A Superradiant Laser and Spin Squeezed States: Collective
Phenomena in a Rubidium Cavity QED System for
Enhancing Precision Measurements
byJustin G. Bohnet
B.S., University of Northern Iowa, 2008
M.S., University of Colorado at Boulder, 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
2014This thesis entitled:
A Superradiant Laser and Spin Squeezed States: Collective Phenomena in a Rubidium Cavity
QED System for Enhancing Precision Measurements
written by Justin G. Bohnet
has been approved for the Department of Physics

James K. Thompson

Jun YeDate

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
Bohnet, Justin G. (Ph.D., Physics )A Superradiant Laser and Spin Squeezed States: Collective Phenomena in a Rubidium Cavity QED
System for Enhancing Precision Measurements
Thesis directed by Prof. James K. Thompson

By allowing a large ensemble of laser cooled and trapped 87 Rb atoms to interact collectively
with an optical cavity, I have explored two phenomena that may prove useful for enhancing precision
measurements: superradiant lasing and spin squeezing.
Superradiant lasers have been proposed as future ultrastable optical frequency references,with predicted linewidths < 1 millihertz. These lasers operate in an unusual regime of laser physics
where collective emission from an atomic ensemble maps the quantum phase stored in the atoms
onto the optical cavity field. I will give an overview of my experimental work using a cold-atom,superradiant Raman laser as a model system to confirm a number of the key predictions concerningsuperradiant lasing, including the possibility of coherent emission with < 1 intracavity photon on
average and greatly reduced sensitivity to cavity frequency noise.
I also present work using cavity-aided, coherence-preserving measurements of the atomic state
population to create entanglement between atoms. The entanglement enables more precise estimation of the quantum phase at the heart of nearly all precision measurements and sensors utilizing
quantum objects. By utilizing a cycling transition for the quantum non-demolition probe, we have
reduced by several orders of magnitude the measurement induced back-action caused by spontaneous Raman transitions. We directly observe, with no background subtraction, a spin squeezed
state with sensitivity to measuring a quantum phase enhanced 10.5 times in variance (i.e. 10.2dB) beyond the standard quantum limit for an unentangled state. This experimental breakthroughdemonstrates that quantum-aided sensing techniques can realize large enough enhancements tohave a substantial impact on precision measurements and may aid advances in technology as well
as searches for new physics.

Dedication

To my wife, Cassy, who has been with me from the beginning of this endeavor.

v

Acknowledgements

One of the primary lessons I learned over the course of my degree was how very rarely great
achievements are the product of an isolated individual. So many people in my life have played
important parts in realizing this doctoral dissertation. First, I would like to thank my parents,
Mark and Denise, who provided a loving, safe, encouraging home where I could feel confident totry many different activities. They impressed upon me the attitude that I would always have towork hard for my achievements, but that they would still love me no matter the outcome of myefforts.
I also need to thank my wife, Cassy, who has been such a steady and encouraging partner
throughout my education. She has always encouraged me to become the best I could be in any
endeavor, and I feel that having her with me has allowed me to achieve much more than I could
have alone. She was willing to move to Boulder with me, without a job in sight for her, and I’mvery thankful for all of the things she has sacrificed for me to pursue graduate school.
I also must give much credit to my PhD advisor, James. As a science mentor, he has been
unmatched, with knack for understanding and explaining complex topics simply and clearly. I
greatly appreciate how closely he worked with me and how much he taught me in these past few
years. I can see in James not only a passion to do great science, but to train great scientists. And
in all of that, I felt like he treated us graduate students with a lot of respect and dignity.
I also want to acknowledge one of my earliest scientific mentors, Paul Shand. While doing myundergraduate work at the University of Northern Iowa, Paul introduced me to what doing science
was really all about by inviting me to work with him on a summer research project, studying thevimagnetic properties of nanocrystalline materials. My love of doing experimental research was reallyborn in that lab, opening up so many exciting opportunities in my life. It is easy to see that theexperiences and the training provided by Paul are culminating in this dissertation.
Of course, so much of what is contained in this manuscript is a joint product with the teamof amazing graduate students I worked with over the years, Zilong, Phoenix, Josh, Kevin and Matt.
In particular, I need to acknowledge Zilong, from whom I learned so much of my atomic physics
knowledge and laboratory techniques. His unwavering precision and patience has been vital for me
to develop while tempering my hasty tendencies.

Contents

Chapter
1 Introduction

1

1.1

Classical and quantum correlations enhancing precision measurements . . . . . . . .

1

1.2

Coherence Preserving Measurements to Create Spin Squeezed States . . . . . . . . .

8

1.2.1

Quantum Projection Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2.2

Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3

Superradiant lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.1

New frontiers in ultrastable lasers

1.3.2

Superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.3

Good Cavity, Bad Cavity: opposite extremes in laser physics . . . . . . . . . 23

1.3.4

A superradiant Raman laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Apparatus

29

2.1

. . . . . . . . . . . . . . . . . . . . . . . . 20

Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1

General ECDL design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1.2

780 nm MOT and state preparation lasers . . . . . . . . . . . . . . . . . . . . 31

2.1.3

780 nm Narrow DBR Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1.4

795 nm and 823 nm Science Cavity lasers . . . . . . . . . . . . . . . . . . . . 36

2.1.5

Details of laser beatnote locks . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.6

Direct Digital Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

viii
2.2

2.3

2.4

Optical Cavity and Science Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1

Optical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.2

Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Magneto Optical Trap and Lattice Trap . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.1

Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3.2

Coils and Drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.3

Lattice Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.4

Loading Sequence, PGC cooling . . . . . . . . . . . . . . . . . . . . . . . . . 51

Detection and Signal Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.1

APD detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4.2

Fast heterodyne detector

2.4.3

IQ demodulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Reducing measurement back-action to generate spin squeezed states with phase sensitivity
10 times beyond the standard quantum limit

56

3.1

Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2

Creating a Squeezed State with a Collective Premeasurement . . . . . . . . . . . . . 58

3.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4

Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1

Atom-Cavity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.2

Collective Population Measurement . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.3

Spin Noise Reduction Measurement Sequence . . . . . . . . . . . . . . . . . . 73

3.4.4

Quantifying Measurement-induced Back Action . . . . . . . . . . . . . . . . . 76

4 A theoretical model for a steady-state superradiant Raman laser
4.1

83

Three-Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1.1

Deriving the Laser Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.2

Steady-state Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

ix
4.2

4.3

4.4

Raman laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.1

Adiabatic Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2.2

Defining Effective Two Level Parameters for the Raman System . . . . . . . 96

Full Model in 87 Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.1

Continuous Superradiant Raman Laser in 87 Rb . . . . . . . . . . . . . . . . . 100

4.3.2

Steady-state Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 A steady state superradiant laser with less than one intracavity photon

104

5.1

Superradiant Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2

Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4

Expt. Measure: Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.6

Additional Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.6.1

Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.6.2

Phasor Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.6.3

Lorentzian Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6 Linear response theory for a superradiant laser

121

6.1

Linear Expansion of Uncoupled Equations . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2

Transfer Function Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.1

6.3

Bloch Vector Analysis of Response . . . . . . . . . . . . . . . . . . . . . . . . 131

Linear Response Theory in 87 Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7 Relaxation oscillations, stability, and cavity feedback in a superradiant Raman laser

144

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.2

Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

x
7.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.4

Additional Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4.1

Technical Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.4.2

Response Function Measurement Sequence . . . . . . . . . . . . . . . . . . . 156

7.4.3

Real time cavity probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8 Active and passive sensing of collective atomic coherence in a superradiant laser

162

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.2

Theory of the Optimal Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.3

Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.4

Conclusion

8.5

Derivation of Optimal Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8.5.1

Photon Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.5.2

Phase Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.5.3

Optimal Phase Estimation with Kalman Filter . . . . . . . . . . . . . . . . . 175

8.5.4

Modifications for Experimental Conditions . . . . . . . . . . . . . . . . . . . 176

Prediction of Time to Return to Steady-state Superradiant Emission . . . . . . . . . 176

9 Summary and Conclusion

178

9.1

Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

9.2

Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Bibliography

185

Appendix
A Details of Raman laser model197

A.1 Adiabatic Elimination of the Optically Excited State . . . . . . . . . . . . . . . . . . 197

xi
A.2 Steady-state Emission Frequency of the Raman Laser . . . . . . . . . . . . . . . . . 200
B Detailed Equations for 87 Rb Model201

B.1 Repumping Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
B.2 Reduced Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
C Full Expressions for the Three-level Model Linear Response Theory

207

C.1 Interesting Limiting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
D Heterodyne Photodiode

210

xii

TablesTable
3.1

Contributions to the observed spin noise reduction R from a fit to the data versus
probe strength Mt that includes photon shot noise, a noise floor, quantum noise,and classical noise. Each value is given at the optimum spin noise reduction at
Mt = 4.1 × 104 . The indented rows show the calculated contributions from thevarious noise sources to each term. The uncertainties are 68.3% confidence intervals.

81

Figures

Figure1.1(A) The laser trapped and cooled sample of neutral atoms in an optical lattice clock
serves as an example of the variety of single particle control techniques used extensively in precision measurements with cold atoms. The optical lattice trap nearly
eliminates all motional perturbations from the system, controlling the atoms’ external states and optical pumping techniques control the internal electronic statesof the atoms. Lattice clocks can also operate with fermions in dilute samples thatcan nearly remove collisional interactions between atoms. (B) By adding a nearlyresonant optical cavity mode, we can introduce controlled atom-atom interactions
and take advantage of this additional degree of freedom in the ensemble. The collective state of the ensemble is more than just many independent realizations of anexperiment, possessing, for example, modified quantum noise properties that produce enhanced sensitivity for precision measurements with the same resources (inthis case, the number of atoms). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

xiv
1.2

Examples of common interactions used for quantum simulation and generation ofinteresting collective states. A) Atomic collisions have a short range of interaction∝ r−6 , and although they can be tuned with techniques such as magnetic Feshbachresonances, can be difficult to quickly and completely extinguish. B) Dipolar forces
are slightly longer range ∝ r−3 , but also anisotropic. C) Coulomb interactions can be
relatively long range ∝ r−1 , but systems that take advantage of Coulomb coupling,
like ion traps, have proved difficult to scale to large atom numbers. . . . . . . . . . .

1.3

5

Atom-cavity coupling is a competition between coherent exchange of excitations√
between the atoms and cavity at a rate N 2g, and loss of excitations through decay
through free space scattering Γ and spontaneous decay through both cavity mirrors
κ. The enhanced atom-cavity coupling rate arises from a constructive interference
for emission and absorption with respect to the cavity mode due to correlations
among the individual atoms, the same physics underlying the phenomena of Dicke
superradiance [43].

1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7Illustration of a Ramsey sequence with a two level system represented by a Bloch
sphere. The first step is optical pumping the atoms all into |↓i (light red arrow),then performing a π/2 rotation about −ŷ axis to place the Bloch vector along x̂.
The Bloch vector then precesses around the ẑ axis at a rate proportional to theenergy difference between |↑i and |↓i. The Bloch vector precession represent theevolution of the quantum phase in the wavefunction |Ψi = cos((π/2 − θ)/2) |↑i +eiφ(t) sin((π/2 − θ)/2) that we ultimately wish to measure. The second π/2 pulse
performs another rotation about −ŷ and maps the φ(t) onto the coefficients of the
population states, or equivalently onto the polar angle of the Bloch sphere. The
quantum phase φ(t) can then be inferred by measuring the state of the two levelsystem. Quantum randomness in the projection onto the population basis results infundamental quantum noise limiting the resolution of the angle φ(t). . . . . . . . . . 10

xv
1.5

The standard quantum limit for a coherent spin state. Quantum noise appears as an
uncertainty in the pointing location of the classical Bloch vector J, represented by
the blurry red region at the tip of the vector. The signal, that is, the change in the z
projection of J for a given angular rotation, increases linearly with N . However, thestandard deviation in a measurement of the z projection from quantum noise grows√
as N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6

A summary of results for the observed entanglement-enhanced sensitivity for measuring a quantum phase across a variety of physical systems and mechanisms. The
solid points indicate results that use coherence preserving pre-measurements, themost relevant results for comparison to this work. The result from this work ishighlighted with a red circle. (A) Absolute observed phase resolution of an ensemble, emphasizing the improvement in the absolute sensitivity from simply startingwith a large ensemble of atoms. Though our result is far from the Heisenberg limit,
the large ensemble size allows improvement for an already highly resolved phase.
(B) The observed enhancement of phase resolution relative to the SQL. The blueline indicates the fundamental Heisenberg limit. Compared to previous work, whichobserved interesting, but small entanglement-enhanced sensitivities, we have observed over an order of magnitude enhancement in a system readily applicable toprecision measurement experiments such as optical lattice atomic clocks. To date,
our result is the largest enhancement in any system using matter as spins. The
other results can be found in Refs. [116, 102, 117] for experiments using ions, Refs.
[49, 130, 62, 28, 108, 69, 104] for experiments using one-axis twisting, and Refs.
[4, 159, 141, 139, 35] for non-demolition measurements. . . . . . . . . . . . . . . . . . 12

xvi
1.7

Squeezing in an optical field, or equivalently a harmonic oscillator, with the two
conjugate variables on the X1 and X2 . Both the coherent and squeezed states can be
represented as a classical phasor with quantum uncertainty. The area of uncertainty
is constant, but can be distributed among the different quadratures. In this case, thesqueezed state has a reduced phase uncertainty ∆φSS at the expense of increased
uncertainty in the amplitude ∆ASS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8

Generating a squeezed state with a coherence-preserving pre-measurement for aRamsey sequence. A coherent spin state has an initial uncertainty in the azimuthal
angle of the Bloch vector ∆θSQL set by quantum noise that defines the SQL for phaseresolution of the state. By performing a measurement on the polar angle θ using
a population readout with resolution below the quantum noise limit, the deviationof the Bloch vector from the equator on that particular trial, δθ, can be recorded.
Another π/2 pulse rotates the quadrature with a known quantum fluctuation into
the the azimuthal quadrature and the Ramsey sequence can begin. A noiseless vector going through the Ramsey sequence is shown as a dashed line. While the final
output of the interferometer will still exhibit quantum fluctuations ∆θSQL about the
average output polar angle δφ, the fluctuations can be subtracted on a trial by trial
basis due to the knowledge of the initial state from the pre-measurement. Effectively the state has an uncertainty set by the measurement resolution, determining
the precision which one can measure the initial quantum fluctuation.

. . . . . . . . 17

xvii
1.9

A cartoon representation of superradiance in an extended ensemble of atoms. Fora collection of atoms radiating with independent phases (A), the light is isotropic
and the total power emitted goes as N P0 , where P0 is the power emitted by a single
atom. For ensembles with spatially correlated radiating phases (B), represented by
the synchronization of the arrows on the atoms, the emitted light can constructively
interfere along certain directions, resulting in an N 2 enhancement of the power radiated along this direction. The cooperativity parameter C serves as a measure of
the range of directions (fractional solid angle) over which the constructive interference occurs. The total power radiated due to this constructive interference is thenproportional to N 2 CP0 , i.e. NC larger than for the total power from non-correlated
radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

xviii1.10 (Left) In the good-cavity limit, the atomic coherence rapidly decays, and the photon field is the primary reservoir of phase information in the laser. As a result,
perturbations which disturb the cavity resonance frequency, such as thermal mirror
vibrations (drawn as curved lines near the mirrors), limit the frequency stability of
the laser. (Right) At the other extreme, the bad-cavity, or superradiant, laser presented here operates in a regime where the atomic coherence decay rate γ⊥ is muchless than the cavity power decay rate κ. In this regime, the atomic gain medium isthe primary reservoir of phase memory in the laser, a fact represented by the aligned
dipoles of the atoms and a cavity mode nearly devoid of light quanta. Because theemission frequency is primarily determined by the atoms, perturbations from fluctuations in the cavity frequency are suppressed. (Middle) Near the crossover regime,
the phase coherence is jointly stored by the atoms and the cavity photons, making ita polariton-like excitation. Most optical lasers operate in the good-cavity limit (oneexample is the cold atom Raman laser of Ref. [156]), with microcavity diode lasers[12] and far infrared (FIR) gas lasers, using Xe [30], NH3 [70, 162], and HeXe/HeNe[97], operating in the vicinity of the crossover, polariton-like regime. Our cold atom
Raman laser is unique both in terms of operating so deeply into the bad-cavity
regime, and also in that the steady-state intracavity photon number can be mademuch less than one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

xix
1.11 The long lived, nearly forbidden 3 P0 to 1 S0 transition in elements like Sr, Yb or Ca,
with excited state decay rate γeg , is mimicked by a two photon Raman transition
between hyperfine ground states in 87 Rb. The Raman dressing laser, with Rabi
frequency Ω, is detuned from an optically excited intermediate state |ii to induce an
optical decay from |ei to |gi at rate γeg , mimicking the naturally occurring decay
on the left hand side. This decay rate can be tuned over a wide range by adjusting
the dressing laser intensity and/or detuning from the intermediate state. In bothcases, population inversion is maintained by incoherently repumping the atoms backto the excited state |ei using another laser that excites atoms in the ground state |gi
to a short-lived optically excited state after which they quickly spontaneously decay
to |ei. The collective blue laser emission in both cases is resonant with the opticalcavity mode, while the single-particle spontaneous emission for repumping (yellow)
is not resonant and is scattered into free space. . . . . . . . . . . . . . . . . . . . . . 26
2.1

A CAD model of the custom ECDL used throughout the laser systems in our lab.
The laser diode is mounted in the lens tube assembly. The diffraction grating (white)provides frequency selective optical feedback to the narrow the laser linewidth. The
grating is mounted on the adjustable pivot. A piezoelectric slab provides a method of
feeding-back to the grating position for wideband laser frequency tuning by pivoiting
the assembly. The grating can also be manually adjusted with the two screws thatpush on the pivot. The 45◦ mirror out-couples the laser light (out-coupling lens notshown). The temperature control Peltier device rests under the mount as it sits inthe external enclosure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

xx
2.2

A schematic of the 780 nm laser subsystem used to generate the magneto-opticaltrap and perform state preparation using the D2 transition of 87 Rb . The system
consists of 3 ECDL lasers, labeled 780 nm Reference, MOT and Repumper. The 780
nm Reference laser is stabilized to an atomic frequency reference using a Dopplerfree polarization spectroscopy. Both the MOT and Repumper lasers are beatnote
stabilized to the Reference laser (not shown). Feedback loops are indicated by theblack line leading back to the lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3

RF beatnote frequencies used to set the MOT and Repumper lasers to resonancefrom their respective hyperfine ground state to the excited states of the 87 Rb D2 line. 33

2.4

A schematic of the 780 nm narrow linewidth DBR laser subsystem used to generate
the probe laser on the D2 transition of 87 Rb for the spin squeezing experiment.
The system consists of 2 Photodigm DBR lasers, labeled DBR Reference and DBR
probe. The DBR lasers have improved short term stability properties due to theoptical feedback provided by the long external free space cavity, of length ∼ 0.5 m.
We can tune the magnitude of the optical feedback using the λ/4 waveplate andpolarizing beam cube in the feedback path. The reference laser provides long term
frequency stability by locking to an atomic reference with MTS. The probe laser isthen beatnote locked to the reference laser, giving us tunable control over the probe
frequency over a range of 3.2 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xxi
2.5

The 795 and 823 nm laser subsystem. The subsystem consists of three ECDLs
at 795 nm (the 795 nm Reference, Dressing, and Transfer lasers) and one ECDL
at 823 nm (the Lattice laser). All are ultimately referenced to the F = 2 − 3 toF = 30 crossover signal of 85 Rb through beatnote locks in the case of the Dressing
and Transfer lasers, and using the transfer cavity in the case of the Lattice laser.
The transfer cavity is locked to the Transfer laser using a PDH lock to stabilize
the cavity length, then the Lattice laser is PDH locked to a longitudinal mode of
the transfer cavity near 823 nm. The Dressing laser is split into three functions,generating the Raman dressing light for inducing superradiance, probing the sciencecavity resonance frequency, and providing the heterodyne reference beam in the
superradiance experiments. The difference in the dressing laser frequency and theprobe frequency is bridged by a EO Space phase modulator driven near the hyperfine
splitting frequency. The Heterodyne beam is path is also phase modulated, so that
RF beatnotes of both the superradiantly emitted light and the dressing light can beobtained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6

The Analog Devices AD9959 direct digital synthesis evaluation board shown fully on
the left and zoomed in on the right. The evaluation board has two modes of control,
either serial control through the header pins at the top of the picture, or through a
USB control on the left side of the picture. To enable pre-programmed ramps of thefrequency, amplitude, or phase of the DDS channels using an external TTL signal,
we modified the evaluation board. The ramps are controlled by profile pins to the
DDS chip, P0, P1, P2, and P3. These pins connect to the serial header, but the
I/O buffer (chip U5, PN: 74LVC5414A) connects the USB profile inputs (PX U) to
the profile pins. By cutting the traces from the IO Buffer (chip U5, pins 11-14), theprofile pins are freed to be controlled by the serial headers. In the photograph on
the right, the traces corresponding to the profile pins are circled in red. Here only
the trace for profile pin P0, corresponding to pin 11 on U5, has been cut. . . . . . . 41

xxii
2.7

Photograph showing the optical cavity spacer, piezos and mirrors inside the glass
cell vacuum chamber. The smaller coils of copper wire are the initial, symmetric
MOT coils. The larger wire coils provide bias magnetic fields in all three directions.

2.8

43

A drawing of the cavity spacer. All dimensions are given in units of cm. The spacer
was machined from a single piece of Zerodur. The spacer is supported by four rods
at the midpoint of the cavity. The holes for the support rods are symmetric on bothends of the spacer for common mode rejection of vibration. The angled cutouts oneither end of the cavity allow for better optical access for the angled MOT beams. . 44

2.9

List of science cavity parameters that are constant across wavelength. . . . . . . . . 45

2.10 Science cavity parameters for the relevant wavelengths in including the spin squeezingprobe at 780 nm, the superradiance at 795 nm, and the trap at 823 nm.

. . . . . . 45

2.11 A drawing of the complete vacuum system (the glass cell surrounding the cavity
spacer is not shown). The vacuum chamber below the reducer fitting (grey) is notshown. The cavity spacer is in magenta, the MOT coil is in light grey, the spacer
supports are in white, and Kimball Physics spherical square in dark red. The damping pendulum hangs off the support structure with the purple rods. The pendulum
weight is colored light green. The three pairs of MOT beams intersect at the centerof the cavity spacer. The cavity axis beam path is shown entering the sphericalsquare and exiting out the top of the optical cavity. The distance to the center ofthe cavity and the end of pendulum are both measured from the top of the optical
table.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

xxiii2.12 (Left) The CAD model of the vacuum chamber, cavity support system, and cavity
spacer. A probe laser beam is shown entering a vacuum port, bouncing off the 45◦
mirror and coupling to the cavity. (Right) A cutaway view of the cavity support
structure. The cylindrical support structure (dark grey) links the cavity spacer
supports (white) to the damping pendulum (purple). The damping pendulum (seenin Fig. 2.11) hangs off the support structure, which rests on 3 Viton spheres (brightgreen) for damping. The 45◦ mirror is mounted on rails connected to the vacuum
chamber with Groove Grabbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.13 (left) A basic schematic of the elongated MOT coils used for most of the experiments
presented in this thesis. The Coil is composed of 128 turns of Belden 20 AWG magnet
wire. The individual turns are held together with Araldite 2011 Epoxy. Each layerof the coil is 12 turns, making the coil 10.7 mm deep. The rounded inside corners
have a radius of curvature of 4 mm. (right) A photograph showing the elongated
MOT coil mounted in the experiment. The coil sits inside the larger bias coil. . . . . 50
2.14 A general time sequence that begins each of the experiments in this dissertation (notto scale). Maybe a chart showing what lasers are on and what the B-field is doing
here would be useful? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.15 An example schematic of the heterodyne data acquisition system. The output of theheterodyne photodiode can have multiple signals separated in RF frequency. We split
the signal after amplification, then have separate paths for each frequency in which
we can separately attenuate the signals to keep the inputs to the IQ demodulators.
The signals are demodulated with RF frequencies from the DDS. The corrections toaccount for imbalances in each IQ demodulator is applied in software (circle with‘C’), giving the corrected signals I 0 (t) and Q0 (t) from which the amplitude A(t) andphase φ(t) of each signal can also be calculated. . . . . . . . . . . . . . . . . . . . . . 54

xxiv
3.1

Spin-squeezing and measurement back-action (A) Atoms collectively interact
with light in an optical cavity. A measurement of the phase of the probe field (red)is sensitive to the total number of atoms in spin up, and projects the ensemble into
an entangled state, conditioned on the measurement outcome. Probe photons canbe scattered into free space, causing atoms to collapse to spin up (orange in A, C,
and D) and can also cause state-changing transitions (blue in A, C, and D). (B)A coherent spin state can be visualized by a Bloch vector (red), with a pointing
uncertainty set by quantum noise, represented by the shaded uncertainty disk. (C)Atoms in |↑i with optical transition frequency ωa couple to the detuned cavity modewith resonance frequency ωc . The coupling results in a dressed cavity mode withresonant frequency ωc0 , so probing ωc0 measures the total number of atoms in |↑i, andhence the Bloch vector’s spin projection Jz , without measuring the state of individual
atoms. Probing on a cycling transition suppresses back-action from scattering events
that change an atom’s state to |↓i (blue), limiting back-action to collapse (orange).
(D) After a pre-measurement, back-action modifies the noise distribution on the
Bloch sphere. Fundamental back-action appears along ŷ. Back-action from nonideal measurements, indicated by dashed lines, include reduction in length J of thecollective Bloch vector and added noise in Jz caused by state-changing transitions.

57

xxv
3.2

Detection of a quantum phase with entanglement-enhanced sensitivity.
We apply a small rotation ψ to the polar angle θ of both a CSS and a spin-squeezed
state, with data and representative Bloch spheres shown on the left and right sides
respectively. Red data points show experimental trials with ψ = 2.3(1) mrad, andblue data points show trials with ψ = 0. The data are represented both as histogramsand Gaussian curves generated from the average and standard deviation of the measurements. The experimental timing sequence consists of probe pulses (black) andmicrowave rotation pulses (green). For the CSS, the rotation ψ is applied immediately after preparing the CSS along x̂. The rotation ψ appears as a change in thequantity N↑ − N↓ , which is normalized to the total projection noise that appears
in this differential quantity. In the case of the spin-squeezed state, we perform therotation ψ after a pre-measurement N↑p . The rotation then appears as a change inN↑f − N↑p , where the projection noise largely cancels. The spin-squeezed state has
a precision W −1 = 7.5(9), even though the change in N↑f − N↑p is slightly smaller
than in the CSS due to free-space scattering during the pre-measurement. The lossof signal is represented by a smaller Bloch sphere for the spin-squeezed state. . . . . 61

xxvi
3.3

Spin-squeezing and probe-induced back-action. (A) Scaling of the spin noise
reduction R (red), loss of signal C 2 /Ci (blue), and the inverse of the spectroscopicenhancement W (black) versus probe intensity Mt for N = 4.8 × 105 . The red, blue,and black curves are fits to the data. The data for W is calculated from R data andthe fit to C 2 /Ci . The 68% confidence band for the W fit and the SQL is in grey. The
dashed red curve shows the fitted R assuming no probe-induced added noise (rc =rq = 0). The light-blue region is the predicted C 2 /Ci due to free space scattering. Allerror bars are 1 std. dev. We use the usual convention for expressing a ratio X indB units, x (dB) = 10 log10 X. (B) Examples of optomechanical oscillations in the
dressed cavity frequency ωc0 . The relative detuning of ωc0 and ωp results in increased
or decreased oscillation damping rates, a source of probe-induced back-action noise(see supplementary text). Each curve is the average of 30 experimental trials. (C)Example data and experimental sequence for the measurement of the contrast C.
Probe pulses (black) are measurements N̂↑ . Microwave pulses (green) rotate thepolar angle θ of the Bloch vector. After the pre-measurement of N↑p , a variablerotation θR is applied and N↑ (θR ) is recorded. The contrast C is determined fromthe amplitude of the N↑ (θR ) fringe (curves are a fit to the data), with two examples
shown in blue and grey for Mt = 3.0 × 104 and Mt = 0 respectively.

3.4. . . . . . . . . 63

Absolute phase sensitivity versus N. The red points are the observed spinsqueezed state phase sensitivities (optimized with respect to measurement strengthMt ) for different atom numbers N . The data show the predicted 1/N 2 scaling
for probing on a cycling transition [134, 34], equivalent to a linear scaling of thespectroscopic enhancement W −1 versus N , shown in the inset. The red line is alinear fit to the data. The SQL is confirmed by measuring the projection noise that√appears in N↓ − N↑p (black points, each 100 trials) and observing 1/ N scaling.
Error bars indicate 68.3% confidence intervals. . . . . . . . . . . . . . . . . . . . . . 64

3.5

The measured detection efficiencies for the spin squeezing experiment. . . . . . . . . 66

xxvii
3.6

Simplified experimental diagram. The one dimensional optical lattice trap at
λl = 823 nm is formed from a standing wave in the cavity (orange). We loadN◦ = 4.0 × 104 to 7.2 × 105 87 Rb atoms into the trap and cool them to 10 µK.
The atomic sample extends ∼ 1 mm along the axis of the L = 1.9 cm long opticalcavity. State preparation is performed using a combination of 780 nm light (purple)for optical pumping and coherent ground state rotations performed with 6.833 GHz
microwaves from the dipole antenna. A uniform magnetic field is applied to provide
a quantization axis and spectrally resolve the ground state Zeeman sub-levels. The
probe electric field forms a standing wave in the cavity, represented by the sinusoidal
red line. The atom-cavity system is probed with 780 nm light (red), set to σ +polarization before entering the cavity. The probe light is separated from trap lightusing a dichroic mirror (grey) on the probe transmission side. The probe lightis detected in both reflection and transmission with a heterodyne beam frequency
shifted by an AOM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.7

Atomic energy level structure. The relevant energy level structure of the 5 2 S1/2
to 5 2 P3/2 transition in 87 Rb. The cycling transition has an optical atomic resonancefrequency ωa . The cavity resonance with no atoms present (dashed light blue) withfrequency ωc is detuned to the blue of atomic resonance. The atom-cavity couplingcreates a dressed cavity resonance (light blue) with frequency ωc0 which we probe
using σ + laser light at frequency ωp (red). The cycling nature of the transition means
scattering primarily maintains population in |↑i (orange). Scattering of the probe
light from atoms in |↓i, detuned by ∼ 6.8 GHz, provides the fundamental limit to
the cycling transition, as atoms can scatter to both |↑i and |1i ≡ |F = 2, mf = 1i
(dark blue). Furthermore, imperfect polarization can lead to transitions for atoms in|↑i to other internal states. An example of scattering from a π-polarized componentof the probe is shown as the dashed red lines, with the state changing transitions indark blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xxviii
3.8

Detailed probe path documenting generation of heterodyne beams, probe quantum
efficiencies to reflection and transmission detectors, and path length stabilization
setup for transmission using 823 nm lattice laser. The 780 nm laser (red) is split
into the heterodyne paths (thick lines) and the probe path (thin lines), which are
eventually overlapped before jointly entering single mode fibers that go to the fast
heterodyne detectors. The 823 nm laser (orange dashed line) that forms the opticallattice trap is also directed to largely follow, in reverse direction, the transmission
heterodyne path (Heterodyne 1). Some of the path length noise is then common
mode, and appears in the heterodyne detection of the 823 light in the probe setup
area. The grey square indicates components on a platform raised 3 ft off the optical
table surface. All other components on table surface. . . . . . . . . . . . . . . . . . . 71

3.9

Timing Sequence.

