then reabsorbed (Fig. 9.1). This process flips one
spin from down to up and another from up to down (although because the interaction is collective,
we do not know which ones).
The Hamiltonian for the system (which describes only the unitary dynamics) can be written
in terms of collective spin operators Jˆx,y,z , with Jˆz corresponding to atomic inversion, acting on
the e-g clock states:Ĥeff = ~χJˆ+ Jˆ− = ~χ[Jˆ2 − Jˆz2 + Jˆz ](9.1)

Here Jˆ± = Jˆx ± iJˆy and χ = 4g 2 ∆/(4∆2 + κ2 ). In the second form, the term ∝ Jˆz induces asmall single-particle rotation and can be safely ignored in the large N limit where our experiment
operates. For now, I will ignore the fact that the atoms couple with different strengths to thecavity mode. This is important for a quantitative understanding of the dynamics, but for now is
not especially important for a qualitative description.
The two other terms each contribute distinct and interesting spin dynamics (see Fig. 9.2).
The χJˆz2 term leads to an effect known as one-axis twisting (OAT). OAT represents a frequency
shift of the atomic transition whose value depends linearly on the atomic inversion Jz . This leads
to a shearing effect of the Bloch sphere, as though it were a rubber ball that you grabbed witha hand on each of the two poles and twisted. If applied to a coherent spin state, represented by
the dashed circle in Fig. 9.2a, OAT will shear the noise distribution, making it narrower along one
direction but wider along the other, generating useful squeezing and entanglement[156, 67, 69].
The final χJˆ2 term leads to an effect known as many-body gap protection or spin-locking[160]. By creating an energy splitting between states of different J, this term suppresses transitions
between states of different J that are caused by static (or slow) frequency shifts to subsets of theatoms. For example, if one prepares a maximally symmetric state with |J| = N/2, this term will

133

Figure 9.2: Illustration of the effects of the two terms in H = ~χ[Jˆ2 − Jˆz2 ] (a) The χJˆz2 term causes a
twisting of the Bloch sphere, referred to as one-axis-twisting (OAT). A quantum noise distribution(represented by the dashed circle) will be deformed by this twisting, leading to a potentially usefulentangled state. (b) The χJˆ2 term leads to a so-called many-body gap protection effect, wheredephasing of the atoms due to inhomogeneous noise sources is supressed.

134

Figure 9.3: A Bloch sphere representation of the atomic ensemble radiating into a detuned cavity.
The phase of the cavity field C (φC ) is locked to the phase of the atomic coherence J + (φ) up to a
fixed offset φr = arctan(2∆/κ) + π/2. The Bloch vector precesses about the cavity field axis, whichis the source of both collectively enhanced decay and of shifts in the output frequency ω` .

suppress transitions to states of lower J, which could be very useful in metrological applications
where maintaining atomic coherence is important.
Spin-locking effects generated by exchange interactions [171, 172] have been observed ina variety of physical systems including NMR experiments in spin polarized hydrogen [173],3 He4 He mixture [174] and in trapped cold atoms [72, 73, 74].

In all of these systems, the exchangeinteractions emerge from direct collisions. In contrast, these effects emerge in our system from theexchange of optical photons in a cavity.

9.2

Classical Description of OAT
I will now present a classical picture for these dynamics using the Bloch sphere. To describe

the system, we need to keep track of expectation values corresponding to the atomic inversion Jz
and the atomic coherence J+ = Jx + ıJy , a complex number which represents the projection of the
Bloch vector onto the equatorial plane of the Bloch sphere, and the cavity field C (another complex
number that we represent on the equatorial plane of the Bloch sphere). As discussed in Chapter 2,

135

Figure 9.4: (a) When the cavity is on resonance with the atomic transition (∆ = 0) the cavity
field C is perpendicular to J + . This causes a rotation of the Bloch vector downwards, and is thephysical origin of superradiant emission. (b) When the cavity is tuned far off resonance from theatomic transition (∆  κ), the cavity field C is nearly aligned with J + . Now, the rotation of the
Bloch vector about C is primarily sideways. Because C follows the Bloch vector as it rotates, thisleads to a frequency shift of the Bloch vector precession, and of the emitted light.

the Bloch vector rotates about the vector that represents the cavity field. When the cavity is onresonance with the atomic transition, the cavity field vector is always perpendicular to the atomic
coherence. This causes the Bloch vector to rotate downwards, and is the origin of the collectively
enhanced emission (Fig. 9.4). The collective emission leads to a decay of the collective excitation
at a rate Γ = 4g 2 κ/(4∆2 + κ2 ).
The situation can be generalized to the case where the cavity is tuned off resonance with theatomic transition by a frequency ∆. In this case, the cavity mode represents a harmonic oscillator
that the atoms are driving off resonance. Like any harmonic oscillator, the response of the cavity
to an off-resonant drive will be shifted in phase relative to its resonant response — it will exhibit a136phase lag if the cavity is driven above resonance and a phase advance if it is driven below resonance.
On the Bloch sphere, this manifests as a phase shift of the rotation axis C. The field’s azimuthal
angle φC follows the azimuthal angle of the Bloch vector’s projection onto the x-y plane φ. The
relative azimuthal angle is φr = φ − φC , and is set by the cavity detuning φr = arctan(2∆/κ) + π/2.
When the cavity is tuned far from resonance and the field becomes nearly parallel with theatomic coherence (φr = 0 or π), depending on the sign of the detuning. The effect of the fieldgenerated rotation on the Bloch vector is no longer a rotation downward, but sideways — the cavity
field drives a rotation of the Bloch vector that changes its azimuthal angle, but not its polar angle(See Fig. 9.4b). This manifests as a modification of the rate at which the Bloch vector accumulates
azimuthal phase. We interpret this additional precession of the Bloch vector’s azimuthal angle as
an inversion-dependent frequency shift ωOAT = −2χJz of the atomic coherence J + ∼ J + (0)eıωOAT t .
Because the cavity mode decays quickly, the cavity field vector will readjust to the new phase of
the Bloch vector, maintaining the same relative phase φr . These effects lead to OAT — the rate
at which the Bloch vector accumulates phase, and the sign of the phase shift, depend on whether
the Bloch vector is above or below the equator, and by how much.
In general, the dynamics will have contributions from both of these effects, with classicalequations of motion for the Bloch vector given by:
dJ +
dt
dJz
dt

= −i(2χ + iΓ)J + (t)Jz (t)(9.2)= −ΓJ + (t)J − (t)(9.3)

In the absence of collective decay (Γ = 0), and assuming homogeneous coupling between theatoms and the cavity, the atomic system evolves according to:dJ += −2iχJz (0)J + .
dt

(9.4)

Because Jz is constant, this simply represents an azimuthal precession of the Bloch vector at a rate
2χJz (0), and is the origin of OAT. The inhomogeneous coupling of the atoms to the cavity makes
things a bit more complicated. I will go through how we treat this here, following derivations byRobert Lewis-Swan.

137
We prepare the atoms by performing a coherent rotation from the bottom of the Bloch sphereusing resonant light coupled through the cavity. Atoms contribute to superradiant emission to adegree that depends on their location along the cavity axis both because their coupling to theradiating mode varies and because the state preparation angle depends on the local intensity of thelaser that performs the initial coherent rotation. We wish to generalize Eq. 9.4 to take these effects
into account. Fortunately, this method of state preparation (in which the inversion and couplingof a given atom are correlated) retains the form of Eq. 9.4 with simple redefinitions of g and Jz (0).
The atom-light coupling for a given atom is characterized as gj = gcos(k0 j), where j indexes
the location of the atom, k0 is the wave-number of the cavity mode at 698 nm, and 2g is the singlephoton Rabi frequency of a maximally coupled atom. Because each atom radiates into the cavity
by an amount proportional to its coupling to the cavity, we define new collective operators that˜+ ≡ 1 P gj ĵ + , where ĵ + represents the coherence of an individual
account for this weighting: Jˆjjg
˜+ that we can infer from the light leaking out of the cavity. The mean-field
atom at location j. It is Jˆequation of motion for J˜+ is given by:XdJ˜+= −iJ˜+
gj2 jjz (t)
dt

(9.5)

j

where jjz (t) = hσ̂jz (t)i/2, the inversion of the atom at location j. We assume here that jjz (t) = jjz (0),valid for times short compared to the characteristic evolution time χN , again assuming Γ = 0. WithP
these assumptions, we can associate ∆ω` = j gj2 jjz (t) with the frequency shift of the radiated light.
We now plug in the known forms for gj and jjz (0): gj = gcos(k0 j) and jjz (0) = cos[θ0 cos(k0 j)],where θ0 is the rotation of a maximally coupled atom (gj = g) and the subscript j now simply
provides an index for the location of the atoms that we will average over. We obtain:g2∆
J1 (θ)∆ωl = 4 2N
− J2 (θ) .
4∆ + κ2
θ(9.6)

Note that the form of this expression follows the homogeneous case, up to a re-definition of√g: g 0 = g/ 2 and Jz : Jz0 = N ( J1θ(θ0 0 ) − J2 (θ0 )). I will call these quantities the “effective coupling”
and the “effective inversion.” As long as we use these quantities instead of the original, unweighted

138
ones, we recover the simple linear dependence of ∆ω` on (effective) inversion. Note that for small
θ0 , Jz0 ' (N/2)cos(θ0 ), and the primary effect of the inhomogeneous coupling is a factor of two
reduction in ∆ω` .

9.3

Experimental Characterization of OAT Term

Figure 9.5: Shifts in frequency of emitted light ω` versus cavity detuning ∆, showing expected
dispersive behavior. Blue line is a fit with cavity linewidth held to its independently measured
value.

Experimentally, we explore the variation of ω` with cavity detuning ∆ (Fig. 9.5) at a given Jz(and Jz0 ). On each trial, we prepare the atoms in the same state near the bottom of the Bloch sphere(for which the shifts are large), and scan the detuning of the cavity mode while recording changesin ω` . Note that in this case because θ0 is small and we are interested in the dependence of ω` on∆, rather than its overall magnitude, we can ignore effects of inhomogeneity for this measurement.
We observe the expected dispersive behavior of the frequency shift versus detuning, as can be seen

139
by the fitted dispersive curve with the cavity linewidth held fixed to its independently measured
value.

Figure 9.6: Main: Pulling coefficient P (black points) and atomic inversion Jz (blue points) versus
state prep pulse duration. Blue line is a fit to the blue points with the expected functional form of
inversion versus state preparation angle, allowing for a fixed fraction of atoms that remain in the
ground state. Black line is predicted value for pulling coefficient given the fit parameters of theblue line, up to an overall scale factor. Inset: the same data can be viewed by plotting P against
the value of Jz0 (0) inferred from the blue fit line, showing the expected linear scaling.

Next, we demonstrate the linear scaling of the frequency shift with effective atomic population`
inversion Jz0 . To do so, it is useful to define the pulling coefficient P = dωd∆ , the sensitivity of emission

frequency to cavity detuning. Measuring P provides a way of measuring frequency shifts of theemitted light that are due to shifts in the cavity frequency while being insensitive to technical drifts
in the stable reference laser. In our system, P is always much less than one, a characteristic featureof superradiant lasers. We expect P to depend linearly on effective atomic inversion.
We vary θ0 by changing the duration of the state preparation pulse and measure the resulting
actual population inversion Jz (Fig. 9.6) using fluorescence detection. Because of the inhomogeneous
coupling to the standing wave cavity mode, we expect Jz /N = 21 J0 (θ0 ). We find that in practice140
somewhat fewer atoms are transferred to the excited state than we would expect. We account forthese by assuming that a certain fraction remain in the ground state, and that the coupling ofthese atoms is representative of the coupling of the atoms in general. The imperfect transfer may
be due to imperfect spin polarization, or to finite laser linewidth. We extract the value of θ0 at agiven preparation pulse duration from a fit to Jz . We allow for an offset in the fit to account for
the finite transfer fraction
On separate trials, we measure P by toggling the cavity detuning ∆ by ±2π × 30 kHz, andmeasuring the resulting change in the emission frequency during the first 8 ms of the superradiance(black points in Fig. 9.6). Vertical error bars reflect statistical uncertainty of each point, whilehorizontal error bars reflect uncertainty in the initial preparation angle due to laser power andfrequency drifts. We expect P to be proportional to Jz0 , so we fit the pulling coefficient data to thefunctional form of Jz0 . We hold fixed the values of θ0 and the fraction of atoms left in the groundstate due to imperfect state preparation to the values returned from the fit to Jz , and allow only an
overall scale factor to be fit to the measured values of P . This is shown as the black line in Fig. 9.6.
There seem to be some additional fluctuations in the data points beyond the pure statistical error
bars, which are of unknown origin. One contribution could be due to small changes in Jz0 during
the 8 ms window that we analyze. Still, the black line clearly captures the main structure of thedata.
We may view the same data by plotting P against the value of Jz0 extracted from the fluorescence measurement Fig. 9.6, inset. This makes clear the linear dependence between P and Jz0 .
The ratio of the measured slope to our independently predicted value is 0.7(3). The error on thismeasurement is due to uncertainty in our atom number calibration and in the actual linewidth ofthe mHz linewidth transition (which determines g).

9.4

The χJ 2 Term
From a classical perspective, we can understand the χJˆ2 term as contributing a rotation of

the collective Bloch vector about itself — there is an energy difference between states aligned with

(a)

(b)

(c)

141

Figure 9.7: Illustration of the effects of the χJ 2 term. We consider two sub-ensembles, represented
by the green and orange Bloch vectors. (a) When the two sub-ensembles have the same transition
frequency, the χJ 2 interactions simply lead to a precession about the total Bloch vector J (bluearrow). (b) When the transition frequencies of the two sub-ensembles are split by δ, the two Blochvectors rotate in opposite directions along the equator when the χJ 2 interactions are absent. (c)When both δ and χ are finite, and χN  δ, the two sub-ensembles follow small circles and neverdevelop a large opening angle. This is the origin of the many-body gap protection.

the collective Bloch vector, and states anti-aligned with it. For a fully symmetric ensemble of atoms(meaning that each atom has the same Bloch vector) this interaction does not lead to an observableeffect on the Bloch sphere. If however the atoms in the ensemble are distinguishable, this term has
important consequences.
The simplest way to see this is to consider two distinguishable sets of atoms inside the cavity(for example occupying two different nuclear spin projections). One of these sets contributes to acollective Bloch vector J~1 , while the other contributes to a collective Bloch vector J~2 . The total
Bloch vector is J~ = J~1 + J~2 . If J~1 and J~2 are prepared with some opening angle between them,~ as shown in Fig. 9.7a.
they will precess in circles about their sum J,
Next, consider a situation where the two sub-ensembles each have a slightly different frequency, split by ±δ/2 and start out aligned on the equator of the Bloch sphere. In the absence ofinteractions, the two Bloch vectors would precess along the equator in opposite directions, leading~ followed by revivals.
to a reduction in the length of J,
In the presence of strong χJ 2 interactions however, the rotation about J~ redirects the precession of the Bloch vectors so that they each follow a small loop, one above the equator and theother below, before returning to their original locations. Because the opening angle remains small,the magnitude of J remains fairly constant, providing supression of the dephasing effects caused

142
by the frequency splitting δ.