The experimental timing sequence showing probe laser pre-alignment and the pre-measurement that prepares a conditionally spin-squeezed
state, followed by a final measurement to quantify the reduction in spin noise. Each
optical pumping step (purple) prepares the ensemble in |↓i. Rotations (green) are
performed by coupling |↑i and |↓i with a coherent microwave source. Probe laser
pulses (red) correspond to individual measurements N̂↑ , with the measurement outcomes labeled as shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xxix
3.10 Comparing measurement noise with and without atoms. The fluctuations in
the difference between two measurements of the dressed cavity frequency ωc0 f − ωc0 p ,used to determine the difference of population measurements N↑f − N↑p , is plottedversus the average number of probe photons Mt . On the left axis, the fluctuations are
expressed as the standard deviation ∆(ωc0 f − ωc0 p )/2π in absolute frequency units.
On the right axis, the same fluctuations are expressed as the ratio of the variance(∆(ωc0 f − ωc0 p ))2 to the variance (∆ωc0 p )2 caused by the quantum projection noise(QPN) of a CSS. For the ensemble of N = 4.8 × 105 atoms here, QPN causes
fluctuations ∆ωc0 = 2π × 144(9) kHz, indicated by the line at 0 dB. Measurement
noise is compared with (red) and without (black) the atoms loaded in the trap. For
the case with atoms, the right axis is equivalent to R. The lines are fit to the data.
The error bars are 1 std. dev.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xxx
3.11 Probe-induced population change.

(A) To measure the sum of transition

probabilities p↓↑ + p↓1 , we first prepare atoms in |↓i with optical pumping (purple).
Next, a scattering probe pulse (red), quantified by the average number of transmitted
probe photons Mt , causes some atoms to change state to |↑i and |1i. Many photons
per atom in |1i are scattered into free space, allowing the atoms in |1i to transition
to |↑i. Thus, the measurement of the dressed cavity frequency ωc0 ↓ gives the total
number of atoms scattered out of |↓i. (B) To measure the transition probability p↑↓ ,we prepare atoms in |↑i using optical pumping and a microwave π pulse (green).
The imperfection in the σ + polarized probe used for the scattering pulse causes
some atoms to change state to |↓i. We again assume all atoms that scatter to |1i
immediately transition back to |↑i. We swap the populations in |↑i and |↓i withanother microwave π pulse, so the measurement of ωc0 ↑ gives the number of atoms
that scattered to |↓i. (C) Measurements of the dressed cavity frequency due to probeinduced internal state-changing transitions, with ωc0 ↓ (red) described in A, and ωc0 ↑(blue) described in B. The lines are fits to the data, yielding a change in ωc0 pertransmitted photon δωc0 ↓ = 2π × 1.11(2) Hz/photon (red) and δωc0 ↑ = −2π × 0.86(5)Hz/photon (blue). Here N = 2.1 × 105 atoms. . . . . . . . . . . . . . . . . . . . . . . 77
4.1

Energy level diagram of a three-level superradiant laser using the optical transition
from |ei to |gi. The emitted optical laser light (blue) is nearly resonant with thecavity mode (dashed lines), detuned from ωc by δbcav . The atoms are incoherently
repumped to a third state at a rate W . Atoms in |gi also Rayleigh scatter at a rateΓR , but leaves them in state |gi. The incoherent repumping from |3i to |ei at rate
Γ3e completes the cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2

Representing the state of the |ei, |gi two level system using a collective Bloch vector. 86

xxxi
4.3

(a) Steady-state photon flux M̄˙ c versus ground state repumping rate W , with a seriesof curves showing the effects of repumping through the additional state |3i.The caseof r = ∞ is the two level model of Ref. [115] (black). (b) Photon flux M̄˙ c versus rwith W = Wopt . For all curves, δ00 = 0 and ΓR = 0, and the photon flux is plottedin units of P2lvl = N 2 Cγ/8.

4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90(a) Steady-state photon flux M̄˙ c versus ground state repumping rate W , with a seriesof curves showing the effects of detuning of the cavity resonance frequency from theemitted light frequency δ00 . (b) Photon flux M̄˙ c versus δ00 with W = Wopt (δ00 ). The
photon flux is plotted in units of P2lvl = N 2 Cγ/8. For all curves, r = ∞ and ΓR = 0. 91

4.5(a) Steady-state photon flux M̄˙ c versus ground state repumping rate W , with a seriesof curves showing the effects of decoherence in the form of Rayleigh scattering fromthe ground state |gi. (b) Photon flux M̄˙ c versus ΓR with W = Wopt (ΓR ). The photon
flux is plotted in units of P2lvl = N 2 Cγ/8. For all curves, r = ∞ and δ00 = 0.

4.6

. . . . 91

Energy level diagram for a superradiant laser enabled by an induced Raman transition. States |ei and |gi are two metastable states separated by a non-optical frequency ωeg . They share an optically excited state |ii that has a linewidth Γ. Using a
Raman dressing laser (green), detuned from |ii by ∆d , we can induced a optical decay 2
to |gi, which, in absence of collective effects, would proceed at rate γ = Γ4 Ω∆d .
Including a single optical cavity mode, coupled to the |ii to |gi transition with coupling constant 2g, gives rise to collective emission. The cavity mode frequency is ωc ,detuned from |ii by ∆c , making the two-photon detuning δ0 = ωc − (ωd + ωeg ). To
complete the laser cycle, the atoms are incoherently repumped from |gi to |ei at a
rate W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xxxii4.7

Lasing transition and Raman dressing scheme on the 87 Rb D1 line (795 nm). The
dressing light (red) and collective emission (blue) are a superposition of σ + and σ −polarizations because the direction of propagation of the light is along the quantization axis defined by the direction of the magnetic field at the atoms. The Raman dressing laser is detuned by ∆ from the atomic transition. The bare cavity
detuning is δ0 = ωc − (ωd + ωhf ). The optically excited state on the D1 line has alinewidth ΓD1 /2π = 5.75 MHz. The effective population decay from |F = 2, mf = 0i 2Ωdto |F = 1, mf = 0i is γ = ΓD1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4
∆

4.8

Two step repumping process on the 87 Rb D2 line (780 nm). The diagram is drawn
showing on only positive Zeeman states, but the process is symmetric with respectto mf = ±1, ±2. The desirable decay branches (magenta) show the most direct
repumping sequence, although any particular repumping sequence could go throughmany ground hyperfine states. The optically excited state on the D2 line has alinewidth ΓD2 /2π = 6.07 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xxxiii
5.1

A steady state superradiant laser. a, (left) In a good-cavity laser far above
threshold, many photons (yellow) circulate inside the cavity, extracting energy fromthe largely incoherent atomic gain medium (blue). Thermal vibrations of the mirror
surfaces modulate the cavity resonance frequency, limiting the linewidth of the laser.
In a superradiant laser (right), the collective atomic dipole stores the coherence, andcontinuous stimulated emission can be achieved even with less than one photon in
the cavity. The stimulation enables phase information to be extracted at a usefulrate, while the small intracavity photon number leads to only weak cavity-induced
backaction on the collective atomic dipole. b, To mimic a narrow optical atomic
transition, we dress the metastable ground state labeled |ei with a laser (blue) to
induce a spontaneous two-photon Raman transition to |gi, with tunable rate γeg .
With no repumping light, a single superradiant pulse is emitted. d, With repumping
light applied, we observe quasi-continuous emission limited by atom loss. The atoms
emit into a single spatial mode of the cavity (TEM00 ) imaged on a CCD (inset).
e, The measured peak power output (black circles) scales as the number of atoms
squared.The red line is a quadratic fit to the data. . . . . . . . . . . . . . . . . . . . 105

5.2

The laser can operate in a quasi-steady-state mode in that each atom can go throughan emission-repumping cycle multiple times. However, the emitted power decreases
over time, eventually going to zero. In (a), we show one example of emission lasting ∼10 ms. Under other conditions, we have observed emission lasting up to 120 ms. We
verify that atoms are being lost from the trap (b) by measuring a changing frequency
shift of the cavity mode consistent with atom loss. The loss is not associated withcollective emission, as shown by the equivalence of the loss observed with (red)collective emission or (blue) induced single particle emission from |ei to |gi via freespace scattering at the same rate as the collectively enhanced decay rate. The atom
number is in units of the initial atom number N0 .

. . . . . . . . . . . . . . . . . . . 108

xxxiv
5.3

Repumping-induced quenching a, The average number of intracavity photons
Mc versus the repumping rate w for different decay rates γeg . The inferred Blochvector is shown for two operating points. b, The repumping rate above which superradiance is quenched wmax (green) and the repumping rate at peak output power
wpk (black) scale linearly with γeg , shown with linear fits to the data.

5.4. . . . . . . . 110

Phase coherence maintained with no intracavity photons. a, The dressing
and repumping lasers are shut off for some variable dark time Tdark . The phase of
the light relative to the dressing laser is measured shortly before and after the darktime. b, The phasor correlation function, defined in Sec. 5.6. At times Tdark < 7 ms,
the phases are correlated because the collective atomic dipole seeds reestablishmentof superradiant emission. At longer times, the collective atomic dipole dephases andsuperradiant emission restarts with a random phase. A gaussian fit is shown in redline. Error bars ±1 s. d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xxxv
5.5

Beyond standard optical laser stability. a, The average power spectrum of theheterodyne signal of the emitted light (black circles) at Mc = 0.20(2) intracavity
photons with a Gaussian fit (red line). b, The average power spectrum of the heterodyne signal of the emitted light (black circles) at Mc = 30.6(5) with a Lorentzian
fit that excludes offset frequencies lower than 4 kHz. c, The comparison of various
characteristic linewidths. Gaussian noise processes are compared to the measured
Gaussian FWHM (dashed), and Lorentzian noise processes were compared to themeasured Lorentzian FWHM (solid). Details of the experimental configurations areincluded in Sec. 5.6. d, As the decay rate γeg decreases, the atomic dipole becomes
more isolated from the mirrors, as shown by directly measuring cavity frequency
pulling P (black circles). Because of the collectively enhanced emission rate, phase
information is still coupled out of the system at a sufficient rate to achieve an ideal
RMS phase noise (red circles) sufficient for spectroscopy below the standard quantum limit for 105 atoms. The fitted curves are consistent with the predicted scaling
with γeg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.6

Primary experimental configuration – dressing beam along cavity axis.
a, The physical arrangement of the apparatus. The cavity is vertically oriented,
with the quantization axis defined by the magnetic field. The linearly-polarized D1
dressing beam is in red, and the π-polarized D2 repumping beams are green (F2)and purple (F1). The emitted light, blue, goes into the cavity mode. It has a linear
polarization that is rotated 90◦ from the dressing beam. b, The energy level diagram
for the D1 and D2 beams. The cavity mode resonance is denoted with a blue dashedline. The repumping dark state is labeled with a gray circle.

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

xxxvi
5.7

Secondary experimental configuration – dressing beam perpendicular tocavity axis. a, The cavity is vertically oriented, with the quantization axis definedby the magnetic field. The linearly-polarized dressing beam is in red, and the repumping beams are green (F2) and purple (F1). The circularly-polarized emitted
light, blue, goes into the cavity mode. b, The energy level diagram for the D1 andD2 beams. The cavity mode resonance is denoted with a blue dashed line. The
repumping dark state is labeled with a gray circle. The polarization of the repumping beams is a linear combination of π-polarization (thick lines) and σ − polarization(thin lines).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xxxvii
5.8

A chart of relevant laser frequencies used in the superradiance experiment. The x-axis
is frequency, but the y-axis is only used to distinguish different beams. The 795nm
reference laser (not shown here) starts the frequency chain, stabilized to the atomic
reference, as described in Chapter 2. The second 795 nm laser is then stabilized withrespect to the reference 1.184 GHz detuned from the F = 2 to F = 20 transition,with the transition indicated by the black arrow. This laser is split into two paths.
The first forms the dressing laser fsp after being shifted 84 MHz lower in frequency
to give the dressing laser detuning ∆ = 1.1 GHz. The other path is ultimately used
for both the heterodyne reference beam fhet and the dressed cavity frequency probe
fprobe . This second path is modulated using an EOspace high frequency modulator
to produce sidebands at fm = 6.800 GHz, placing the higher frequency sidebands
near the superradiant emission frequency fγ , which is fixed to the be fsp +fhf , where
fhf is the hyperfine splitting frequency. After high frequency modulation, the probe
beam is split from the heterodyne beam, and frequency shifted lower by 100 MHz.
The setup produces 3 frequency separated signals on the heterodyne detectors, theemitted light at 50 MHz, the dressing laser leakage at 84 MHz, and the cavity probe
at 100 MHz. The high frequency modulation fm can be swept using the DDS toscan the probe component over the cavity resonance, while keeping the probe signal
at a constant 100 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.1

Output photon flux transfer function for different ground state repumping rates,with r = ∞, δ 0 = 0, α = 0 and ΓR = 0.

6.2

. . . . . . . . . . . . . . . . . . . . . . . . . 125

Output photon flux transfer function for different repumping ratios r, with W =Wopt , δ 0 = 0, α = 0 and ΓR = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xxxviii
6.3

Output photon flux transfer function for different dressed cavity detuning from emitN Cγ
ted light frequency δ, with W = Wopt (δ) = 2(1+δ02 ) , r = ∞, and ΓR = 0. The solid(dashed) lines show α = N Cγ × 10−3 (α = 2N Cγ × 10−3 ) to demonstrate the effects
of increased cavity feedback.
6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Stability plot using γ0 stability condition of Eqn. 6.9 as a function of detuning δ 0
and the cavity shift parameter α, assuming N = 106 . The stability condition also
assumes W = Wopt (δ 0 , ΓR ). The region of stability is exact for the two-level model(r = ∞), and a good approximation for all values of r. The blue region shows where
the real part of all the poles of the ⊥ solution are negative, indicating a damped
return to steady-state conditions for a perturbation. The red region shows where
any of the real parts of the poles become positive, making J⊥ unstable, with nosteady-state solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.5

Output photon flux transfer function for different Rayleigh scattering rates ΓR , with
W = Wopt (ΓR ), r = ∞, α = 0 and δ 0 = 0. The dashed line shows the transfer
functions when W is held to N Cγ/2, not varied to remain at Wopt , and ΓR = 0.3.
A dashed red curve is not shown, as with ΓR = 0.6 and W = N Cγ/2 the maximum
repumping rate threshold has been exceeded and the output photon flux is zero. . . 130

6.6

Response of the 2D Bloch vector to external modulation of the repumping rate. The
steady-state Bloch vector, i.e. J¯⊥ and J¯z from Eqns. 4.18-4.19, is indicated by theblue line, plotted on the axis with units of N , so N/2 is the maximum value. The
ellipse is the trajectory of the Bloch vector responding to the modulation of therepumping rate w(t) = Re[eiωt ], described by the small signal responses ⊥ and z
in Eqns. 6.4 and 6.5. The parameters are  = 0.1, r = 5, δ 0 = 0, α = 0, and ΓR = 0.
The black arrow indicates the direction of the trajectory, starting from the blue dot
at t = 0. The values of ω are chosen to show ω  ωres , ω ≈ ωres , and ω  ωres . . . . 132

xxxix
6.7

Parametric plots of response of the three degrees of freedom Jz , J⊥ , and A, highlighting effect of cavity frequency tuning on response of atomic coherence and output
light field. The blue line represents the steady-state atomic Bloch vector, J¯z and J¯⊥ ,from Eqns. 4.18-4.19. The blue ellipse shows small single response of the Blochvector to a modulation of the repumping rate W , given by z and ⊥ from Eqns. 6.4
and 6.5. The parametric response is plotted with units of N , so N/2 is the maximum
value. The red dashed line is the trajectory formed by the response of the cavity
field a (Eqn. 6.8) and z . The cavity field is plotted as a fraction of the average
field, then centered on the steady-state Bloch vector to compare with the atomic
response. The arrows indicate the direction of the response with respect to a modulation W (t) = W(1 + eiωt ). Here N Cγ = 104 s−1 , r = 5, ΓR = 0, W = Wopt (δ 0 ),
 = 0.1, and ω = 0.02 N Cγ, chosen to show the stable J⊥ response. (a) When δ 0 < 0,
the cavity feedback can be positive, leading to larger oscillations compared to thecase of no feedback δ 0 = 0 shown in (b). Because of the coupling of Jz to the cavity
mode frequency, A is not locked to the J⊥ response, as in (b), but is anti-correlatedwith Jz . In (c), where δ 0 > 0, the negative feedback reduces the response amplitudes
in all quadratures. The cavity tuning again shifts the cavity amplitude response,
but with the opposite phase relationship due the change in sign of the slope of theLorentzian, so A follows Jz . The inset shows a close up of the response. . . . . . . . 133

6.8

Cavity tuning stabilizing the cavity field amplitude. By changing the average repumping rate W from Wopt to 0.44 × 104 for the same parameters as Fig. 6.7c
(N Cγ = 104 s−1 , r = 5, ΓR = 0,  = 0.1, δ 0 = 1, and ω = 0.02 N Cγ), the response of
the Bloch vector (blue ellipse) becomes primarily perpendicular to the steady state
Bloch vector (blue line). Under these conditions, the cavity field response A (reddashed ellipse) has the smallest fractional deviation of the three degrees of freedom.
The inset shows a close up of the response. . . . . . . . . . . . . . . . . . . . . . . . 134

xl
6.9

Response for different repumping rates. (a) Light field amplitude response transfer
function TA versus repumping rate W . (b) Resonant response TA (ωmax ) and (c)the resonant modulation frequency ωmax as a function of repumping rate W . HereN Cγ = 4 × 105 s−1 , δ 0 = α = 0, and r = 0.71. . . . . . . . . . . . . . . . . . . . . . . 137

6.10 Response for different repumping ratio. (a) Light field amplitude response transfer
function TA versus repumping ratio r. (b) Resonant response TA (ωmax ) and (c)the resonant modulation frequency ωmax as a function of repumping ratio r. HereN Cγ = 4 × 105 s−1 , δ 0 = 0 and W = Wopt . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.11 Response for different detunings. (a) Light field amplitude response transfer function
TA versus detuning δ 0 . The transfer function is not plotted in regions of instability.
(b) Resonant response TA (ωmax ) and (c) the resonant modulation frequency ωmax
as a function of δ 0 . The red shaded regions indicate parameters in which the system
is unstable and no steady-state solutions exist. Here N Cγ = 4 × 105 s−1 , r = 0.71
and W = Wopt . The cavity shift parameter is given by α
~ + . These parameter values
reflect the conditions of the experimental system in Ref. [21]. . . . . . . . . . . . . . 139
6.12 Stability diagram for the full model of a superradiant Raman laser in 87 Rb, plotting
the region any of the real parts of the poles of the ⊥ solution are positive. Whenany pole becomes positive, the system is unstable and has no steady-state solutions.
The stability regions are shown versus the detuning of the cavity from the emission
frequency δ 0 and the detuning of the bare cavity frequency from the atomic frequency∆. The critical contour (bold) marks where the pole changes sign. The dashed line
indicates the detuning ∆ of experimental work (Refs. [24, 21]). For the calculation
we use N Cγ = 10−4 s−1 , W = Wopt , and r = 0.71. . . . . . . . . . . . . . . . . . . . 140

xli
6.13 Critical stability conditions for variable values of the repumping ratio r. Each line
shows the contour as a function of ∆ and δ 0 separating stable lasing from unstable.
The unstable region is defined as any set of parameters that results in a positive value
for the real part of any pole of the J⊥ response solution. The stability conditions
change as a function of the repumping ratio r. (a) As r increases from 0, the unstableregion gets smaller until it reaches some value between 0.4 and 0.5, after which (b)the unstable region grows to its asymptotic value. . . . . . . . . . . . . . . . . . . . . 142
6.14 The value of ∆ for the critical contour versus r, assuming δ 0 = −1. Lower ∆ indicatesthat more of the parameter space has stable, steady-state solutions. Here we assume
N Cγ = 10−4 s−1 and W = Wopt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.1(a), (b) Physical setup and energy level diagram. The trapping light (orange) and
Raman dressing laser (red, power ∝ Ω2d ) are injected along the cavity axis. The
repumping light (purple, green) is applied perpendicular to the cavity axis. The
emitted optical laser light (blue) is nearly resonant with the cavity mode (dashedlines) detuned from ωcav by δ. The repumping is accomplished through a pair of
two-photon transitions through intermediate optically excited states |IIi and |IIIiwith incoherent decay rates Γ. We individually control the two-photon rates W andΓ3 with the repumping laser powers ∝ Ω21,2 . |3i represents other metastable groundstates besides the laser levels. (c) Example emitted laser photon flux |A(t)|2 versus
time showing spiking and relaxation oscillations at turn-on. . . . . . . . . . . . . . . 146

xlii
7.2

(a) Parametric plots of the Bloch vector components Jz (t) and J⊥ (t) over a single
cycle of modulation of the repumping rate W for modulation frequencies below, near,and above resonance or ω/2π = 0.11, 2.2, 8 kHz, from left to right. The black points
are the measured small-signal deviations about the measured steady-state Blochvector (red arrow). The blue curve is the predicted deviation from steady-state given
the experimental parameters N = 1.3 × 106 , r = 0.71, δ̄ = 1, W̄ = 0.35 N Cγ, andN Cγ = 125 × 103 s−1 . The modulation depth  for data below and above resonancewas doubled to make the response more visible. (b) The amplitude (upper) and phase(lower) response transfer functions TA (ω), Tφ (ω) of the light field for three values of
the repumping rate W . The points are measured data, and the lines are zero freeparameter predictions of the response. (Inset) ωres versus W (points) and a fit to ω0(line) showing the expected frequency dependence of the relaxation oscillations on
repumping rate.

7.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Effects of finite ratio of repumping rate r (a) Comparison at two values of r ofTA (ω)versus modulation frequency. The points are measured data in good agreementwith the zero-free parameter fit (lines). (b) Plot of the resonance frequency ωres(black) and the peak value TA (ωres ) (red) versus r.

7.4. . . . . . . . . . . . . . . . . . 151

Evidence of negative and positive cavity feedback. (a) Amplitude transfer functionsof the emitted electric field for three detunings from cavity resonance. The data
points are the average of 4 experimental trials. The lines are fitted transfer functions
with N as a free parameter. (b) Cavity damping of collective atomic degrees of
freedom. The response going from δ̄ = 0.2 (left) where the γ0 is small to δ̄ = 0.9(right) where the system is expected to be critically damped. The red lines are
sinusoidal fits to the data (circles). The dashed lines highlight the damping in bothJ⊥ and Jz .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

xliii
7.5(a) The energy level diagram for the D1 Raman transition used for lasing. The linearly polarized Raman dressing laser is shown in red and the superradiantly emitted
light in blue. The cavity mode resonance frequency ωcav is denoted with a blue
dashed line. With the quantization axis defined by a 2.7 G magnetic field oriented
along the cavity axis, the linearly polarized light is a linear combination of σ+ andσ− polarizations. (b) The energy level diagram for the D2 repumping beams F2(green) and F1 (purple). The dark state with respect to the repumping lasers islabeled with a gray circle and corresponds to |↑i. . . . . . . . . . . . . . . . . . . . . 154

7.6

Measurement setup and sequence timing diagram. (a) The physical setup for measuring the light field amplitude response A(t) and the atomic responses Jz (t), J⊥ (t)to a small modulation of the repumping rate W (t). Optical beams are colored witharrows; RF and microwave signals are in black. (b) Timing diagram for measurements. Superradiance is started at t1 , and the repumping rate W (t) is modulated(with r constant) starting at time t2 . At t3 , the repumper and dressing lasers areshut off, freezing the atomic populations. The microwave π-pulse corresponds to a
12 µs pulse of 6.834 GHz microwaves resonant only with the ground hyperfine statetransition |↑i to |↓i. The pulse completely swaps the populations Ne and Ng of states|↑i and |↓i. The cavity frequency probe windows M1 and M2 used to determine thedressed cavity frequency ωcav1 and ωcav2 , and from which we can determine Jz (t3 ). . 157

7.7

A schematic of the real time cavity probe RF setup. The two ZASWA-2-50DR+
switches provide the modulation of the signals, controlled by the two SRS frequency
generators locked to the atomic clock reference. The signal generated by the lighttransmitting the cavity is then demodulated to produce the signals used to determine
the dressed cavity detuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

xliv
7.8

Data showing the correlation of amplitude stability of the laser with the cavity
detuning from the laser emission frequency. The data in particular highlights thetransition from stable operation on when the cavity is positively detuned, but thetransition in instability for negative detuning. Here the IQ amplitude of emitted
photons is proportional to the emitted light amplitude, fc is the dressed cavity, andfγ is the emitted light frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.1(a) Superradiant emission (blue) mapping the collective atomic coherence, represented by a Bloch vector J~ with phase φ and equatorial projection J⊥ , onto thephasor representing the emitted light field defined by a phase ψ and amplitude A.
(b) The experimental setup. 87 Rb atoms are trapped in a 1D optical lattice (dashed,orange) within an optical cavity. Optical repumping light is applied perpendicular
to the cavity axis. The emitted light is detected in heterodyne, then demodulatedusing a direct digital synthesis frequency reference to obtain both field quadratures,I(t) and Q(t), and calculate the light’s amplitude A and phase ψ. (c) An energy
level diagram of a superradiant Raman laser. The Raman dressing laser (red) detuned from an intermediate state |ii induces optical decay at rate γ between ground
hyperfine states |↑i and |↓i. The atoms are incoherently repumped back to |↑i at
rate w. When the dressing and repumping lasers are off, the atoms remain in asuperposition of |↑i and |↓i, and the quantum phase φ(t) that evolves at fhf . (d)Example data from a Ramsey-like sequence in a superradiant laser. Superradiantemission continuously maps φ(t) onto ψ(t). We measure a differential light phase
ψ(T ) − ψ(0) over a free evolution time T . . . . . . . . . . . . . . . . . . . . . . . . . 165

xlv
8.2

(a) Example time trace of the emitted light amplitude with an evolution time T = 0.4
ms. A(0− ) and A(T + ) are average amplitudes measured in the blue time windows
before and after the evolution time. The characteristic rise time t80 is the time when
the amplitude has returned to 80% of the steady-state value. (b, left) The decay ofJ⊥ as estimated by (blue squares) the ratio A(T + )/A(0− ) and (blue line) a fit to lossof contrast c(t) of traditional Ramsey microwave spectroscopy. (b, right) Measured(red circles) and predicted (line) recovery time t80 . The solid curve is the prediction
accounting for the low-pass filter that was applied to the data. The band around thedata indicates 1 s. d. on each side of the data point. We attribute the fluctuations
observed at short times to finite measurement precision. Each point is the averageof 20 trials. (c) Standard deviation of the phase accumulated in time T as estimatedby (diamonds) superradiant mapping and (circle) population mapping. The lines
are predictions based on the observed loss of contrast c(T ). . . . . . . . . . . . . . . 169

8.3

Demonstration of non-demolition superradiant measurements enabling a hybrid active/passive oscillator for two different duty cycles shown in (a) and (b). The measured light phase ψ(t) is shown when the superradiance is switched on (black). The
periods of evolution time, when the superradiance is switched off, are gray regions(random phase data not shown). The blue and red exponential curves in (a) correspond to the ideal optimal estimate weight functions before and after an evolution
period respectively, calculated with our experimental parameters of N , Γc , and q.
(b) The emission amplitude (red) returns to the steady-state value (dashed line)
during active oscillation, reflecting restoration of the coherence J⊥ .

. . . . . . . . . 171

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9.1

Frequency spectrums (horizontal axis) of the superradiantly emitted light, shown as
a function of an applied magnetic field (vertical axis) that tunes the lasing transition
frequency. Warmer colors indicate more power at a particular frequency. With
the atomic ensembles pumping to magnetic field sensitive hyperfine states, as in
the secondary configuration discussed in Chapter 5, Sec. 5.6, the laser emission
frequency can be controlled with the magnetic field. (A) In the case where all atoms
are optically pumped to the same state F = 2, mF = +2 (see Fig 9.2a), the laser
emission frequency follows nearly linearly with the change in the transition frequency.
(B) When two ensembles are pumped to the opposite Zeeman states F = 2, mF = +2
and mF = −2, we can observe two distinct superradiant emission frequencies, eachassociated with one ensemble. As the magnetic field is tuned to bring the laser
emission frequencies closer to degeneracy, the ensembles begin to synchronize andeventually lock up, even though their transition frequencies may not exactly the
same.

9.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Results from a demonstration of a superradiant magnetometer [160]. (A) By operating the superradiant Raman laser in the secondary configuration described in in
Chapter 5, Sec. 5.6, the laser emission frequency is sensitive to a magnetic field.
(B) We can continuously readout the effects of the oscillating magnetic field appliedin the lab during superradiant emission by observing the phase of the emitted light(red). However, for high frequency modulations, a large phase deviation has no timeto accumulate, reducing the signal and thus the sensitivity of the magnetometer.
By turning off the superradiant emission (blue region), and applying microwave πpulses that coincide with the know frequency of magnetic field oscillations, the phaseaccumulated by the atoms can build up. Then at a later time, the superradiancecan be restored and magnetic field can be inferred from the difference in the lightphase before and after the turn off. The ‘lock-in’ detection scheme can theoretically
recover the DC sensitivity of the magnetometer.

. . . . . . . . . . . . . . . . . . . . 183

xlvii
D.1 A schematic of the transimpedance amplifier circuit for the Hamamatsu S5973 photodiode used as the Heterodyne detector in this thesis. Crucial notes about thecircuit are noted in text on the schematic. The most important features for a photodiode with a flat gain response up to 200 MHz was robust connections to the groundplane, even adding additional ground connections near the AD8015 transimpedance
amplifier. The noise properties of the circuit were also improved by making the capacitance up to the input of the AD8015 low with short, in-air connections from theS5973 output pin to the input of the AD8015. The circuit layout was designed by
Terry Brown in the JILA Electronics Shop. . . . . . . . . . . . . . . . . . . . . . . . 211

Chapter 1

Introduction

1.1

Classical and quantum correlations enhancing precision measurements
In this dissertation, I describe fundamental investigations of collective effects in atomic en-

sembles, contributing to the fields of quantum control and precise quantum measurements. My
studies took the form of two different projects that both utilized an ensemble of laser cooled andtrapped 87 Rb atoms collectively interacting with a single mode of an optical cavity.
The first project focuses on non-destructive, coherence-preserving measurements of the collective pseudo-spin of the 87 Rb atoms. We use this measurement to generate entangled states, called
spin squeezed states, with modified quantum noise properties. In spin squeezed states, the quantum
noise of one atom can partially cancel quantum noise in another, due to the quantum correlations
between atoms, making the collective entangled state useful for improving quantum sensors. The
primary advance in my work is the implementation of a measurement scheme that nearly eliminates
the dominant source of measurement back-action that limited our previous attempts to create spinsqueezed states. With this advance, we achieve a direct observation of ensembles with phase noise
10.5 times lower in variance than the limit set by the uncorrelated quantum noise for an equivalentensemble of unentangled atoms. The enhancement in phase sensitivity is nearly an order of magnitude greater than our previous proof-of-principle experiments [35] and to our knowledge represents
the largest entanglement-enhancement in matter systems observed to date.
In the second project, I studied collectively-enhanced emission, known as superradiance, inthe ensemble of 87 Rb atoms. Ultimately, my work has led to realizing a Raman laser operating

2
in an unusual regime of laser physics where the coherence of the gain medium, in this case theatoms, is several orders of magnitude better than the coherence of the cavity field. This Raman
laser serves as a good model system for exploring the physics of proposed ultranarrow linewidthlasers [114, 33], such as their insensitivity to cavity vibration, quantum-limited linewidth, andamplitude stability. Using this model system as a framework, we also investigated the link between
the quantum-limited linewidth of a laser (the well known Schawlow-Townes linewidth [136]) and thequantum limits on the precision of the phase of the field radiated by an ensemble of atoms. With
the potential to posses a sub-millihertz quantum-limited linewidth and many orders of magnitude
reduced sensitivity to cavity vibrations, superradiant lasers may be an important next-generation
optical frequency reference.
These two projects share the common connection of applying collective phenomena to thefield of precision measurement. The collective phenomena arise from the effective interactions
between the atoms mediated by a cavity mode. Such interactions between quantum systems can beseen as another degree of freedom in the system that, if properly controlled and manipulated, canresult in new tools for science and technology that fundamentally cannot be realized with individual
quantum systems.
Historically, research to improve precise quantum measurements has focused on controllingand manipulating quantum objects to purify their internal states, control their motional states,
and remove perturbations associated with the external environment and also interactions between
individual quantum objects. For example, optical pumping and NMR techniques were developed toallow precise initialization and control of the internal quantum states of atoms. Later, laser coolingand trapping techniques were developed to gain control over the external atomic motional degrees
of freedom with the goal of decreasing doppler broadening and enhancing measurement coherence
times. To mitigate interactions between atoms, precise quantum measurements have been donewith single ions or atoms, or using dilute gases where collisions are sufficiently reduced to avoid
systematic errors.
Eliminating interatomic interactions, along with mastery over the internal and external de-3grees of freedom of these quantum objects has resulted in some of the most precise and accurate
measurements ever made [132, 74, 19]. These precision quantum measurements have led to a number of advances in science, including tests of the standard model [83] and searches for new physics[16, 8, 128, 75]. The impacts of precision measurement tools based on quantum sensors also impacts
everyday technologies such as GPS, cellular telephones, biological sensors, and the internet [89, 44].
At the heart of many of these advances is precision time or frequency standards.
Quantum systems serve as excellent time standards because their quantum nature guarantees
that all atoms or ions are identical, the transitions can be precisely measured, and systems can bechosen that are insensitive to external perturbations. Laser-cooled atoms can realize the current
definition of the second [165], and along with the invention of the femto-second laser frequency
comb [153, 40], new trapped atomic optical frequency standards (so called optical lattice clocks)
can have fractional frequency uncertainties approaching 1 part in 1018 [74, 19]. In the future, opticallattice clocks may be able to use the sensitivity of time to gravity as predicted by the theory ofgeneral relativity to serve as gravimeters for advancements in geodesy [38].
The optical lattice clock serves as a pinnacle example of the paradigm of controlling boththe internal and external degrees of freedom of single atoms (Fig. 1.1). Interactions between theatoms are kept to a minimum by working at low densities or possibly in the future by utilizing 3D
optical lattices with at most one atom per lattice site.
However, as optical atomic clocks continue to mature, they are approaching fundamental
limitations, which serve as a good examples of the fundamental limitations for all atomic sensors.
One such limitation is the quantum noise that arises from the projection of the atomic wavefunction
from a superposition into a discrete atomic state. Another limitation comes in the form of noise in
the frequency of the laser light used to probe the atoms in the optical lattice trap. Looking forward,
one possible way to address these limitations is to leverage the additional degree of freedom provided
by inter-atomic interactions.
It has long been recognized that particular interactions in the ensemble can give rise tointeresting and useful collective states with enhanced measurement properties. Interactions between

4

Figure 1.1: (A) The laser trapped and cooled sample of neutral atoms in an optical lattice clock
serves as an example of the variety of single particle control techniques used extensively in precision
measurements with cold atoms. The optical lattice trap nearly eliminates all motional perturbations
from the system, controlling the atoms’ external states and optical pumping techniques control theinternal electronic states of the atoms. Lattice clocks can also operate with fermions in dilute samples that can nearly remove collisional interactions between atoms. (B) By adding a nearly-resonant
optical cavity mode, we can introduce controlled atom-atom interactions and take advantage of thisadditional degree of freedom in the ensemble. The collective state of the ensemble is more than justmany independent realizations of an experiment, possessing, for example, modified quantum noiseproperties that produce enhanced sensitivity for precision measurements with the same resources(in this case, the number of atoms).

5

Figure 1.2: Examples of common interactions used for quantum simulation and generation ofinteresting collective states. A) Atomic collisions have a short range of interaction ∝ r−6 , andalthough they can be tuned with techniques such as magnetic Feshbach resonances, can be difficult
to quickly and completely extinguish. B) Dipolar forces are slightly longer range ∝ r−3 , but also
anisotropic. C) Coulomb interactions can be relatively long range ∝ r−1 , but systems that takeadvantage of Coulomb coupling, like ion traps, have proved difficult to scale to large atom numbers.

the quantum objects, such as collisions, are often treated as a perturbation to be eliminated froma precision measurement system, because of the systematic errors they can introduce. Thus a key
advance in the field of quantum information is to extend the precise manipulation and control overthe quantum system to include inter-particle interactions in order to both create and apply theseuseful collective states while controlling or avoiding systematic errors.
Controlled interactions open up a wide range of interesting quantum many-body phenomena
for study and for applications that cannot be realized in purely classical systems. As an ensembleof quantum objects is allowed to interact, correlations build up throughout the system so that theensemble can no longer be described just as many copies of a single quantum object, but must
instead be described by a new collective state. In addition to applications in precise quantum
measurements, controlled interactions are at the heart of the fields of quantum simulation [18, 15],
quantum computation [100], quantum memories [80], quantum teleportation [5, 129], and quantum
cryptography [59]. Of primary interest in this thesis will be collective states with enhanced emission
properties (superradiance) and reduced quantum noise (spin squeezed states).
Useful inter-particle interactions can take many forms, but for the precision measurement
applications envisioned here, the interactions should be both strong and long range. Strong interactions allow the useful correlations to develop quickly compared to any decoherence effects that6
often tend to destroy the collective state. Long range interactions allow the correlations to build up
homogeneously throughout the ensemble. In addition, the interaction should have rapid dynamic
control. This allows one to use interactions to create useful collective states, but then quickly
extinguish the interactions to allow the collective state to evolve during a sensing period free fromsystematic errors produced by interactions.
The atoms collectively interacting with a cavity mode can meet these criteria for generating
useful collective states. The cavity provides a common communication bus between the atoms,
creating infinite range, effective interactions between the atoms (Fig. 1.1). In principle, atoms
can interact through the a cavity mode, even when separated by many millimeters or centimeters,a much longer range than the typical ranges of interactions for s-wave collisions or dipole-dipole
interactions. The optical cavity can greatly enhance the interaction of atoms with the light field byconfining the light to a small volume. The strength of the interactions can be varied in real time as
simply as turning off the optical field providing the coupling, with the time scale limited only by
the cavity ring down time, typically < 1 microsecond. The tunability and range of interactions offer
distinct advantages compared with other types of interactions that are intrinsically much shorterrange in nature and/or more difficult to control, such as direct collisions [49, 130, 62, 28, 108, 69],
magnetic and electric dipolar interactions [148, 168], and Coulomb interactions between ions [164,117] illustrated in Fig. 1.2.
As another point of comparison, the top-down nature of the collective states realized in cavity
QED systems means that interesting collective phenomena can emerge in large ensembles composedof hundreds to millions of neutral atoms. Collective cavity QED technicques have also recently been
applied to ensembles of ions [71]. The emergent nature of the correlations that develop in the cavity
QED systems mean that the ensemble size is easier to scale to a large number of quantum objects
than the deliberate correlations that are built in through pair-wise quantum logic gate operations
in ensembles of trapped ions, for example. In both the spin squeezing and superradiance studies,
we are going to leverage the ability to use large ensembles to enhance the collective interaction withthe cavity mode required to create the correlations.

7

Figure 1.3: Atom-cavity coupling√
is a competition between coherent exchange of excitations between
the atoms and cavity at a rate N 2g, and loss of excitations through decay through free spacescattering Γ and spontaneous decay through both cavity mirrors κ. The enhanced atom-cavity
coupling rate arises from a constructive interference for emission and absorption with respect tothe cavity mode due to correlations among the individual atoms, the same physics underlying thephenomena of Dicke superradiance [43].

Collective coupling means that the ensemble of atoms interacts much more strongly with thecavity than a single atom. One atom in the cavity absorbs and reemits a photon into the cavity
at a characteristic frequency 2g, given by the atomic dipole moment and the electric field of aphoton in the cavity mode. The strength of the coupling relative to the coupling of the atom to2

all other non-cavity modes is characterized by the single particle cooperativity C ≡ 4g
κΓ , where κis the spontaneous decay rate of the cavity field and Γ is the spontaneous decay rate of the atomic
excited state. The relevant rates are illustrated in Fig. 1.3. The single particle cooperativity can beinterpreted as a ratio of the atom-cavity coupling rate to the rates of losing a photon through thecavity mirrors or through spontaneous free-space scattering, after which the photon never returns
to interact with the atoms. Another useful interpretation of C is as the fraction of the time thata single atom in the excited state decays by emitting a photon into the cavity mode. Our system
operates in the weak single particle coupling limit with C ∼ 0.5 × 10−3 .
However, if N atoms are in the cavity, then the atoms can collectively absorb a cavity photon√
g2
and reemit it at frequency N 2g. The system then has a collective cooperativity N C = 4NκΓ which8can be much greater than 1, meaning the coherent exchange of excitations between the atoms andcavity field can dominate over the decay processes. The collective cooperativity N C, or the effective
optical depth, also gives the fraction of time that excitations from the ensemble are emitted into
the cavity mode.
As the ensemble of atoms exchanges photons with an optical mode collectively, strong correlations can build up between atoms. The correlations arise from a fundamental ambiguity of
the atoms interacting with the optical mode, as there is no way to assign photon emission andabsorption to a single atom. These correlations can be simply classical, or quantum, indicating
entanglement.
Collective emission and correlations play the central role in both of the experiments describedin this thesis. The enhanced emission into the cavity mode in a spin squeezing experiment allows
for more collective information to be extracted for the same amount of free space scattering thatcauses decoherence. Non-classical correlations generated from such a measurement reduce thequantum noise of the resulting entangled state, as correlated fluctuations can cancel. In the caseof the superradiant laser, the classical correlations between atoms spontaneously synchronize theiremission, providing the collective enhancement in the ensemble’s scattering rate necessary to extract
a useful amount of output power. In this thesis, we will study these two collective effects indetail, particularly how they can advance precision measurements. As we will see, the correlations
between atoms introduced by their interactions in the atom-cavity system may enable precision
measurements that go beyond the limits to the single particle precision measurement paradigm
that has already been so influential and fruitful. The rest of this introduction will introduce these
two topics in detail and give a basic outline of the content of the thesis.

1.2

Coherence Preserving Measurements to Create Spin Squeezed States
The first body of work I will describe is the generation of conditional spin squeezed states inensembles of laser cooled 87 Rb atoms confined in an optical cavity. The squeezed state generation
depends on coherence-preserving, collective measurements of atomic populations enhanced by the9optical cavity. The key experimental advance described in my work is the reduction of measurement
induced back-action on the population states of the atoms. We reduced measurement back-action
by probing the atom-cavity system near an optical cycling transition, suppressing probe-induced
Raman scattering between internal atomic states by several orders of magnitude. The processof performing a collective measurement on the ensemble can be thought of as building quantum
correlations, or entanglement, between the atoms through their effective interaction with the opticalcavity. The measurement is used to record the value of the correlation of quantum noise in a given
trial. The information gained can then be used to essentially subtract out the quantum noise in
subsequent measurements within the same trial.

1.2.1

Quantum Projection Noise

Quantum noise is the price we pay for the benefits associated with metrology using quantum
objects. While quantum mechanics guarantees that each quantum object, like a 87 Rb atom, will
have the same, well-defined transition frequencies, we must cope with an unavoidable source ofnoise that provides the fundamental limitation to many state of the art precision measurements.
One way to understand the source of quantum noise in atomic sensors is by considering theprobabilistic nature of the measurement process. Atomic sensors encode their information in aquantum phase φ(t), a relative phase that accumulates in the wavefunction of a two level system
placed in the superposition of the two quantum states |Ψi = cos((π/2 − θ)/2) |↑i + eiφ sin((π/2 −θ)/2) |↓i. The state of such a two level system can be directly mapped onto a Bloch-vector[51] oflength |J| = J = 1/2, whose tip lies on a sphere and whose orientation is given by the azimuthal
angle φ and a polar angle θ, here measured from the equator as shown in Fig. 1.4. The value of φis most often estimated by measuring the population of atoms in different quantum states after φ
is mapped onto θ at the end of a Ramsey sequence (see Fig. 1.4). The angle θ can then be simply
estimated by measuring the probability that the atom is in |↑i P↑ or |↓i P↓ , i.e. the projection of
the Bloch vector onto the z-axis Jz = (P↑ − P↓ )/2, as in the small angle limit θ ≈ Jz /J.
The quantum phase φ evolves continuously while an atom or qubit is in a superposition of

10

Prepare	  Evolve	  Readout	  θ (t) =π
− φ (t)
2

φ (t)
ẑ
x̂

ŷ

Figure 1.4: Illustration of a Ramsey sequence with a two level system represented by a Bloch
sphere. The first step is optical pumping the atoms all into |↓i (light red arrow), then performing
a π/2 rotation about −ŷ axis to place the Bloch vector along x̂. The Bloch vector then precesses
around the ẑ axis at a rate proportional to the energy difference between |↑i and |↓i. The Blochvector precession represent the evolution of the quantum phase in the wavefunction |Ψi = cos((π/2−θ)/2) |↑i+eiφ(t) sin((π/2−θ)/2) that we ultimately wish to measure. The second π/2 pulse performs
another rotation about −ŷ and maps the φ(t) onto the coefficients of the population states, or
equivalently onto the polar angle of the Bloch sphere. The quantum phase φ(t) can then be
inferred by measuring the state of the two level system. Quantum randomness in the projection
onto the population basis results in fundamental quantum noise limiting the resolution of the angle
φ(t).

11

ẑ
ŷ

x̂

Noise:	  Δ(J ⋅ ẑ) = ΔJ z = N / 2
Signal:	  	
  J = J = N /2

Figure 1.5: The standard quantum limit for a coherent spin state. Quantum noise appears as an
uncertainty in the pointing location of the classical Bloch vector J, represented by the blurry red
region at the tip of the vector. The signal, that is, the change in the z projection of J for a given
angular rotation, increases linearly with N . However,the standard deviation in a measurement of√
the z projection from quantum noise grows as N .

its two spin states, but the measurement process always finds an atom in one of the two population states |↑i or |↓i, or equivalently θ = ±π/2. Thus for an atom in an equal superpositionof spin states, the readout has fundamental fluctuations that require repeated measurements overidentical experiments in order to precisely estimate the probabilities P↑ and P↓ . This noise is often
called quantum projection noise[76], associated with the notion of the measurement projecting thesuperposition state onto one of the population states.
Using ensembles of N identical atoms is equivalent to running many independent experimentsin parallel in order to more precisely determine the probabilities P↑ and P↓ . We can conveniently
represent the measurement resolution of the collective state using a collective Bloch vector oflength |J| = N/2, as shown in Fig. 1.5. Averaging the outcomes of the N parallel experiments,
quantum projection noise limits the uncertainty in the estimate of θ (and therefore φ) to a variance(∆θ)2 ≥ (∆θSQL )2 = 1/N , a limit known as the standard quantum limit (SQL) for a coherent

10

-1

St
an

-2

da

-3

rd

da

10

BEC Mean Field Interactions
Heidelberg - Oberthaler
Basel - Treutlein
Vienna - Schmiedmayer
Georgia Tech - Chapman
Hannover - Klempt

Q
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nta

-4

lH

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Nonlinear Cavity Feedback
MIT - VuleticLim

Non-demolition Measurements
Bohr Institute - Polzik
MIT- Vuletic
ICFO - Mitchell
JILA- Thompson

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-5

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rg

-6

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Atom or Ion Number

10

5

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6

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2

B	
  
12
JILA/Thompson

11(Cavity QND)

Heisenberg	
  Lim
it	
  

2

10

n
Fu

Variance of Phase Estimation [radians ]10

Qubit Logic Gates
NIST Boulder - WinelandInnsbruck - Blatt

0

Spectroscopic Enhancement in Variance ( ΔθSQL/ Δθ )

A	
  

10
9
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GaTech/Chapman(Collisions)6
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Innsbruck/Blatt(Ion Logic/Cat States)

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MIT/VuleticBasel/Treutlein(Cavity One-Axis)(Collisions)

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Vienna/Schmiedmayer(Collisions)

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Atom or Ion Number

Figure 1.6: A summary of results for the observed entanglement-enhanced sensitivity for measuring
a quantum phase across a variety of physical systems and mechanisms. The solid points indicate
results that use coherence preserving pre-measurements, the most relevant results for comparison
to this work. The result from this work is highlighted with a red circle. (A) Absolute observed
phase resolution of an ensemble, emphasizing the improvement in the absolute sensitivity from
simply starting with a large ensemble of atoms. Though our result is far from the Heisenberg limit,
the large ensemble size allows improvement for an already highly resolved phase. (B) The observed
enhancement of phase resolution relative to the SQL. The blue line indicates the fundamental
Heisenberg limit. Compared to previous work, which observed interesting, but small entanglementenhanced sensitivities, we have observed over an order of magnitude enhancement in a system
readily applicable to precision measurement experiments such as optical lattice atomic clocks. To
date, our result is the largest enhancement in any system using matter as spins. The other results
can be found in Refs. [116, 102, 117] for experiments using ions, Refs. [49, 130, 62, 28, 108, 69,104] for experiments using one-axis twisting, and Refs. [4, 159, 141, 139, 35] for non-demolition
measurements.

spin state (CSS). Throughout the thesis the notation ∆X indicates the standard deviation of thequantity X over many independent trials. Thus, the primary way to mitigate quantum projection
noise is to use as many atoms, or photons in the case of optical interferometry, as possible.
However, scaling up the system size also has limitations. In the case of photons, this limitation
may not be immediately apparent as the photons don’t interact with one another. However, highpower effects like parametric instabilities [50] and radiation pressure back-action (recently observedin optomechanical systems [123]) can make photon shot noise a limitation in precision measurement

13experiments such as gravitational wave detectors LIGO [1] and GEO600 [2].
In atomic sensors, the limitations to system size are much more apparent. Collisions between
atoms can cause perturbations to atomic transitions [29], affecting the accuracy of the sensor. Other
times, the desire for high spatial resolution of a sensor can constrain the trap dimensions, limiting
the ensemble size.
Ultimately, for an optimal number of quantum objects, quantum projection noise sets thefundamental phase precision. Both atomic sensors and the optical interferometers used for gravitational wave detection are nearly or already limited by quantum noise, so entanglement-enhanced
metrology would improve some of the most precise measurements of external fields [93], rotations[66], and time [74], and will advance searches for new physics [120].

1.2.2

Squeezed states

By entanglement-enhanced metrology, we mean the use of non-classical correlations between
individual members of the ensemble to improve the fundamental precision. It has long been recognized that non-classical states can have modified quantum noise properties, which could enhance
the resolution of sensors and interferometers beyond the SQL [31, 164].
Squeezed states are non-classical states with modified quantum noise properties first investigated in the field of optics. It was theoretically shown and experimentally demonstrated thatentanglement in the optical field could reduce quantum noise in one quadrature of the field at theexpense of added noise in the orthogonal quadrature. For instance, the quantum noise can besqueezed out of the field’s phase quadrature for enhanced phase measurements in an interferometer. The quantum noise is squeezed into the amplitude quadrature, but this quadrature is not
relevant for making phase measurements. Represented on a phasor diagram, shown in Fig. 1.7,
the quantum uncertainty of the most classical state, a coherent state, is represented by a classicalprobability distribution that smears out the tip of the classical vector. Here the distribution isvisualized as a circle with equal quantum noise in both quadratures. The entangled state has anelliptical shape. The Heisenberg uncertainty relationships require that the total area defining the14
Coherent	  State	
  

X1	
  

Squeezed	
  State	
  

X1	
  

ΔACS

ΔASS

Δφ SS

ΔφCS

X2	
  

X2	
  

Figure 1.7: Squeezing in an optical field, or equivalently a harmonic oscillator, with the two conjugate variables on the X1 and X2 . Both the coherent and squeezed states can be represented
as a classical phasor with quantum uncertainty. The area of uncertainty is constant, but can be
distributed among the different quadratures. In this case, the squeezed state has a reduced phase
uncertainty ∆φSS at the expense of increased uncertainty in the amplitude ∆ASS .

size of the quantum noise must remain constant. So the uncertainty in one quadrature can be
pushed into the other, giving the squeezed state its name.
Squeezed states in optics were long considered a solution looking for a problem, as many times
the quantum noise of a coherent state could more easily be mitigated by using more optical power
than implementing squeezing protocols. Currently however, squeezed states are actively being used
to improve searches for gravitational waves using optical interferometers [1]. Another possible useof squeezed light is microscopy in biological systems where intense light can damage biologicalsystem [150]. As discussed previously, the situation is different for sensors that use ensembles ofatoms or ions.
Spin squeezed states, referring to the squeezing of the pseudo-spin formed by any two levelsystem, were first produced and shown to operate below the projection noise limit for two ions[116]. Spin squeezed states have a direct analogy to optical squeezing, except that the noise in one
spin projection, say Jz is reduced at the expense of increased noise in an orthogonal spin projection,
Jy for example. The impressive degree of coherence and state control allowed ion system to realize
other non-classical states (such as NOON or cat states) with altered noise properties and/or phase15
accumulation rates [102, 117] realizing the fundamental noise limit even for entangled states, known
2 = 1/N 2 (see Fig. 1.6B).
as the Heisenberg limit, where ∆θ2 ≥ ∆θH

Though entangled states of a few ions have been demonstrated at the fundamental Heisenberg
limit, their phase precision can be surpassed by classical states that simply use more atoms (Fig.
1.6). The atomic fountain clocks that realize SI second use of order 105 atoms per experimental trial,and optical lattice clocks, good candidates for future time standards, use of order 103 to 105 neutral
atoms. Sensors with many neutral atoms are promising because they possess an intrinsically small
quantum-limited noise floor. In addition, the potential enhancement in measurement precision froma squeezed state can be much larger in systems with large ensemble size because of the 1/N relative
scaling in phase variance between the SQL and the Heisenberg limit. As a result, techniques thatcan can generate a large degree of squeezing systems such as optical lattice clocks [74, 19], could
make a large impact on precision metrology.
One approach to generating spin squeezing in atomic systems is to non-destructively premeasure the quantum fluctuation in Jz before each Ramsey sequence, and then subtract the measurement outcome from the measurement outcome for Jz obtained at the end of the Ramsey
sequence. The quantum noise will be common to the two measurements, while any phase accumulated during the Ramsey sequence will not be common [99]. While the resulting state of theensemble is sometimes termed a conditional spin-squeezed state, the reduction in noise is essentially deterministic with no discarding of trials necessary. My work here uses this pre-measurement
approach. The essential idea is illustrated in Fig. 1.8.
The pre-measurement scheme stands in contrast with other techniques to generate squeezed
states where appropriate interactions between atoms generate correlations without the need fornoise subtraction. For comparison, these results are included in Fig. 1.6. In systems of ions,interactions mediated by the Coulomb force produce the entanglement [164, 117]. In neutral atoms,squeezed states have been generated using atomic collisions in ensembles of degenerate gas toproduce the desired interactions [49, 130, 62, 28, 108, 69]. Notably, it was also demonstrated that
effective atom-atom interactions mediated by cavity-based optical feedback could also produce

16squeezed states [104]. Note that there is an intellectual connection with the pre-measurement
technique presented in this thesis and the cavity feedback technique in that cavity feedback can beunderstood as the cavity measuring the Jz spin-projection and then creating a ẑ rotation of thespin proportional to the measured value. This rotation causes a shearing of the state uncertainty,
similar to the original one-axis twisting proposal put forth by Kitagawa and Ueda [87].
The pre-measurement approach relies on measurement back-action, the ability of a measurement to change the state of the system being measured. For example, a measurement of a system ina superposition of two states causes the system to project, or collapse, into one of the two discrete
states, resulting in quantum projection noise. Measurements performed on an ensemble, however,can project the ensemble into an entangled state when only collective quantities are measured. For
example, using two spins each in a superposition state, a joint measurement that only revealed
that one of the two spins was in spin-up, but not which one, is projected into the entangled state√|Ψi = (|↑1 ↓2 i + |↓1 ↑2 i) / 2. Any information about the spin-state of a single atom that leaks tothe environment due to imperfections in the collective measurement reduces entanglement as thecollapse of state of the individual atoms means the previous entangled state becomes either one of
the two unentangled states |↑1 ↓2 i or |↓1 ↑2 i. Such collective or joint measurements arise in a wide
range of applications, including quantum teleportation [119], quantum information protocols [91],studies of strongly-correlated quantum systems [48], Dicke superradiance [43], and entanglement
generation in optical [65], solid state [131] and atomic systems [39].
A collective pre-measurement of the spin state of an ensemble can precisely measure thequantum noise, while preserving the coherence between states that is required for sensing a quantum
phase [99]. The collective pre-measurements invoked here are intimately tied to the notion ofquantum state preparation and quantum non-demolition measurements. To make a non-destructive
measurement of the atomic state, often an optical beam is passed through the ensemble. The lightatom interactions cause the state of the field (for instance, it’s polarization state or quadratureamplitudes) to become entangled with the state of the atoms, such that a subsequent destructive
measurement of the light field can give the state of the ensemble. After the destructive measurement

17

Ini$al	  State	  Start	
  Ramsey	
  Squeezed	
  State	  Pre-­‐measurement	
  π/2	
  along	  	
  − x̂δθΔθ SQL

δθ

ẑ
x̂

ha
ve	
  P
Evol

ŷ

se	
  
Readout	
  π/2	
  along	
  	
   x̂

δθ
δφ
δφ

Figure 1.8: Generating a squeezed state with a coherence-preserving pre-measurement for a Ramsey
sequence. A coherent spin state has an initial uncertainty in the azimuthal angle of the Bloch vector∆θSQL set by quantum noise that defines the SQL for phase resolution of the state. By performinga measurement on the polar angle θ using a population readout with resolution below the quantum
noise limit, the deviation of the Bloch vector from the equator on that particular trial, δθ, can berecorded. Another π/2 pulse rotates the quadrature with a known quantum fluctuation into thethe azimuthal quadrature and the Ramsey sequence can begin. A noiseless vector going throughthe Ramsey sequence is shown as a dashed line. While the final output of the interferometer will
still exhibit quantum fluctuations ∆θSQL about the average output polar angle δφ, the fluctuations
can be subtracted on a trial by trial basis due to the knowledge of the initial state from thepre-measurement. Effectively the state has an uncertainty set by the measurement resolution,
determining the precision which one can measure the initial quantum fluctuation.