9.5

Experimental Characterization of Effects from the χJ 2 Term
While it is conceptually nice to understand the gap protection in terms of Bloch vectors near

the equator, we found it experimentally easier to observe the effects of the χJ 2 term for states near
the south pole of the Bloch sphere. We perform two distinct types of experiment that both show
experimental manifestations of the effect of the χJ 2 term of the Hamiltonian.
The first experiment mimics the model that I presented in the previous section, with two
distinct spin sub-ensembles. To achieve this in practice, we prepare half of the atoms in mF = −9/2
(described by a Bloch vector J~1 ) and the other half in mF = 9/2 (described by a Bloch vector J~2 ).
The two ensembles experience a differential Zeeman energy shift ±~δ/2 proportional to an appliedmagnetic field.
It is useful to consider the behavior of both the sum and difference of the two Bloch vectors.
ˆ
ˆ
ˆ
ˆ ˆ
ˆ
We define operators J~ = J~1 + J~2 and ~j = J~1 − J~2 that represent the sum and difference respectively.
The Hamiltonian can then be written as Ĥ = χJˆ+ Jˆ− + δ ĵz (neglecting Γ, as it does not effect theazimuthal phase of the Bloch vectors).
The mean-field equations of motion (neglecting Γ for simplicity) are given by:

dJ +
dt
dj +
dt
d2 jz
dt2

= −i2χJ + (t)Jz (0)−iδj + (t)(9.7)= −i2χJ + (t)jz (t)−iδJ + (t)

(9.8)



2
= −4 δ 2 + Jz2 (0) + J⊥(0) jz + 6δχjz2 + 2δχ|J + (0)|2(9.9)

The detuning δ converts the amplitude of J + into amplitude in j + and back. The presence ofinteractions (χ 6= 0) suppresses this interconversion through an effect that we may describe as anenergy gap.
In general, the variation of jz with time makes the interpretation of these equations somewhat
complicated. However, when the total Bloch vector is prepared near a pole of the Bloch sphere,

143
energy conservation requires that |jz |  |Jz |. If we neglect the term proportional to jz , then it
is straightforward to analyze the normal modes of the system. When δ = 0, we find that J + andj + represent eigenstates — a system prepared with full coherence (j + (0) = 0) will maintain fullcoherence, while a system prepared with no coherence (J + (0) = 0) will always have no coherence.
On the other hand, if δ  χN , then we know intuitively that the degree of coherence will oscillatewith time as the Bloch vectors of the two ensembles align and cancel, and the eigenstates will beequal combinations of j + and J + .
In general, we can write the eigenstates as B+ = cos(α/2)J + + sin(α/2)j + and B− =sin(α/2)J + − cos(α/2)j + where the mode mixing angle α is tan α = 2δ/χN . The normal modepfrequencies associated with these eigenstates are ω± = χN ± (χN )2 + 4δ 2 /2.
The frequency splitting at zero detuning ω+ − ω− = χN is exactly the frequency gap thatsuppresses decoherence from inhomogeneous noise sources. For values of δ  2χN , the normal
mode characters remain nearly either fully coherent or fully incoherent. The energy gap between
the two modes supresses interconversion between coherent and incoherent states.
In our experiment, we detect the field radiated into the cavity, which is proportional to J + .
In contrast, the j + atomic coherence does not emit light. We thus think of the J + and j + as brightand dark modes. For small δ, we thus expect that the mode B+ will be bright, while the modeB− will be dim, while for large δ the two modes should be equally bright. Further, the two modes
should undergo an avoided-crossing type behavior.
We experimentally observe these effects by recording the power spectra from simultaneous
lasing on the mf = ±9/2 transitions for atoms prepared near the bottom of the Bloch sphere. We
scan the value of the the inhomogeneous frequency shift δ between trials. The experimental results
are displayed in the spectrograms of Fig. 9.8b with the cavity detuned 150 kHz above resonance(upper panels), and 150 kHz below resonance (lower panels). The center panels show spectrogramsfrom a simulation of the mean-field equations using our estimated experimental parameters, which
qualitatively reproduce the features of the experimental data, espcially an imbalance of the powerin the two radiating branches that switches sign with the cavity detuning. The far-right panels

(a)

(b)

experiment

simulation

simulation

144
(c)

(Hz)

200

0

-200
Spectral Power(arb. units)

(Hz)

200

0

normalized
spectral
power

-200

Fourier frequency (Hz)Figure 9.8: (a) We introduce controlled inhomogeneity by simultaneously populating the mF =±9/2 states and applying a magnetic field to split the Zeeman sublevels by a frequency δ. (b)Output power spectra for lasing on the two transitions, with atoms prepared near the south pole of
the Bloch sphere. In the presence of a large detuning δ, the two branches correspond to light emitted
independently on the mF = ±9/2 transitions. When the cavity mode is detuned from atomic
resonance by ∆/2π = 150 kHz ' κ/2π (top row), we experimentally observe an asymmetry in thetwo branches (left panel). This asymmetry is reversed for opposite cavity detuning (middle row).
A simulation with the full Hamiltonian Ĥ = ~χ[Jˆ2 − Jˆz2 ] qualitatively reproduces these features(center column), while a simulation that includes only Ĥ = ~χJˆz2 does not (right column). This
indicates the important role of the term χJ 2 , which is associated with many-body gap protection.
Power spectra at fixed δ (bottom row) highlight this comparison,taken at a detuning δ indicatedby the horizontal green lines. (c) Cartoon projections of the Bloch vectors associated with the two
spin ensembles onto the xy plane for pure OAT Hamiltonian (upper) and full Hamiltonian (lower)
in reference frame of total Bloch vector. For pure OAT, the two ensembles precess independently,
leading to a large opening angle. For full dynamics, the two Bloch vectors synchronize, preventing
a large opening angle from developing.

show the results of a simulation with only the OAT contribution to the Hamiltonian Ĥ = ~χJˆz2 .
These do not reproduce the asymmetry observed in the data, highlighting the role of the χJˆ2 termin Ĥ.
The second feature that we expect to see is a repulsion of the normal modes near δ = 0,
corresponding to the many-body energy gap. Instead of using two separate spin states to introduce
controlled dephasing of the atomic ensemble, our next experiment relies on inhomogeneous lightshifts to controllably destroy and restore atomic coherence. We apply a laser near resonance with

145
the 1 S0 to 3 P1 transition, whose intensity varies over the axial extent of the atoms in the lattice(Fig. 9.9a). In practice, we achieve this by simply misaligning the center of the roughly 3 mmdiameter laser spot from the center of the atoms in the direction along the cavity axis. When weapply this laser with a detuning of typically 14 MHz and a duration of typically 1 ms, we see that theinhomogeneous dephasing destroys superradiance. We can then reverse the sign of the detuning
and apply a second pulse of equal duration to reverse the dephasing and restore superradiance(Fig. 9.9b).
This procedure allows us to controllably swap the atoms between the bright state (nondephased) and the dark state (dephased). While we are not working with the two spin-ensembles
that we used to derive the forms of the bright and dark states, the concept is general and canbe meaningfully applied here anyway. We are interested in measuring the frequency differences
between these two states. However, because the dark state does not radiate light for us to detect,
we need to be a bit more sophisticated to measure its frequency.
We prepare a bright state near the south pole of the Bloch sphere using our usual method.
We then measure the optical phase (ψ1 ) of the emitted light relative to the reference laser. We
next dephase the atoms using the inhomogeneous laser, populating a dark state. After a variable
amount of time Thold , we rephase the atoms, restoring population to the bright state and make asecond phase measurement of the emitted light (ψ2 ). We may then interpret the slope of the phasedifference ψ2 − ψ1 versus Thold as the frequency of the dark state. If we follow exactly the sameprocedure, but do not apply the dephasing and rephasing beams, we instead interpret this same
slope of the phase difference as the frequency of the bright mode.
In practice, we actually measure how the phase difference changes when we toggle the cavity
between two detunings ∆ = ±30 kHz: ∆ψ = (ψ2 − ψ1 )|30 kHz − (ψ2 − ψ1 )|−30 kHz , which is related
to the pulling coefficient (Fig. 9.10). For the atom number and inversion used here, we find thatwhen we leave the atoms in the bright state by not applying the dephasing or rephasing pulses,
the slope of ∆ψ versus Thold represents a frequency shift of the bright state due to cavity-mediatedinteractions of ±4.5 Hz. When the dephasing and rephasing steps are introduced, the slope of

146

Figure 9.9: (a) We introduce controlled, reversible dephasing by illuminating the atoms with lightnear resonance with the 1 S0 to 3 P1 transition. The beam is offset from center with the atomic
cloud, so the atoms experience a spatially varying intensity, and therefore a spatially varying
light shift. (b) Applying the beam for roughly 1 ms scrambles the phases of the atoms, reducing
atomic coherence and superradiant emission. Applying the laser again, this time with the opposite
detuning from the 1 S0 to 3 P1 transition unscrambles the atomic phases, restoring atomic coherence
and superradiance.

∆ψ versus Thold is consistent with zero, indicating no shift of the dark state due to the cavity.

147
The difference in frequency between the bright and dark states is the direct manifestation of the
many-body gap for a system near the south pole of the Bloch sphere.

Figure 9.10: We measure the change in the difference between optical phase measurements before dephasing (ψ1 ) and after rephasing (ψ2 ), as the cavity is toggled between opposite detunings∆/2π = ±30 kHz: ∆ψ = (ψ2 − ψ1 )|30 kHz − (ψ2 − ψ1 )|−30 kHz . Red (black) points correspond to trials where the atoms were dephased (not dephased) during Thold , corresponding to the bright (dark)
normal modes. ∆ψ shows no dependence on Thold for the dephased case, and a phase shift thatscales linearly with Thold when the atoms are not dephased, confirming a frequency shift between
the bright and dark portions of the atomic coherence.

To make this connection explicit, we may consider the effect of dephasing and rephasing ina Dicke basis for states prepared near Jz = −N/2 under the effect of a Hamiltonian Ĥ = ~χJˆ2 .
We neglect the effects of the χJˆz2 term for now, as this commutes with both Jˆ2 and Jˆz so we canaccount for its effects at the end. States in this basis |J, mj i are eigenstates of Jˆ2 and Jˆz , andare labelled by the magnitude of their spin J and their eigenvalue mj . A maximally symmetriccoherent spin state |θ, φi with J = N/2 (and corresponding Bloch vector with polar angle θ andazimuthal angle φ) is represented by a superposition of Dicke states as|θ, φi =N/2
Xc(mj , θ)e−ıφmj |N/2, mj i

(9.10)

mj =−N/2

where the coefficients c(mj , θ) are real. The constant relative phase between adjacent values of mj
sets the azimuthal angle φ of the coherent spin state.

148

Figure 9.11: The connection between our observed phase shift and an energy gap can be understood
explicitly by considering the effect of dephasing in a Dicke basis for states prepared near Jz = −N/2,
subject to a pure χJ 2 Hamiltonian. Atoms begin in a superposition of maximally symmetric stateswith J = N/2 (far left states). Dephasing (red arrows) transfers the atomic ensemble to states ofminimal coherence for a given Jz . Because the difference in J between the states of maximal andminimal coherence for a given Jz is Jz + N/2 for states with Jz ' −N/2, the energy difference
between these states is approximately χ(Jz + N/2). When coherence is restored in the rephasing
step (blue arrows), the different Jz states will have accumulated a phase proportional to this energy
shift. Because a linear phase shift between adjacent Jz states corresponds to an azimuthal rotationof the Bloch vector, the measured phase shifts of (b) provide a direct measurement of the energy
splitting between manifolds of different J, which is the origin of many-body gap protection.

The relevant Dicke states for the three lowest values of mj are shown in Fig. 9.11, along withthe energy gaps created by a ~χJˆ2 Hamiltonian. When ideal dephasing is applied, Dicke stateswith constant mj are coupled such that the maximally symmetric Dicke state |N/2, mj i is mapped
to the the state ||mj |, mj i, the Dicke state with the smallest possible value of J = |mj | compatible
with the initial spin projection mj of each Dicke state.
For Dicke states near the bottom of the Bloch sphere, the energy gap between ||mj |, mj i and|N/2, mj i scales linearly with mj as E(mj ) ≈ ~χN (mj + N/2). After an ideal rephasing step, thecomplex amplitudes that evolved phase during Thold will have been mapped back from ||mj |, mj ito the original Dicke states |N/2, mj i. Because of the time spent in the minimally symmetricstates, each Dicke state’s amplitude will have acquired a phase shift ∝ χN Thold mj , equivalent toa change of the coherent state’s azimuthal phase φ by χN Thold . Therefore, we can understandthat measuring the difference in phase accumulated by the Bloch vector during Thold with andwithout dephasing/rephasing is a direct spectroscopic measure of the many-body energy gap. The
additional OAT dynamics changes which configuration appears to have a measurable phase shift,149
but importantly, the two configurations retain a relative phase shift caused by the many-bodyenergy gap.
Finally, we can consider what happens if we only partially dephase the atoms (Fig. 9.12). This
corresponds to populating a mix of the bright and dark states. Will the frequency of the emitted
light be determined by the bright mode frequency, or the dark mode, or some combination of thetwo? We vary the power in the dephasing beam, and infer the fractional remaining coherence fromthe amplitude of the emitted field (which is proportional to the coherence J+ ). We then measure
the pulling coefficient P associated with the partially dephased state by measuring the frequency
shift of the emitted light when the cavity detuning is toggled between ∆ = ±2π × 30 kHz. We findno dependence of the pulling coefficient on the degree of coherence, as the coherence is modified bya factor of roughly three — the frequency of the emitted light is determined by the bright modeeven when the system is partially dark.
This observation is consistent with the form of the Hamiltonian Ĥeff = ~χJˆ+ Jˆ− ≡ ~χ[Jˆ2 −Jˆz2 + Jˆz ]. The J 2 term does not contribute here, as J is unmodified during the relevant part of theexperiment. All we see is the effect of the Jz2 term, which explicitly does not depend on the degreeof atomic coherence.