18of the light field, the atomic ensemble is projected into an eigenstate of the measurement. As
described above, a measurement of a collective quantity can then result in projection into an
entangled state.
While measurement back-action can work to our benefit in the case of preparing a state with
reduced quantum noise properties, this does not need to be the case. Fundamental measurement
back-action requires added noise in the orthogonal quadrature to the measurement. Fortunately, inspin systems, the orthogonal degree of freedom does not couple back into the measured quadrature,unlike in mechanical systems, where a position measurement adds uncertainty to the momentum,
which affects the position at a later time [26].
However, there can be additional sources of back-action in a spin system. One example isatoms individually scattering probe light out of the cavity mode, projecting them into a definite
population state and taking them out of the collective ensemble, leading to a shortening of the
Bloch vector J. This back-action makes the ensemble less sensitive to a quantum phase, as theobservable signal Jz is proportional to Bloch vector length in the small squeezing limit Jz = |J|θfor the same rotation θ. In addition, a scattering event can also cause an atom to change its spinstate, resulting in added noise in the quantity Jz being measured. The added noise in Jz from
population diffusion can reduce the correlation between the pre- and final measurements of Jz ,limiting the degree of noise reduction in the differential spin projection measurements. Ultimately,
the spectroscopic enhancement W −1 relative to the SQL that can be achieved, quantified here as
the ratio of the SQL to the observed angular variance W −1 = (∆θSQL /∆θ)2 , is a combination of
reduction in noise and the loss of signal caused by the pre-measurement.
The degree to which atoms appear indistinguishable to the measurement, i.e. only collective
information is obtained, sets the limit on these two forms of back-action. Thus one figure of meritfor the measurement is the degree of indistinguishability, quantified by the collective optical depth
N C. Free space ensembles with high optical depth provided by trapping and/or high densities havebeen shown to be able to generate conditional spin squeezed states [4, 159, 141].
An optical cavity enhances the optical depth by a factor of the cavity finesse F . Compared19
to free space measurements, significant enhancements on the optical depth can be achieved at lower
atomic densities, critical for sensors to avoid inaccuracies associated with collisions in high density
samples. The collective scattering of the atomic sample into the cavity mode means that the rate ofobtaining collective information is enhanced relative to the rate of single particle scattering. Thus
operation deep in the strong collective coupling regime, N C  1, is important.
First proof-of-principle experiments, both in free space [4, 159, 141] and in optical cavities[139, 35], have generated conditional spin-squeezed states by using collective pre-measurements,
with results shown in Fig. 1.6. However, measurement-induced back-action has limited direct
observations of spectroscopic enhancement to only W −1 ≤ 1.4 [159]. Note that larger statedimprovements appear in these papers, but most often these values include background subtractions
and do not represent the actual obtained spectroscopic enhancement. Such modest improvements
are not useful for precision measurements, as the added cost and complexity to a device are likely
not worth such small improvement. If entanglement-enhanced metrology is to continue to advance,
we must show it is possible to realize metrologically relevant degrees of spectroscopic enhancement
using entanglement, such as the order of magnitude improvement realized in this work. Importantly,
the results presented here can be mapped directly onto several atomic species used in optical latticeclocks, such as Sr and Yb.
In Chapter 2, I describe our atom-cavity system, giving technical details regarding the opticalcavity and trapping and cooling the atoms. This chapter serves as a foundation for both thesqueezing experiment and the superradiant laser experiment. Then in Chapter 3, I describe thegeneration of spin squeezed states of ∼ 5 × 105 atoms with precision enhanced by W −1 up to 10.5
beyond that of an entangled state using a method compatible with state of the art optical latticeclocks. I give the specific setup of the squeezing experiment, focusing on the improvements to theprevious implementation of measurement-induced spin squeezing that allowed us to improve thedirectly observed spectroscopic enhancement by nearly an order of magnitude beyond our previous
result.

20

1.3

Superradiant lasers
The second class of experiments covered in my thesis is the development of a superradiant

Raman laser using the laser-cooled 87 Rb cavity QED system. Our system operates very deep in thebad-cavity regime where the atomic linewidth γ⊥ is much less than the cavity linewidth κ, similar toan active hydrogen maser [60]. In this regime, the correlations between individual atomic dipolesthat result in the collectively enhanced emission, or superradiance, arise spontaneously throughemission into the cavity mode, building a collective coherence that can exceed the coherence of asingle particle alone. As a result of operating in this unusual regime of laser physics, we observea variety of interesting phenomena, including lasing and linewidth narrowing with on average lessthan 1 photon inside the laser cavity. Such a collectively-emitting ensemble of cold atoms is at
the heart of proposals for future millihertz linewidth lasers where the gain medium would consistof alkaline-earth atoms with narrow optical transitions[115, 33], and we confirm a number of key
predictions for superradiant lasing relevant for these proposed millihertz linewidth lasers.

1.3.1

New frontiers in ultrastable lasers

Frequency stable lasers are our primary rulers for space and time. In particular, narrow
linewidth lasers are crucial ingredients in optical clocks [45, 118, 74, 19] and their comparison[132, 118], timing signal distribution [54], and precision tests of fundamental physics [38]. Most
neutral atom optical clocks work by stabilizing the frequency of a probe laser to the nearly forbidden
optical transition in alkaline-earth atoms using high-resolution laser spectroscopy. In state of theart clocks, the frequency stability of the probe laser is currently the primary limitation to theclocks’ precision. The finite frequency stability of the laser both limits the spectroscopy interaction
time and introduces additional noise during the dead time in the measurement cycle while atoms
are being re-loaded in the clock. Consequently, enabling a next generation of ultra-stable, narrowlinewidth lasers would impact a wide range of technical and fundamental scientific applications,
including communications, navigation, geodesy [38], determining the fundamental constants of

21
nature [53] and searches for physics beyond the standard model [17].
Lasers stabilized to highly engineered Fabry-Perot cavities have achieved remarkable levelsof frequency stability. However, even in state-of-the-art stable lasers, the mirrors of the reference
cavity to which they are stabilized vibrate due to thermal noise, causing time-integrated phasedrifts that limit the laser linewidth to between 0.04 and 0.3 Hz [78, 84].
Superradiant lasers offer an alternative, complementary path forward. Recent proposals[115,33, 114, 113] suggest that superradiant laser oscillators utilizing very narrow optical transitions inatoms can be orders of magnitude more spectrally pure than current lasers. When κ  γ⊥ , theoptical atomic transition acts as the repository for phase coherence and the laser frequency andlinewidth is set by the atomic transition. In essence, the man-made artifact of the reference cavity
is replaced with a quantum reference. Deep in this bad-cavity regime, the laser can be primarily
viewed as a very coherent ensemble of atoms where the emission rate of the gain medium is enhanced
by superradiance [43, 63]. Therefore, I use the terms bad-cavity regime and superradiant regime
interchangeably.

1.3.2

Superradiance

Superradiance is a constructive interference effect where the correlations among the individual
atomic dipoles results in an enhancement by N of the excited state decay rate, and thus the total
power emitted by the ensemble. For an ensemble of 2-level atoms confined within an optical
wavelength, the indistinguishability of the atoms with respect to the emitted radiation results inthe ensemble emitting collectively. The ensemble can store many excitations, making it a sourceof multiple photons with a well-defined phase relationship as they reflect the coherence of thecorrelations stored in the ensemble.
Even in an extended sample, correlations between atoms (in this case, a spatially dependent
phase relationship between the optical dipoles of the atoms) can result in constructive interferencealong preferred directions. The correlations spontaneously form as excitations are emitted into
optical modes in which the sample has high optical depth. This can be viewed again as a condition

22

Figure 1.9: A cartoon representation of superradiance in an extended ensemble of atoms. For acollection of atoms radiating with independent phases (A), the light is isotropic and the total power
emitted goes as N P0 , where P0 is the power emitted by a single atom. For ensembles with spatiallycorrelated radiating phases (B), represented by the synchronization of the arrows on the atoms, theemitted light can constructively interfere along certain directions, resulting in an N 2 enhancementof the power radiated along this direction. The cooperativity parameter C serves as a measure of
the range of directions (fractional solid angle) over which the constructive interference occurs. The
total power radiated due to this constructive interference is then proportional to N 2 CP0 , i.e. NC
larger than for the total power from non-correlated radiation.

23
for indistinguishability of the emitted excitation giving rise to the correlations between atomic
dipoles. As a result, superradiance in extended samples have a directionality to the enhanced
emission (Fig. 1.9).
The addition of an optical cavity provides a very high effective optical depth mode for thecollective emission. Although the visual representation of correlations present in the atomic dipoles
becomes more complex, the addition of the cavity mode actually greatly simplifies the system as the
single cavity mode easily becomes preferred over all other spatial modes. Thus, in a superradiantlaser, the cavity serves primarily as a way to establish and maintain the synchronization between
the atoms.

1.3.3

Good Cavity, Bad Cavity: opposite extremes in laser physics

Previously, most optical lasers have operated in the good-cavity regime of laser physics defined by the condition κ/2  γ⊥ (see the left side Fig. 1.10). In other words, the cavity field
is more coherent than the gain medium. High rates for excited-state decay in optical transitionsin atoms, combined with transition broadening from repumping, lattice interactions in solid state
gain mediums, or Doppler effects in gases, have prevented lasers from operating with κ/2  γ⊥ .
In the good-cavity regime, the phase coherence is primarily stored in the cavity light field, and thepolarization of the gain medium adiabatically follows the state of the cavity field. As a result, thelaser frequency and linewidth is determined primarily by the cavity properties, with the familiar
quantum-limited linewidth given by the Schawlow-Townes limit [136] as ∆fGC ∝ Mκc where Mc isthe average number of intracavity photons. The Schawlow-Townes limit can be made quite small
in optical lasers, and the linewidth is instead limited by the laser cavity stability or the stability ofan external optical reference cavity.
Deep in the bad-cavity limit, the roles of the atoms and photons have been reversed insome ways. Here the cavity field rapidly decays, and the phase coherence resides primarily in theatom’s robust collective coherence. The cavity field adiabatically follows the polarization. In fact,
such lasers are predicted to operate with a very small intracavity field – just enough to maintain24
the synchronization between the individual atomic dipoles. This means that the laser frequency
properties are determined largely by the atoms (right side of Fig. 1.10). In this limit, the SchawlowTownes linewidth is set by atomic properties ∆fBC ∝ γN⊥ , analogous to the good-cavity case, butwith the number of atomic quanta N replacing the photonic quanta Mc and the cavity damping
rate κ replaced by the atomic decoherence rate γ⊥ .
Additionally, the laser frequency is mainly determined by the atomic transition frequency,with the sensitivity to cavity frequency changes being reduced by the ratio P = 2γ⊥ /κ. The laser
emission frequency is fl ≈ fa + P (fc − fa ), where fa is the atomic transition frequency and fc is thecavity resonance frequency. This insensitivity could be important for operating optical frequency
sources in high vibration applications such as communications, geodesy, space-based interferometry,
and GPS.
The correlation among dipoles in the gain medium exists in both good-cavity and bad-cavity
lasers, but the difference is largely how the correlations arise. In a good cavity laser the very strong,coherent cavity field locks the ‘forgetful’ atomic dipoles into a well defined phase relationship, forcing
them to emit in phase with the cavity field. However, the cavity field can be greatly impacted byvibrations of the cavity mirrors. Conversely, in the superradiant laser, the atoms are very coherent
and will largely stay in phase on their own, sustained by the spontaneous build up of correlations
as a result of the ensemble collectively emitting into the cavity mode. In the superradiant case, it is
the cavity mode that is ‘forgetful’, so the intracavity field is directly locked to the atomic coherence.
At microwave frequencies, masers typically operate in the bad-cavity regime with a long history of contributions to precision spectroscopy [60, 90, 127]. Because the system actively oscillates,
the linewidth is narrowed by the presence of many atoms oscillating and can be much narrower
than both the broadened linewidth and potentially the atomic decay linewidth. Because of thethermal photon occupation at microwave frequencies, cryogenic operation is required for optimal
stability [9]. In contrast, an optical system can potentially access quantum limits on performance,
can operate in a novel regime with less than single quanta on average in the cavity-field, and canovercome fundamental thermal limitations on current optical light sources.

25

Photonic-like

Polariton-like“Good”
Cavity

-4

10

-3

10

FIR gas lasers

Microcavity
Diode
Laser
-2

10

10

{

Cold Rb

Atomic-likeNH3 HeXe
HeNe
Xe

-1

1

1

10

Cold Rb[this work]
2

10

3

10

4

10

“Bad”
CavityFigure 1.10: (Left) In the good-cavity limit, the atomic coherence rapidly decays, and the photon
field is the primary reservoir of phase information in the laser. As a result, perturbations which
disturb the cavity resonance frequency, such as thermal mirror vibrations (drawn as curved lines
near the mirrors), limit the frequency stability of the laser. (Right) At the other extreme, the badcavity, or superradiant, laser presented here operates in a regime where the atomic coherence decayrate γ⊥ is much less than the cavity power decay rate κ. In this regime, the atomic gain medium
is the primary reservoir of phase memory in the laser, a fact represented by the aligned dipolesof the atoms and a cavity mode nearly devoid of light quanta. Because the emission frequency
is primarily determined by the atoms, perturbations from fluctuations in the cavity frequency aresuppressed. (Middle) Near the crossover regime, the phase coherence is jointly stored by the atoms
and the cavity photons, making it a polariton-like excitation. Most optical lasers operate in the
good-cavity limit (one example is the cold atom Raman laser of Ref. [156]), with microcavity
diode lasers [12] and far infrared (FIR) gas lasers, using Xe [30], NH3 [70, 162], and HeXe/HeNe[97], operating in the vicinity of the crossover, polariton-like regime. Our cold atom Raman laser
is unique both in terms of operating so deeply into the bad-cavity regime, and also in that thesteady-state intracavity photon number can be made much less than one.

26

87Rb	
  

Sr,	
  Yb,	
  Ca…	
  
3

i

S1

w
3

e

γ eg

P0

Δ
Raman	  	
  Dressing	  	
  Laser	
  

Ω2

w

γ eg
g

1

S0

e
g

Figure 1.11: The long lived, nearly forbidden 3 P0 to 1 S0 transition in elements like Sr, Yb or
Ca, with excited state decay rate γeg , is mimicked by a two photon Raman transition between
hyperfine ground states in 87 Rb. The Raman dressing laser, with Rabi frequency Ω, is detuned
from an optically excited intermediate state |ii to induce an optical decay from |ei to |gi at rate
γeg , mimicking the naturally occurring decay on the left hand side. This decay rate can be tuned
over a wide range by adjusting the dressing laser intensity and/or detuning from the intermediate
state. In both cases, population inversion is maintained by incoherently repumping the atoms backto the excited state |ei using another laser that excites atoms in the ground state |gi to a shortlived optically excited state after which they quickly spontaneously decay to |ei. The collective
blue laser emission in both cases is resonant with the optical cavity mode, while the single-particle
spontaneous emission for repumping (yellow) is not resonant and is scattered into free space.

27
Until recently, attempts to observe the features of optical lasers in the bad-cavity regime havebeen limited to operation near the cross-over regime (i.e. where the cavity linewidth is approximately equal to the linewidth of the gain medium). In the cross-over regime, the excitation of theoscillator is described best as a polariton, possessing significant atomic and photonic contributions(Fig. 1.10 center). Kuppens et al. [97, 98] studied HeNe and HeXe infrared gas lasers in the goodcavity to bad-cavity crossover regime 2γ⊥ /κ = 6 to 0.2, with 2γ⊥ ≈ 2π × (100 to 500) MHz, andwith Mc  1 intracavity photons. They observed linewidths up to an order of magnitude below
the naive linewidth limit expected in the good cavity regime ∆fGC . Other work with far infrared
gas lasers investigated the mapping of the Lorenz model onto the laser equations, observing signatures of chaos [67, 162]. Microcavity lasers, such as VCSELs, have operated near the bad-cavity
regime [12], but their usefulness as a state-of-the-art phase reference is limited due to the linewidth
broadening inherent in semiconductor gain media.
The utilization of nearly forbidden optical atomic transitions, along with advances in laser
cooling and trapping, have opened up the possibility to access the superradiant regime in the
optical domain. To minimize inhomogeneous broadening of the atomic transition, proposed narrowlinewidth superradiant lasers would use trapped, laser-cooled atoms as the gain medium [115, 33].
The first use of cold atoms as a laser gain medium was reported in Ref. [73]. Recently, the spectral
properties of a cold-atom Raman laser were studied in a high finesse cavity, deep into the so-called
good-cavity regime [156]. Clouds of cold atoms can also simultaneously provide gain and feedback
for distributed feedback lasing [138] and random lasing [6]. Cold atoms have also been used as thegain medium in four-wave mixing experiments [61, 7, 13] and in collective atomic recoil lasing [94].

1.3.4

A superradiant Raman laser

In my thesis work, we use our 87 Rb cavity QED system to realize a superradiant Raman
laser as a controllable, model system to explore the physics of proposed superradiant optical lasers.
By dressing a meta-stable ground state, we use a two-photon Raman transition to demonstrate
operation deep into the optical bad-cavity regime, achieving 2γ⊥ /κ ∼ 10−4 . An energy level diagram

28
showing the analogy between the two systems is shown in Fig. 1.11. Previously, Raman lasers using
laser-cooled, untrapped atoms have operated in the good-cavity limit [156], or in the bad-cavity
limit without studying the spectral properties of the light, with Mc  1 intracavity photons, andwith no distinction between the Raman dressing laser and repumping laser[73, 64, 32]. Additionally,
optomechanical forces have made interpretation of previous results difficult [32]. Four-wave mixing[64, 13] was used to observe a discrete quantum symmetry-breaking phase transition [7].
Starting in Chapter 4, I introduce a theoretical model of a superradiant laser, deriving thesteady-state behaviors of the laser’s atomic inversion, atomic polarization, and light field amplitude.
Then in Chapter 5, I give the first experimental results using the apparatus described in Chapter 2,
where we demonstrate lasing deep in the bad cavity regime with less than one intracavity photon
and observe cavity frequency pulling coefficients P as small as 5 × 10−5 . In Chapter 7, I identify
relaxation oscillations and dynamic cavity feedback that can serve to damp or enhance oscillatory
behavior of the laser field’s amplitude, as well as the atomic inversion and polarization. The research
into laser amplitude stability is supported by the linear response theory of a superradiant laser
contained in Chapter 6. Finally, in Chapter 8, I connect the two topics of this thesis by exploring
the connection between the Schawlow-Townes linewidth and the standard quantum limit for phaseestimation. Because the coherence of a superradiant laser is stored primarily in the atomic ensemble,
the laser light can be viewed as a continuous, non-destructive readout of a collective quantum phase.
In this last chapter, I theoretically explain the optimal estimator of the quantum phase from thelight field measurement and experimentally demonstrate using the superradiant readout for Ramsey
spectroscopy, which has lead to additional work as a superradiant magnetometer [160].

Chapter 2

Apparatus

In this chapter, I describe the apparatus used to perform the experiments discussed in thelater chapters, common to both the spin squeezing and superradiant experiments. First, I describe
the laser systems, including the lasers, frequency stabilization, and frequency distribution to theexperiment. Next, I describe the details of the science chamber, including the optical cavity used
for the experiments. I give the schemes used for the optical trapping that prepares the atoms forthe subsequent experiments. Finally, I give an overview of the detectors and electronics that areused to detect and process signals from the experiments. Technical details for specific experiments
will be provided in subsequent chapters.

2.1

Laser Systems
Lasers are the primary way we control and manipulate the atoms’ internal and external

degrees of freedom in our experiment. The laser system used can be subdivided into three modules
that operate mostly independently from one another. Two laser subsystems operate on the D2line of 87 Rb at 780 nm. The final subsystem uses lasers operating at both 795 nm (the D1 line in
87 Rb ) and 823 nm (for an off-resonant optical lattice trap), combined into a single system using

a stabilized Fabry-Perot cavity to transfer frequency stability to the trap laser and ultimately tothe science cavity. In this section, I first describe the general external cavity diode laser (ECDL)
design used throughout the experiment, then give the details on the individual laser subsystems.
The section concludes with a description of the laser beatnote phase locks and RF signal generation

30

Figure 2.1: A CAD model of the custom ECDL used throughout the laser systems in our lab.
The laser diode is mounted in the lens tube assembly. The diffraction grating (white) provides
frequency selective optical feedback to the narrow the laser linewidth. The grating is mounted on
the adjustable pivot. A piezoelectric slab provides a method of feeding-back to the grating position
for wideband laser frequency tuning by pivoiting the assembly. The grating can also be manually
adjusted with the two screws that push on the pivot. The 45◦ mirror out-couples the laser light(out-coupling lens not shown). The temperature control Peltier device rests under the mount as itsits in the external enclosure.

used to control the laser frequencies.

2.1.1

General ECDL design

All the lasers, except for the 780 nm narrow DBR lasers, are the same ECDL originallydesigned by graduate student Shannon Sankar and machined in the JILA instrument shop, shownin Fig. 2.1. The laser diodes are mounted in the internal mount, which is housed in the external
enclosure. The different laser diodes used are described in their respective sections. The mountholding the laser diode is temperature controlled with a Melcor Peltier device (PN: UT8-12-25-F2)
separate from the laser diode itself. The laser mount has an adjustable diffraction grating providing
optical feedback to the laser diode. The gratings can be adjusted manually to optimize feedback into
the diode which is critical for robust single mode operation of the lasers. The optimum alignment31
is associated with the lowest threshold current for lasing. A separate piezoelectric control provides
tuning of the angle of the grating, adjusting the frequency of the optical feedback into the laser
diode to provide wideband, mode-hop-free tuning when used in conjunction with a feedforward to
the laser current. We observe at least 6 GHz mode-hop free tuning for the laser frequency for timescales on the order of seconds, turning the laser lock point by hand. The feedforward is the limiton the tuning range. We have observed rapid single mode frequency tuning (1.184 GHz per ms)
for short frequency sweeps, on the order of 70 MHz, used in the experiments described in Ref. [35].
For these sweeps we assume only the current is changing and the grating tuning is not needed.
After exiting the ECDL enclosure, the beams pass through an opto-isolator, either from
Electro Optics Technology (PN: WT-04-I-780-MP) for the 780 nm MOT and state preparation
lasers or a Optics For Research (OFR) opto-isolator (PN: IO-3-795-HP). The output is coupled
into single mode optical fibers for distribution throughout the experiment. All fiber optics used forlaser signals are OZ optics (PN: PMJ-3AF3AF-780-5/125) single mode optical fibers. The fibers
have an 8◦ angled output to avoid back reflections off the fiber tip. In addition, fibers that directly
face the ECDL have an in-house AR coating applied to the tips. We avoid using fiber patch cordconnectors as we noticed back-reflections that can modify the laser linewidth for the better orworse, but in a manner that drifts in time.

2.1.2

780 nm MOT and state preparation lasers

One laser system in our experiments serve primarily to generate the magneto-optical trap
(MOT), to perform optical state preparation, and to maintain population inversion through incoherent pumping in the superradiant Raman laser experiments. The trapping and state prep laser
system operates on the D2 line in 87 Rb near 780 nm and consists of three ECDLs. Each one use a
non-AR coated Sanyo laser diode (PN: DL-7140-201S) nominally selected for a lasing at 780 nm.
An overview of the MOT and state preparation laser system is shown in Fig. 2.2.
The system starts with with the 780 nm Reference laser, which serves only as a reference
for the other lasers and does not interact with atoms in the science chamber. The 780 nm Ref-

32

Figure 2.2: A schematic of the 780 nm laser subsystem used to generate the magneto-optical trap
and perform state preparation using the D2 transition of 87 Rb . The system consists of 3 ECDL
lasers, labeled 780 nm Reference, MOT and Repumper. The 780 nm Reference laser is stabilized
to an atomic frequency reference using a Doppler-free polarization spectroscopy. Both the MOT
and Repumper lasers are beatnote stabilized to the Reference laser (not shown). Feedback loops
are indicated by the black line leading back to the lasers.

State
F=0’
F=1’
F=2’
F=3’

MOT (F = 2) BN [MHz]
–
934
777
510

Repumper (F = 1) BN [MHz]
810
738
581
314

33

Figure 2.3: RF beatnote frequencies used to set the MOT and Repumper lasers to resonance fromtheir respective hyperfine ground state to the excited states of the 87 Rb D2 line.

erence laser frequency is stabilized to an absolute atomic reference using polarization saturation
spectroscopy [122], locked to the F=2 to F=3’ transition of 87 Rb. The lasers we call the MOT and
Repumper lasers operate on the F=2 ground hyperfine state to excited state and F=1 ground hyperfine state to excited state respectively. The lasers are stabilized to the atomic reference using alaser beatnote lock, described in Section 2.1.5. We can control the frequency of both the MOT and
Repumper laser using the beatnote reference frequency, giving us the ability to address the entire
excited state structure from the ground hyperfine state associated with each laser. The beatnote
frequencies needed for the two lasers are given in Table 2.3.
The MOT laser is amplified with an Eagleyard tapered amplifier (PN:EYP-TPA-0780-010003006-CMT03). After passing through Isomet AOMs (PN: 1205C-1) which act as a fast switch,
the beam is split to produce 6 beams each containing 3 mW of power. The beams form 3 pairsof counter-propgating beams to trap in all three spatial dimensions. The details of the MOT aredescribed in Section 2.3.
The Repumper laser is combined with the MOT beam just before being split to 6 beams tomaintain separate control over the Repumper beam power. The light co-propagates with the MOT
laser to provide repumping for atoms in the F=1 hyperfine ground state. There is enough powersuch that each of the six paths contains 0.5 mW of repumping light.
In an attempt to vary the density of the atomic ensemble in the trap, we added an additional
set of MOT trapping beams that only confine the atomic ensemble in two dimensions. These beams
are generated by picking off 30% of the MOT light and directing it into two beam perpendicular toeach other and the cavity axis. The beams are retro-reflected to provide the 2D confinement. These34
beams were used for the superradiant laser experiments and were not used for the spin squeezing
results. The orientation of the beams is given in Section 2.3.

2.1.3

780 nm Narrow DBR Lasers

The key technical advance for the coherence preserving, non-destructive population measurement in this work was the implementation of probing the atomic population on a cycling transitionof 87 Rb , which only exists for the D2 transition. However, the hyperfine ground states associatedwith the cycling transition have a significantly reduced coherence time compared to so-called clocktransition used in our previous work [35]. The reduced coherence time requires a faster measurement sequence, which then imposes constraints on the laser frequency noise beyond what we could
achieve with the ECDL setups described previously.
We have implemented a second D2 laser system to produce a narrow linewidth probe laser(Fig. 2.4. The system uses two distributed Bragg grating reflector (DBR) lasers with an extra
long external cavity feedback setup to generate a narrow linewidth probe laser. We label the twolasers as the DBR Reference and DBR Probe. Both lasers start with a Photodigm DBR laser ina long eternal cavity setup [106] to provide optical feedback to the laser diode, resulting in goodshort term frequency stability properties. Measurements indicated both lasers have linewidths of∼ 5 kHz. The DBR Reference laser is stabilized to an atomic rubidium reference frequency forlong term stability using modulation transfer spectroscopy (MTS) [110]. The MTS locks the laser
to the F = 3 to F = 40 transition in 85 Rb and provides both high frequency feedback to the laser
diode current and low frequency feedback to the length of the external cavity through a Thorlabs
piezo stack (PN: AE050D08F).
The DBR Probe laser is frequency offset locked, or beatnote locked, to the DBR reference
laser, providing us the flexibility to set DBR Probe laser frequency to any F = 2 to excited statetransition. A beatnote of −1.246 GHz places the probe laser on resonance with the F = 2 toF = 30 transition in 87 Rb. In the squeezing experiments where a DBR probe laser detuning fromthe transition by +200 MHz was used, the beatnote was set to −876 MHz to account for the AOMs

35

Figure 2.4: A schematic of the 780 nm narrow linewidth DBR laser subsystem used to generate theprobe laser on the D2 transition of 87 Rb for the spin squeezing experiment. The system consists of
2 Photodigm DBR lasers, labeled DBR Reference and DBR probe. The DBR lasers have improved
short term stability properties due to the optical feedback provided by the long external free spacecavity, of length ∼ 0.5 m. We can tune the magnitude of the optical feedback using the λ/4
waveplate and polarizing beam cube in the feedback path. The reference laser provides long term
frequency stability by locking to an atomic reference with MTS. The probe laser is then beatnote
locked to the reference laser, giving us tunable control over the probe frequency over a range of 3.2
GHz.

36
and EOM in the probe path, described in detail in Chapter 3. We can also sweep the laser frequency
by sweeping the beatnote lock reference RF frequency, controlled by our direct digital synthesis
system, described in Section 2.1.6.
Tuning of the long external cavity feedback is largely empirical. We start by looking at the
beatnote on a wideband spectrum analyzer, with either the 780 nm Reference ECDL laser in thecase of the DBR Reference, or with the DBR Reference in the case of the DBR Probe. As the opticalcavity feedback is increased by changing the λ/4 waveplate, peaks start to appear in the beatnote,
separated by the external cavity free spectral range of 50 MHz. The feedback is increased until
the point where the laser oscillates at multiple longitudinal modes of the external cavity when the
feedback loop is engaged. Then the waveplate is turned slightly to decreases the optical feedback so
the laser can operate on a single mode. Fine tuning of the laser spectrum is performed by forming
the probe heterodyne signal with the bare cavity (described in detail in Chapter 3), and minimizing
the detection noise by changing the feedback loop parameters.
The presence of multiple external cavity modes impacts the laser spectrum by adding small
noise bumps appearing at ∼ 50 MHz from the optical carrier. In our heterodyne measurementof the probe beam, the signal can appear at an RF frequency between 0 and 200 MHz, so caremust be taken to avoid the noise bump near 50 MHz to preserve the highest signal to noise of theheterodyne detection.

2.1.4

795 nm and 823 nm Science Cavity lasers

In this section, I described the laser system that generates the beams used for trapping theatoms in the optical lattice, frequency stabilizing the science optical cavity, probing the cavity
resonance frequency on the D1 line at 795 nm, and creating the effective optically excited state bydressing the ground hyperfine state used for the superradiant Raman laser. This system consistsof four ECDL lasers with the same design as was described earlier. We call these lasers the 795 nm
Reference, Dressing, Transfer, and Lattice lasers.
The 795 nm Reference laser provides an absolute frequency reference for long term frequency

37
stability. The frequency stability is extended to the lattice trap laser, and ultimately the sciencecavity, using the Transfer laser and a second Fabry-Perot optical cavity called the transfer cavity.
A simplified schematic is shown in Fig. 2.5. The basic idea is that the Transfer laser is beatnote
locked to the 795 nm Reference laser, and the transfer cavity length is stabilized with the Transfer
laser. The Lattice laser is then frequency locked to a lower longitudinal mode of the transfer cavity,
and the science cavity is finally locked to the lattice laser, stabilizing it with respect to atomic
resonance. Therefore, changing the Transfer laser beatnote ultimately changes the science cavity
frequency. The details are described below.
The Reference, Dressing, and Transfer lasers all use Eagleyard anti-reflection (AR) coatedlaser diodes at 795 nm (PN:EYP-RWE-0840-06010-1500-SOT02-0000) in the ECDL setup. The 795
nm Reference laser is stabilized to the F = 3 to F = 20 − 30 crossover signal of 85 Rb with frequency
modulation saturation spectroscopy [68]. The Dressing and Transfer lasers are stabilized to the 795
nm Reference laser using a beatnote lock. The Dressing laser is a versatile laser which functions
as both the Raman dressing laser and a cavity resonance frequency probe in the superradianceexperiments described in subsequent chapters. The output of the Dressing laser has three important
outputs to the science chamber. The first is the Dressing beam path, which after passing throughan Isomet AOM to act as a fast switch, goes directly to the science cavity. The beatnote lock of
845 MHz places the Dressing beam on resonance with the F = 2 to F = 20 transition on the D1line of 87 Rb . While running superradiance, the beatnote was most often set to −255 MHz to place
the dressing laser +1.1 GHz detuned from the F = 2 to F = 20 transition, though the beatnote
allowed for flexible changing of the detuning ranging from ∼ ±3 GHz.
Early in the dressing laser path, half of the power is split off to form the the probe andsuperradiance heterodyne beams. The light passes through an EO Space phase modulator (PN:
PM-OK5-PFU-800-UL). The modulator can generate frequency components more than ±6.8 GHz
from the optical carrier, bridging the hyperfine splitting so that the probe beam can be resonantwith the optical science cavity resonance that is usually ∼ 1.1 GHz detuned from the F = 1 toF = 20 atomic transition. The two frequency components on the heterodyne beam also allow us to38

Figure 2.5: The 795 and 823 nm laser subsystem. The subsystem consists of three ECDLs at 795 nm(the 795 nm Reference, Dressing, and Transfer lasers) and one ECDL at 823 nm (the Lattice laser).
All are ultimately referenced to the F = 2 − 3 to F = 30 crossover signal of 85 Rb through beatnote
locks in the case of the Dressing and Transfer lasers, and using the transfer cavity in the case of
the Lattice laser. The transfer cavity is locked to the Transfer laser using a PDH lock to stabilize
the cavity length, then the Lattice laser is PDH locked to a longitudinal mode of the transfer
cavity near 823 nm. The Dressing laser is split into three functions, generating the Raman dressing
light for inducing superradiance, probing the science cavity resonance frequency, and providing theheterodyne reference beam in the superradiance experiments. The difference in the dressing laser
frequency and the probe frequency is bridged by a EO Space phase modulator driven near thehyperfine splitting frequency. The Heterodyne beam is path is also phase modulated, so that RF
beatnotes of both the superradiantly emitted light and the dressing light can be obtained.

39
form detection signals of both the dressing light (∼ +1.1 GHz detuned from the 2 − 20 transition)and the emitted light (∼ +1.1 GHz from the 1 − 20 transition). The probe light is recombined withthe dressing light for most of the superradiance results, as both go to the science cavity.
The optical lattice trap is generated by injecting a laser that is far red-detuned from atomic
resonance along the cavity cavity axis to build up a standing wave inside the science cavity. This
Lattice laser uses another Eagleyard AR coated laser diode that is nominally wavelength selected
to 820 nm. Measurements with an optical wavemeter indicate that the wavelength we use is near
823 nm. The frequency of the laser is stabilized with respect to the 795 nm reference laser using
a transfer cavity, as shown in Fig. 2.5. The optical lattice trap confines the atoms to the waist of
the optical cavity TEM00 mode, near the center of the cavity as described later in Section 2.3.3.
The transfer cavity is a Fabry-Perot cavity with a length of 3 cm and a finesse of 5000.
The cavity mirrors are glued to piezo-electric stacks which are mounted on a Zerodur spacer. The
transfer cavity length is stabilized using a Pound-Drever-Hall (PDH) lock [46, 14] to the frequencystabilized Transfer laser. When in lock, tuning the reference frequency of the Transfer laser beatnote
lock allows us to change the transfer cavity length. We found that adjusting the resonance frequencyof the cavity by more than κ put the cavity outside of the PDH lock range, and the cavity would
become unlocked from the laser and unable to re-capture lock without manual tuning.
With the transfer cavity frequency stabilized to the atomic reference at 795 nm, the cavity
provides a series of resonances separated by the free spectral range, about 7 GHz. We again use
a PDH lock to now stabilize the 823 nm laser to one of these resonances, fixing the lattice trap
laser in frequency with respect to an atomic transition. Finally, we use another PDH lock tostabilize the science cavity (described in Section 2.2), controlled by Piezomechanik piezoelectric
stacks (PN: PZT-5H). We can use the transfer cavity to then ultimately change the optical cavity
resonance frequency, setting it to the desired detuning from atomic resonance. The change in thetrap frequency is small enough to be negligible for the purposes of the off-resonant trapping light.

40
2.1.5

Details of laser beatnote locks

Laser beatnote locks are used to stabilize trapping and probing lasers with respect to thereference lasers. The laser beatnote locks give agile control over the laser frequencies during theexperiment. The beatnote typically is detected with high speed Hamamatsu GaAs photodiodes(PN: G4176-01, TO-18 package) connected to a Mini Circuits Bias Tee (ZX85-12G-S+) providing+5 V of DC bias (exceeding 10 V kills the diode immediately). The signal is amplified using a
microwave frequency SBW5089 MMIC amplifier. This detector provides the VCO signal for a
Hittite HMC 440S16G phase frequency detector, with a reference signal provided by the Analog
Devices DDS boards (PN: AD9959) described in Section 2.1.6. Although the minimum reference
frequency specified by the HMC 440QS16G is 10 MHz, we found a square wave reference frequency
was needed to operate below 20 MHz.

2.1.6

Direct Digital Synthesis

We use a series of versatile, phase coherent Analog Devices direct digital synthesis (DDS)
chips (PN: AD9959) to generate RF frequencies ranging from a few MHz to 200 MHz for use
throughout the experiment. The chips are integrated in evaluation boards purchased from Analog
Devices. In principle, the DDS can produce frequencies up to 500 MHz using the digital image
frequencies, but a high-order 200 MHz low pass filter is included on the output of our boards toreject the high frequency image signal. The DDS boards share a common 500 MHz clock derived
from the 5x multiplication of a Wenzel 100 MHz crystal oscillator (PN: 501-16843), with long term
stability provided by a rubidium atomic clock from Stanford Research Systems (PN: FS725). The
5x multiplier is constructed using surface mount PIN diodes (PN: BPF-B503+) [151]. The same 100
MHz oscillator also provides the phase stable microwave frequencies used to manipulate the groundstates in 87 Rb using the microwave source described in detail in Ref. [36]. For RF frequencies that
do not need sweeps of the frequency or ramps of the power, we control the DDS chips via serial
communication from our Labview experimental control and data acquisition software.

41

Figure 2.6: The Analog Devices AD9959 direct digital synthesis evaluation board shown fully on
the left and zoomed in on the right. The evaluation board has two modes of control, either serial
control through the header pins at the top of the picture, or through a USB control on the left sideof the picture. To enable pre-programmed ramps of the frequency, amplitude, or phase of the DDS
channels using an external TTL signal, we modified the evaluation board. The ramps are controlled
by profile pins to the DDS chip, P0, P1, P2, and P3. These pins connect to the serial header, but
the I/O buffer (chip U5, PN: 74LVC5414A) connects the USB profile inputs (PX U) to the profile
pins. By cutting the traces from the IO Buffer (chip U5, pins 11-14), the profile pins are freed tobe controlled by the serial headers. In the photograph on the right, the traces corresponding to the
profile pins are circled in red. Here only the trace for profile pin P0, corresponding to pin 11 on
U5, has been cut.

For measuring cavity resonance frequencies, a probe laser frequency is often swept in time byusing a triggered sweep of a DDS frequency to vary a laser beatnote lock reference frequency. We
have two DDS boards dedicated to sweeping, each with four output channels. These boards are notcontrolled with serial communication, but use the evaluation software included with the evaluation
board and interfaced through a USB controller. In principle, we should be able to control sweep
functions by serial commands from Labview as well, but for unknown reasons, repeated attempts to
add this functionality failed. To allow the chips to have sweeps triggered by an external TTL signalthat did not come through the USB channel, we disconnected the DDS profile pin connections to
the USB sweep trigger lines. This allows us to preprogram sweeps using the USB software, while
triggering them with TTL signals provided by our Labview data acquisition system. The details
are described in Fig. 2.6.

42

2.2

Optical Cavity and Science Chamber
In this Section, I give an overview of the Fabry-Perot optical cavity used in both of the

projects described in the thesis. This includes details of the cavity mirrors, their mounting andstabilization, and the vacuum chamber that houses the optical cavity. The optical cavity is shownin Fig. 2.7, and the vacuum chamber is shown in Fig. 2.12.

2.2.1

Optical Cavity

The optical cavity is a nearly confocal Fabry-Perot cavity. The mirrors are dielectric coatedmirrors produced by Advanced Thin Films. The super-polished substrates are 7.75 mm in diameterwith a 5 cm radius of curvature. The dielectric coating produced highly reflecting mirrors at
wavelengths ranging from 750 nm to > 850 nm. The mirrors had a specified transmission coefficientof 2500 ppm, and the delivered mirrors were measured to have transmission coefficients at 780 nm
and 795 nm of T780 = 2011 ppm and T795 = 2122 ppm respectively.
The length of the optical cavity is controlled using piezoelectric stacks from Piezomechanik(PN: PZT-5H), one on each side of the cavity. The piezo stacks have a tubular geometry. Mirrors
fit inside cylindrical piezo tubes and are held in place with the Torr Seal. The piezos have freestanding axial/radial resonance frequency of 80/75 kHz. After one set of wires that apply voltage
to the piezo stacks shorted, arcing through vacuum, we restricted the operating range of the voltage
applied to the piezo stacks. In the future, we could include separate vacuum feedthroughs for eachpiezo. Then in the case that one piezo fails, we can still drive the other. After the arc, we noticed anincrease in the cavity linewidth associated with increased losses from the cavity mirrors, reflectedin the cavity finesse F in Table 2.10.
In Tables 2.9 and 2.10, I give a table summarizing the important properties of the cavity
modes for the three wavelengths of interest. The numbers are based primarily on the specifications
given by ATF for the mirrors and measurements performed of the cavity mode resonant structure
at 795 nm.

43Figure 2.7: Photograph showing the optical cavity spacer, piezos and mirrors inside the glass cell
vacuum chamber. The smaller coils of copper wire are the initial, symmetric MOT coils. The larger
wire coils provide bias magnetic fields in all three directions.

44

Figure 2.8: A drawing of the cavity spacer. All dimensions are given in units of cm. The spacer
was machined from a single piece of Zerodur. The spacer is supported by four rods at the midpointof the cavity. The holes for the support rods are symmetric on both ends of the spacer for common
mode rejection of vibration. The angled cutouts on either end of the cavity allow for better opticalaccess for the angled MOT beams.

ParameterFree spectral range fF SR
Transverse mode spacing
Cavity length L
Mirror radius of curvature
Rayleigh range ZR

457828(1) MHz
2257(1) MHz
1.9149(2) cm
5.00(1) cm
1.967(2) cm

Figure 2.9: List of science cavity parameters that are constant across wavelength.
Parameter
Measured Linewidth [MHz]Finesse F
Mode waist [µm]Mode Volume [cm3 ]780 nm11.8 × 106
663
69.9
7.35 × 10−5795 nm11.1 × 106705
70.6
7.49 × 10−5

823nm
–
–
71.8
7.75 × 10−5

Figure 2.10: Science cavity parameters for the relevant wavelengths in including the spin squeezingprobe at 780 nm, the superradiance at 795 nm, and the trap at 823 nm.

The mirrors and piezo tubes are mounted to a spacer machined from a solid piece of Zerodur.
The vertical cavity orientation and spacer support points located midway between the mirror givecommon-mode rejection of vibrations to passively stabilize the cavity length. The spacer rests on
Viton spheres for additional vibration damping. A drawing of the cavity spacer is shown in Fig.
2.8. The machining was performed by Hans Greene in the JILA machine shop.
The spacer is supported by four support rods that attach to a vibration damping systemin the vacuum chamber, seen in Fig. 2.12. The spacer supports connect to the support damping
structure inside the spherical cube. The support structure also rests on three Viton spheres for
damping. The long pendulum that hangs down off the support structure serves to lower the
resonance frequency of of any vibrations the would shift the cavity position. We found that inpractice, low frequency vibrations in the floor of the building, caused by an air conditioning fan,would cause 17 Hz vibrations in the cavity damping structure, and modulating the coupling of ourprobe beam to the cavity. This appeared as fluctuations in the reflected and transmitted power,
as the coupling into single mode optical fibers from the cavity was very sensitive to laser pointing
stability.

46
2.2.2

Vacuum System

The heart of the vacuum system is the spherical square from Kimball Physics (PN: MCF450SS20400) that sits just above the optical table. The bottom of the cube connects to the vacuum
pumps, and the top connects to a metal-to-glass transition that holds the glass cell containing thecavity spacer. One of the side ports is occupied with a valve connected to the source of rubidium,
which is kept between 40 and 50 C◦ to provide a rubidium vapor throughout the vacuum chamber.
A second port of the optical cube is used for electrical feedthroughs to drive the piezoelectric tubes
that position the cavity mirrors. Finally, the third port is a window to allow laser light to bounce
off a 45◦ mirror at the center of the spherical cube and propagate upwards to the cavity. The
system was initially pumped down with Varian turbo pump (PN: Turbo V 81-M, IDP-3 Dry Scroll
Pump). The vacuum is maintained with a Varian VacIon Starcell 50L/s ion pump (PN: 9191340).
We measure pressure with a Varian Ion gauge (PN: UHV-24P).

2.3

Magneto Optical Trap and Lattice Trap
In this section, I describe the procedure for trapping and cooling approximately 106 atoms

near the middle of the optical cavity and localizing them near the radial center of the cavity mode
as well. The procedure starts with a magneto-optical trap (MOT) [124] which is then loaded into
a 1D optical lattice formed by the resonant standing wave at 823 nm created by the optical cavity.
After the lattice trap is loaded, the sample is sub-Doppler cooled using polarization gradient cooling(PGC) and the experiments begin.

2.3.1

Beams

Optical molasses for the MOT is provided by the six MOT beams, derived from the MOT
laser described in Section 2.1.2. Each beam has a characteristic diameter of 1.2 cm. Repumping
light co-propagates with each MOT beam to close the cycling transition. The six beams intersect
at the center of the optical cavity as shown in Fig. 2.11.

47

4.0	
  cm	
  

28	
  cm	
  
8.7	
  cm	
  

52.5	
  cm	
  

6ROLG:RUNV6WXGHQW(GLWLRQ)RU$FDGHPLF8VH2QO\Figure 2.11: A drawing of the complete vacuum system (the glass cell surrounding the cavity spacer
is not shown). The vacuum chamber below the reducer fitting (grey) is not shown. The cavity
spacer is in magenta, the MOT coil is in light grey, the spacer supports are in white, and Kimball
Physics spherical square in dark red. The damping pendulum hangs off the support structure withthe purple rods. The pendulum weight is colored light green. The three pairs of MOT beams
intersect at the center of the cavity spacer. The cavity axis beam path is shown entering thespherical square and exiting out the top of the optical cavity. The distance to the center of thecavity and the end of pendulum are both measured from the top of the optical table.

48

6ROLG:RUNV6WXGHQW(GLWLRQ)RU$FDGHPLF8VH2QO\ROLG:RUNV6WXGHQW(GLWLRQ
RU$FDGHPLF8VH2QO\Figure 2.12: (Left) The CAD model of the vacuum chamber, cavity support system, and cavity
spacer. A probe laser beam is shown entering a vacuum port, bouncing off the 45◦ mirror andcoupling to the cavity. (Right) A cutaway view of the cavity support structure. The cylindricalsupport structure (dark grey) links the cavity spacer supports (white) to the damping pendulum(purple). The damping pendulum (seen in Fig. 2.11) hangs off the support structure, which restson 3 Viton spheres (bright green) for damping. The 45◦ mirror is mounted on rails connected to
the vacuum chamber with Groove Grabbers.

49
One additional pair of MOT and Repumping beams were added perpendicular to the original
horizontal beams. By leaving on only these two pairs of beams, we found that we could create a
2D MOT which would allow atoms to spread out along the cavity axis and load into about twice
as many lattice sites as with the 3D MOT alone. In the loading sequence, this 2D MOT step was
on for 5 ms after initial 3D lattice step, and then the atoms were loaded into the lattice.

2.3.2

Coils and Drivers

The addition of the magnetic field provides the true confinement to the MOT. In the experiments described here, we have primarily used a pair of asymmetric coils of wire to provide themagnetic quadrapole field needed for trapping. The MOT coils are elongated with the goal of providing an elongated MOT to allow more atoms to be confined in the optical lattice with the sameatomic density by spreading them over more lattice sites utilizing the 2D MOT beams. A schematicof the coils are included in Fig. 2.13. The current in the coils is ∼ 2 Amps for the circular coilsand ∼ 3 Amps for the elongated coils. The current is driven with high voltage power MOSFETs
from Advanced Power Technology (PN: APL501J). The gradient established by the MOT coils is∼ 34 G/cm.
Three pairs of additional coils are used to apply small bias fields, such that the total current
flowing is 0.75 to 1 Amp. The bias coils are seen in Fig. 2.7. The bias field typically applied todefine a quantization axis is 3 Gauss.

2.3.3

Lattice Trap

The optical lattice trap is operated with intracavity lattice powers ranging from 0.3 W circulating in the spin squeezing experiment to up to 1 W circulating in the superradiance experiments.
In the spin squeezing experiments, 26 mW was delivered to the 50/50 beamsplitter that directs
lattice light to both the cavity and the path length fluctuation measurement (see Chapter 3 fordetails, particularly Fig. 3.8). In the superradiance experiments, all the lattice light was directedto the cavity alone, allowing us to reach 1 W circulating powers only directing ∼ 40 mW to the50

Figure 2.13: (left) A basic schematic of the elongated MOT coils used for most of the experiments
presented in this thesis. The Coil is composed of 128 turns of Belden 20 AWG magnet wire. The
individual turns are held together with Araldite 2011 Epoxy. Each layer of the coil is 12 turns,
making the coil 10.7 mm deep. The rounded inside corners have a radius of curvature of 4 mm.
(right) A photograph showing the elongated MOT coil mounted in the experiment. The coil sits
inside the larger bias coil.

51

Figure 2.14: A general time sequence that begins each of the experiments in this dissertation (notto scale). Maybe a chart showing what lasers are on and what the B-field is doing here would be
useful?

dichroic mirror that overlapped the Dressing laser and the Lattice laser. The given intracavity
powers result in lattice trap frequencies of 180 kHz and 335 kHz for the squeezing and superradiance experiments respectively. The intracavity power is stabilized by detecting the transmitted
light and feeding back to the power of the RF drive of an AOM acting as an amplitude modulator.

2.3.4

Loading Sequence, PGC coolingFig. 2.14 shows a general time sequence that contains the usual experimental sequence that
prepares the atoms in the optical lattice trap and in a well-defined internal state to begin theexperiments. Over the course of the experiments, some timings or details of the applied laser
power may vary slightly, but this diagram gives the overall procedure.
We start by forming the MOT directly from the rubidium vapor between the optical cavity
mirrors. The MOT laser was set to be slightly red detuned from the 2-3’ transition, and the
Repumper laser set slightly red of the 1-2’ transition. Both laser frequencies are empirically tuned
to maximize the atom number loaded in the trap. For the usual 1 Hz repetition rate for theexperiments, this step was about 350 ms, though to vary the atom number N ultimately loaded
into the lattice, we would increase or decrease this time. We also varied N by changing the powerin the MOT beams during this step.
With a healthy MOT formed, we then load into the lattice trap by eliminating the magnetic
quadrapole field and using the bias coils to zero the magnetic field at the location of the atoms. We
jump the MOT light to 50 MHz to the blue of the 2-2’ transition. During this time, the intensity

52of the MOT and Repumping light is stepped downward. Then the F=2 MOT light is stepped evenfurther down for the remaining loading time. Note that the lattice light is always on all throughout
the process. The optimal time to load the lattice has varied from 30 to 50 ms over the course ofthis and is manually tuned for a given experiment.
After the lattice trap loading period, the MOT laser is adjusted to +20 MHz detuned fromthe 2-2’ transition for the polarization gradient cooling step. All light is extinguished for 1 ms
while the laser frequencies settle. Polarization gradient cooling [41] allows us to reach measured
temperatures of 10-25 µK.
After the final cooling step in the lattice, all light except for the trap is extinguished and abias magnetic field specific to the particular experiment is applied to the ensemble, usually with amagnitude of about 3 G. We usually allow about 30 to 50 ms for all magnetic fields and induced
eddy currents to settle after this change, especially for experiments with magnetic field sensitivetransitions. The final state preparation using optical pumping takes about 1 ms, although thedetails of the state preparation are particular to each experiment and described in detail in theirrespective chapters.

2.4

Detection and Signal Chain
Here I describe components common to both experiments used in detecting signals from thescience cavity, electronically processing those signal and recording them into the computer.

2.4.1

APD detector

A simple avalanche gain photodiode provides a good first look at the light transmitted throughthe cavity and superradiant emission. For detecting these small signals, we used a Hammamatsu
avalanche photodiode (PN: S2381). The avalanche gain was ∼ 98 at a bias voltage of -150 V. The
detector uses a transimpedance amplifier (PN: OPA656U) with 0.8 MΩ of gain. The detector hasa bandwidth of about 1 MHz, allowing us to see the initial superradiant pulses that were taking
place on timescales of a few microseconds.

53
2.4.2

Fast heterodyne detector

For low noise detection that includes sensitivity to the frequency of the detected light, we
also implemented heterodyne detection schemes in the various experiments, described in detail inthe chapters of their respective experiments. In each experiment, the heterodyne detection uses a
Hamamatsu photodiode (PN: S5973) with 1 GHz intrinsic bandwidth. We removed the uncoatedglass cover to gain about 8% in quantum efficiency. The photodiode sensitivity without glass is
0.54 A/W at 795 nm corresponding to internal photodiode quantum efficiency of 84%.
The photocurrent generated at the photodiode is converted to electric voltage with a 240
MHz Analog Devices transimpedance amplifier (PN: AD8015). In assembling the photodiode circuit, we found that all the ground connections for the AD8015 must be connected close to theground pins. When not properly grounded, we observed oscillations around 300 MHz in the output
voltage. We obtained the best high bandwidth performance when the path from the photodiodeto the transimpedance amplifier was made as short as possible to reduce parasitic capacitance andinductance. The design used in the experiment has the output of the S5973 connect to the AD8015
input pin in free space through an ac coupling capacitor, bypassing the surface mount trace on thecircuit board. A circuit diagram of the photodiode is included in Appendix D.

2.4.3

IQ demodulators

The signal from the high frequency photodiode is split and frequency filtered as we often
have more than one signal contained in different frequency beatnotes. For example, in addition
to a signal that passed through the atomic sample, we may also have a path length reference
signal that did not interact with the atoms but contains information on path length fluctuations.
Each separate signal gets its own IQ demodulation channel and reference frequency derived fromthe DDS. Each demodulator splits the signal again and demodulates to near DC using an Analog
Devices IQ demodulator (PN: ADL5385) or a Linear Technologies IQ demodulator (PN: LT5517)
for RF frequency signals below 50 MHz. The IQ demodulators provide two orthogonal quadratures

54

Figure 2.15: An example schematic of the heterodyne data acquisition system. The output of
the heterodyne photodiode can have multiple signals separated in RF frequency. We split thesignal after amplification, then have separate paths for each frequency in which we can separately
attenuate the signals to keep the inputs to the IQ demodulators. The signals are demodulated with
RF frequencies from the DDS. The corrections to account for imbalances in each IQ demodulator
is applied in software (circle with ‘C’), giving the corrected signals I 0 (t) and Q0 (t) from which theamplitude A(t) and phase φ(t) of each signal can also be calculated.

of the signal with respect to the reference frequency. These two orthogonal quadratures, which
we call I(t) and Q(t) are read into the computer with a National Instruments analog to digitalconverter (PN: PCI-6133).
In software, we apply corrections to the I(t) and Q(t) channels to correct for small imperfections in the demodulation hardware, including average offsets between the channels, scale factors
between the channels, and higher order corrections. After applying corrections to the individualI(t) and Q(t) traces, we form the amplitude A(t) and the phase φ(t) quadratures. We tune thecorrections by applying test signals from the DDS boards. First, an overall voltage offset is removed
by applying no test signal to the RF input. Then the average voltage on the I(t) and Q(t) channels give the individual DC offsets yI and yQ . The exact corrections applied to the I(t) and Q(t)quadratures are given below. First, each quadrature is corrected for an offset y and an amplitude
K error, then a curvature β correction is applied

55

I 0 = KI (I + yI )(2.1)

Q0 = KQ (Q + yQ )(2.2)

I 00 = I 0 − βI (I 02 + Q02 )(2.3)

Q00 = Q0 − βQ (I 02 + Q02 )(2.4)

Finally, a relative angle error θ between the two quadratures can also be corrected

I 000 = I 00
Q000 =

Q00 − I 00 sin(θ)
cos(θ)

(2.5)
(2.6)

The coefficients are choosen empirically by driving the RF input with a test signal that differsfrom the LO frequency by ∼ 1 kHz. Imbalance between the two quadratures appears as oscillations
pin amplitude quadrature A(t) = I 000 (t)2 + Q000 (t)2 as the actual input voltage oscillates from I(t)to Q(t) and back. The deviations show up as peaks in the Fourier transform of the amplitude atharmonics of the frequency offset between the two test signals. Tuning the coefficients, we can push
the size of these peaks into the detection noise floor. Higher order corrections, i and φi are also
applied to the phase quadrature φ(t) as 000 

X
Q000
Q
φ(t) = arctan( 000 ) +i cos i arctan+ φi .
I
I 000

(2.7)

i=3

We normally apply only the i = 3 correction term.
The corrected amplitude A(t) and phase φ(t) then reflects the electric field coming from thecavity, either as a result of a probe laser or superradiant emission. We calculate the scale factor to beable to quantify A(t) in units such as average intracavity photons Mc or photons transmitted froma probe beam Mt . The calibration also depends on the average power contained in the heterodyne
reference beam. This beam is responsible for the DC current generated by the photodiode that we
detect separately from the IQ detection, and read in through the analog to digital converter.

Chapter 3

Reducing measurement back-action to generate spin squeezed states withphase sensitivity 10 times beyond the standard quantum limit

3.1

Experimental System
In this chapter, I describe collective measurements of a cavity field that is entangled with thetotal number of spin-1/2 atoms in spin up (Fig. 3.1A). The measurement uses an optical cyclingtransition to reduce the measurement back-action on the atomic system. The high readout precisionand small measurement back-action allows us to use the measurement to generate a spin squeezed
state by subtracting the pre-measurement result from subsequent measurements. The degree towhich the measurement precision has been improved is quantified by the spectroscopic enhancement
W −1 =∆θSQL∆θ , which is equal to the ratio of the phase variance at the standard quantum limit (SQL)

to the measured phase variance ∆θ. We directly observe a state with spectroscopic enhancement
W −1 = 10.5(1.5) times beyond the SQL for phase estimation of a coherent spin state composed of
N = 4.8 × 105 87 Rb atoms. This result reflects no background subtraction or corrections for finite
probe detection efficiency, which is critical for the realization of practical applications of entangledstates.
The experimental system consists of an ensemble of N pseudo-spin-1/2s formed by the hyperfine ground states |↑i ≡ |F = 2, mf = +2i and |↓i ≡ |F = 1, mf = +1i in 87 Rb, separated by
6.8 GHz. The quantum state of the ensemble can be approximated as a single collective spin or
Bloch vector J ≡ hĴi, in an abstract space defined by the collective spin operator Ĵ = Jˆx x̂+Jˆy ŷ+Jˆz ẑ
(Fig. 3.1B). The spin projection operator Jˆz = N̂↑ − N/2 can be written in terms of the measurable

57

Figure 3.1: Spin-squeezing and measurement back-action (A) Atoms collectively interact
with light in an optical cavity. A measurement of the phase of the probe field (red) is sensitive tothe total number of atoms in spin up, and projects the ensemble into an entangled state, conditionedon the measurement outcome. Probe photons can be scattered into free space, causing atoms tocollapse to spin up (orange in A, C, and D) and can also cause state-changing transitions (blue in
A, C, and D). (B) A coherent spin state can be visualized by a Bloch vector (red), with a pointing
uncertainty set by quantum noise, represented by the shaded uncertainty disk. (C) Atoms in |↑iwith optical transition frequency ωa couple to the detuned cavity mode with resonance frequency
ωc . The coupling results in a dressed cavity mode with resonant frequency ωc0 , so probing ωc0
measures the total number of atoms in |↑i, and hence the Bloch vector’s spin projection Jz , without
measuring the state of individual atoms. Probing on a cycling transition suppresses back-action
from scattering events that change an atom’s state to |↓i (blue), limiting back-action to collapse(orange). (D) After a pre-measurement, back-action modifies the noise distribution on the Bloch
sphere. Fundamental back-action appears along ŷ. Back-action from non-ideal measurements,
indicated by dashed lines, include reduction in length J of the collective Bloch vector and added
noise in Jz caused by state-changing transitions.

58
quantities: total atom number N and the spin up population operator N̂↑ = ΣN
i=1 |↑i i h↑i |, where i
D E
labels individual atoms. The length of the vector is J = Ĵ . For an unentangled CSS, J = N/2.
The quantum projection noise and standard quantum limit can be understood as arisingfrom uncertainty in the orientation of the Bloch vector (Fig. 3.1B). This quantum uncertainty
can be visualized as a quasi-probability distribution perpendicular to the mean vector. When the
Bloch vector is oriented along x̂, the degree of uncertainty in the orthogonal spin projections isconstrained by a Heisenberg uncertainty relationship ∆Jz ∆Jy ≥ N/4, where ∆X indicates thestandard deviation of repeated measurements of X. For a CSS of atoms, ∆Jz = ∆Jy = ∆NCSS =√N /2. For the polar angle θ ≈ Jz /J = 2N↑ /N − 1 measured from the equator of the Bloch sphere,√
the SQL is set by the projection noise fluctuations to ∆θSQL = 1/ N .