Figure 9.12: We may vary the power in the dephasing beams to only partially dephase the atoms.
We can infer the residual coherence from the amplitude of the superradiant emission. We observethat the pulling coefficient is uneffected by this process, maintaining the same value as the fully
coherent case (to within our experimental uncertainty).

150

9.6

Conclusion
Apart from being potentially useful in the context of quantum simulation, the interactions

that we have explored here will likely have immediate impacts in the field of precision opticalmetrology. Understanding the inversion dependence of the pulling coefficient is key for the designand specification of superradiant lasers. Further, it is important to understand the nature of cavity
pulling effects if one wishes to use an optical cavity in a clock — for example to provide nondestructive atom counting. Further the same dynamics that we observe here could in principle beused to create spin-squeezed states (as independently proposed in ref.[175]).
Creating useful spin squeezing requires operating in a far-detuned regime ∆  κ to ensurethat the squeezing dynamics occur before superradiant decay destroys coherence. In our current
system, background dephasing at a rate γ⊥ ' 2π × 4 Hz, due at least partly to atomic collisions,becomes dominant over collective decay in limiting the pulse duration at a detuning comparable
to the cavity linewidth. This prevents us from winning by going into the far-detuned regime, as χ
decreases while γ⊥ remains constant. In addition to challenges associated with observing any spinsqueezing that did occur, generating useful entanglement would require substantial experimental
upgrades. Besides addressing the issue of background dephasing γ⊥ , it would be useful to operatewith a higher cooperativity cavity in order to speed up the dynamics relative to any remaining
dephasing mechanisms, and to operate with a somewhat stronger transition such as the Yb clocktransition considered in Ref. [175].

Chapter 10

Sawtooth-Wave Adiabatic Passage (SWAP) Cooling

Most implementations of laser cooling of atoms and molecules has utilized broad-linewidth
optical transitions. Because the excited state of such transitions is short-lived, it is typical totreat the atoms as generally occupying one or more ground states. The excited state population istypically treated only as a proxy for spontaneous emission, or as a means of modifying the energy
and character of the ground state. The same cooling techniques used on broad-linewidth transitions
have been extended to narrow-linewidth dipole-forbidden optical transitions [176], including the
7.5 kHz linewidth transition in strontium [75, 76, 77, 78, 79, 80]. Even in these experiments
however, the atoms are generally thought of as primarily occupying the ground state, and thecooling mechanisms are fundamentally very similar to their broad-linewidth counterparts.
This chapter concerns a fundamentally new mechanism for cooling that is enabled by the long
lifetime of the excited state of a dipole-forbidden transition. We call this mechanism SawtoothWave Adiabatic Passage (SWAP) cooling. SWAP cooling explicitly relies on transferring atomic
population to the excited state, and storing it there for significant periods of time. In addition, thiscooling mechanism relies heavily on the unitary dynamics of stimulated emission and absorption.
Spontaneous emission plays an important, though greatly reduced role in the cooling process relative
to many previous cooling methods. This mechanism thus brings up interesting questions relating
to entropy redistribution, and may be applicable to the cooling of particles for which spontaneousemission causes severe problems, such as molecules.

152

10.1

How it Works

10.1.1

Adiabatic Passage

As the name suggests, SWAP cooling relies on a mechanism called adiabatic passage, also
known as a Landau-Zener crossing. A detailed explanation can be found in many places, forexample in Chapter 5 of ref. [87]. The basic idea of is that a drive (in our case a laser) that couples
two atomic states is swept in frequency over the transition frequency between the two states. If
the strength of the drive is sufficiently high compared to the rate of the frequency sweep, then theatomic population may be transferred from one state to the other with very high efficiency.
Several intuitive pictures can be used to illustrate this mechanism. The first picture involves the concept of dressed states and an avoided crossing, and is illustrated in Fig. 10.1. The
Hamiltonian, written in the reference frame of the drive is given by:

Figure 10.1: Illustration of Landau-Zener crossing in a dressed state picture. Black lines represent
energy eigenstates in the presence of drive coupling Ω. An atom that begins in |gi when the drive
has a large negative detuning δd will be transferred to |ei as the drive sweeps to large positive
detuning (red arrows).

153











−δd Ω






H=









 Ω 0

(10.1)

δd = ωd − ωa , where ωd and ωa are the frequencies of the drive and atomic transitions, respectively,and Ω is the Rabi frequency associated with the drive. This Hamiltonian acts on the state vector:








 g, n 
















e, n − 1

(10.2)

where g, e represent the ground and excited states of the atom and n, n − 1 indicate the numbersof photons in the drive field associated with these states. The eigenvalues of this Hamiltonian are
shown in Fig. 10.1, along with the character of the eigenstates at large detuning |δd |. An atomin the ground state with a large negative detuning corresponds to the left side of the lower modeon the plot. As the drive frequency is swept, we move to the right on the plot. So long as thegap between the two states remains sufficiently large, the atom will stay in the lower-energy state,ultimately being transferred to the bottom right branch, which represents an atom in the excitedstate with large positive drive detuning.
The condition for the gap being big enough is that Ω2  α, where α is the rate of thefrequency sweep, α = dδd /dt. If this condition is not fulfilled, the chance of ending up on the wrong
2

branch is given by e−πΩ /2|α| .
The second picture involves the Bloch sphere. The atoms start in the ground state, and thedrive starts out with a large negative detuning. This drive is represented by a vector that points
near the south pole of the Bloch sphere (at an angle θ = tan−1 (Ω/δd )) about which the Bloch vector

154
rotates at the effective Rabi frequency Ω0 =q
Ω2 + δd2 . As the detuning of the drive is swept overresonance, the drive vector sweeps up the Bloch sphere. If the rate of precession of the Bloch vector
about the drive vector is high enough, then the Bloch vector will follow the drive vector up the
Bloch sphere, making tiny circles about it. When the drive ends at a large positive detuning, both
the drive vector and the Bloch vector will point near the north pole, indicating nearly complete
transfer of the atomic population.

10.1.2

Basic Cooling Mechanism in One Dimension

a.

b.

v

P(e)

laser
frequency

c.

1
0

time

Figure 10.2: Illustration of basic cooling mechanism. (a) Atoms moving in one dimension withvelocity v are illuminated by counter-propagating laser beams. (b) Atoms have a single relevant
atomic transition at frequency ωa . In our case, this transition is dipole-forbidden, so |ei has arelatvely long lifetime. (c) Representation of frequencies in lab frame. Because of Doppler shifts
resulting from atomic motion, the co-propagating (counter-propagating) laser beams are resonantwith |gi to |ei transition when the laser frequency ω = ωa − kv (ωa + kv). Because ω increases withtime, the counter-propagating laser sweeps over the transition before the co-propagating laser. If
the atom starts in |gi, the counter-propagating laser transfers the atom from |gi to |ei as is sweeps
over resonance, and the co-propagating laser transfers the atom back to |gi. This results in thetransfer of two photon recoils of momenta to the atom in the correct direction to cool.

To understand our cooling mechanism in its simplest form, we may consider a two-level

155
atom with a long-lived optically excited state |ei and ground state |gi moving in one dimension
with velocity v, as shown in Fig. 10.2a. Two counterpropagating laser beams with wave-number k
and frequency ωd are linearly ramped in frequency from below to above the atomic transition
frequency ωa , with full sweep range ∆s . This ramp is repeated to form a sawtooth pattern intime, as in Fig. 10.2b. Each laser interacts with the two-level atom with a Rabi frequency Ω,which is tuned to be larger than the spontaneous decay rate γ from |ei to |gi. The sweep range isadjusted such that ∆s > Ω, 4kv. We control the frequency sweep rate α to ensure that it fulfills
the Landau-Zener condition α  Ω2 for adiabatic transfer of atomic populations between groundand excited states. Lastly, the jump in laser frequency at the end of each ramp is considered to beinstantaneous, and therefore perfectly diabatic.
In the reference frame of the atom, both cooling beams start below resonance with the atomic
transition. Doppler shifts due to the atomic motion cause the beam counter-propagating to theatom’s velocity to appear kv higher in frequency and the co-propagating beam to appear kv lower
in frequency. As the beams sweep upward in frequency, the counter-propagating beam sweeps overresonance first, and adiabatically transfers the atom from |gi to |ei. Because of the long lifetimeof the excited state, the atom remains in |ei until the co-propagating beam sweeps over resonanceand adiabatically transfers it back to the ground state. In this process, the atom has absorbed
one photon from the beam propagating against its motion, and emitted one photon into the beam
propagating along its motion, resulting in a net momentum transfer of 2~k against its motion. The
laser frequency is then diabatically jumped back to its start frequency such that the atom remains
in |gi, and then the process is repeated. Because the direction of the net momentum kick is dictatedby the time-ordering of the adiabatic transfers, which is in turn determined by the atom’s velocity,
these momentum kicks will always be in the correct direction to reduce the velocity of the atom
regardless of the direction of its initial velocity. This leads to a compression up of the atomic
velocity towards zero.

156
10.1.3

Where Does the Entropy Go?

So far, the dynamics that I have introduced are completely unitary, and thus cannot in fact
lead to true cooling as defined by an increase in the phase-space density of the atoms. The missing
ingredient in the preceeding section is that I assumed the atom always starts each sweep in the
ground state. If the transition being used has an infinite excited-state lifetime, this assumption
would be problematic. As the velocity of the atoms decreases, the time-ordering of the two adiabatic
transfers becomes ambiguous, and the atom may end up in the excited state at the end of the sweep.
If this happens, the atom will then receive momentum kicks in the direction of its motion rather
than against its motion, and will begin to heat.
The presence of even small amounts of spontaneous emission can be sufficient to allow steadystate cooling to occur, by ensuring that the atoms preferentially begin each sweep in the groundstate. So long as the total time during a sweep before the first adiabatic transfer and after thesecond adiabatic transfer exceeds the time between the two adiabatic transfers, excited-state decay
will ensure that the atom is more likely to start each sweep in the ground state than in the excitedstate. This breaks the time-reversability of the otherwise unitary dynamics, and allows this coolingmechanism to remove entropy from the atoms.
Intuitively, how should we think about where this entropy goes? Imagine an idealized scenario
where an atom starts with an arbitrary initial velocity. It is then slowed by repeated sweeps, witheach removing 2~k of momentum, until it reaches some low velocity. At that point, it will begin
to scatter photons. For the sake of this example, let us then assume that the atom decays intosome other state with high probability during the first scattering event, and stays there forever,decoupled from the cooling process. Is this single photon enough to remove the initial entropy of
the atom? To answer that question, we need to ask whether the photon carries enough information
to reconstruct the initial state of the atom. In this case, it does (up to a possible ambiguity in thesign of the initial velocity). If the atom were initially moving with high velocity, it would take more
sweeps to slow it to the point where it scatters a photon than if it were moving slowly. The initial

157
momentum of the atom is thus encoded in the time-delay before the first emitted photon.
In a realistic system, things are obviously more complicated. There is not some well-defined
cutoff at which the atom begins to scatter, and the atom will likely emit at least several photons
during the cooling process. This idealized version illustrates a limiting case, and should hopefully
make the idea that large amounts of entropy can be carried away by a single photon seem acceptable,
when combined with carefully engineered unitary dynamics. Similar ideas have been pursued in
the context of optical pumping in large spin systems [177] and cooling with small numbers of
spontaneous emission events [178].

10.1.4A Dressed-State Picture

We can also understand our cooling mechanism in a dressed-state picture, as illustrated inFig. 10.3.
We imagine that the left and right propagating lasers represent counter-propagating modesof a ring cavity, and label the number of photons in each mode m for the left going mode, and n
for the right going mode (the ring cavity is just a conceptual tool to justify a conserved numberof photons.) We may then label the state of the combined atom-light system as |m, n, ai, wherea = e, g labels the state of the atom. In this simplified model, we will neglect spontaneous emission,
so the total number of photons n + m is conserved when the atom is in |gi, and is reduced by one
when the atom is in |ei. The atom can transfer photons from one mode to the other via absorption
and stimulated emission. Each of these processes is accompanied by the transfer of momentum±~k to the atom. By counting changes in the relative population of the two modes, we implicitly
track the change in momentum of the atom.
We plot the energies of the combined atom-field eigenstates (in the atom’s rest frame) versus
the detuning of the light modes from atomic resonance (as measured in the lab frame). At eachlaser detuning, we reference the energy of the combined atom-light states to the atom frame energyof the initial photon number. With this subtraction, the uncoupled states (for which Ω = 0)|m, n, gi appear as horizontal lines of constant energy, while the states |m, n, ei appear as slanted158
lines because one of the photons has been turned into an atomic excitation, whose energy does notchange with laser frequency. These uncoupled states are represented by black lines in Fig. 10.3. For
this analysis, we assume that the atom is moving to the right, so the left-going photons (labeled m)have energies increased by ~kv relative to the lab frame, while right-going photons have energies
decreased by ~kv. This means that states with different photon number are shifted vertically fromeach other on our plot.
When we turn on coupling between the atoms and light at frequency Ω, the eigenvalues
resemble those shown in color in Fig. 10.3. Avoided crossings with energy splitting ~Ω occur when
two states for which either m or n (but not both) change by exactly one cross.
We consider the system to start in |M, N, gi with a large negative detuning of the lasers
from atomic resonance (point 1 in Fig. 10.3). As the detuning of the cooling beams is swept
to the blue, the system adiabatically follows the dressed eigenstate, until the atom encounters acrossing with |M − 1, N, ei. Now, the atom will adiabatically transfer to |M − 1, N, ei, and then to|M − 1, N + 1, gi, ultimately ending up at at point 2. When the detuning of the lasers diabaticallyjumps back to the red, the system stays in |M − 1, N + 1, gi but is now at point 3. In one cycle ofthe sweep, we see that a photon has been transferred from the left-going mode to the right-goingmode, implying a transfer of 2~k of momentum to the atom.
If Ω is increased, or kv is decreased, avoided crossings where more than one photon is transferred become significant. At crossings such as the one between |M, N, gi and |M − 2, N + 1, ei,
the energy splitting is ~Ω3 /16(kv)2 . In the regime where kv  Ω, these avoided crossings arevery small and for a range of laser frequency sweep rates, the atom will diabatically traverse thesecrossings (for example remaining in |M, N, gi). When this inequality no longer holds, as will be
the case once the atom is sufficiently cold, then the atom may adiabatically follow the transition
from |M, N, gi to |M − 2, N + 1, ei. When this occurs, the atom may then follow a path that leads
to the exchange of more than one photon between the two light modes, and as a result experience
multiple photon recoils worth of momentum change. This is the physical mechanism for the onsetof multiphoton processes, which we observe in both experiment and simulation at low velocity or

159high Rabi frequency.