3.2

Creating a Squeezed State with a Collective Premeasurement
The squeezed state is created by first preparing a CSS along x̂ and then making a collective

pre-measurement N̂↑ , with measurement outcome labeled N↑p , and subtracting the result from asubsequent final measurement N̂↑ , labeled N↑f . The differential quantity N↑f − N↑p can possess
reduced noise relative to the projection noise fluctuations ∆NCSS appearing in the two separate2
measurements. The spin noise reduction is calculated as R = (∆(N↑f − N↑p ))2 /∆NCSS. By

making a collective or joint measurement, any rotation of the vector’s polar angle θ that occurs
between the two measurements will still modify the differential quantity N↑f − N↑p , leading to thedesired enhancement in the estimation of applied phase shifts.
To measure the collective state population N↑ , the atomic ensemble is coupled to the TEM0,0
mode of an optical cavity. The coupling is characterized by an effective single-atom couplingg = 2π × 450(20) MHz. The details of inhomogeneous coupling to the probe in our standing wave
cavity are handled as in Refs. [139, 35]. With no atoms present, the cavity has a resonant frequency
ωc and decay rate κ = 2π ×11.8(1) MHz. We detune the cavity frequency from an atomic transitionby δ = ωc − ωa = 2π × 200 MHz, where ωa is the frequency of the atomic transition from |↑i toan optically excited state |ei ≡ |F = 30 , mf = +3i. The radiative decay rate of |ei in free space is59Γ = 2π × 6.07 MHz. Atoms in |↑i produce a dressed atom-cavity resonance at frequency ωc0 , suchpthat ωc0 − ωc = ( δ 2 + 4g 2 N↑ − δ)/2 (Fig. 3.1C). We measure ωc0 with a probe laser (frequency ωp )to determine N↑ . The strength of the collective measurement is characterized by the average numberof probe photons Mt transmitted through the cavity along with the average number of photons
scattered into free space Ms . The transmitted probe provides collective population information
corresponding to the total number of atoms in |↑i, without providing individual atomic state
information. Detailed schematics of the probe measurement scheme are provided in section 3.4.
The information gained from a pre-measurement N̂↑ causes back-action on the system, illustrated in Fig.

3.1D. First, the measurement reduces the collective spin projection uncer-

tainty to ∆Jz = ∆N↑m , where ∆N↑m is the measurement imprecision. The Heisenberg uncertainty relationship requires fundamental back-action to appear in the orthogonal spin projection∆Jy ≥ (N/4)/∆N↑m , referred to as anti-squeezing. Because Jz is not coupled to the back-action
quadrature Jy , the ideal measurement is intrinsically back-action evading[26].
However, real systems experience at least two additional sources of probe-induced backaction, also illustrated in Fig. 3.1D. Both are caused by photons spontaneously scattered fromthe probe into free space, with average number of scattered photons Ms scaling linearly with themeasurement strength Ms ∝ Mt . One source of back-action arises from free-space scattered photons
leaking single-atom information to the environment, projecting an individual atom into |↑i or |↓ifor every free-space scattered photon. The result is a shortening of the Bloch vector such that asubsequent angular deflection θ will produce a reduced change of the measured population N↑f .
Another source of probe-induced back-action is spontaneous Raman transitions between
ground states driven by the same free-space scattering. Quantum randomness in the number oftransitions between states adds noise to the measurement of N↑ as the population diffuses amongst
ground states. The added noise ∆N↑D scales as (∆N↑D )2 ∝ pMt , where p is the probability an
atom changes state if it scatters a photon into free-space. The optimum spin-noise reduction R is√
fundamentally limited by the need to balance the decrease in measurement noise ∆N↑m ∝ 1/ Mt√
versus the increase in diffusion noise ∆N↑D ∝ Mt . This balancing is analogous to radiation

60
pressure back-action that sets the SQL for measurements of mechanical position[26].
The key experimental advance enabling the results presented here is the elimination of statechanging transitions as a limitation on the spectroscopic enhancement W −1 . This is achieved bycreating a system in which collective coupling to the probe mode is enhanced relative to singleatom processes. The approach uses the medium finesse optical cavity F = 660 to enhance thecollective coupling, with the figure of merit N C ≈ 6 × 103 , where C = 1.1(1) × 10−2 is the singleatom cooperativity [139, 35]. In addition, we suppress state-changing transitions by using σ +polarized probe light on a cycling transition [134, 4, 170, 34]. If a photon is scattered into freespace, the ground-state transition probability p is ∼ 1/150 that of our previous work [35]. As aresult, previously ignored noise sources, described in Sec. 3.4.4, now dominate the probe-induced
back-action on the measurement N̂↑ .

3.3

Results
In Fig. 3.2, we directly sense an externally-applied phase shift with resolution below the

SQL. We apply a small rotation ψ of the polar angle θ using a microwave pulse. In one case, therotation is applied to a CSS with no pre-measurement. In a second case, the rotation is appliedjust after the pre-measurement of N↑ prepares a conditional spin-squeezed state. The deflectionof N↑f is slightly smaller for the spin-squeezed state due to probe-induced collapse during the
pre-measurement N↑p . However, the reduction in noise in the quantity N↑f − N↑p allows therotation angle to be estimated with an enhancement W −1 = 7.5(9) in this example data set withmeasurement strength Mt = 2.7(1) × 104 and N = 4.3 × 105 . No background subtractions orcorrections are applied. In a single-shot, the fractional error rate in determining whether the phaseshift of ψ = 2.3(1) mrad was applied is reduced from 0.27(1) without the pre-measurement to0.022(7) with the pre-measurement.
More generally, we can identify an optimum spectroscopic enhancement by measuring boththe spin noise reduction R and the fractional shortening of the Bloch vector C = J/(N/2) asa function of measurement strength Mt , shown in Fig. 3.3A. First we consider the spin noise

61

Figure 3.2: Detection of a quantum phase with entanglement-enhanced sensitivity. We
apply a small rotation ψ to the polar angle θ of both a CSS and a spin-squeezed state, with data andrepresentative Bloch spheres shown on the left and right sides respectively. Red data points show
experimental trials with ψ = 2.3(1) mrad, and blue data points show trials with ψ = 0. The data
are represented both as histograms and Gaussian curves generated from the average and standard
deviation of the measurements. The experimental timing sequence consists of probe pulses (black)and microwave rotation pulses (green). For the CSS, the rotation ψ is applied immediately after
preparing the CSS along x̂. The rotation ψ appears as a change in the quantity N↑ − N↓ , which isnormalized to the total projection noise that appears in this differential quantity. In the case of thespin-squeezed state, we perform the rotation ψ after a pre-measurement N↑p . The rotation thenappears as a change in N↑f − N↑p , where the projection noise largely cancels. The spin-squeezed
state has a precision W −1 = 7.5(9), even though the change in N↑f − N↑p is slightly smaller than in
the CSS due to free-space scattering during the pre-measurement. The loss of signal is represented
by a smaller Bloch sphere for the spin-squeezed state.

62
reduction. The maximum R−1 observed, with no background subtraction, is R−1 = 16(2) at
Mt = 4.1 × 104 . The contributions of various noise sources are quantified using a fit to the observed
R versus Mt . The model, R = rP SN /Mt + Rtf + rq Mt + rc Mt2 , includes four noise contributions:−1photon shot noise rP SN , a technical noise floor Rtf= 73(34) independent of Mt , probe-induced

quantum back-action rq , and probe-induced classical back-action rc . Photon shot noise dominates
at low Mt so that R−1 initially increases as Mt increases. However, the rise in classical back-action
rc Mt2 eventually limits R−1 . At the optimum Mt , the classical back-action rc alone would limit
R−1 to 67(15). The quantum back-action rq is statistically consistent with zero.
Raman transitions between hyperfine ground states have been effectively eliminated as a
substantial source of back-action in the current experiment. The noise added due to population
diffusion from state-changing transitions, described in Sec. 3.4.4, is estimated to only limit R−1 to
1.7(3) × 103 as measured by probe-induced optical pumping between different ground-states. Also,
the inferred contribution to rc due to the observed classical fluctuations in probe power would
only limit R−1 to 3.2(4) × 104 . The equivalent transition probability is p ≤ 4.4(8) × 10−3 . For
comparison, the clock states in our previous work had a transition probability p = 2/3 which, in
our current system, would limit R−1 to 1.9(2). With population noise considerably suppressed,
other sources of back-action, including optomechanical effects (Fig. 3.3B), appear to dominate theprobe induced back-action on R.
Free-space scattering also leads to a reduction in the Bloch vector length J, and the resulting
loss of signal must be accounted for to determine the spectroscopic enhancement. To determine J,
the polar angle of the Bloch vector θ is varied after the pre-measurement N↑p using a microwave
pulse. The population N↑ is then recorded versus the rotation angle, shown in Fig. 3.3C. The
fractional reduction in length of the Bloch vector is determined from the fitted contrast C = 2J/Nof the observed fringe. The initial contrast at Mt = 0 is Ci = 0.97(3), and C monotonically decreases
as a function of Mt , close to the limit from wave function collapse due to free space scattering (Fig.
3.3A). We believe uncanceled inhomogeneous probe light shifts are responsible for the additional
small loss of contrast.

63

A

B

Population N↑(θR)/N

C

1.0
0.5
0.0

0

Probe
Rotate:

180
360
540
720
Rotation Phase θR (degrees):
θR

Figure 3.3: Spin-squeezing and probe-induced back-action. (A) Scaling of the spin noise
reduction R (red), loss of signal C 2 /Ci (blue), and the inverse of the spectroscopic enhancement
W (black) versus probe intensity Mt for N = 4.8 × 105 . The red, blue, and black curves are fits
to the data. The data for W is calculated from R data and the fit to C 2 /Ci . The 68% confidenceband for the W fit and the SQL is in grey. The dashed red curve shows the fitted R assuming
no probe-induced added noise (rc = rq = 0). The light-blue region is the predicted C 2 /Ci due tofree space scattering. All error bars are 1 std. dev. We use the usual convention for expressing
a ratio X in dB units, x (dB) = 10 log10 X. (B) Examples of optomechanical oscillations in the
dressed cavity frequency ωc0 . The relative detuning of ωc0 and ωp results in increased or decreased
oscillation damping rates, a source of probe-induced back-action noise (see supplementary text).
Each curve is the average of 30 experimental trials. (C) Example data and experimental sequence
for the measurement of the contrast C. Probe pulses (black) are measurements N̂↑ . Microwave
pulses (green) rotate the polar angle θ of the Bloch vector. After the pre-measurement of N↑p ,a variable rotation θR is applied and N↑ (θR ) is recorded. The contrast C is determined from theamplitude of the N↑ (θR ) fringe (curves are a fit to the data), with two examples shown in blue andgrey for Mt = 3.0 × 104 and Mt = 0 respectively.

64
10

S

ta

d

a

rd

W

n

1-

5

Q

u

a

n

tu

m

-4

10

0

L

im

0

it

200

N ( x10

400

3

)

-5

s

ie

H

10

n
b
m
ti

iL

g

re

-6

10

e

2

) dar( ytivitisneS devresbO yltceriD

-3

10

-7

10

3

10

4

10

Atom number

5

10

6

10

N

Figure 3.4: Absolute phase sensitivity versus N. The red points are the observed spin-squeezed
state phase sensitivities (optimized with respect to measurement strength Mt ) for different atom
numbers N . The data show the predicted 1/N 2 scaling for probing on a cycling transition [134, 34],
equivalent to a linear scaling of the spectroscopic enhancement W −1 versus N , shown in the inset.
The red line is a linear fit to the data. The SQL is confirmed by measuring√ the projection noisethat appears in N↓ − N↑p (black points, each 100 trials) and observing 1/ N scaling. Error bars
indicate 68.3% confidence intervals.

Taken together, the decrease in spin noise and loss of contrast quantify the spectroscopicenhancement of the spin-squeezed state W −1 = R−1 C 2 /Ci , as calculated in Refs. [4, 139, 35]. The
optimum observed improvement corresponds to W −1 = 10.5(1.5) or 10.2(6) dB. This value includes
no measurement background noise subtraction, and thus represents the actual realized improvementin phase sensitivity.
Further confirmation that our collective measurement is near the cycling transition limit
is the observed ∆θ2 ∝ 1/N 2 scaling of the absolute phase resolution[4, 34], shown in Fig. 3.4.
For comparison, the optimal phase resolution when state-changing processes are the dominant
limitation on W scales as N −3/2 . This more favorable scaling with N is important for practicalapplications where absolute phase resolution is the figure of merit.

65
When the spectroscopic enhancement W −1 ≥ 1, the ensemble is guaranteed to be entangled.
We also compute from Ref. [144] the multi-partite entanglement witness ξ = 100 >> 1 with thephysical interpretation that a system of consisting of 100 spin-1/2 atoms that undergo ideal two-axis
twisting[88] would exhibit the same spin-noise reduction and contrast as we observe in our system.
2
Maximally entangled ensembles can achieve phase estimation precision of ∆θHL
= 1/N 2 , known as

the Heisenberg limit, which has been realized with small ensembles[117]. Our system is far from theHeisenberg limit for N = 4.8 × 105 atoms. However, the absolute phase sensitivity is equivalent to∼ 44000 copies of a maximally-entangled 11 atom ensemble, emphasizing the massive parallelism
achievable by collective measurements to generate entanglement in neutral atom ensembles.
The optical nature of our approach, among others[4, 139, 159, 104, 141], offers the advantagethat the probe or squeezing laser can be completely extinguished after the squeezed state is generated. In contrast, a potential challenge for squeezing generated using atomic collisions is whether
the interactions that generate entanglement can be sufficiently reduced to avoid loss of accuracy
and precision during the subsequent sensing period.
The collective measurement approach is compatible with a wide array of atomic sensors but
is particularly appealing for optical lattice clocks[74], where systematic errors and atom loss mayplace a limit on the ensemble size. The probe also provides a low-noise, non-destructive readout
method, similar to Ref. [170]. We achieve a readout sensitivity at the SQL while only imparting
0.007(1) photon recoils per atom, and hence very little loss of atoms from the trap due to heating.
This may provide a purely classical improvement for sensors limited by measurement dead timeassociated with reloading the atomic ensemble in between trials, a crucial issue for advancing opticallattice clock stability [163].
Straightforward technical improvements could both decrease the technical measurement noise
floor and increase the total probe detection efficiency from 8(5)% (see Table 3.5) to > 50%, allowing
us to reach W −1 ∼ 100 in our current system with only a medium finesse optical cavity. However,to realize even further spectroscopic enhancement, previously unimportant forms of probe-induced
back-action will likely require additional study.

Common to both paths
Cavity mirror losses
Heterodyne detection
Photodetector (S5973)
Transmission Path
Path efficiency
Detector electronic noise
Added noise from 823 nm path length reference
Total transmission path efficiency
Reflection Path
Path efficiency
Detector electronic noise
Added noise from carrier path length reference
Total reflection path efficiency

66
0.42
0.5
0.865
0.69
0.80
0.9
0.046
0.54
0.83
0.9
0.037

Figure 3.5: The measured detection efficiencies for the spin squeezing experiment.

3.4

Experimental Details

3.4.1

Atom-Cavity System

The optical cavity used for the collective measurements has a measured free spectral rangeof 7.828(1) GHz and a measured transverse mode spacing of 2.257(2) GHz, determining the cavity
length to be L = 1.9149(3) cm and the mode waist to be w780 = 69.90(4) µm at 780 nm. Withno atoms in the cavity, the cavity frequency is denoted ωc and the measured cavity linewidth or
power decay rate is κ = 2π × 11.8(1) MHz, giving a cavity finesse F = 663(5). The power decayrate from factory-specified mirror transmission alone is κ◦ = 2π × 5.02 MHz.
The cavity is also used to generate the one-dimensional, intra-cavity optical lattice trap atwavelength λl = 823 nm, with a mode waist w823 = 71.78(4) µm and an axial trap frequency ωax =2π × 150 kHz. The atoms are polarization-gradient-cooled to 10(2) µK in the trap, putting themin the Lamb-Dicke regime along the cavity axis. A magnetic field with magnitude |B| = 0.73(1) G
is oriented along the cavity axis. A simplified diagram of the cavity setup is shown in Fig. 3.6, anda more detailed schematic of the system is shown in Fig. 3.8.
The atomic ensemble is prepared and probed using the D2 line in 87 Rb at 780 nm (Fig S2).
The hyperfine ground states |↓i ≡ 5 2 S1/2 , F = 1, mf = 1 and |↑i ≡ 5 2 S1/2 , F = 2, mf = 2

67

Heterodyne Beams

Reflection Detector

AOM
Probe
(780 nm)

L = 1.9 cm
λ/4

B = 0.74 G

Transmission
Detector

Trap (823nm)
6.833 GHz

State Prep
(780 nm)

Figure 3.6: Simplified experimental diagram. The one dimensional optical lattice trap at
λl = 823 nm is formed from a standing wave in the cavity (orange). We load N◦ = 4.0 × 104 to
7.2×105 87 Rb atoms into the trap and cool them to 10 µK. The atomic sample extends ∼ 1 mm along
the axis of the L = 1.9 cm long optical cavity. State preparation is performed using a combinationof 780 nm light (purple) for optical pumping and coherent ground state rotations performed with
6.833 GHz microwaves from the dipole antenna. A uniform magnetic field is applied to provide
a quantization axis and spectrally resolve the ground state Zeeman sub-levels. The probe electricfield forms a standing wave in the cavity, represented by the sinusoidal red line. The atom-cavity
system is probed with 780 nm light (red), set to σ + polarization before entering the cavity. The
probe light is separated from trap light using a dichroic mirror (grey) on the probe transmission
side. The probe light is detected in both reflection and transmission with a heterodyne beamfrequency shifted by an AOM.

68
form the pseudo-spin-1/2 system. Coherent rotations of the spin system are accomplished bycoupling the ground states with a 6.833 GHz microwave field[36]. The relevant optically excitedstate |ei ≡ 5 2 P3/2 , F 0 = 3, mf = 3 has a decay rate Γ = 2π × 6.06 MHz.
The coupling between the cavity mode and a single atom on the |↑i to |ei transition isparametrized by the single-photon Rabi frequency 2g0 = 2π × 1070(30) kHz, given at an antinode of the standing wave probe field at the center of the cavity both in the transverse and axial
dimensions[85]. The probe coupling varies sinusoidally as g(z) = g0 cos (2πz/λp ), where z is thespatial coordinate along the axis of the cavity. The probe field at wavelength λp = 780 nm is
incommensurate with the trapping site spacing set by λl , so we define an effective single photon
Rabi frequency 2g for the ensemble, accounting for both the axially and radially varying coupling
as in [139, 35]: 2g = 2π × 894(46) kHz (or cooperativity parameter C = 4g 2 /(κΓ) = 1.1(1) × 10−2 ).
The total number of atoms confined in the optical lattice is N◦ , irrespective of their coupling to thestanding-wave probe mode. An effective atom number N = 0.663(4) × N◦ with uniform coupling
2g produces the observed projection noise fluctuations. Throughout the main text, N refers to theeffective atom number.

3.4.2

Collective Population Measurement

The collective measurement of atomic population in |↑i, N̂↑ , is made by measuring the resonance frequency of the coupled, or dressed, atom-cavity system ωc0 . The collective coupling of thepatoms in |↑i to the cavity mode is Ω↑ = N↑ 2g, where N↑ is number of atoms in |↑i. The barecavity resonance frequency ωc is detuned δ = ωc − ωa = 2π × 200 MHz to the blue of the |↑i → |eitransition at frequency ωa . The collective coupling results in a dressed cavity resonance frequency
that is shifted by an amount1
ωc0 − ωc =
2

q

2
δ 2 + N↑ (2g) − δ .

(3.1)

We measure the frequency of the dressed atom-cavity system using σ + polarized light from a narrow(< 5 kHz) linewidth probe laser similar to that in Ref. [106], described in Chapter 2. The probe

69

F’ = 3F’ = 2F’ = 1

e
ωp

ωc’

ωc
ωa

F=2

1

↑
6.833 GHz

F=1↓

mf = 1

mf = 2

mf = 3Figure 3.7: Atomic energy level structure. The relevant energy level structure of the 5 2 S1/2
to 5 2 P3/2 transition in 87 Rb. The cycling transition has an optical atomic resonance frequency ωa .
The cavity resonance with no atoms present (dashed light blue) with frequency ωc is detuned to theblue of atomic resonance. The atom-cavity coupling creates a dressed cavity resonance (light blue)
with frequency ωc0 which we probe using σ + laser light at frequency ωp (red). The cycling natureof the transition means scattering primarily maintains population in |↑i (orange). Scattering of theprobe light from atoms in |↓i, detuned by ∼ 6.8 GHz, provides the fundamental limit to the cyclingtransition, as atoms can scatter to both |↑i and |1i ≡ |F = 2, mf = 1i (dark blue). Furthermore,imperfect polarization can lead to transitions for atoms in |↑i to other internal states. An exampleof scattering from a π-polarized component of the probe is shown as the dashed red lines, with thestate changing transitions in dark blue.

frequency ωp is tuned nominally to resonance with the dressed cavity mode ωc0 . The phases of thetransmitted and reflected probe signals give the resonance frequency ωc0 , described in detail below.
The measured frequency is then used in Eqn. 3.1 to determine N↑ . The relevant cavity and probe
frequencies are schematically represented in Fig. 3.7.
The collective measurement strength is quantified by Mt , the average number of photons

70transmitted through the cavity in a single measurement N̂↑ . For each measurement, the probe
light is on for 43 µs, but the measurement is an average of ωc0 over only 40 µs to avoid edge effects
in the data acquisition. Probe-induced back-action from free space scattering is quantified by Ms ,the number of photons scattered into free space during each measurement. Ms is related to Mt byN↑ 4g 2
Ms = Mt 2Γ
[34]. For the data with optimum squeezing at N = 2N↑ = 4.8 × 105 , the2κ◦ 4(ω 0 −ωa )ccalculated number of photons scattered into free space is Ms = 1.0(1) × Mt .
Both the transmitted and reflected probe light are detected in heterodyne with respect tofrequency-shifted light derived from the same probe laser (Fig. 3.8). In both cases, the probe
signal appears at the photodiode output as a 50 MHz signal that is quadrature demodulated,with the in-phase I and quadrature-phase Q digitized and recorded by a computer. The I and Q
amplitudes during a measurement are used to compute the phase of the detected probe light inboth transmission φpt and reflection φpr as described in Chapter 2.
Fluctuations in the relative paths of the probe and heterodyne beams adds fluctuations to themeasured phases, equivalent to variances two times projection noise in transmission and 15 times
projection noise in reflection. We use reference beams traversing the same optical paths to remove
the added phase fluctuations without introducing significant additional atomic decoherence.
The probe light at frequency ωp that resonantly excites the dressed cavity mode is generated
by phase modulating a carrier 170 MHz to the blue of the probe frequency. The modulationis performed with the EOspace phase modulator shown in Fig. 3.8. In reflection, this carrier
follows the same path as the reflected probe light, and therefore can be used to cancel path length
fluctuations. Because this reference beam is detuned from the dressed cavity resonance ωc0 bymany cavity linewidths, it is primarily reflected from the cavity input and does not interact withthe atoms. The carrier produces a signal in the reflection photodetector at 120 MHz, which isthen quadrature demodulated, digitized, and a carrier phase is computed φcr . Path length noise isremoved in the differential quantity φr = φpr − φcr , which is found to be very stable over time.
The carrier frequency component is not transmitted through the cavity and cannot be used toremove path length noise in the transmission heterodyne detection. Instead, we exploit the 823 nm

71

Figure 3.8: Detailed probe path documenting generation of heterodyne beams, probe quantum
efficiencies to reflection and transmission detectors, and path length stabilization setup for transmission using 823 nm lattice laser. The 780 nm laser (red) is split into the heterodyne paths (thicklines) and the probe path (thin lines), which are eventually overlapped before jointly entering single
mode fibers that go to the fast heterodyne detectors. The 823 nm laser (orange dashed line) that
forms the optical lattice trap is also directed to largely follow, in reverse direction, the transmission
heterodyne path (Heterodyne 1). Some of the path length noise is then common mode, and appears
in the heterodyne detection of the 823 light in the probe setup area. The grey square indicatescomponents on a platform raised 3 ft off the optical table surface. All other components on tablesurface.

72
optical lattice light passing through the cavity to remove path length noise from the transmission
phase φpt . The details are shown in Fig. 3.8. A 50/50 beam splitter takes the light from the823 nm lattice laser path to send back through the heterodyne arm of the transmission path to
the 823nm heterodyne detector, a Hamamatsu S5973 photodiode with a AD8015 transimpedance
amplifier like those described in Chapter 2. The 823 nm trapp light that transmits through thecavity travels back along the path of the probe beam, with enough light leaking through the opticalisolator to form a signal in heterodyne. This leakage light is then frequency shifted by an AOM toform a beatnote signal on the 823 nm photodiode at 50 MHz. The detected signal is demodulated
and sampled to yield a phase φlt . Path length noise in the transmission probe phase is removed by
calculating the differential quantity φt = φpt − φlt (λl /λp ), where λl = 823 nm and λp = 780 nm.
The two heterodyne interferometers for the transmitted probe light and tracer light are made to
be completely common up to two short free space paths of ∼ 0.5 m total running along the optics
table. While fluctuations in the differential path length lead to long term drifts (1 radian at 20ms) in φt , the drifts are not relevant to the differential measurements of populations made at shorttime scales (100 µs) in this work.
The measured phases are converted to detunings between the probe and cavity mode δt,r =φt,r /St,r using separate scale factors St,r = dφt,r /dωp . The scale factors are directly measured
during the experiments by jumping the probe frequency by 0.5 MHz (much less than the cavity
half linewidth) and recording the change in the measured phases. In the case of transmission, an
additional linear phase shift appears in φt and must be accounted for, because the lattice or tracer
beam frequency is not changed in the scale factor calibration. St must be adjusted accordingly. In
contrast, in reflection both the probe and carrier frequency components are jumped in frequency
together.
For measurements of the crucial differential quantity N↑f − N↑p , we construct differential
detunings δrf p ≡ δrf − δrp = (φrf − φrp )/Sr and similarly for transmission. Long terms drifts
in φt cancel from this differential quantity. The final estimate for the differential detuning is
calculated from a weighted average of the transmission and reflection measurements as δf p =

73
2 δ
2
2
2
2
(σrf
p tf p +σtf p δrf p )/(σrf p +σtf p ). The measured noise variances in transmission σtf p and reflection2
σrf
p are estimated from the variance in measurements of δtf p and δrf p performed over 100 trials

at the same atom number N and probe photon number Mt . Ideally, the weights are equal, but we
2 /σ 2 = 1.65(4) indicating different detection efficiencies for the two paths. We calculate
find σrfp
tf p
the differential atom number N↑f − N↑p = δf p dωc0 /dN↑ |N↑rp , where the derivative is calculatedusing Eqn. 3.1 and evaluated at an atom number N↑rp in |↑i that is calculated from Eqn. 3.1 andthe reflection measurement δrp .

3.4.3

Spin Noise Reduction Measurement Sequence

Spin noise reduction R is determined from the directly measured noise in the difference of
two measurements of the population N↑ . The experimental sequence for measuring R is shown in
Fig. 3.9. We first pre-align the probe frequency ωp to the dressed cavity frequency ωc0 , describedin detail below. Next, we prepare a coherent spin-state on the equator of the Bloch sphere by
optically pumping all atoms to |↓i (J along −ẑ), then applying a π/2 microwave pulse to orient
the Bloch vector along x̂, equivalent to placing each atom in a superposition of |↑i and |↓i. We
infer the number of atoms in |↑i by switching on the probe laser to make the first measurementof ωc0 , and label the result N↓ . After phase-coherently swapping the populations in |↑i and |↓iusing a microwave π-pulse, we perform two successive measurements of the population in |↑i andlabel the results N↑p and N↑f . The experiment is repeated more than 100 times and the varianceof the difference between the final measurement and the pre-measurement (∆ (N↑f − N↑p ))2 is
calculated from the measured phases as described above. The spin noise reduction R is computed2

as R =(∆(N↑f −N↑p ))N/4.

In Fig. 3.10, we compare measurement noise versus Mt both with and without atoms in
the cavity. With atoms, the contribution from photon shot noise is 4.5 dB lower because thepresence of atoms creates a narrower dressed cavity resonance [34]. The difference is visible forsmall Mt where photon shot noise dominates. However, measurement back-action with atoms
present contributes noise that increases with Mt and limits the maximum-achievable spin noise

74
Pre-alignment

Probe

Pre-measurement

Set ωp ~ ωc’:Opt. Pump:
Rotate:time
2.5 ms6 µs100 µs98 µs600 µs

43 µs

Figure 3.9: Timing Sequence.
The experimental timing sequence showing probe laser prealignment and the pre-measurement that prepares a conditionally spin-squeezed state, followed bya final measurement to quantify the reduction in spin noise. Each optical pumping step (purple)
prepares the ensemble in |↓i. Rotations (green) are performed by coupling |↑i and |↓i with acoherent microwave source. Probe laser pulses (red) correspond to individual measurements N̂↑ ,with the measurement outcomes labeled as shown.

5

0

p'c

p2/) w - w(D

100
-5

80

60

f'c

-10
40

)Bd( NPQ ot evitaler ecnairaV)zHk(

esioN tnemerusaeM ycneuqerF SMR

200

30
-15
4

10

5

10

Average Number of Probe Photons Mt

Figure 3.10: Comparing measurement noise with and without atoms. The fluctuationsin the difference between two measurements of the dressed cavity frequency ωc0 f − ωc0 p , used todetermine the difference of population measurements N↑f − N↑p , is plotted versus the average
number of probe photons Mt . On the left axis, the fluctuations are expressed as the standard
deviation ∆(ωc0 f − ωc0 p )/2π in absolute frequency units. On the right axis, the same fluctuations
are expressed as the ratio of the variance (∆(ωc0 f − ωc0 p ))2 to the variance (∆ωc0 p )2 caused by thequantum projection noise (QPN) of a CSS. For the ensemble of N = 4.8 × 105 atoms here, QPN
causes fluctuations ∆ωc0 = 2π × 144(9) kHz, indicated by the line at 0 dB. Measurement noise iscompared with (red) and without (black) the atoms loaded in the trap. For the case with atoms,
the right axis is equivalent to R. The lines are fit to the data. The error bars are 1 std. dev.