Eigenfrequency(k v)
Eigenfrequencykv

e
-2,
1, N
M+

0

,e
N-1
M,

,e
1, N
M-

,e
+1
2, N
M1

Singlelaser Rabi Frequency 1
Detuningbetweenlasers 2.5

M, N, g

1

1

2

3

3

2 M-1, N+1, g

Ω

4

0

2

2

4

Laser Detuning (kv)averagefrequencylaser from resonance
2 Detuningof
2

Ω /4Δ/16(kv)
3

Figure 10.3: Eigenstate diagram of coupled atom-light modes versus detuning of the lasers fromatomic resonance. Uncoupled eigenstates are labeled as |m, n, ai (defined in text). Atoms start at
point (1), then are adiabatically transferred to (2) by the laser sweep. When the laser detuning
jumps, they non-adiabatically project to (3), having given up 2~k of momentum by transfering aphoton from the left-going to right-going modes. Higher-order crossings, which lead to the exchangeof multiple photons become significant when kv ∼ Ω

10.1.5

Regimes of Applicability

There are several conditions that need to be fulfulled for this cooling mechanism to work.
First, as mentioned before, there is the requirement that the frequency ramp must be slow enoughto be adiabatic. This means that we need α  Ω2 . In addition, in order to be in the regime in which2~k of momentum is transferred per sweep, which is also likely the regime in which spontaneousemission is highly suppressed, we need the ramp to be fast enough to cross the smaller level crossingsin Fig. 10.3 diabatically. This means that α  (Ω3 /16(kv)2 )2 . In practice, violation of this second
condition does not prevent the cooling mechanism from working. What we see instead is that much

160
larger amounts of momentum are transferred in a single sweep. This appears to lead to larger forces
and faster cooling than would be possible if only 2~k of momentum were transferred per sweep,
but has uncertain implications for the final temperature of the atoms and for the degree to which
spontaneous emission is suppressed.
Fulfilling both of the above requirements at the same time requires 4kv  Ω, which will
always be violated at some point if the atoms cool far enough. Intuitively, we can understand thisrequirement as follows: the characteristic frequency range over which adiabatic transfer occurs is
set by Ω. The frequency difference between the co-propagating and counter-propagating laser frequencies is 2kv. So, when kv . Ω, the time-ordering of the adiabatic transfers becomes ambiguous.
Interestingly, it appears that rather than the effects from the two beams cancelling out, it turns
out that in this regime the typical momentum transfer is much larger than in the regime withwell-resolved adiabatic transitions. Very roughly, in the dressed state picture of Fig. 10.3 we cansee that the lines generally go down and to the right, so if the atom makes all available transitions
it will end up quite a ways down on the graph, indicating a large change of momentum.
The next condition is much simpler: we don’t want the atom to decay between absorbing aphoton from the counter-propagating beam and emitting one into the co-propagating beam. This
simply means that 1/γ  2kv/α, where γ is the excited state decay time. This condition is mucheasier to achieve with a dipole-forbidden transition than a dipole-allowed transition.
Finally, there is an important though slightly subtle requirement on the total sweep range∆s . Clearly, we must sweep far enough to address the two Doppler-shifted transitions, which would
imply ∆s > 2kv. However, the requirement is actually more strict than that. Cooling relies on theatom preferentially beginning each sweep in the ground state than the excited state. This meansthat the time between adiabatic transfers in the middle of the sweep must exceed the total time
before and after the adiabatic transfers at the end of the sweep, requiring ∆s > 4kv. The net force
experienced by the atoms goes to zero when ∆s = 2kv, then switches sign for ∆s > 2kv.

< p̂ > ( k)

161

40
20
0
20
40

0

2

4

6

8 10 12time (sweep cycles)

Figure 10.4: Simulated trajectories for atoms with different initial momenta. Produced by a simulation based on a Monte Carlo wave function trajectory method written by Murray Holland andJohn Bartolotta.

10.1.6

Simulated Trajectories

Colleagues John Bartolotta and Murray Holland have developed a simulation of SWAP cooling based on a Monte Carlo wave function trajectory method. Everything but the external electricfield associated with the cooling lasers is treated quantum mechanically. The atomic wave function
contains ground and excited internal states in addition to a discrete family of external momentum
states, which are equally spaced in units of ~k up to a maximum cutoff. Unitary dynamics are
dictated by the Hamiltonian

Ĥ =



p̂2
1
+ ~ωa σ̂ z + ~Ω cos(kẑ) σ̂ + eiη(t) + h.c. ,2m 2(10.3)

where the instantaneous phase of the applied coherent field η(t) is the time integrated instantaneous
frequency (a phase), where the frequency is ramped in the sawtooth pattern previously described.
Because this is an open quantum system, a decay operator proportional to the spontaneous emission
rate is also included to fully simulate the master equation.
Fig. 10.4 shows simulated trajectories for atoms that begin at a spread of different initialmomenta. Atoms are prepared in a pure quantum state specified by their internal ground state

162
and fixed momentum. The average momentum over 50 trajectories is calculated at each time step.
At high momentum, each sweep lowers the momentum of the atom by roughly 2~k. At lower
momentum, we find that the change in momentum per sweep can greatly exceed 2~k. This is amanifestation of the multi-photon exchanges that occur when the atomic velocity is relatively low
compared to the Rabi freuncy Ω.

10.2

Experimental Demonstration
We experimentally explore this cooling mechanism in one dimension using a cloud of freely

falling 88 Sr. We first cool the atoms to typically around 10 µK using a version of SWAP coolingthat creates a restoring force, discussed later in this section. We then apply laser beams to theatoms from both sides. Each beam passes through a separate AOM, which is used to modulate thefrequencies of the beams either independently or identically. After applying the beams for someperiod of time, we turn off all light and allow the cloud of atoms to freely fall and expand for avariable amount of time. We then image the cloud using fluorescence from the 461 nm transition
and a camera that views the cloud in a direction perpendicular to that of the cooling beams. We
can extract the average velocity and temperature of the atoms by looking at the displacement of
the cloud center and the cloud width as a function of time. We use the states |gi ≡ 1 S0 , mj = 0
and |ei ≡3 P , m = 0 , where m labels the magnetic zeeman sub-level, and apply a magnetic
1jjfield of several Gauss along the direction of the cooling beam polarization to split out the mj = ±1
sublevels by more than our typical sweep range, so the cooling beams nominally interact with asingle excited state.

10.2.1Forces in 1D

First, we wish to characterize the average forces on the atoms as a function of their velocity.
The easiest way for us to accomplish this experimentally is to apply a frequency offset ∆ between
the two cooling beams, as shown in Fig. 10.5a. This creates a moving reference frame F moving at
velocity vF = ∆/2k in which the two beams appear to have equal instantaneous frequency. From

163

frequencya.

left beamtimeright beamb.

Figure 10.5: Cooling into a moving reference frame. (a) We apply a frequency offset in the lab
reference frame between the two beams, which creates a moving reference frame in which the beams
have equal frequency. (b) If laser frequency is swept upwards, (green points) the atoms settle into
the moving reference frame, which has a velocity indicated by the dashed black line. If frequency
is swept downwards (orange points), the atoms accelerate in the other direction, without cooling.
If the frequency sweep is symmetric (purple points), atoms experience a small acceleration due toradiation-pressure imbalance.

the perspective of F , atoms at rest in the lab frame appear to have a velocity v = −vF .
We apply the frequency-swept beams for a variable amount of time Ta , then measure theresulting velocity of the atoms. This is shown in Fig. 10.5b for several sweep configurations.
The green points represent the cooling configuration described above, with the laser frequency
adiabatically swept from low to high and diabatically jumped from high to low. The two beams
are offset by a detuning ∆/2π = 400 kHz, and are swept by ∆s /2π = 8 MHz every 33 µs. We
observe that the atoms undergo initial acceleration until they reach equilibrium with the velocityof the moving frame vF = 0.14 m/s, as one would expect for cooling into this moving frame
To rule out the interpretation that the atoms are merely being dragged by the moving standing
wave formed by the detuned lasers, we reverse the direction of the sawtooth sweep without changing
the relative detuning ∆ of the two beams. The standing wave still moves in the same direction

164
as before, but now the atoms accelerate in the other direction. While the upwards sweep causes
saturation of the atomic velocity at vF , no such saturation is apparent for a downwards sweep.
This confirms that the downwards sweep configuration does not lead to cooling.
Finally, when we apply a symmetric triangle-ramped frequency sweep to the lasers, the atoms
undergo a much smaller acceleration, likely due to residual radiation pressure from an intensity
imbalance between the two beams.

Figure 10.6: Acceleration versus polarization angle between two beams. Acceleration is enhanced
when the polarizations of the two beams are aligned. The blue line represents the maximum
acceleration for a saturated two-level atom without stimulated emission. The dashed blue line
represents the measured acceleration with a single beam. Black points represent acceleration withboth beams applied. The dashed black line represents the acceleration that two photon recoils per
sweep of the laser frequency would produce.

Because our mechanism relies on stimulated emission, much larger accelerations can beachieved than would be possible with Doppler cooling on such a narrow transition. To quantify this acceleration, we apply the beams for a time much shorter than it takes the atoms to reachequilibrium velocity, and measure the resulting change in velocity. If we apply only one of the twocooling beams, the atoms experience an acceleration consistent with (γ/2)(~k/m) = 155 m/s2 , theexpected value for a maximally saturated atom.
We see a far more dramatic effect when we apply both beams at the same time (Fig. 10.6).
When the polarizations of the two beams are aligned, we observe a maximum acceleration of around
600 m/s2 for a sweep period of 20 µs, a factor of almost 4 above both the observed accelerationfrom the leftward beam alone and the maximum expected acceleration for a two-level atom without

165
stimulated emission. This measured acceleration is within ten percent of the value we would expect
if each sweep of the laser frequency led to two photon recoils of momentum transfer, though thisagreement may partly be due to a cancellation between the effects of imperfect adiabatic transfersand multi-photon processes. When the polarizations are made orthogonal, the acceleration returns
to near the single-beam value, as only one of the beams interacts with |ei.
In contrast to Doppler cooling, the presence of the laser co-propagating with atomic motion
enhances slowing. Interestingly, because this laser interacts with inverted atoms, the associatedradiation serves to pull the atom toward toward the laser source.

acceleration (m/s 2 )

800
600
400
200
0
200

3

2
1 0
1
2
center frequency (MHz)3

Figure 10.7: Measured acceleration versus the sweep center frequency. As long as the lasers sweep
over both atomic resonance (blue region), the acceleration is roughly constant. The sweep range
here was 4 MHz.

One advantage of this cooling mechanism is that while it operates on a narrow linewidth transition, its performance is largely insensitive to drifts in the cooling laser frequency. To demonstrate
this insensitivity, we apply a shift to the average frequency of the two laser beams and measure
the resulting acceleration when both beams are applied. We find that as long as the lasers sweep
over resonance, the measured acceleration is largely unmodified. In our case, this allows a large
acceleration to be achieved over a range of roughly 4 MHz, even though the natural linewidth of
the transition is 7.5 kHz.
Interesting features emerge when we attempt to measure the atomic acceleration as a functionof their apparent velocity in the moving refernence frame F . For the previous force characteriza-

acceleration (m/s 2 )

166

600
400
200
0
200
400
600

0.6

0.4 0.2 0.0 0.2 0.4
apparent velocity (m/s)

0.6

Figure 10.8: Force versus velocity of moving reference frame, as set by the relative detuning /Delta
between counterpropagating cooling beams. Red points correspond to an upwards frequency sweep,
while black points correspond to a downwards frequency sweep. Blue line represents half of thesum of the red and black points, while the green line represents half of the difference. We attributethe contributions to the acceleration represented by the blue line to transient effects not directly
associated with our cooling mechanism.

tions, we were able to fix the apparent velocity of the atoms to be large enough to be well-describedby our simple picture of subsequent adiabatic transfers, and to allow time for several sweeps before
the atoms reach equilibrium with the moving reference frame. When we lower the velocity of themoving frame, we reach a regime where the number of sweeps required for the atoms to reach equilibrium becomes very small, forcing us to work with very short acceleration times Ta . This makes
the results very sensitive to transient dynamics. Further, for fixed Ω, we reach a regime where therequirement kv  Ω no longer holds, and we observe strong effects associated with multi-photon
processes.
The results of this measurement are shown in Fig. 10.8. While the center region is contaminated by these transient artifacts, for larger velocity we do observe a roughly dispersive acceleration
profile typical of the kind of velocity-dependent force that leads to cooling. The sign of the force
switches sign the direction of the sweep is reversed from increasing to decreasing.

167

Figure 10.9: Experimental study of an atomic ensemble using one-dimensional cooling by adiabatic
transfer. (a) An ensemble of atoms, precooled to around 600 µK, is illuminated by frequencyswept counterpropagating beams for a variable amount of time. The one-dimensional temperature
is decreased to a steady-state value of 45 µK by the cooling lasers. (b) The phase space density
in one dimension (grey trace) and three dimensions (black trace) is increased during the coolingprocess.