75
reduction R, as described later in the supplementary text.
The role of the first measurement, labeled N↓ in Fig. 3.9, is to cancel inhomogeneous lightshifts caused by the standing-wave probe light during the pre-measurement N↑p , with the π-pulse
forming a spin-echo sequence. Light shift cancellation is critical for restoring coherence (i.e. lengthof the Bloch vector) so that a Bloch vector rotation through an angle θ after the pre-measurement
will produce an observable change of the final measurement N↑f .
In principle, the first measurement can be used to enhance the spin noise reduction. However,
we found that the π-pulse degraded the observed spin noise reduction when the information gained
from the first measurement was utilized. The noise added by the microwave rotation results fromboth fluctuations in the transition frequency |↓i to |↑i, as well as amplitude and phase noise of theapplied microwave field. Neglecting the information in N↓ avoids these sources of rotation-added
noise at the cost of a potential factor of 2 improvement in R.
Fluctuations in the atom number N produce significant fluctuations in the detuning δp =
ωp − ωc0 of the probe from the dressed cavity resonance from one trial to the next. For scale,the atom number changes by ∼ 1% peak to peak on 1 minute time scales, and can drift by 2.5%
over 30 minutes. Fluctuations in δp can create additional technical noise that limits R. First, thenumber of transmitted photons Mt changes with the detuning of the probe, modifying the small
observed chirping of ωc0 described below. Secondly, the point of maximum measurement sensitivity
is achieved when δp  κ/2.
To mitigate these two effects, we implement a scheme to pre-align the probe laser frequency ωp
to the dressed cavity frequency ωc0 at the start of each trial (Fig. 3.9). After we prepare a coherent
spin-state on the equator of the Bloch sphere by optical pumping and a π/2 rotation, we actively lock
ωp to ωc0 , then hold ωp fixed at the final frequency for all subsequent measurements within a trial.
After this pre-alignment, the ensemble is optically repumped to |↓i and the measurement sequencedescribed previously is performed. Pre-alignment of the probe laser reduced the trial to trial
standard deviation of the probe-cavity detuning during the pre-measurement N↑p to ∆δp = 0.045 ×
κ/2. For comparison, the fundamental limit set by the uncorrelated projection noise appearing in

76both the pre-alignment and the pre-measurement is ∆δp = 0.034 × κ/2.

3.4.4

Quantifying Measurement-induced Back Action

To verify that the collective measurements generate a conditionally spin-squeezed state withenhanced phase sensitivity, we must determine the fractional reduction in the Bloch vector length
J caused by measurement back-action. The degree to which the measurement avoids collapse anddephasing is quantified by the contrast C = J/(N/2). To measure C, we apply a variable polar angle
rotation θR to the Bloch vector using an additional π/2 microwave pulse after the measurement ofN↑p . The polar angle is changed by varying the phase of the final microwave π/2 pulse relative tothat of the initial π/2 pulse. We then measure the population of |↑i with measurement outcome
labeled N↑ (θR ). The contrast is determined from fitting the data to its expected dependenceN↑ (θR ) = (N/2)(1 + C cos θR ). Assuming each free space scattered photon causes a single atom
to collapse into spin up, reducing J by one unit, the predicted contrast is Cpred = Ci e−Ms /N . The
initial contrast Ci = 0.97(3) is the contrast measured with no probe light (Ms = Mt = 0) during
the pre-measurement. When Ms is increased such that C falls below ∼ 0.9, the contrast begins todecrease more rapidly than predicted by free space scattering, as seen in Fig. 3.3A. We believe thedeviation is likely due to light shifts that are not fully canceled by the spin-echo formed by the
combination of measurements labeled N↓ and N↑p .
In this section, we quantify measurement back-action due to spontaneous scattering thatchanges the internal state of an atom. We consider limitations to the spin noise reduction fromboth the fundamental quantum noise, which we call population diffusion, and noise from classicalfluctuations in the average change in internal state populations. For a perfect probing scheme ona cycling transition, only Rayleigh scattering is allowed, and so no change of the internal state of
the atom would take place. However, as shown in Fig. 3.7 and discussed in ref. [34], atoms in|↓i can non-resonantly Raman scatter to both |↑i and |1i ≡ |F = 2, mf = 1i. Another possible
measurement imperfection we consider is impure probe light polarization allowing population to
Raman scatter out of |↑i.

77

A

Opt. Pump:

B

Prepare in ↓time
Scattering pulse

Probe:

Opt. Pump:
Rotate:C

Scattering pulse

Probe:

Prepare in ↓time

Figure 3.11: Probe-induced population change.
(A) To measure the sum of transition
probabilities p↓↑ + p↓1 , we first prepare atoms in |↓i with optical pumping (purple). Next, ascattering probe pulse (red), quantified by the average number of transmitted probe photons Mt ,causes some atoms to change state to |↑i and |1i. Many photons per atom in |1i are scattered into
free space, allowing the atoms in |1i to transition to |↑i. Thus, the measurement of the dressedcavity frequency ωc0 ↓ gives the total number of atoms scattered out of |↓i. (B) To measure thetransition probability p↑↓ , we prepare atoms in |↑i using optical pumping and a microwave π pulse(green). The imperfection in the σ + polarized probe used for the scattering pulse causes some atoms
to change state to |↓i. We again assume all atoms that scatter to |1i immediately transition backto |↑i. We swap the populations in |↑i and |↓i with another microwave π pulse, so the measurementof ωc0 ↑ gives the number of atoms that scattered to |↓i. (C) Measurements of the dressed cavity
frequency due to probe-induced internal state-changing transitions, with ωc0 ↓ (red) described in
A, and ωc0 ↑ (blue) described in B. The lines are fits to the data, yielding a change in ωc0 pertransmitted photon δωc0 ↓ = 2π × 1.11(2) Hz/photon (red) and δωc0 ↑ = −2π × 0.86(5) Hz/photon(blue). Here N = 2.1 × 105 atoms.

78
Our model for the added noise in R considers transitions between three states, |↑i, |↓i, and|1i. The probabilities per free space scattered photon to make a transition are denoted as p↑↓ , p↓↑ ,p↑1 , and p↓1 for the four transitions |↑i → |↓i, |↓i → |↑i, |↑i → |1i, and |↓i → |1i respectively.
To determine the transition probabilities, we do two experiments, shown in Fig. 3.11. In eachexperiment, we measure an average change in ωc0 per transmitted probe photon. We then convert
the frequency change into a transition probability using the calculated dressed cavity shift perdωatom added to each state αs = dNcs0 , where s =↑, ↓, 1. In one experiment, shown in Fig. 3.11A,
we measure the sum p↓↑ + p↓1 and use known branching ratios to get the individual probabilitiesp↓↑ = 7.3(7) × 10−4 and p↓1 = 3.6(4) × 10−4 . In the other experiment, shown in Fig. 3.11B, we
directly measure p↑↓ = 8(1) × 10−4 . We assume p↑↓ results from imperfect polarization of the probe
laser. This measurement of p↑↓ constrains the fraction of probe power in the non-cycling polarizationto < 6.7(8)%, if the imperfection is assumed to be σ − -polarized light, and < 1.7(2)%, assuming
π-polarized light for the imperfection. Direct measurements more tightly constrain the fraction of
power in the probe that is not in the σ + mode to < 0.5%. The final transition probability p↑1 can becalculated using branching ratios, known probe detuning from atomic resonance ωp − ωa , and p↑↓ ,assuming a specific ratio of σ − to π polarization for the probe polarization imperfection. For thefull range of possibility for the polarization of the probe light imperfection (i.e. the ratio of σ − toπ polarized light in the probe imperfection ranging from 0 to 1), the transition probability doesn’tchange within its uncertainty, p↑1 < 3.9(5) × 10−3 . In all cases, these independent measurements
show that the transition probabilities are small p  1, approaching the ideal cycling transition
limit of p = 0.
The fundamental limitation on R from internal state-changing scattering events comes fromquantum noise in the scattering process. Specifically, if the average total number of transitions
from |↑i to |↓i is N↑↓ = p↑↓ Ms , then on a given trial there will be quantum fluctuations in the totalpnumber of transitions with the standard deviation of N↑↓ given by ∆N↑↓q = p↑↓ βMs . Here thefactor of β = 2/3 accounts for the unweighted time averaging of the measurement records during
the two measurements N̂↑ that form the desired differential quantity N↑f − N↑p . In the limit that79
Ms /(N/2) is small, the noise from each process is uncorrelated, and the noise added to R is thesum of the individual variance of each process

Rpop,q =βMs hp↑↓ (α↓ − α↑ )2 + p↑1 (α1 − α↑ )2N/4i+ p↓↑ (α↑ − α↓ )2 + p↓1 (α1 − α↓ )2 .

(3.2)

We assume no multiple Raman scattering, a good assumption for the low average probe photon
number Mt used here.
Classical fluctuations in the probe photon number Mt , and hence Ms , also introduce classicalnoise in the number of Raman transitions between ground states which adds noise to N↑f − N↑p ,limiting the spin noise reduction. Also, an average population change can add technical challenges tothe experiment as the dressed cavity mode frequency ωc0 changes or chirps during a measurement.
For our experiment, the classical rms fluctuations in Ms are ∆Ms = 0.04Ms . These classicalfluctuations contribute a term to the total spin noise reduction

Rpop,c =2 h∆Mscp↑↓ (α↓ − α↑ ) + p↑1 (α1 − α↑ )N/4i2
+ p↓↑ (α↑ − α↓ ) + p↓1 (α1 − α↓ ) .

(3.3)

The noise added by classical probe power fluctuations exhibits a fortuitous cancellation in
the current experiment. The added classical noise due to the population change is reduced by afactor of ∼ 6.5 in variance due to a cancellation of the two terms containing (α↓ − α↑ ) and as seenin Fig. 3.11.
The results of both the quantum and classical noise models, combined with the determination of the four transition probabilities, predict that internal-state changing transitions have beeneffectively eliminated as a source of probe induced back-action in this system (see Table 3.1 forconstraints). Our measurements also indicate the added noise is small enough to allow for anothertwo orders of magnitude improvement in spectroscopic enhancement for future experiments.

80
By suppressing internal-state transition noise, we reveal sources of probe-induced back-actionthat were not relevant in our prior measurement-induced spin-squeezing experiments [35]. Here we
consider two sources that contribute noise to population measurements, both of which arise from thestanding wave probe field in the cavity. The first source of back-action results from dispersive forces
exerted on the atoms by the probe, which cause coherent oscillations of ωc0 (previously observedin ref. [170]). The second source results from free-space scattering of probe photons leading tophoton recoil heating, which causes a change of the dressed cavity frequency ωc0 . In this section,
we describe how these two effects contribute both classical and quantum noise to our measurementof N↑f − N↑p . Note that neither source of back-action acts as an additional source of decoherence(i.e. a loss of contrast C). While these sources of back-action do not fully account for the measured
back-action rc in our experiment, they are interesting as sources of back-action that may limit the
spin noise reduction R in future work.
In our standing wave cavity, the probe induces a spatially varying light shift on the atoms,
proportional to g 2 (z). The gradient of this light shift imparts a force on atoms not at a node oranti-node of the standing wave. When the probe is turned on, the atoms experience an impulsive
momentum kick along the axial direction and oscillate at ωax = 2π × 150 kHz in the confinementof the trapping lattice at 823 nm. Because the coupling strength g(z) is position dependent, theatomic motion causes ωc0 to oscillate at the same frequency. The oscillations damp in approximately
10 µs due to a spread in axial oscillation frequencies caused by the spread in radial position of theatoms in the confining optical lattice.
The oscillation is phase coherent between trials, and each measurement N̂↑ largely averages
over the oscillations. Nonetheless, variation in initial conditions lead to measurement noise in the
differential quantity N↑f − N↑p by causing variation in the collective oscillation. For example,
classical and quantum fluctuations in N↑ cause variation in the probe detuning from the dressedcavity mode δp = ωp − ωc0 . The detuning δp determines the nature of dynamic optomechanicaleffects on the atoms [140]. If ωc0 > ωp , the collective oscillation experiences optomechanical damping. If ωc0 < ωp , the oscillation will experience optomechanical anti-damping that lengthens theNoise Source
Observed Optimum
Photon Shot Noise rP SN
Technical Noise Floor Rt
Laser Linewidth
Classical Noise rc
Variable Damping Ro
Photon Recoil Rext,cPopulation Change Rpop,cQuantum Noise rq
Photon Recoil Rext,qPopulation Diffusion Rpop,qR−1 (uncertainty)
16(2)
32(4)
73(34)
520(250)
67(15)
620(60)4.5(5) × 103
3.2(4) × 104> 7 × 105
4.8(5)) × 105
1700(300)

81

Table 3.1: Contributions to the observed spin noise reduction R from a fit to the data versus probe
strength Mt that includes photon shot noise, a noise floor, quantum noise, and classical noise. Eachvalue is given at the optimum spin noise reduction at Mt = 4.1 × 104 . The indented rows show thecalculated contributions from the various noise sources to each term. The uncertainties are 68.3%
confidence intervals.

characteristic decay time of the observed oscillations of ωc0 (Fig. 3.3B).
A change in the damping rate can alter the degree of cancellation of the optomechanicalringing in the differential measurement N↑f − N↓p . Pre-alignment of the probe frequency to thedressed cavity resonance, described previously, was important for reducing fluctuations in the probe
detuning δp that drive these variations.
We estimate the noise contribution of probe-induced collective oscillations by comparing datawith and without the oscillations. We make the comparison by applying the probe continuously,
and defining measurement windows that are time shifted with respect to the turn on of the probe
where the ringing of ωc0 is largest. We first calculate ∆(N↑f − N↑p ) measured using the first 86µs of data, as was done in the spin-squeezing experiments. Then we calculate ∆(N↑f − N↑p ) using
data starting 90 µs after the probe turn on when the collective oscillations have largely damped
away. From the comparison of the two variances, we estimate that variable damping contributes
a term Ro to the spin noise reduction that empirically scales as Mt2 and limits R−1 to 620 (Table
3.1).
The second source of optomechanical back-action we consider here arises from free-spacescattered photons heating the ensemble in a position-dependent manner. Importantly, atoms that82
are more strongly coupled to the cavity mode are more strongly heated due to free-space scattering.
This process can be thought of as an external state-changing transition, paralleling the internal
state-changing transitions discussed previously (i.e. Eqns. 3.2 and 3.3). The rise in temperature
while probing changes the atoms’ spatial distribution, reducing the overall coupling to the cavity andshifting ωc0 . The average shift of ωc0 caused by the increase in temperature is 1.3 Hz per free spacescattered photon, making it the largest average probe-induced frequency shift of ωc0 in our system.
Thus, quantum and classical fluctuations in Ms adds noise to measurements N̂↑ , contributing therelatively small quantum Rext,q and classical Rext,c terms to the spin noise reduction R (see Table
3.1).
Again, the two identified sources of back-action, spatially-dependent photon-recoil heating
and variable damping of the optomechanical ringing, fail to predict the entirety of the measured
probe-induced back-action rc that scales as Mt2 for data in Fig. 3.3 and 3.10. We believe theremaining added noise results from the details of the collective optomechanical oscillations, which
are difficult to quantify and require further characterization. This additional noise will be importantto understand for future advances but is beyond the scope of this work.

Chapter 4

A theoretical model for a steady-state superradiant Raman laser

In this chapter, I develop a theoretical model of a steady-state Raman laser that makes threeextensions to the two-level superradiant laser model presented in Ref. [115]. These extensions
are motivated by repumping and cavity tuning effects present in the experimental work of Refs.
[24, 21, 23, 160], described subsequently in this thesis. The extensions include: (1) an imperfectatomic repumping scheme in which some population remains in an intermediate third level, (2)additional decoherence caused by Rayleigh scattering during repumping, and (3) a tuning of thecavity mode frequency in response to the distribution of atomic populations among the available
atomic states. We initially make no assumptions about operation in the good or bad cavity regime.
I explicitly show that a Raman lasing transition involving three levels can be reduced to a twolevel lasing transition, making the Raman system a relevant model system to explore the physics
of superradiant lasers. Finally, I extend the simple Raman system to the experimental system in
87 Rb, incorporating all eight atomic ground hyperfine states.

4.1

Three-Level Model

4.1.1

Deriving the Laser Equations

We begin by equations for a general three-level laser, making no assumptions about a goodcavity or bad cavity regime. The three-level model for the laser presented here is pictured in Fig.
4.1. It consists of two lasing levels denoted by excited state |ei and ground state |gi separated by
optical frequency ωeg , a third state |3i which the atoms must be optically pumped to before they

3

Γ3e
δbcav

84

e
W

γ

ωc

ΓR

g
Figure 4.1: Energy level diagram of a three-level superradiant laser using the optical transition from|ei to |gi. The emitted optical laser light (blue) is nearly resonant with the cavity mode (dashedlines), detuned from ωc by δbcav . The atoms are incoherently repumped to a third state at a rate
W . Atoms in |gi also Rayleigh scatter at a rate ΓR , but leaves them in state |gi. The incoherent
repumping from |3i to |ei at rate Γ3e completes the cycle.

can be optically pumped back to |ei, and a single optical cavity mode with resonance frequency ωc .
The cavity resonance is near the |ei → |gi transition frequency, with δbcav = ωc − ωeg . We describe
the atoms-cavity system using the Jaynes-Cummings Hamiltonian [77]Ĥ = ~ωc ĉ† ĉ + ~g(ĉ† Jˆ− + ĉJˆ+ ).

(4.1)

Here 2g is the single atom vacuum Rabi frequency that describes the strength of the coupling of theatoms to the cavity mode, set by the atomic dipole matrix element. The operators ĉ and ĉ† are thebosonic annihilation and creation operators for photons in the cavity mode. We have introduced
P
P
(q)
(q)
e(q) , and Jˆ+ = N
g (q) for the |ei to |gi
the collective spin operators Jˆ− = N
q=1 g
q=1 e
transition, assuming uniform coupling to the cavity for each atom. The index q labels the sum over
N individual atoms. We also define the number operator for atoms in the state |ki, k ∈ {e, g, 3},
P
P
(q)
(q)
(q)
as N̂k = N
k (q) and the collective spin projection operator Jˆz = 21 N
q=1 k
q=1 (N̂e − N̂g ).
P P∞
The density matrix for the atom cavity system is ρ̂ =kl
m,n=0 |k, ni hl, m| where thesecond sum is over the atomic basis states k, l ∈ {g, e, 3}, and the third sum is over the cavity

85field basis of Fock states. The time evolution of ρ̂ is determined by a master equation for the atom
cavity system1ρ̂˙ = [Ĥ, ρ̂] + L[ρ̂].
i~

(4.2)

Dissipation is introduced through the Liouvillian L[ρ̂] [115]. Sources of dissipation and associated characteristic rates include the power decay rate of the cavity mode at rate κ, the spontaneous
decay from |ei to |gi at rate γ, the spontaneous decay from |3i to |ei at rate Γ3e , and Rayleighscattering from state |gi at rate ΓR . The repumping is treated as “spontaneous absorption” atrate W , analogous to spontaneous decay, but from a lower to higher energy level. Note that here
I redefine the symbol W to refer to the repumping rate, which holds throughout the rest of thethesis. Physically, this is achieved by coupling |gi to a very short lived excited state that decays to|3i. The Liouvillian is written as a sum of contributions from the processes above respectively as
L[ρ̂] = Lc [ρ̂] + Leg [ρ̂] + L3e [ρ̂] + LR [ρ̂] + Lg3 [ρ̂]. The individual Liouvillians areκ
Lc [ρ̂] = − (ĉ† ĉρ̂ + ρ̂ĉ† ĉ − 2ĉρ̂ĉ† )
2

(4.3)

N

Leg [ρ̂] = −γ X (q) (q)(q) (q)(q)(q)(σ̂eg σ̂ge ρ̂ + ρ̂σ̂eg
σ̂ge − 2σ̂geρ̂σ̂eg
)
2

(4.4)

q=1

N

Γ3e X (q) (q)(q) (q)(q)(q)
L3e [ρ̂] = −(σ̂3e σ̂e3 ρ̂ + ρ̂σ̂3e σ̂e3 − 2σ̂e3 ρ̂σ̂3e )
2

(4.5)

q=1

N

LR [ρ̂] =ΓR X (q)
(q)
(q)
(q)
(q)
(q)
((σ̂ee − σ̂gg)ρ̂(σ̂ee− σ̂gg) − (σ̂ee+ σ̂gg
)ρ̂).
4

(4.6)

q=1

N

Lg3 [ρ̂] = −W X (q) (q)(q) (q)(q)(q)(σ̂g3 σ̂3g ρ̂ + ρ̂σ̂g3 σ̂3g − 2σ̂3g ρ̂σ̂g3 )
2

(4.7)

q=1

We obtain equations of motion for the relevant expectation values of the atomic and field
˙ Complex expectation values are indicated with script notation, while
operators using Ȯ = Tr[Ôρ̂].

86

Figure 4.2: Representing the state of the |ei, |gi two level system using a collective Bloch vector.

˙ and J˙− = Tr[Jˆ− ρ̂].
˙ We assume thereal definite expectation values are standard font, so C˙ = Tr[ĉρ̂]
unknown emitted light frequency is ωγ and factor this frequency from the expectation values for the
˘ −iωγ t and J− = J˘− e−iωγ t . The symbol ˘ indicates acavity field and the atomic polarization, C = Cequantity in a frame rotating at the laser frequency. The set of coupled atom-field equations is then˙C˘ = −(κ/2 + i(ωc − ωγ ))C˘ − ig J˘−

(4.8)

˙
˘ z
J˘− = −(γ⊥ + i(ωeg − ωγ ))J˘− + i2g CJ

(4.9)

N3
Jz
+ (2Γ3e − W + γ)J˙z = − (W + γ)
24N˘
+ (W − γ) + ig(J˘− C˘∗ − J˘+ C)
4(4.10)

Ṅ3 = − (Γ3e + W/2) N3 + W (N/2 − Jz )(4.11)In the above equations, we have combined the broadening of the atomic transition into asingle transverse decay γ⊥ = γ/2 + W/2 + ΓR /2. We have assumed no entanglement between
the atomic degrees of freedom and the cavity field in order to factorize expectation values of theform hσ̂kl ĉi = hσ̂kl i hĉi. The equations make no assumptions about the relative sizes of the various
rates, making them general equations for a three level laser, but one of the distinct differences incold-atom lasers versus typical lasers is that the transverse decay rate is often dominated by the87repumping rate γ ∼ W/2.
It is useful to represent the two-level system formed by |ei and |gi as a collective Blochvector (Fig. 4.2). The vertical projection of the Bloch vector is given by the value of Jz , and is
proportional to the laser inversion. The projection of the Bloch vector onto the equatorial plane
2 = |J˘ |2 . We refer to J as theJ⊥ is given by the magnitude of atomic polarization |J˘− |, with J⊥−⊥

collective transverse coherence of the atomic ensemble.

4.1.2

Steady-state Solutions

To understand how extending to this three-level model affects the fundamental operation of
the laser, we now study the steady-state solutions with respect to repumping rates, cavity detuning,
and Rayleigh scattering rates. The steady-state solutions assume γ  W , the regime of operation
for proposed superradiant light sources [115, 33] and the experiments of Refs. [24, 21], but makeno approximations based on the relative magnitudes of γ⊥ and κ. Thus γ⊥ ≈ W/2 + ΓR /2, butotherwise the results in the section hold for both good-cavity (κ  γ⊥ ) and bad-cavity (κ γ⊥ ) lasers. We first determine the steady-state oscillation frequency, starting by setting the time˘
derivatives in Eqn. 4.8 and Eqn. 4.9 to zero. After solving for C,C˘ = −igJ˘−
κ/2 + iδ0(4.12)

where δ0 denotes the cavity detuning from the laser emission frequency δ0 = ωc − ωγ . Substituting
the result into Eqn. 4.9, we have2g 2 Jz = (γ⊥ + i(ωeg − ωγ ))(κ/2 + iδ0 ) .

(4.13)

Since Jz is always real, the imaginary part of Eqn. 4.13 must be zero. This constrains the frequencyof oscillation to

ωγ =

ωeg
ωc
,
κ +
1 + 2γ⊥
1 + 2γκ⊥(4.14)

88
a weighted average of the cavity frequency and the atomic transition frequency.
We solve for the steady-state solutions of Eqns. 4.8-4.10 by setting the remaining time
derivatives to zero and substituting Eqn. 4.12 for C in all the equations. In this work, the amplitude
properties are our primary interest (as compared to the phase properties studied in Refs. [115, 24]),
so we further simplify the equations, at the expense of losing phase information, by considering the
2 ismagnitude of the atomic polarization |J˘− |. The equation for the time derivative of J⊥d 2ddJ⊥ = |J˘+ || J˘− | + |J˘− || J˘+ |
dt
dt
dt

(4.15)The steady-state output photon flux is just proportional to the square of the equatorial
projection

˘ 2 = J 2 Cγ .
M̄˙ c = κ|C|⊥
1 + δ002(4.16)

Here we have also defined a normalized detuning δ00 = δ0 /(κ/2) and a single particle cavity cooperativity parameter

C=

(2g)2
κγ

(4.17)

that gives the ratio of single-particle decay rate from |ei to |gi for which the resulting photon isemitted into the cavity mode, making C equivalent to the Purcell factor [149].
After these substitutions and simplifications, the steady-state solutions (denoted with a bar)
are

02

2
J¯⊥
=



N
2

2 

2γ⊥ (1 + δ0 )
J¯z =
2Cγ

(4.18)W (1 + δ002 )N Cγ(4.19)

2r
1/2 + r2γ⊥ (1 + δ002 )
1−
N Cγ

89


2 
J¯⊥
Cγ
N̄3 =
r
W (1 + δ002 )M̄˙ c =



N
2

2 

2r
1/2 + r



W
N



2γ⊥ (1 + δ002 )1−
N Cγ(4.20)(4.21)

written in terms of the repumping ratio r ≡ Γ3,e /W . Note that r also determines the steady-state
2 andbuild up of population in |3i as N̄3 /N̄g = 1/r. To succinctly express the modification of J¯⊥M̄˙ c due to inefficient repumping, we define the reduction factor

R(r) ≡r.
1/2 + r(4.22)

which appears in Eqns. 4.19 and 4.21 above.
Next we discuss the behavior of these solutions for the characteristic parameters of the threelevel model: W , r, δ00 , and ΓR . The results are illustrated in Figs. 4.3-4.5.
First, we focus on the impact of repumping on the steady-state behavior. The photon fluxM̄˙ c follows a parabolic curve versus the ground state repumping rate W (Fig. 4.3). In the limitδ00 → 0, r → ∞, and ΓR → 0, Eqn. 4.21 reduces to the result for the simple two-level model of Ref.
[115]. This limit is shown as the black curve in part (a) of Figs. 4.3-4.5. At low W , the photon
flux is limited by the rate at which the laser recycles atoms that have decayed to |gi back to |ei.
At high W , the photon flux becomes limited by the decoherence from the repumping, causing theoutput power to decrease with increasing W . When the atomic coherence decays faster than thecollective emission can re-establish it, the output power goes to zero. This decoherence limit isexpressed in the condition for the maximum repumping threshold, above which lasing ceases:

Wmax =N Cγ− ΓR .
1 + δ002(4.23)The output photon flux is optimized at Wopt = Wmax /2. Notice that the maximum repumping rate
is not affected by r. However, the additional decoherence (here in the form of Rayleigh scattering)

90
(a)

1.0

r

]lvl2

¥

P[ cM xulF notohP

0.8
1

0.6
0.5

0.4
0.2

0.2
0.1

0.0
0.0

0.2

0.4

0.6

0.8

Ground State Repumping Rate

(b)

1.0

W [NCg]1.0]lvl2

P[ cM xulF notohP

0.8
0.6
0.4
0.2
0.0
0.01

0.1

1

Repumping Ratio

10

100

r

Figure 4.3: (a) Steady-state photon flux M̄˙ c versus ground state repumping rate W , with a seriesof curves showing the effects of repumping through the additional state |3i.The case of r = ∞ is
the two level model of Ref. [115] (black). (b) Photon flux M̄˙ c versus r with W = Wopt . For all
curves, δ00 = 0 and ΓR = 0, and the photon flux is plotted in units of P2lvl = N 2 Cγ/8.

N Cγlowers the turn-off threshold. If ΓR > 1+δ
02 , the decoherence will prevent the laser from reaching0superradiant threshold regardless of W .
In Figs. 4.3-4.5, we plot Eqn. 4.21 emphasizing (a) the modification to the photon flux
parabola, and (b) the optimum photon flux as a function of the population in the third state(as parameterized by the repumping ratio r), detuning of the cavity resonance from the emission
frequency δ, and additional decoherence from Rayleigh scattering ΓR . The photon flux is plottedin units of the optimum photon flux in the two-level model of Refs. [115, 114], P2lvl = N 2 Cγ/8.
As the repumping process becomes more inefficient and population builds up in |3i, parameterized by r as N̄3 /N̄g = 1/r, we see from Eqns. 4.19 and 4.21 that the photon flux M̄˙ c decreases(Fig. 4.3). A repumping ratio r = 10 ensures the laser operates within a few percent of its maximum output power. Notice that M̄˙ c saturates after r is greater than ≈ 2. Although inefficient
repumping suppresses M̄˙ c , the optimum and maximum repumping rates Wopt and Wmax are notmodified.

91
(a)

]lvl2

1.0
d'0

0

P[ cM xulF notohP

0.8
0.25

0.6

0.5

0.4
1

0.2
2

0.0
0.0

0.2

0.4

0.6

0.8

Ground State Repumping Rate

(b)

1.0

W [NCg]1.0]lvl2

P[ cM xulF notohP

0.8
0.6
0.4
0.2
0.0
0.0

0.5

1.0

1.5

2.0

Detuning d'0

Figure 4.4: (a) Steady-state photon flux M̄˙ c versus ground state repumping rate W , with a seriesof curves showing the effects of detuning of the cavity resonance frequency from the emitted lightfrequency δ00 . (b) Photon flux M̄˙ c versus δ00 with W = Wopt (δ00 ). The photon flux is plotted in unitsof P2lvl = N 2 Cγ/8. For all curves, r = ∞ and ΓR = 0.

(a)

1.0
]lvl2P[ cM

GR

0

0.8
0.6

0.2

xulF notohP

0.4

0.4

0.2

0.6
0.8

0.0
0.0

0.2

0.4

0.6

0.8

1.0

Ground State Repumping Rate W [NCg]

(b)

1.0
]lvl2P[ cM

0.8
0.6

xulF notohP

0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

Rayleigh Scattering Rate GR [NCg]Figure 4.5: (a) Steady-state photon flux M̄˙ c versus ground state repumping rate W , with a seriesof curves showing the effects of decoherence in the form of Rayleigh scattering from the groundstate |gi. (b) Photon flux M̄˙ c versus ΓR with W = Wopt (ΓR ). The photon flux is plotted in unitsof P2lvl = N 2 Cγ/8. For all curves, r = ∞ and δ00 = 0.