10.2.2

Cooling and Phase-Space Compression in 1D

To characterize how these forces lead to cooling in one dimension, we eliminate the detuning
between the two laser beams. We prepare an atomic ensemble with a temperature of roughly
600 µK, which we achieve by briefly turning on the 461 nm MOT beams after initial cooling. We
then apply the sawtooth-ramped cooling beams for a variable amount of time before measuring
the temperature of the atoms using our usual method. The frequency of the two beams is swept
upwards by ∆s = 2π × 6.6 MHz every 50 µs. In the direction of beam propagation, we find that forthese parameters the atoms are cooled from their initial temperature of roughly 600 µK to 45 µK
in of order 300 µs or 6 sweeps. It should take roughly 25 photon recoils to slow an atom from therms velocity associated with a temperature of 600 µK to that of 45 µK, which should take 12.5
sweeps if an average momentum of 2~k were transferred per sweep. The fact that we observe faster

168
cooling than this prediction is most likey due to the presence of multi-photon processes transferringlarger amounts of momentum per sweep.
Importantly, we directly observe an increase in phase-space density during the cooling process,and not simply velocity reduction. We measure atom loss during cooling to be negligible and therelative increase in phase space density is ρ/ρ◦ = ∆x◦ ∆v◦ /(∆x∆v), where ∆x and ∆v (∆x◦ and∆v◦ ) are the measured cloud size and velocity spread after (before) cooling. In Fig. 10.9b, thisquantity is shown as the grey line. The black line in Fig. 10.9b accounts for measured heating in
the orthogonal directions (which for simplicity we have assumed to be equal in the two directions).
Note that because the atoms are not confined and the cooling is only applied along one dimension,
this increase of phase space density is much smaller than one would obtain with the same decreasein temperature for three-dimensional cooling in a loose harmonic trap. In that scenario, the spatial
extent ∆x in each direction would decrease with the square root of the temperature, as woulld thevelocity. The same temperature increase in a three-dimensional trap would lead to an increase inphase-space density of more than a factor of 2000.
We study the scaling of temperature with Rabi frequency by varying the power in the coolingbeams. An analytic or intuitive prediction of the final temperature has proved difficult, especiallyin the presence of multi-photon processes. We compare our experimentally measured temperature
to John and Murray’s simulation in Fig. 10.10.
Both experiment and simulation show a minimum temperature as a function of Rabi frequency. At low Rabi frequency, the Landau-Zener condition α  Ω2 breaks down, leading toinefficient adiabatic transfers. For the parameters used in (Fig. 10.10), α = Ω2 when Ω = 130 kHz,roughly the point at which temperature is minimized in both the simulation and experiment.
At larger Ω, both the experimentally measured and simulated temperatures rise with Ω,
though the predicted and observed values appear to diverge at large Ω.
Several experimental factors could lead to this divergence. Because an acousto-optic modulator (AOM) is used to sweep the frequency of the cooling lasers, there are significant power
variations during the sweep. Including this effect in our simulation results in lower temperatures

169
at high Rabi frequency, and this effect can explain much of the discrepancy (as indicated by the
red band in Fig. 10.10). Furthermore, imperfect laser polarization and finite bias field may lead tocooling on the transitions to 3 P1 , mj = ±1, especially at high Rabi frequency.
The exact scaling of the equilibrium temperature is as yet an open question. Because of
the limited experimental tunability of important parameters such as atomic mass and photon
momentum, we will rely on simulation to provide more details on temperature scalings. This isTemperature (µK)

currently an ongoing project.

70
60
50
40
30
20
10
0
0.0 0.2 0.4 0.6 0.8 1.0
Rabi frequency Ω (MHz)Figure 10.10: Temperature versus Rabi frequency for a sweep range of 7 MHz and repetition rateof 15 kHz. Red line represents the results of our simulations, and black points are experimental
data. The red band indicates uncertainty of prediction due to statistical uncertainty in simulation
results, experimental calibration of Ω and variations in Rabi frequency during the sweep. The
vertical line at 130 kHz represents the approximate point at which we expect adiabaticity to break
down: Ω2 = α.

10.2.3Practical Advantages

SWAP cooling has several key advantages over more common Doppler cooling techniques.
For Doppler cooling, the minimum achievable temperature, known as the Doppler temperature
TD is given by (TD = ~γ/2kB ), where kB is Boltzmann’s constant. The maximum force, set by
the rate at which atoms can be returned to the ground state so they can scatter more photons is170
set by Fmax = ~kγ/2. Both are thus set by immutable properties of a given atomic transition.
In order to achieve a low temperature, one would like to use a narrow-linewidth transition. This
leads to challenging technical requirements on laser stability, and requires substantial pre-coolingto compensate for the small achievable forces. For even narrower transitions, this force can be sosmall that atoms cannot be supported against gravity using standard Doppler cooling techniques[176].
SWAP cooling provides tunability in that quantities like the peak force and final temperaturedepend on parameters that are under the control of the experimentalist, such as the sweep repetition
rate and laser power. This allows large forces to be generated using transitions that are also
compatible with reaching low temperatures. Further, the requirements for long-term laser stability
are greatly reduced compared to Doppler cooling, as the laser must now be stable relative to theentire sweep range, rather than the atomic transition linewidth. In a similar manner, SWAP coolingalso provides a great deal of insensitivity to shifts of the atomic transition frequency, for example
due to Doppler and Zeeman shifts when capturing from a relatively warm sample, or due to lightshifts associated with deep optical traps. So long as these shifts are small relative to the range ofthe sweep ∆s , they do not have a major impact on the cooling performance.

10.3

Creating a Restoring Force
The mechanism that I have presented so far generates a viscous force, but does not lead tospatial confinement. A related mechanism allows us to generate both viscous drag and a spatiallyvarying force, allowing us to form a new type of magneto-optical trap (MOT). I will call this a
SWAP MOT. From a practical perspective, this mechanism has proved very useful to us, and isused in the everyday operation of our experiment. Advantages include an increased atom number,a simplified laser system and greater robustness against experimental drifts.

σ(from left)

σ+(from right)
g (mj = 0)

Laser
Frequency

e2
e1

1

P(e1)
P(e1)

e1 (mj = 1)e2 (mj = -1)

171

Time

0

Figure 10.11: Mechanism for realizing a confining force with swept lasers. Counter-propagating
beams with opposite circular polarizations illuminate the atoms in the presence of a magnetic field
gradient, as in a standard MOT (left). The atoms are located near the zero of the magnetic field.
The MOT lasers are swept upwards in frequency in a sawtooth manner, as before. In the simplest
system, each laser connects the ground state to a unique excited state, which shift in opposite
directions in an applied magnetic field. The lasers sweep over the transition to the state thatis shifted to lower energy first, causing acceleration in the direction of the beam that addresses
that state, and depleting the population in the ground state (right). Because the excited state isrelatively long lived, the atom is less likely to be in the ground state when the other laser sweeps
over its resonance. This leads to an imbalance in the number of photons absorbed from each beam,
and causes a magnetic field dependent force that can be used to spatially confine the atoms near amagnetic field zero.

10.3.1

How it Works

The experimental configuration for a SWAP MOT is very similar to that of a standard
MOT. Three pairs of counter-propagating beams, each pair containing beams with opposite circularpolarizations, intersect at the location of a magnetic field zero. Photon recoil forces, which aremodified by the presence of a magnetic field gradient, lead to a restoring force towards the fieldzero. The only important difference between our SWAP MOT and a conventional MOT using anarrow-linewidth transition is that our laser frequencies are swept in a sawtooth manner all theway across atomic resonance.
While in practice we use this in a three-dimensional configuration, I will focus on a single
dimension here for simplicity. Also, while this mechanism is appropriate for species with more
complex internal structure, I will focus on the simplest applicable level structure, which happens
to represent 88 Sr. The atom has a single ground state, which connects to a trio of optically172
excited states by dipole-forbidden transitions labeled by different angular momentum projections
mj = −1, 0, 1, as shown in Fig. 10.11. In the presence of a magnetic field, the states experience
Zeeman shifts ∝ Bmj . The two lasers are polarized with opposite circular polarizations — in ourcase the laser beam impinging on the atom from the right has σ+ polarization and couples theground state to the mj = 1 excited state. The laser beam incident from the left has σ− polarization,and couples the ground state to the mj = −1 excited state. The two laser beams are swept infrequency in a sawtooth fashion as before, with the sawtooth ramp centered on the frequency ofthe mj = 0 state (though the exact centering is not critical).
We will assume that the atom starts in the ground state at the beginning of a sweep. As thetwo lasers sweep upwards in frequency, the laser that addresses the transition to the state that was
shifted downward by the magnetic field sweeps over resonance first. This transfers the population
to that excited state, where it will stay for a while because the transition is long-lived. The otherlaser will cross resonance with its own transition shortly after. At this point, the atoms are still
stored in the other excited state, and thus cannot absorb a photon from the second beam. We
assume for the SWAP MOT that the remainder of the sweep is long enough for the atom to decay
back down to the ground state before the next sweep begins. This process leads to an imbalance inthe number of photons that the atom absorbs from the two beams, and in turn a force on the atom.
The direction of the imbalance is set by the sign of the magnetic field, so will switch directions on
opposite signs of a magnetic field zero, as desired. While the direction of the magnetic field andsweep direction can both be reversed and still lead to a restoring force, only for an upward sweep
do we also get a damping force that enables a stable MOT. For atoms near the magnetic field zero,
where the Zeeman shifts are small, the Doppler shift associated with atomic motion can break thetie to determine which beam the atom absorbs a photon from.
In contrast to SWAP cooling, the SWAP MOT relies heavily on spontaneous emission to reset
the atom to its ground state after every absorbed photon. I think one could combine symmetricallyswept laser frequencies with a rotating magnetic field to transfer the atoms back down though, but
this is a project for another time.

173
10.3.2Demonstration in One Dimension

Freq. ramp:

Figure 10.12: Atomic acceleration in the configuration described in Fig. 10.11. The direction of
the force switches sign depending on the sign of the magnetic field applied along the direction of
the counter-propagating beams, and depending on the direction of the sweep (blue and red points).
This is the necessary criteria for generating a restoring force toward a zero in a spatially dependent
magnetic field. A symmetric frequency sweep generates a smaller force, which does not switch signwith the magnetic field (black points), likely due to an intensity imbalance between the two beams.

Using time-of-flight images as before, we demonstrate an atomic acceleration whose direction
depends on the sign of an applied magnetic field (Fig. 10.12, blue and red points). This is thenecessary condition to generate a restoring force when combined with an appropriate spatiallyvarying magnetic field. Whereas before we set the polarization of the two lasers to be in the
direction of the magnetic field, we now set the two beams to have opposite circular polarizations
with respect to a direction along which we apply a variable magnetic field. The direction of theacceleration also depends on the direction of the asymmetric frequency ramp (Fig. 10.12, blue andred points). When we instead apply a symmetric frequency ramp (black points), we see a smaller
acceleration whose sign does not depend on the direction of the magnetic field, presumably due toradiation-pressure imbalance between the two beams.

10.3.3

Application to a Real Experiment

We utilize the SWAP MOT in the everyday operation of our experiment. As discussed in
Chapter 3, our experimental sequence consists of an initial phase where atoms are accumulated and

174

0.8

1.0
88 Sr

0.6
0.4
0.2
0.0
10 1 2 3 4 5 6Center detuning (MHz)Capture Fraction

Capture Fraction

1.0

0.8

87 Sr

0.6
0.4
0.2
0.0
10 1 2 3 4 5Center detuning (MHz)Figure 10.13: Fraction of atoms transferred from the broad-line MOT at 461 nm to the narrowline MOT at 689 nm versus center frequency of the sweep for 87 Sr and 88 Sr for different sweep
profiles. Red points represent an upwards-swept asymmetric sawtooth ramp, which we refer to as
the SWAP MOT configuration. Black points represent the more commonly used symmetric triangle
ramp. Blue points represent a downwards-swept asymmetric sawtooth ramp, which captures almost
no atoms.

pre-cooled in a broad-line MOT formed using the dipole-allowed 1 S0 to 1 P1 transition at 461 nm(blue MOT). This stage prepares the atoms with a temperature of roughly 1 mK. At this point,
the atoms are transferred to a narrow-line MOT formed using the 7.5 kHz linewidth 1 S0 to 3 P1transition (red MOT).
In Fig. 10.13 we explore the efficiency with which we can recapture the atoms from the blue
MOT into the red MOT. For this step, it is common practice to frequency-modulate the 689 nm
laser to cover the range of Doppler shifts experienced by the atoms coming from the relatively warm
blue MOT. Typically, people use a symmetric triangle ramp for this modulation. This configurationis represented as the black points in Fig. 10.13.
If instead we apply an upward-swept sawtooth ramp profile, we capture substantially more
atoms, and over a broader range of laser frequencies. The improvements are more pronounced inour system for 87 Sr than for 88 Sr. This technique both allows us to capture more atoms than thestandard approach, and makes our experiment very robust to drifts in the frequency of our 689 nm

175
laser. With our 689 nm laser locked to our reference cavity, we can go for multiple weeks without
tuning its offset frequency with only minor reductions capture and cooling performance.
After recapturing the atoms in the SWAP MOT, the next step in our experiment is to cool andload the atoms into the optical lattice. We typically operate with a deep optical lattice, sometimes
up to 200 µK. Because the 813 nm lattice is not a magic wavelength for the 1 S0 to 3 P1 transition,
it causes substantial shifts (up to several hundred kHz) to the cooling transition. This would
cause challenges for conventional Doppler cooling due to the narrow linewidth of the transition.
Because SWAP cooling and the SWAP MOT are largely insensitive to differences between theatomic transition frequency and average laser frequency, we observe no decrease in the efficiency
with which we can load the lattice when we work at high depths.
The whole transfer process from the blue MOT to the lattice can be performed with a single
set of sweep parameters and magnetic field gradient. We typically sweep the MOT beams byroughly 7 MHz every 50 µs, in the presence of a roughly 2 G/cm magnetic field gradient in thevertical direction. For 87 Sr, the laser addresses 1 S0 to 3 P1 , F = 11/2 transition. We typicallytransfer up to 50% of the atoms from the red MOT to the lattice. For typical operating conditions,with a lattice depth of 100 µK, we typically achieve temperatures of roughly 10 µK, sufficient forour needs.
In other experiments that use standard Doppler cooling to form a narrow-line MOT, a separate set of beams called “stirring beams” are used that address the 1 S0 to 3 P1 , F = 9/2 transitionin addition to the MOT beams that address the 1 S0 to 3 P1 , F = 11/2 transition. Because themagnetic moment of the F = 9/2 states is smaller than the F = 11/2, the stirring beams provide
more robust Doppler cooling than the the MOT beams. The stirring beams also prevent populationfrom becoming trapped in unconfined Zeeman sublevels of the ground state [179].
The SWAP MOT does not require stirring beams. Because the laser sweeps over all mf
transitions, even if they are shifted by a large amount, the relatively large magnetic shifts of theF = 11/2 states (compared to the F = 9/2 excited states) do not present a problem, and there are
no unconfined ground states. This substantially simplifies the laser systems required to perform

176
narrow-line cooling on 87 Sr, removing one of the main experimental barriers to working with thisisotope.
Overall, the use of the SWAP MOT has greatly simplified our experimental sequence andhardware, and has lead to far more robust operation. While it is likely that lower ultimate temperatures could be obtained with standard Doppler cooling (though we do not yet understand thetrue limits of SWAP cooling), the temperatures that we obtain with our SWAP MOT are sufficientfor our application.

10.4

Extensions and Applications
We can envision a host of extensions and application for SWAP cooling and related techniques.

Perhaps the most exciteing would be as a means to aid in the cooling of molecules that have narrowlinewidth transitions. Because molecules have many vibrational and rotational state to which they
can decay when a photon is spontaneously emitted from an excited state, Doppler cooling techniques
have proven very difficult for molecules. The reduced reliance on spontaneous emission for SWAP
cooling could remedy this issue.
A promising application could be to the cooling of a molecule like YO, which has a narrowlinewidth optical transition with a linewidth of roughly 160 kHz [180]. In a typical experiment[181], the molecules emerge from a cryogenic buffer gas source with a longitudinal velocity of order
70 m/s. They must then be slowed substantially in order to be captured in a MOT. Because of thelarge forces and low scattering possible with SWAP cooling, it could be used either for longitudinalslowing of the molecules, or for a transverse cooling stage prior to longitudinal cooling, that would
reduce the divergence of the molecular beam during the longitudinal slowing stage.
As another extension, SWAP cooling techniques can be applied using Raman transitions.
This allows their application to species that do not have narrow-linewidth optical transitions, suchas rubidium, or a broad class of molecules. Further, because spontaneous emission can be addedin a controlled and tunable manner, because the effective excited state does not naturally decay,
we can imagine using it as a tuning parameter to improve the performance of the cooling. On the177rubidium side of the Thompson lab, Graham Greve and Baochen Wu are currently studying SWAP
cooling using Raman transitions in rubidium, and have demonstrated initial cooling results.
From a theory perspective, much work still remains to be done in terms of understanding thefinal temperature achievable with SWAP cooling, and its dependence on the various experimental
parameters.

Chapter 11

A new perspective on gravitational wave detection with cold atoms in space

Recently, several proposals have been published that involve using cold atoms with narrow
linewidth optical transitions to detect gravitational waves (GWs) [81, 82, 84, 182]. I found these
proposals very appealing as a way to apply the kinds of technologies we are working on towards avery exciting problem in physics.
The proposals fall into two apparently (at first) disconnected categories: atom interferomenters (AI) [82, 84, 182], and optical lattice clocks [81]. I became very interested in understanding
the fundamental similarities and differences between these two kinds of proposals, which lead to
the work presented in this chapter. Several specific questions arise from this quest:• The AI proposals involve spatially separation of the atom’s spatial wave-function, while
the clocks do not. Is this important?
• AI proposals include an enhancement from so-called ’Large Momentum Transfer’ (LMT)
sequences. Is there a way to mimic these in a clock?
The short answers to these questions are ’No,’ and ’Yes,’ but the rest of this chapter will
outline a perspective that should hopefully make this clear and provide further intuition into what
is possible with atomic detectors.
In addition to co-authors James and Julia, conversations with Shimon Kolkowitz, Jason
Hogan and Peter Graham were key in developing these ideas.
In both types of proposals, an ensemble of atoms with a long-lived optically excited state is179
prepared in a drag-free reference frame in or near each of two satellites separated by a large distance.
A passing gravitational wave leads to a periodic stretching of space between the two satellites
that create phase shifts on light transmitted between the satellites. The laser light interacts withthe atoms in each satellite, imprinting the gravitational wave-imposed optical phase shift on the
distantly located atoms. The technical laser phase noise is imprinted on the atoms both satellites.
As a result, the gravitational wave signal can be retained but the technical laser phase noise canceledby looking at the difference in the phases imprinted by the laser pulses at the two satellites.
In the clock community, the gravitational wave detection is described as a frequency measurement of Doppler shifts that result from the stretching of space between the satellites. In the AI
community, the stretching of space is described as causing a phase shift between the laser phase andatomic coherence. A key element of AI proposals is enhanced sensitivity to gravity waves throughthe use of Large Momentum Transfer (LMT) pulses in which the atoms acquire many photon recoil
momentum kicks. In contrast, in optical lattice clocks the atoms are tightly confined in an opticallattice and photon recoil is suppressed. This difference in particular, as well as the language used
to describe the devices, would seem to point toward the two sensors somehow being fundamentallydifferent.
In what follows, we provide a comparison of these two sensors to show that the fundamental
mechanism for sensitivity is in fact the same between the clock and AI type detectors. In both typesof sensors, an internal quantum superposition state of the atoms has imprinted on it a relative phaseset by the local phase of the laser pulses that couple a ground and a long-lived optically excited
atomic state. The atoms primarily act as a highly coherent phase memory that keeps track of thesephase imprints, and allows them to be read out through atomic observables.
By viewing clock-like detectors as phase memories, rather than simply as clocks whose solecapability is to measure frequency, we show that they support the implementation of LMT-likeprotocols with enhanced sensitivity even when negligible momentum is transferred to the atom.
We further show that the relevant phase information can actually be stored in stable ground statesthat evolve phases at acoustic rather than optical frequencies. This enables useful coherent evolution

180
times beyond the lifetime of the optically excited state.

Confined Atoms‘Clock’

Gravitational Wave

Unconfined Atoms‘Atom Interferometer’

Figure 11.1: Optical lattice clock and atom interferometer detectors for gravitational waves. Each
type of proposal relies on two ensembles of atoms, one in each satellite. Lasers transmitted between
the satellites encode phase shifts due to changing optical path lengths caused by a gravitational
wave. The key difference between clock and atom interferometer proposals is that in clocks, atomic
recoils due to the momentum of absorbed/emitted photons are supressed by tightly confining theatoms, while it atom interferometers the atoms are free to recoil.

11.1

Optical Path Length Changes due to Gravitational Waves
For both types of proposals, the important effect of a gravitational wave is a modification of

the optical path length L between the two satellites. We will describe the effects of a gravitational
wave at angular frequency ωg by replacing the gravitational wave with a phase modulator that fills
the space between the two satellites (e.g. an electro-optic modulator or EOM) sinusoidally driven
at frequency ωg . The drive applied to the modulator leads to a phase shift α(t) on laser light that islaunched from one satellite and detected at the other satellite. Physically, α(t) is the relative phasethat one would measure at the location of the far atom between a laser pulse that passes throughthe driven modulator (i.e. mimicing the gravitational wave) and a second laser pulse that runs
parallel to the first pulse but passes through an identical but undriven modulator (i.e. mimicing

181

a

Time

(b) Confined Atomsabb(c) Unconfined Atoms
a

b

Time

(a)

Figure 11.2: A toy model for gravitational wave detection. (a) Two atoms (a and b) are addressedby a single laser resonant with a transition between the ground state |gi and a long-lived excitedstate |ei. A phase modulator mimics the effect of a gravitational wave by modifying the optical
path length between the two atoms. (b) For tightly confined atoms (clocks), a simple Ramsey
sequence can be used to detect changes in optical path length. A first π/2 pulse imprints the laser
phase on the two atoms when the path length is shifted by L1 , and a second π/2 pulse acts when
the path length is shifted by L2 . Random variations in the laser phase (φ1 and φ2 ) are common to
the two atoms, while phase shifts resulting from changes in path length are not. The signal fromthe phase modulator manifests as phase shift Φs = k(L2 − L1 ) between the excited state probability
oscillations of the two atoms. (c) For unconfined atoms (atom interferometers), transitions between|gi and |ei are accompanied by momentum kicks, which necessitate an additional laser π pulse in
the middle of the sequence. The phase shift between excited state oscillations of the two atoms is
now Φs = k(L3 − 2L2 + L1 ).

182
no gravitational wave.) The light is launched from the first satellite at time t, but arrives at thesecond satellite where it is detected at time t + Td , where Td is the nominal delay or transit time
between satellites.
The total phase shift α(tj ) for a pulse launched at time tj is calculated by integrating along
the optical path. We can define a net effective path length for the pulse L + Lj = L + α(tj )/kwhere the small change in optical path length is

Lj =

hc
(sin (ωg (tj + Td )) − sin (ωg tj )) .
2ωg(11.1)We have approximated that the gravitational wave’s strain is very small h  1. In this expression,c is the speed of light in the undriven modulator, k = 2π/λ is the wavenumber of the laser light,
and λ is the laser wavelength.

11.2

Detecting Changes in Optical Path Length
The goal is to understand how AI and clock-like sensors can be used to precisely estimatechanges in the optical path length Lj . In our model, shown in Fig. 11.2, we consider two cold atoms
labeled a and b that are separated by distance L. The atoms may either be cooled in free space(atom interferometers) or tightly confined (clocks). The atoms have a long-lived optical transition
such as exist in alkaline-earth and similar atoms such Sr and Yb. A laser located near atom a canlaunch pulses of light that interact with both atoms with equal intensity.
The laser pulses interact with the atoms by applying so-called π/2 and π pulses between
ground |gi and excited |ei states. We will assume that the coupling or Rabi frequency is muchlarger than the atomic decay rate from |ei and any relevant Doppler shifts due to atom motion.
We will also assume for simplicity that the laser frequency is exactly equal to the atomic transition
frequency. These assumptions rule out some of the capabilities of atom interferometers (suchas addressing atoms that have experienced different recoils independently), but retains the basic
mechanism for sensitivity. The effect of the laser pulses on the atoms written in a rotating frame

183
at the atomic transition frequency can then be expressed using the operators:










−i(φ+kx)
 1

−e



1 


R̂π/2 (φ) = √ 

2





 i(φ+kx)

e

1


(11.2)









−i(φ+kx)
 0

−e






R̂π (φ) = 







 i(φ+kx)e
0(11.3)

These operators act on the basis states :
 
 
 
 
1
 
 

|gi ⊗ Ψg (x) =   ⊗ Ψg (x)
 
 
 
 
0

(11.4)

 
 
 
 
0
 
 

|ei ⊗ Ψe (x) =   ⊗ Ψe (x)
 
 
 
 
1

(11.5)

that are a product of an internal state label |ei or |gi, and an external state wavefunction Ψe,g (x).
Here x represents the distance from a fixed plane where the laser’s phase is defined as φ. The effectof these interactions is both to transfer amplitude between the internal states of the atoms and184
also to imprint the laser’s local phase upon the transferred portion of the atom’s wave function.
The fact that Ψg (x) may differ from Ψe (x) indicates the possibility of entanglement between theinternal and external degrees of freedom of the atom.
For simplicity, we account for laser frequency noise by allowing the laser phases to vary
between pulses, but taking the actual laser frequency to be fixed at the atomic transition frequency
such that k is constant. One can relax this assumption by accounting for an additional phase that
results from the lasers detuning from atomic resonance and cancels the effect of associated changes
in the wave-vector k. This leads to the laser frequency noise insensitivity demonstrated in previous
proposals [81, 82], but doing this careful accounting does not alter the main results here.
First consider the effect of these rotations for a clock-like sensor in which the atoms are
confined to much less than the laser wavelength (known as the Lamb Dicke regime.) In this limit,
we can think of the ei(φ+kx) term as imprinting a spatially constant phase onto the atom, whose
value is determined by the location of the atom along the laser’s path. For example, a π pulse
applied to an atom tightly confined in a trap centered at position x = A and in the ground state|Ψ0 i = |gi ⊗ Ψ0 (x) transfers the atom to the state |Ψi ≈ |ei e−i(φ+kA) ⊗ Ψ0 (x). The external state
wave-function is to very good approximation unmodified by the pulse, but its internal wavefunction
can be seen to have acquired a net phase φ + kA.
In contrast, in AI sensors the atoms are not confined and a change in the internal state isaccompanied by a change in the external state. In particular, one cannot neglect the variation of
the optical phase factor over the spatial extent of the atomic wavefunction. For concreteness, a πpulse applied to an unconfined atom centered at x = A and in the ground state |Ψ0 i = |gi ⊗ Ψ0 (x)transfers the atom to the state |Ψi = |ei e−iφ ⊗ e−ikx Ψ0 (x). As written, the internal portion of the
wavefunction appears identical to that of the confined case, but now the external wavefunction has
a spatially varying phase corresponding to one photon’s worth of momentum recoil.
A change in the optical path length Lj between the atom and the laser producing the π-pulse
manifests in the confined case as |Ψi = e−i(φ+k(Lj +A)) |ei ⊗ Ψ0 (x) and in the unconfined case as|Ψi = e−i(φ+kLj ) |ei ⊗ Ψ0 (x)e−ikx . The sensitivity to changes in path lengths from gravitational

185
waves is due to the imprinting of an additional phase kLj , which is the same whether the atoms
are confined or not. In the case of AI sensors one must add additional pulses to become insensitive
to the terms associated with the photon recoil.

11.3

Clock-like Detectors
A clock-like gravitational wave detector with confined atoms could be used to detect thegravitational wave phase shift using a basic Ramsey sequence [81], pictured in Fig. 11.2b. A change
in the optical path length Lj leads to a modification of the laser phase experienced by atom b of
kLj for the pulses of light that are launched by the laser near atom a. The phase experienced byatom a is unmodified by the change in optical path length. In this sequence, the role of atom a isthen simply to record any variation of the phase of the laser itself due to technical sources of noise
so that this laser phase noise can be subtracted out from the final measurement.
Stepping through the Ramsey measurement, atoms in both locations are initially preparedin |gi. At time t1 atom a interacts with a pulse of light created by laser a which drives the first
π/2 pulse with phase φ = φ1 (we keep track of this phase only to demonstrate insensitivity to its
value.) The rotation applied to atom a is Ra = Rπ/2 (φ1 ). When the same pulse arrives at atomb it creates a rotation Rb = Rπ/2 (φ1 + kL1 ). At a later time t2 , a second π/2 pulse is applied to
atom a with a laser phase φ = φ2 : Ra = Rπ/2 (φ2 ). The same pulse of light travels to atom b todrive a π/2 pulse Rb = Rπ/2 (φ2 + kL2 ).
The final signal that is detected to estimate the gravitational wave’s signal is the difference inthe probability for finding the atom in its excited and ground states. For atom a this observable canbe parameterized as Sa = Pea − Pga = cos Φas and for atom b as Sb = Peb − Pgb = cos Φbs . For theabove Ramsey sequence, the signal phases are given Φas = (φ2 −φ1 ) and Φbs = kL2 −kL1 +(φ2 −φ1 ),The gravitational wave signal is given by extracting the difference of the phase of these two signals[183]

Φs = Φbs − Φas = k(L2 − L1 )(11.6)

186
The key result is that the technical laser phase noise is canceled by having been recorded on atoma, and only the gravitational wave’s signal remains.

11.4

Atom Interferometer Detectors
In an atom interferometer, the atoms are unconfined and a slightly more complicated sequenceis needed. The initial π/2 pulse entangles internal and external degrees of freedom of the atom
by imparting a momentum kick to the portion of the atomic wavefunction transferred to |ei. If
this momentum kick is not reversed, the portions of the atomic wavefunction will not be spatiallyoverlapped to interfere at the time of the second π/2 pulse. The simplest solution is to add a π pulsein the middle of the sequence (Fig. 11.2c). More complex sequences, with enhanced performancein different regimes are presented in [82, 84], though the core mechanism for sensitivity is the sameas this simple version.
Stepping through the simplest AI sequence, the first π/2 pulse, π pulse and second π/2 pulse
are launched from the laser at times t1 , t2 , and t3 , respectively. When the three pulses arrive at
atom b they will have experienced optical path length differences L1 , L2 , and L3 .
The signal that is extracted is the same as above, and depends only on the optical path
lengths asΦs = k(L3 − 2L2 + L1 )(11.7)

where the random fluctuations in laser phase φ3 , φ2 , and φ1 are again cancelled because they arecommon to both atoms. A similar analysis for the atom interferometer is presented in [?].

11.5

Enhanced sensitivity analogous to large-momentum-transfer
We now propose a mechanism by which the signal recorded by the clock-like sensor canbe enhanced. In atom interferometry in general, and AI based GW detectors in particular [82],large momentum transfer (LMT) is a crucial tool used to enhance the size of measured signals. Our

(a)

a

b
a

b

Repeated
M timesTime

Repeated
M times

(b)

187

(c)

Figure 11.3: LMT-like enhanement with confined atoms. (a) A toy model to demonstrate enhancement mechanism. Atoms are addressed by counter-propagating lasers, both of which pass throughthe phase modulator that sits between the atoms. (b) Two blocks of M pairs of π pulses (blueboxes) are inserted between the π/2 pulses of the standard Ramsey sequence. Each pair of π pulses
contains a pulse originating from each direction. The duration of each block is assumed here to be
short relative to the GW period, while a long evolution time (grey box) between the two blocks may
be comparable to the GW period. These additional pulses lead to a factor of 4M + 1 enhancementin the signal phase Φs compared to simple Ramsey sequence.

enhancement mechanism is very similar to LMT, but is applied to confined atoms, so no momentumis actually transferred to the atoms. Instead, by allowing for multiple interactions between the lasers
and atoms, a phase is repeatedly written in to the atomic coherence in a constructive manner toenhance the signal size Φs .
The key to this enhancement sequence is to apply the laser pulses from alternating sides (as188
is proposed for LMT protocols for single-photon transitions with unconfined atoms [82, 182, 84])
using one laser near atom a and one laser near atom b, as illustrated in Fig. 11.3. Because the signof the laser phase shift imprinted on the atoms is opposite in sign when the atoms are driven from|ei to |gi versus |gi to |ei, if the same laser were used to try to imprint its phase multiple times, itwould simply unwrite the phase that it had just written in. By interleaving π pulses from a second
laser, which does not encode the same gravitational wave phase shift as the first, we can ensurethat the phase shift corresponding to the GW is always written in with the same sign. The phasesof the laser pulses are always referenced to a plane near the laser from which they originate so that
there is negligible GW contribution to this phase.
For simplicity, we treat the case where a set of pulses is applied in quick succession at the
beginning of the measurement sequence, with launch times roughly equal to t1 and with optical
path length changes all equal to L1 . During a subsequent free evolution period Te , much longer
than the time over which the rotations were applied, the optical path length may change. At time
t2 = t1 + Te , a second set of rotations is quickly applied with launch times all roughly equal to t2
such that all pulses experience an optical path length change equal to L2 (a discussion of the effectsof time delays is included in the appendix).
Consider the first group of pulses. From the perspective of one of the two atoms, the laser
pulse that arrives from the distant laser encodes in its phase kL1 the change in path length. This
phase is then imprinted on the atomic superposition with a different sign between the portion thatwas transferred from |ei to |gi versus |gi to |ei such that the difference phase is 2kL1 . The next
laser pulse that is launched from the locally situated laser resets the two portions of the atomic
superposition to their original internal states. It does imprint a phase shift of its own, but thisphase does not encode information about the change in optical path length L1 , and the laser’s
phase noise cancels in the final differential signal phase Φs . Because the two portions of the atomic
superposition have been reset to their original internal states, a subsequent rotation from the distantlaser will imprint a phase encoding the same path length difference kL1 that adds constructively
with the previously imprinted phase so that the total imprinted phase on the atom is now 2 × 2kL1 .

189
If we apply M pairs of π rotations after the first π/2 pulse and M pairs of π rotations before
the last π/2 pulse (i.e. 4M total launched π pulses), with each pair containing a rotation originatingfrom both the left and right sides, the differential phase shift between the two output channels isenhanced by a factor of (4M + 1) relative to the simple Ramsey sequence presented above so thatnowΦs = (4M + 1)k(L2 − L1 )(11.8)In the regime we consider here, even a single pair of LMT pulses (i.e. 4 total π-pulses), improves theestimate of (L2 − L1 ) by a factor of 25 in variance. As a result, the same precision can be achieved
with a reduction in required resources such as atom number or averaging time by a factor of 25.
Alternatively, one could achieve the same sensitivity to gravitational waves with a 5 times shorter
satellite separation or evolution time Te for reduced technical complexity or enhanced bandwidthfor detecting gravitational waves, respectively.
Previous proposals for detectors using confined [81] and unconfined atoms [84] include socalled dynamical decoupling (DD) sequences. A DD sequence would amount to applying a π pulse
from the same laser every time tj at which the magnitude of the optical path length change |Lj |is maximal. Because the sign of Lj alternates between pulses, the resulting imprinted phase shifts
add constructively. This is conceptually similar to our LMT-like enhancement mechanism, exceptthat we can switch the sign of the phase shift by alternating which laser applies the pulse insteadof waiting for the sign of the GW to switch.
Dynamical decoupling is useful when the evolution time Te greatly exceeds the GW period
Tg . In this regime the enhancement in signal scales as Te /Tg  1. This reflects the fact that thephase shifts from Te /Tg cycles of the gravitational wave may be added constructively. There is noconstraint on delay time Td .
LMT-type sequences are useful when the period of the GW Tg and total evolution time Te
are long compared to the pulse transit time Td . This is necessary in order to send multiple pulses

190back and forth between the two satellites before the path length changes caused by the gravitational
wave switches sign. The full signal response versus gravitational wave frequency is considered in the
appendix. The main result is that one can build a long baseline experiment and set Te = Td using astraightforward Ramsey sequence, or one can use LMT-type sequences to dramatically shorten thebaseline such that Te is the same but now Te  Td . In either case, the sensitivities are comparable,
but LMT-like enhancement allows one to address technical constraints that might be relaxed byoperating with shorter baselines.
Finally, as analyzed in [84], LMT can be combined with DD when both Te  Tg and Tg  Td .
Doing so simply amounts to varying from which satellite the π pulses are sent.

11.6

Evolution Times Beyond the Excited State Lifetime
We now present a further enhancement that can be realized by treating the atoms as a phasememory, rather than a clock. In a typical clock, the atoms are considered to be a two-state system.
Because external degrees of freedom ideally remain unchanged, clocks lack additional quantum
labels to specify other states. Any pulse that interacts with one ‘arm’ of the clock interferometer(|gi) also interacts with the other (|ei): the quantum mechanical amplitudes may be swapped
between ground and excited states, but the portion of the wave-function in the ground state cannot
be manipulated without also modifying the portion of the wave function in the excited state.
The same is not true of matter wave interferometers, if we relax the earlier assumption that thepulses were short enough to neglect Doppler shifts. When the atoms are unconfined, the momentum
recoil that accompanies any change in internal state allows the two arms of the interferometer to be
manipulated independently, as the laser frequency can be tuned to the Doppler-shifted resonanceof one arm but not the other.
While in clock type proposals, a fraction of the atomic wave-function must always be in the
excited state to achieve sensitivity, AI type detection sequences may include useful periods where
both arms of the interferometer are in the ground state [82], allowing for evolution times greaterthan the excited state lifetime, particularly useful for enhancing the signal in the regime where the

(b)

a

191

b

Time

(a)

or
Figure 11.4: Use of three-state system to allow atoms to stay in ground state for much of evolution
time. (a) Our procedure utilizes two ground states, |g1 i and |g2 i and a single long-lived optically
excited state |ei. |g1 i and |g2 i can independently be coupled to |ei using laser pulses with different
frequency and/or polarization (red and blue arrows). (b) After preparing an atomic superpositionof |g1 i and |ei using a π/2 pulse from the left (atoms begin in |g1 i), the portion of the atoms in|ei is transferred to |g2 i using a π pulse from the right. The atoms are then in a superposition of|g1 i and |g2 i with phase dictated by the lasers and the path length L1 . The sequence of pulses isreversed after a long evolution time, during which time the atoms are not susceptible to excitedstate decay. The phase shift in the excited state probabilities of the two atoms Φs = 2k(L2 − L1 )now encodes changes in the optical path length that occurred while the atoms were in the groundstates.

period of the GW exceeds the lifetime of the excited state.
In a recoil-free GW detector, additional internal states of the atoms can provide additional
quantum labels that allow for the independent manipulation of only one arm of the interferometer.
Here, we consider an atomic species like 171 Yb with nuclear spin I = 1/2. There are two groundstates with nuclear spin projections mI = ±1/2 that we we will label as |g1 i and |g2 i. For our
purposes, we care about a single excited state |ei, as shown in Fig. 11.4a. A small magnetic field
is applied to define a quantization axis. Transitions may be driven between either |g1 i and |ei or|g2 i and |ei by applying laser light with different polarizations or frequencies.
An experimental sequence for such a protocol is shown in Fig. 11.4b. A superposition isprepared by driving a π/2 pulse originating from the left between |g1 i and |ei. A π pulse originatingfrom the right then transfers the population of |ei to |g2 i. The atoms remain in this configuration
while the path length is allowed to change during an evolution time Te which we take to be much

192
larger than the delay time Td . The pulse sequence is then reversed and the change in path length isread out via a population measurement. The resulting signal is Φs = 2k(L2 − L1 ), and again withno contribution from laser phase noise. Memory storage in ground states could be combined with
LMT-type enhancement by inserting π pulses on the |g1 i to |ei transition in between the π pulses
on the |g2 i to |ei transition and the initial and final π/2 pulses to realize further enhancement.
In this sequence, the differential phase signal for a given change in path length is twice thatof a simple Ramsey sequence, and the laser phase noise associated with all pulses is common modefor both atoms and cancels. In thinking about the signal size it is help to remember that compared
to a π pulse applied to the |g1 i to |ei transition, the π pulse applied to the |g2 i to |ei transition
contributes only half the relative phase, as it is applied to only one arm of the interferometer.
Because the atoms are in superpositions of ground states, there is no spontaneous emission
during Te . Instead, the relative phase that would normally be exist between |g1 i and |ei, is now
stored between |g1 i and |g2 i during the evolution period Te . In addition to extending the evolution
time beyond the spontaneous-decay limited coherence time of the atoms, this technique of shelving
the atoms in ground states provides insensitivity to collisional effects that may limit the atomic
density or coherence time achievable in practice. Inelastic collisions between excited state atoms,
which contribute to atom loss in dense systems, would be eliminated by storing the atoms in
the ground states [184, 185]. Because elastic collisional properties of alkaline-earth-like atoms are
independent of nuclear spin state [114, 186], mean-field shifts due to elastic collisions are common
to the two ground states, and will not lead to noise or dephasing. Other bias errors in clocks suchas black-body, electrostatic, and lattice polarizability and hyperpolarizability shifts would all also
be suppressed.
The key result is that it is not necessary for the phase memory to exist in a frame accumulatingphase at an optical frequency. It is only necessary that the interactions with the laser pulses happen
between optical transitions, but these interactions can in principle represent a small fraction of thetotal evolution time.

193

11.7

Conclusions
Gravitational waves create phase shifts on optical pulses, and the challenge is to to detect

these phase shifts as sensitively as possible. To detect these changes, at its most basic level, one
would like to store an optical pulse of light sent at time t1 with phase shift kL1 until a second
pulse of light arrives with a different phase shift kL2 at time t2 , then compare the phases of thetwo pulses. By providing a highly coherent phase memory with which the two pulses may interact,
the atoms allow one to do just this.
Further, multiple pulses can be made to constructively imprint their phases onto the atoms
by properly alternating their launch direction (LMT) or by synchronizing their launch times withthe frequency of the gravitational wave one wishes to detect (DD). From a fundamental perspective,
the ability to achieve LMT-type signal enhancement appears to have nothing to do with the transfer
of momentum to the atoms, or that the two lasers originate from different directions, although forcancellation of technical sources of noise, the latter is required. In certain regimes, LMT allows
the signal size to be increased such that the uncertainty in the estimate of the phase of interest√k(L2 − L1 ) can be greatly reduced to well below the atomic standard quantum limit 1/ N rad.
For gravitational wave detection, we do not need to measure the absolute frequency of alaser relative to an atomic transition frequency, as one would do in a clock. As we have shown, thisallows the construction of a ground state shelving protocol for reduced sensitivities to perturbations
and evolution times Te greater than the optical transition lifetime. While in certain situations it is
helpful to think of these systems as clocks, in other regimes such a picture is incomplete.
If we view a gravitational wave detector as a clock whose only function is to make frequency
measurements, we would expect neither of the approaches presented in sections III and IV to work.
Instead, these mechanisms suggest that instead of viewing the atom as a frequency reference, weshould think of it as a phase memory. Often these two interpretations are equivalent: frequency isthe derivative of phase, so one can infer frequency from two time-separated phase measurements.
However, this view does not always give the complete picture.

194
The key point here is that we are not trying to measure the absolute frequency of a laser
relative to an atomic transition, as one would do in a clock. Rather, we are using the atoms as anintermediary to measure the differential phase between various laser pulses (defined by their launchtimes and the direction from which they are applied).
To illustrate this point, let us imagine that we are trying to use an ensemble of atoms tomeasure the frequency difference between two lasers (lasers 1 and 2) in an allotted time T . There
are two ways one could proceed — one could either use use laser 1 to perform a Ramsey sequenceof duration T on half of the atoms and laser 2 to perform a Ramsey sequence on the other half.
This would represent a standard clock approach, and would independently yield the frequency ofeach laser, which could then be subtracted.
Alternatively, one could use the atoms to make a differential phase measurement between
the two lasers at time t = 0, then again at time t = T . The difference in these differential phasemeasurements can then be used to extract the frequency difference between the two lasers. One
can do much better with this approach, and not just because each laser is addressing all of theatoms instead of just half.
Instead of performing a simple Ramsey sequence that spans the time T , one could performa short sequence at t = 0 and again at t = T in which π pulses that alternately originate fromeach of the two lasers are applied between two π/2 pulses. These π pulses greatly enhance thephase imprinted on the atoms relative to the phase difference between the two lasers (ignoring
issues of this phase exceeding 2π). Rather than independently measuring each lasers frequency
over the time T , this protocol performs a direct measurement of the differential phase of the lasers
at the beginning and end of the time T with enhanced precision, leading to a better estimate of thefrequency difference between the two lasers than would be accomplished with independent clocks.
The point here is that by performing separate differential phase measurements of two lasers,
we can do much better than we could if we tried to do independent frequency measurements of
the lasers. This is exactly what LMT enhancement enables. Rather than making an absolute
frequency measurement of a single laser as in a clock, LMT should be thought of as making

195
differential phase measurements between two lasers — the local laser that has not experienced apath length modification due to a GW, and the laser from the other satellite, which has. The
use of two ensembles of atoms and pulses transferred back and forth between two satellites simply
allows one to do this measurement in a manner that is insensitive to laser phase noise. From afundamental perspective, the ability to achieve LMT-type enhancement has nothing to do with thefact that momentum may be transferred to the atoms, or that the two lasers originate from different
directions (opposite sign on k). It also has nothing to do with the fact that we are performing ameasurement that can be described in the language of a change in position. Fundamentally all thatmatters is that we are interested in making a differential phase measurement between two lasers. It
is the photon shot noise on the lasers, and not the atom projection noise that sets the fundamental
limit on such a measurement.
Ultimately, the decision to use confined or unconfined atoms will depend on a myriad oftechnical considerations that I have no business weighing in on. From a fundamental perspective,
both methods have the same mechanism for sensitivity: atom interferometers and clocks bothsense changes in phase that result from changes in optical path length between the two satellites.
This interpretation should help distinguish to what degree differing expected sensitivities for future
proposals are due to the choice of sensor architecture versus the specific parameters considered.

11.8

Accounting for Time Delays
In the main text, we considered an LMT pulse sequence in which the time delay Td is much

shorter than the evolution time Te and gravitational wave period Tg . Rather than launching a seriesof LMT pulses at the beginning and at the end, one could in principle launch a continuous seriesof alternating π-pulses during the entire period of time between the π/2 pulses. For continuous
LMT, we find that the signal size averaged over all phases of the gravitational wave is to good
approximation given by:196√ ωl |sin(ωg Td /2)|Φ̄s = 4 2hsin2 (ωg Te /4)
ωg2 Td(11.9)

where ωl is the laser angular frequency. When Td  Tg , the sensitivity falls of rapidly as 1/ωg2 .
However, In the limit Td  Tg the scaling changes to√ ωlΦ̄s ≈ 2 2h sin2 (ωg Te /4)
ωg(11.10)For comparison, if the spacing of the satellites is increased until Td = Te then one cannot use LMT
and the Ramsey sequence yields an averaged signal:

Φ̄s =

√

2h

ωl
sin2 (ωg Te /2)
ωg(11.11)The oscillations in the signal versus ωg differ by a factor of 2 and the envelope of the signal
size is larger by a factor of 2 for LMT. Most importantly, approximately the same signal can beobtained using continuous LMT with of order Te /Td π pulses and an approximately a Te /Td times
shorter satellite satellite spacing. It is likely that decreasing the spacing of the satellites will beadvantageous for technical reasons, for instance, reduced requirements on laser power and pointing
stability.

Chapter 12Conclusion and future outlook

In this thesis, I have demonstrated several ways in which narrow-linewidth optical transitions
can be used to create useful tools for precision metrology. These tools include a new method of laser
cooling, a preliminary demonstration of nondestructive atom counting in a system suitable for an
optical lattice clock, and the demonstration of the most precise active atomic frequency reference
to date. While encouraging, this work really represents a step towards several long-term goals,
which include the application of spin squeezing to an optical lattice clock and the development ofa narrow linewidth, steady-state superradiant laser. In this chapter, I will give a perspective on
the future of these challenges, and present what I think is an encouraging way to proceed towards
each.

12.1

Spin squeezing in an optical lattice clock
The application of spin squeezing to an optical clock has been a long-standing motivation in

the development of cavity QED systems for metrology. The ability to create useful spin squeezing
using nondestructive measurements with an optical cavity is now well-demonstrated using states
separated by microwave frequencies [14, 15]. In ref. [20], I demonstrated that the measurement
portion of these techniques can be extended to a system with a narrow-linewidth optical transition
suitable for use in an optical lattice clock.
The remaining challenge is to combine this measurement approach with the coherent rotations
needed for clock operation. While this has been demonstrated in the microwave domain, these198
rotations are far more difficult in the optical domain. The most fundamental challenge here is
the difference in wavelengths between optical fields and microwave fields (micron vs centimeterscales). This means that while the Doppler shifts of the microwave transitions may be small
compared to typical Rabi frequencies for coherent rotations, they are prohibitively large for the
optical transitions (remember, Doppler shifts scale as v/λ, where v is velocity and λ is wavelength).
For optical systems, this leads to the requirement that the atoms are tightly confined in the direction
along which the clock laser (the one that addresses the narrow-linewidth clock transition) is applied.
In our current system, this presents a problem. The lattice is supported by a cavity that is
highly reflective at the clock wavelength of 698 nm. Clock light would have to be applied along
the cavity axis, through the cavity mirrors, as this is the only direction in which Doppler shifts are
not present. This means that the clock light that interacts with the atoms forms a standing wave
with inhomogeneous Rabi frequencies. This is not ideal for clock operation, as any rotation would
be accompanied by a shortening of the collective Bloch vector.
In principle, the issue of inhomogeneous coupling could be overcome by preselecting atoms
that couple homogeneously to the mode of the cavity at the clock transition wavelength. However,a second challenge is that the cavity should simultaneously have a mode on resonance with the clocktransition (in order to couple the clock light to a well-defined mode), and a mode nearly resonantwith the stronger transition used for atom counting (in our case at 689 nm). Achieving this would
require precise predetermination of the absolute cavity length, which would be challenging.
Another solution that has been considered is to use cavity mirrors that are highly reflectiveat 689 nm and minimally reflective at 698 nm [187]. This requirement was found to be very difficult
with current mirror coating technologies.
I currently think that the best solution to this problem is to confine the atoms in two dimensions, as shown in Fig. 12.1. This means that the atoms can be addressed in a Doppler-free mannerby a clock laser incident from any direction within a plane. The two-dimensional confinement could
be realized by intersecting the modes of two cavities, each of which supports an optical lattice near
the magic wavelength. One of these cavities would have similar parameters at 689 nm to the one

199

Lattice

Lattice + Squeezing Probe

P1
P0

3

Trans
ition

Clock
Laser

b.

Clock

Atoms

Measurement Transition

a.

3

S0

1

Figure 12.1: A proposed architecture for spin squeezing in an optical lattice clock. Atoms are tightly
confined in two dimensions by intersecting lattices supported by two crossed optical cavities. Oneof these cavities is used to perform atom counting measurements on the 7.5 kHz linewidth 3 P1transition. Clock rotations are performed by a 698 nm laser applied in the plane defined by thetwo cavity modes, along which direction the atoms are tightly confined and Doppler-shift free.

used in this thesis, which I have already demonstrated to be sufficient for measurements compatible
with spin squeezing, and would be used to perform atom counting.

12.2

Towards a steady-state superradiant laser
While the spectral properties of the pulsed superradiance explored here are encouraging, fora technologically useful device it would be desirable to have a laser that operates in steady state.
This requires that inversion is constantly maintained as atoms emit photons. Two approaches may
be employed: the atoms that have emitted a photon can be repumped back to the excited state as
demonstrated in [63, 24], or fresh atoms already prepared in the excited state can be introduced
as atoms that have decayed to the ground state are discarded. Even if the repumping approach
is applied, replacement atoms will be required to compensate for atom loss due to heating andcollisions with background gas.
The repumping approach would most likely involve exciting the atoms from 1 S0 to 3 P1 witha laser at 689 nm, then to 3 S1 with a laser at 688 nm. From there, the atoms will decay into allof the 3 P states. Additional lasers at 707 nm and 679 nm would then be used to depopulate the

200
3P

2 states and undesired

3 P sublevels. I am currently not optimistic about this approach for 87 Sr.
0

The number of photon recoils imparted on the atoms to reach a particular state within 3 P0 is high(several 10’s, depending on details), meaning that the atoms would heat out of the lattice withina few cycles. One could attempt to cool during the repump process, but these schemes have thierown problems. Raman sideband cooling within 3 P2 would not address the loosely confined radial
directions, and other sub-Doppler cooling methods require high intensities that may perturb thelasing transition frequency. We are considering cooling on the 50 kHz linewidth 3 P2 to 3 D3 transition
at 2.9 µm, but this has yet to be demonstrated. Besides heating issues, the 679 nm laser used to
optically pump within 3 P0 would likely lead to large perturbations to 3 P0 that would degrade thestability of the superradiant laser. Because of these issues, and the fact that a continuous streamof new atoms will be required anyway to compensate for losses, I think the first approach for 87 Srshould be to figure out how to continuously load fresh atoms into the lattice. If this works wellenough, then repumping may not be required.
Steady-state operation will require a fundamentally different mode of experimental operation
than used in our current apparatus. Like most cold atom experiments, the current apparatus
operates in a time-sequenced manner. Many atomic manipulations occur in the same location,
separated by time. In the continuous version, the operations that are currently time-sequenced will
have to be performed in spatially separated regions. This leads to challenges, as it is much easier tocreate a magnetic field or laser beam whose properties vary abruptly in time than in space. Below,
I will outline one possible architecture, shown in Fig. 12.2.
As in our current setup, atoms emerge from an oven and are slowed using a Zeeman slower.
They are then confined in a 2D blue MOT, perhaps with additional beams present to plug the twoends of the MOT region. The magnetic field for this MOT has vertically oriented minimum. The
MOT is operated without repump lasers, so the atoms eventually fall into 3 P2 . Some of the sublevels within 3 P2 are magnetically trapped, and will fall under gravity, guided along the magnetic
field zero.
At some point, the field gradient that provides confinement must fall off in order to allow for201
superradiant laser operation. At this point, the atoms are pumped back to the ground states, andtransversely cooled and perhaps compressed using the 7.5 kHz transition. The atoms continue tofall into a region where the optical cavity resides.
In order to load into the optical cavity, dissipation is required. Most dissipation schemes
compatible with low enough temperatures for superradiant laser operation would perturb the lasingtransition and be incompatible with simultaneous laser operation. To get around this, we divide thecavity mode into a loading region and a lasing region, probably separated by a carefully engineered
divider. The cavity has a ring geometry, which allows us to create a moving lattice by driving thebeams propagating in opposite directions with slightly different frequencies.
In the loading region, the atoms are cooled into the lattice using cooling light at 689 nm.
There may be a magnetic field gradient present here to facilitate spatial compression. 679 nm lightin the loading region provide strong dephasing of the clock transition and prevent the atoms frominteracting with the superradiant light circulating the cavity. Once trapped and cooled within thelattice, the atoms are transferred to 3 P2 and transported to the lasing region, where 707 nm and
688 nm beams transfer them to 3 P0 . While travelling through the lasing region, each atom emits
up to a single photon, before being removed from the participating ensemble by shelving in 3 P2 .
The velocity of the moving lattice should be sufficiently high relative to the cavity linewidth that
one direction of circulating light is shifted off resonance with the cavity, and the superradiant lightpropagates in a well-defined direction.
At the time of writing this thesis, we have begun to design and build a new experimental
apparatus compatible with such a plan for steady-state operation. I am confident that getting thisapparatus to work will be an exciting and unique challenge.

202

B-field Zero
Zeeman Slower

Blue 2D MOT

Oven

Zeeman Slower Beam

707, 679nm depumps
Red 2D
Cooling
Loading Region

Lasing Region

Moving Lattice

Figure 12.2: A proposed architecture for a continuously loaded superradiant laser. Atoms emitted
from oven are slowed by a Zeeman slower, then captured in a 2D MOT formed on the 461 nm
transition. Some atoms that fall into 3 P2 are magnetically trapped and fall guided along themagnetic field zero. They are then depumped to the ground state and cooled transversely with
689 nm light before falling into the mode of the optical cavity. Here, they are loaded and cooled
into a moving optical lattice, which transports them to the lasing region, where each atom emits
up to one photon.

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