92
The preservation of the operating range can be important, as in practice large values of r
can lead to added decoherence (due to intense repumping lasers for example), which does reduce
the operating range. Lowering the value of r allows some flexibility as some output power can besacrificed to keep the laser operating over a wider range of W.
Cavity detuning modifies both the M̄˙ c and Wopt (Fig. 4.4). The modification arises from theδ00 dependent cavity cooperativity

C0 =

C
.
1 + δ002(4.24)The modified cooperativity C 0 originates from the atomic polarization radiating light at ωγ , which
non-resonantly drives the cavity mode with the usual Lorentzian-like frequency response. Thus,
the output photon flux M̄˙ c , turn-off threshold Wmax , and optimum repumping rate Wopt all scale
like 1/(1 + δ002 ). This effect is symmetric with respect to the sign of δ00 . Physically, the rate a single
atom spontaneously decays from |ei to |gi by emitting a photon into the cavity mode is Γc ≡ C 0 γ,
which we use to simplify some later expressions.
Finally, we examine the effect of additional atomic broadening through ΓR in Fig. 4.5.
Additional broadening linearly reduces Wopt and Wmax , but because we require the repumping rate
to remain at Wopt in Fig 4.5b, M̄˙ c has a ( NΓΓRc − 1)2 dependence.
The key insight from the steady-state solutions for our three-level model is that imperfectionsin the lasing scheme can quickly add up, greatly reducing the expected output power of the laser.
A repumping scheme should be chosen to minimize Rayleigh scattering ΓR and maximize therepumping ratio r. Added decoherence, as well as the detuning δ00 are especially problematic
because they restrict the possible range of W for continuous operation. The results of this section
were crucial for understanding the operation of the experiments described in Chapters 5, 7, and 8.

93

4.2

Raman laser system
In the previous section, I presented a model for a three level laser for qualitatively describing

the results from recent experiments that use laser cooled 87 Rb as the gain medium [24, 21, 23].
However, the 87 Rb system also relies on a two-photon Raman lasing transition between hyperfine
ground states, instead of a single optical transition. To address this difference, here we provide amodel that has a two-photon Raman lasing transition, but a simple one-step repumping scheme
directly from |gi to |ei. Then in Sec. 4.3, we present a full model of the bad-cavity laser in 87 Rb
that has both the two-photon Raman transition and a more complex repumping scheme.
In Sec. 4.2.1, we derive equations of motion for the expectation values in the Raman model,
then explicitly adiabatically eliminate the optically excited intermediate state in the Raman transition. In Sec. 4.2.2 I establish the equivalences (and differences) between the Raman and non-Raman
models. We will find that the Raman transition is well described as a one-photon transition witha spontaneous decay rate γ, an effective atom-cavity coupling g2 , and with a two-photon cooperativity parameter C2 equal to the original one-photon cooperativity parameter. The Raman systemdiffers in the appearance of two new phenomena: differential light shifts between ground states andcavity frequency tuning in response to atomic population changes. The latter effect was inserted
by hand in Sec. 4.1. As in Sec. 4.1, we first derive equations without assuming a good-cavity or
bad-cavity laser, only specializing to the bad-cavity limit at the end of the section.

4.2.1

Adiabatic Elimination

To establish the connection between two-photon Raman lasing and one-photon lasing, westart by defining the Hilbert space for a three-level Raman system with two ground states denoted|gi and |ei (separated by only 6.834 GHz in 87 Rb) and an optically excited intermediate state|ii (Fig. 4.6). The Hilbert space also includes a single cavity mode that couples |gi to |ii. TheP
P P∞ (q)density operator for the Hilbert space is ρ̂ = N
l(q) , m . The first sum is
q=1kl
mn k , nover individual atoms, the second is over the atomic basis states k, l ∈ {i, e, g}, and the third sum

94

δ0
∆d

∆c
Γ

ωc, 2ggWiωd, Ωdωeg

eFigure 4.6: Energy level diagram for a superradiant laser enabled by an induced Raman transition.
States |ei and |gi are two metastable states separated by a non-optical frequency ωeg . They sharean optically excited state |ii that has a linewidth Γ. Using a Raman dressing laser (green), detuned
from |ii by ∆d , we can induced a optical decay to |gi, which, in absence of collective effects, would
 2proceed at rate γ = Γ4 Ω∆d . Including a single optical cavity mode, coupled to the |ii to |gi
transition with coupling constant 2g, gives rise to collective emission. The cavity mode frequency
is ωc , detuned from |ii by ∆c , making the two-photon detuning δ0 = ωc − (ωd + ωeg ). To complete
the laser cycle, the atoms are incoherently repumped from |gi to |ei at a rate W .

95
is over cavity Fock, or photon-number, states. Raising and lowering operators for the cavity field
and atoms are defined as in Sec. 4.1. The state occupation operators for atoms in the state |ki areP(q)again N̂k = N
k (q) , where the index q denotes a sum over individual atoms. We also
q=1 k
P
(q)
l(q) .
define collective atomic raising and lower operators Jˆkl = N
q=1 k
We describe the system via the semi-classical Hamiltonian

H = ~ωc ĉ† ĉ + ~ωi N̂i + ~ωe N̂e + ~ωg N̂gΩd (t) ˆ(Jei + Jˆie ) + ~g(ĉ† Jˆgi + ĉJˆig ).
+~2(4.25)The Raman dressing laser at frequency ωd is described by the coupling Ωd (t) = Ωd (e−iωd t + eiωd t ),and the atoms are uniformly coupled to the dressing laser. The rotating wave approximation will beapplied so that only near-resonant interactions will be considered. The dressing field is externallyapplied, and we assume it is unaffected by the system dynamics (i.e., there is no depletion of the
field).
To reduce the Raman transition to an effective two-level system, we derive the equations of
motion for expectation values of the operators that describe the field and the atomic degrees of
freedom. As was done in Sec. 4.1, we use the time evolution of the density matrix obtained fromthe master equation (Eqn. 4.2) to derive the equations of motion Ȯ = Tr[Ôρ̂]. The details areincluded in Appendix A.
After adiabatic elimination of the optically excited state, we have the set of three coupledequations analogous to Eqns. 4.8-4.10:


 2

g
gΩd
−κ/2 − iNg + ωc
C−i
Jge e−iωd t
∆
2∆

 2

Ωd
g 2 |C|2
˙
−
+ ωeg
Jge
Jge = −γ⊥ − i
4∆
∆
gΩd
+ i2
Jz Ceiωd t
2∆gΩd ∗J˙z = W (N/2 − Jz ) + i(C Jge e−iωd t − CJeg eiωd t )2∆
C˙ =(4.26)(4.27)(4.28)

96
Here γ⊥ = W/2 and ∆ = ∆d + (δ0 /2), which is also the average of ∆d and ∆c . The Eqns.
4.26-4.28 are general equations, valid without assuming a good-cavity or bad-cavity laser.

4.2.2

Defining Effective Two Level Parameters for the Raman System

We can now identify the effective two-photon atom-cavity coupling constant

g2 =

gΩd
.
2∆

(4.29)The effective Rabi flopping frequency between |ei and |gi is just 2g2 .
Using this coupling constant, we can also construct an effective cooperatively parameter forthe two-photon transition using C2 = (2g2 )2 /κγ, where

Γ
γ=
4



Ωd
∆

2
(4.30)

is the decay rate for an atom in |ei to |gi induced by the dressing laser, calculated for large
detunings. Substituting Eqns. 4.29 and 4.30 into the above expression for C2 , one finds that thetwo-photon cooperatively parameter and the one-photon cooperatively parameter (Eqn. 4.17) are
identical C2 = C = (2g)2 /κΓ. This is explained by the geometric interpretation of C, a ratio
which is determined by the fractional spatial solid angle subtended by the cavity mode and theenhancement provided by the cavity finesse F which enters through the value of κ ∝ 1/F [149].
The adiabatic elimination yields the two-photon differential ac Stark shift of the frequency
difference between |ei and |gi

ωac =

Ω2d
g 2 |C|2−,
4∆∆(4.31)

seen in Eqn. 4.27. The two contributions to ωac correspond to virtual stimulated absorption anddecay. The same virtual process also acts back on the cavity mode creating a cavity frequency as
seen in Eqn. 4.26. The shift corresponds to a modification of the bare cavity resonance frequency,
leading to a new dressed cavity resonance ωD given by

97ωD = ωc + Ng

g2.
∆

(4.32)

This is the cavity frequency tuning in response to atomic populations artificially introduced in Sec.
4.1. We have assumed that only an atom in |gi couples to the cavity mode, but in reality both
g2

2

ge
, where we havestates may couple to the cavity mode such that in general ωD = ωc + Ng ∆gg + Ne ∆especified independent populations, coupling constants, and detunings for the two states |ei and |gi
denoted by subscripts. For tractability in Sec. 4.1’s three-level model, we assumed the pre-factors
g 2 /∆ were equal in magnitude but opposite in sign so that cavity frequency tuning could be written2

as g∆ (Ne − Ng ) = αJz .
As in Sec. 4.1, we also determine the steady-state frequency of the laser

ωγ =

2γ⊥
κ
ωD +
(ωeg + ωd − ωac ) .
2γ⊥ + κ2γ⊥ + κ(4.33)and define δ = ωD − ωγ as the detuning of the emission frequency ωγ from the dressed cavity mode
ωD .
Here we see that, in general, both the atomic transition frequency tuning from ωac and thecavity frequency tuning in ωD are important for the laser amplitude dynamics. Comparing theexpressions for ωac and ωD , both scale with g 2 /∆, and the determining degrees of freedom are therelative number of atomic to photonic quanta. In good-cavity systems, a large number of photons
can build up in the cavity, and the frequency tuning dynamics are dominated by the ac Stark shift[156]. Superradiant lasers, operating deep in the bad-cavity regime, can operate with less than one
intracavity photon on average [24], resulting in a system with amplitude dynamics dominated bydispersive tuning of the cavity mode from population [21]. Additional energy levels that couple tothe dressed cavity mode ωD can result in a proliferation in the degrees of freedom for dispersive
cavity tuning, resulting in a much richer system than one dominated by ac Stark shifts, which
depend only on |C|2 .
To complete the analogy to the non-Raman lasing transitions from the previous section and98arrive at equations for the bad-cavity laser dynamics, here we make the bad-cavity approximationκ  2γ⊥ . We again adiabatically eliminate the cavity field amplitude, assuming that it varies
slowly compared to the damping rate. We define a normalized detuning δ 0 = δ/(κ/2), and use thecooperativity parameter C to describe the coupling. After simplifying, we have a two-level systemanalogous to Eqns. 6.1 and 6.2 in Sec. 4.1

d2Cγ
Jz |Jge |2|Jge |2 = −2γ⊥ |Jeg |2 +dt
1 + δ 02dCγ
Jz = W (N/2 − Jz ) −|Jge |2 .
dt
1 + δ 02(4.34)(4.35)

Note that in the bad-cavity limit, the detuning of the dressed cavity mode from the emission
frequency δ is to good approximation the difference of the dressed cavity resonant frequency andthe dressed atomic frequency, modified by a small cavity pulling factorWδ ≈ (ωD − (ωeg − ωac + ωd )) 1 −.
κ

(4.36)

Our conclusion is that a Raman superradiant laser can perform as a single-photon superradiant laser with C2 = C, but with a transverse collective coherence that evolves a quantum phaseat a frequency set by the separation of the two ground states. This means that while superradiant
Raman lasers based on hyperfine transitions may not be useful for optical frequency references,
their tunability and control make them excellent physical “test-bed” systems for studying cold
atom lasers [156, 24, 21]. In addition, the switchable excited state lifetime in a Raman system
introduces the possibility of dynamic control in the superradiant emission, useful for novel atomic
sensors [23, 160].

4.3

Full Model in 87 Rb
In this section, we give the results of a model for a superradiant Raman laser using the groundstate hyperfine clock transition (|gi = 52 S1/2 , F = 1, mf = 0 , |ei = 52 S1/2 , F = 2, mf = 0 ) in
87 Rb, including all eight ground state levels for repumping.

The results include specific values

99

F’=2
F’=1

δ0

∆

ΓD1
ωd
ωc

γ
F=2

ωhf
F=1
mf= -2

-1

0

1

2

Figure 4.7: Lasing transition and Raman dressing scheme on the 87 Rb D1 line (795 nm). The
dressing light (red) and collective emission (blue) are a superposition of σ + and σ − polarizations
because the direction of propagation of the light is along the quantization axis defined by the
direction of the magnetic field at the atoms. The Raman dressing laser is detuned by ∆ fromthe atomic transition. The bare cavity detuning is δ0 = ωc − (ωd + ωhf ). The optically excitedstate on the D1 line has a linewidth ΓD1 /2π = 5.75 MHz. The effective population decay from 2Ωd|F = 2, mf = 0i to |F = 1, mf = 0i is γ = ΓD1.
4∆

Step 1

Step 2

F’=2

ΓD2

F’=1

F=2
F=1

mf= 0

1

2

mf= 0

1

2

Figure 4.8: Two step repumping process on the 87 Rb D2 line (780 nm). The diagram is drawn
showing on only positive Zeeman states, but the process is symmetric with respect to mf = ±1, ±2.
The desirable decay branches (magenta) show the most direct repumping sequence, although any
particular repumping sequence could go through many ground hyperfine states. The optically
excited state on the D2 line has a linewidth ΓD2 /2π = 6.07 MHz.

100
of parameters taken from the superradiance experiments in this thesis. The model combines thethree-level repumping from the model in Sec. 4.1 and the Raman transition between |ei and |gi
of Sec. 4.2. After summarizing the key steady-state results, we use linear response theory similar
to Sec. 4.1 to examine the stability of the laser, identifying the important parameters for stable
operation in superradiant Raman lasers.

4.3.1

Continuous Superradiant Raman Laser in 87 Rb

We model steady-state superradiance in the full 87 Rb Raman system by first including incoherent repumping among the eight ground 5 2 S1/2 hyperfine populations NF,mF (Fig. 4.8). We use
λ = {F, mF } to refer to a generic set of population quantum labels as Nλ .
The repumping is performed using single particle scattering off optically excited states toresult in Raman transitions to move population from F = 1 to F = 2 (Fig. 4.8). The repumping
has a clear analogy to the three-level model from Sec. 4.1 because population cannot be directlytransfered from |ei to |gi, meaning some finite population accumulates outside the coherent lasinglevels. Separate lasers repump atoms in the F = 1 state (green) and the F = 2 state (purple).
The lasers are characterized by Rabi frequencies Ω1,0,2,0 and Ω2,1,2,1 respectively. The rate of
population transfer out of |gi is proportional to the total scattering rate W , which includes theRayleigh scattering rate. The transverse decoherence rate γ⊥ = W/2 is dominated by the necessary
scattering from repumping. In analogy with the model in Sec. 4.1, the repumping rates out of thestates in F = 2 are proportional to rW , where r = Ω22,1,2,1 /Ω21,0,2,0 . The detailed equations for the
repumping are given in Appendix B.
To include the collective emission in our 87 Rb Raman laser model, we reduce the Raman
transition dynamics to an effective two-level model by eliminating the optical intermediate state(see Sec. 4.2). The hyperfine ground states |gi and |ei form the effective two-level transition shownin Fig. 4.7. The optical transition is induced by a 795 nm dressing laser with Rabi frequency Ωd
far detuned from the |ei → 52 P3/2 , F 0 = 2, m0F transition (∆ is typically 1-2 GHz). The dressing
 2Ωdlaser creates an effective spontaneous scattering rate from |ei to |gi γ = ΓD1(see Eqn. 4.30
4
∆

101
in Sec. 4.2). The fraction of this single particle scattering that goes into the cavity mode is givenby the cooperativity parameter C = 8 × 10−3 .
2Single particle scattering in the cavity mode results in a build up of collective coherence J⊥

between |ei and |gi. The collective emission has an enhanced scattering rate which dominates the
population transfer from |ei to |gi. We include the population transfer from collective emission
along with the equation for the collective coherence Eqn. 4.34 with the population equations fromrepumping to form the set of equations used to obtain the steady-state solutions and perform thelinearized analysis. We give the details in Appendix B.

4.3.2

Steady-state Solutions

In analogy to the model in Sec. 4.1, we are concerned with steady-state values of the inversion
2 , and the population that occupies energy
Jz = 21 (N2,0 −N1,0 ), the collective transverse coherence J⊥

levels outside the laser transition Nother = N −N2,0 −N1,0 . The steady-state solutions of the systemequations are

2
J¯⊥
=



N
2

2

3
13 r
27
104 + r!

2W (1 + δ 02 )
N Cγ



W (1 + δ 02 )1−
N Cγ

(4.37)02

W (1 + δ )
J¯z =
2Cγ


N̄other = N

4
13



27
32 + r27
104 + r2M̄c = J¯⊥!W (1 + δ 02 )1−
N Cγ

(4.38)

Cγ.
1 + δ 02

(4.39)(4.40)

Here δ 0 is the detuning of the dressed cavity mode from the laser emission frequency.
As in Sec. 4.1, there is again both a repumping rate that maximizes the coherence (alongwith the output photon flux) and a repumping threshold for laser turnoff

102

N Cγ
,
2(1 + δ 02 )(4.41)

Wmax = 2Wopt .

(4.42)

Wopt =To understand the effect of repumping in the full 87 Rb model, we compare Eqn. 4.37 to
the steady-state coherence in the three-level model, Eqn. 4.19 in Sec. 4.1. While the form of theexpression versus the ground state repumping rate W is the same as the three-level model, thescale factor associated with the repumping ratio is modified. The power reduction factor

3
RRb (r) ≡13

r
27
104 + r!
(4.43)

is the modification to the steady-state photon flux compared to the ideal model in Refs. [115]. RRb
has a maximum value of 3/13 contrasted with R, Eqn. 4.22, which has a maximum of 1.
While the repuming ratio r 6 1, most of the population remains in N2,0 and N1,0 (Eqn.
4.39). The inversion Jz is the same as the model from Sec. 4.1 (Eqn. 4.18), as here W corresponds
to 2γ⊥ . Thus, the results of Figs. 4.3-4.5 give a good qualitative understanding of the steady-statebehavior of the 87 Rb system as well.

4.4

Conclusion
We have developed a minimal model for a steady-state, superradiant laser that includes

key features of observed in recent experiments using 87 Rb [24, 21, 23, 160]. The model describes
the reduction in the laser output power with the repumping ratio r, the cavity-atomic transition
detuning δ, and an additional source of decoherence, such as caused by Rayleigh scattering ΓR .
The explicit elimination of an intermediate excited state in our Raman laser theory shows that a
Raman laser can serve as a good physics model for lasers operating deep into the bad-cavity regime.
The adiabatic elimination also reveals the source of the crucial atomic and cavity frequency tunings
that can play a key role in the amplitude stability of Raman lasers, both in the bad-cavity [21] andgood-cavity [156] regimes.

103
In addition to explaining experimental observations in previous work, this paper serves as a
guide for the design of other cold-atom lasers and superradiant light sources that utilize nearlyforbidden optical transitions [115, 33]. Our minimal model includes a multi-step repumping process
and shows the path to adding more energy levels or repumping steps as required for realistic
experimental systems. Many of the results here do not assume a good-cavity or bad-cavity laser,
making them general results that can be followed until simplified expressions based on a particularlaser regime are required.
In general, superradiant laser designs should strive to eliminate sources of decoherence, suchas Rayleigh scattering or differential ac Stark shifts from repumping light, while maintaining efficient repumping that avoids accumulation of population outside the atomic energy levels of thelasing transition. The steady-state and amplitude stability properties of cold-atom lasers can besignificantly modified by their repumping scheme.
Future designs may also apply optical dressing techniques to induce decay of the excited state[160, 23]. In such Raman systems, the dressing of the cavity-mode can provide positive or negative
feedback for stabilizing the output power of the laser. The dressed cavity mode also can pull
the laser emission frequency, serving as an amplitude noise to phase noise conversion mechanism.
Future theoretical and experimental work, beyond the scope of this paper, can extend the linear
response theory presented here to incorporate quantum noise in the repumping process. Cavity
frequency pulling and quantum noise in the dressing of the cavity mode are possible sources of thelaser linewidth broadening observed in Ref. [24], where the observed linewidth exceeded the simple
Schawlow-Townes prediction [115].

Chapter 5

A steady state superradiant laser with less than one intracavity photon

In this chapter, I describe the experimental implementation of a superradiant Raman laser
source in which spontaneous synchronization of more than 106 87 Rb atomic dipoles is continuously
sustained by as few as 0.2 photons on average inside the optical cavity. By operating at low intracavity photon number, I demonstrate isolation of the collective atomic dipole from the environment
by a factor > 104 . The emitted light has a frequency linewidth, measured relative to the Raman
dressing laser, below that of single-particle decoherence rates and more than 104 times below thequantum linewidth limit typically applied to good-cavity, optical lasers [136]. These results demonstrate several key predictions relevant for future superradiant lasers that may be used to improve
the stability of passive atomic clocks [78] and have the potential to open up searches for new physics[16, 55]. This chapter is based on the work in Ref. [24].

5.1Superradiant Lasers
The fundamental linewidth of a laser is given by a generalized expression for the Schawlow-

Townes full width at half maximum (FWHM) ∆fST in Hertz by Kolobov et al. [92] and in a
simplified form by Kuppens et al. [97] as

1 hf∆fST =
4π Pout2γ⊥ κ2γ⊥ + κ

2
,

(5.1)

where Pout is the power exiting the cavity, f is the oscillation frequency, h is the Planck constant, andκ is the cavity power decay rate. The transverse decoherence rate of the lasing optical transition is

105Superradiant Laser

Standard Laser

a

b
Phase
Information

Back-action

Thermal Vibrations

15
10
5
0
0.0

0.5

1.0

1.5

Time (µs)

2.0

25
20
15
10
5
0

e

e

0

10

20

30
Time (ms)

40

50

60

Power Out (photons x109 s-1)dIntracavity Photons Mc

Intracavity Photons Mc(x103)

c

γ eg

i

g

6.8 GHz

4
3
2
1
0
0.0

0.5

1.0

1.5

Atom Number N (x106)

Figure 5.1: A steady state superradiant laser. a, (left) In a good-cavity laser far above
threshold, many photons (yellow) circulate inside the cavity, extracting energy from the largely
incoherent atomic gain medium (blue). Thermal vibrations of the mirror surfaces modulate thecavity resonance frequency, limiting the linewidth of the laser. In a superradiant laser (right), thecollective atomic dipole stores the coherence, and continuous stimulated emission can be achieved
even with less than one photon in the cavity. The stimulation enables phase information to be
extracted at a useful rate, while the small intracavity photon number leads to only weak cavityinduced backaction on the collective atomic dipole. b, To mimic a narrow optical atomic transition,
we dress the metastable ground state labeled |ei with a laser (blue) to induce a spontaneous
two-photon Raman transition to |gi, with tunable rate γeg . With no repumping light, a single
superradiant pulse is emitted. d, With repumping light applied, we observe quasi-continuous
emission limited by atom loss. The atoms emit into a single spatial mode of the cavity (TEM00 )imaged on a CCD (inset). e, The measured peak power output (black circles) scales as the numberof atoms squared.The red line is a quadratic fit to the data.

106γ⊥ = γeg /2 + 1/T2 , where γeg is the decay rate from excited to ground state and 1/T2 parameterizes
additional atomic dephasing mechanisms.
Optical lasers have typically operated in the good-cavity limit 2γ⊥ >> κ (Fig. 5.1a) due toDoppler broadening or broadening due to interactions with crystal lattices, for instance. In thislimit, the generalized linewidth expression reduces to ∆fGST = κ/(4πMc ), where Mc is the average
intracavity photon number (see Sec. 5.6).
If the cavity and atomic transition frequencies, fcav and fatomic , are not identical, the systemoscillates at a weighted average frequency fγ = (2γ⊥ fcav +κfatomic )/(2γ⊥ +κ). The cavity frequency
pulls the oscillation frequency away from the atomic transition frequency by an amount P ≡df /dfcav = 2γ⊥ /(2γ⊥ + κ). In the good cavity limit, the emission frequency tracks the empty cavity
frequency as P ≈ 1.
2 /(πκM ), a
In the bad-cavity limit (2γ⊥  κ), the FWHM linewidth reduces to ∆fBST = γ⊥cresult intimately connected to cavity narrowing using slowed light [97, 158]. The collective atomlight excitation is stored predominantly in the gain medium, making the atoms the primary carrier ofphase information (Fig. 15.1a). The weak intracavity photon field acts mainly as a communication
bus to drive spontaneous synchronization of the atomic dipoles and to extract information about
the phase stored in the collective atomic dipole. The synchronized atomic dipoles radiate at anenhanced rate, a phenomenon known as superradiance or superfluorescence [43]. In addition, thefrequency pulling becomes P ≈ 2γ⊥ /κ  1, dramatically reducing the impact of noise in the cavity
frequency. This isolation of the oscillator from the environment is the key to reducing the sensitivityof such a laser to thermal and technical noise.

5.2

Experimental System
Our experimental system, described in Sec. 5.6 and Chapter 2, operates deep into the opticalbad-cavity regime, with 2γ⊥ /κ = 2 × 10−5 to 10−3 . The effective excited state scattering rate
γeg ≈ 2 to 60 s−1 is comparable to some optical clock candidates, and the average intracavity
photon number Mc ranges from 0.2 to 60. The system consists of an ensemble of N ≈ 106 87 Rb

107
atoms confined to the TEM00 mode of a medium finesse (F = 710) optical cavity with cavity
power decay rate κ = 2π × 11 MHz. Optomechanical effects are suppressed by tightly trapping theatoms using a one-dimensional intracavity optical lattice at 823 nm. We observe no lasing at thevibrational sideband frequencies.
The cavity is coupled to an optically dressed atomic ground state engineered to imitate
a long-lived, optically excited atomic state [115, 33] (Fig. 15.1b). The effective excited |ei andground |gi states are the magnetic field insensitive ground hyperfine clock states. The upper state|ei is metastable and separated from |gi by only 6.8 GHz, but we induce an optical Raman decay
from |ei to |gi by applying a linearly polarized 795 nm dressing laser tuned 1 to 2 GHz higher infrequency than the optically excited intermediate state |ii. The cavity is tuned to resonance withthe spontaneously emitted photon. The combination of detuning and dressing laser power sets atunable single particle decay linewidth γeg , the rate from |ei to |gi. The cycle is completed using
780 nm light to incoherently repump atoms from |gi to |ei at a rate proportional to the single
particle rate,w, out of the ground state.
The superradiant linewidth ∆fBST reduces to Cγeg /π at a peak repumping rate wpk =N Cγeg /2. The cooperativity is C = (2g2 )2 /κγeg , where g2 is the single particle, two-photon
coupling of the atoms to the cavity emission mode after adiabatic elimination of the intermediate
state |ii. It should be possible to obtain ∆fBST  γeg by operating in the single particle, weakcoupling regime C  1, a limit that contrasts with the trend in quantum optics to engineer systems
with single-particle, strong-coupling C  1. It is predicted that strong collective coupling N C  1
is sufficient for self-sustained stimulated emission. In this work, we operate with C = 7.7(4) × 10−3
and N C ≈ 104 (all reported uncertainties describe 68% confidence intervals).

5.3

Results
A laser operating in the superradiant regime assumes continuous light generation, but opti-

cal superradiance is regularly regarded as an intrinsically pulsed process with no phase coherence
between the pulses. A single superradiant pulse of photons is emitted if all atoms are initial-

(a)

108

(b)

Figure 5.2: The laser can operate in a quasi-steady-state mode in that each atom can go throughan emission-repumping cycle multiple times. However, the emitted power decreases over time,
eventually going to zero. In (a), we show one example of emission lasting ∼ 10 ms. Under otherconditions, we have observed emission lasting up to 120 ms. We verify that atoms are being lostfrom the trap (b) by measuring a changing frequency shift of the cavity mode consistent with atomloss. The loss is not associated with collective emission, as shown by the equivalence of the loss
observed with (red) collective emission or (blue) induced single particle emission from |ei to |gi via
free space scattering at the same rate as the collectively enhanced decay rate. The atom number
is in units of the initial atom number N0 .

ized in |ei and the dressing laser is applied (Fig. 5.1c). This pulse contains approximately 0.9(1)photons/atom. Without a pump to recycle the atoms to |ei, this pulse is the full extent of thesuperradiance.
If instead we apply continuous optical repumping from |gi to |ei along with the dressing laser,
we observe a transition to quasi-continuous superradiance. We verify that the signal is not just asingle pulse as the integrated number of photons emitted into the cavity mode is approximately
35(4) photons/atom for the data in Fig. 5.1d. By varying the excited state lifetime and repumping
rates, we were also able to observe superradiant emission lasting as long as 120 ms. By measuring
a temperature of the atoms both before and after the superradiant emission, we see a increase intemperature from 16(1) µK to 30(2) µK. Given the trap depth of 351 µK, this is only a minor
increase in temperature. The light source shuts off as atoms are lost from the lattice, most likely

109
due to light assisted atomic collisions.
We measure the atom loss by starting superradiant emission and then switching off the
dressing and repumping laser after a variable emission time. Turning off the lasers stops the emission
and freezes the state populations. Immediately afterwards, we probe the resonance frequency ofthe cavity, which is dispersively shifted by the atoms. The non-destructive probing is similar tothe probing used in spin squeezing experiments described in Chapter 3, and the details for thisexperiement are included in Section 5.6.1.
Fig. 5.2(a) shows the emitted power approaching zero after lasing for ∼ 10 ms. The measured
cavity shift decreases the longer the laser is allowed to oscillate, indicating that either atoms arebeing lost from the trap or are having their coupling to the cavity mode reduced. We verify that
atoms are actually being lost from the trap by also performing fluorescence measurements in thesame experiment. The decay of the fluorescence signal is consistent with loss of atoms from thetrap. The loss of atoms is shown in Fig. 5.2(b).
The lasing quenches after only half the atoms are lost. This is explained by the fact that the
fixed repumping rate W was initially tuned to an optimum value W ≈ N0 Cγeg /2 for the initial N0
atoms in the trap. The maximum repumping rate Wmax = N Cγeg , above which lasing is quenchedas discussed below, then crosses below the fixed repumping rate W when the changing atom number
N meets the condition N < N0 /2.
Lastly, in Fig. 5.2(b), we present data showing that the loss of atoms is not enhanced by
the superradiant process itself. To observe the loss of atoms without superradiant emission, wetune the dressing laser closer to atomic resonance, increasing the single particle scattering to arate comparable to the collectively enhanced rate. We also move the cavity resonance frequency
far from the atomic transition frequency to inhibit any collective enhancement of the decay. Afterrunning this non-collective emission for a variable time, we measured the atom number (Fig. 5.2(b),blue circles). The atom loss was the same as in the case with collective emission (Fig. 5.2(b), redcircles). Because we observe loss from the trap that is not connected to collective emission, we
suspect that light assisted collisions, either from the dressing laser or the repumping lasers, are

110

a

30

20

25
20

15

15

10

10

5

5
0

0

Repumping Rate, w(x103 s-1)b

Decay rate , γeg (s-1)

Intracavity Photons, Mc

25

25

0

50

75

100

125

Repumping Rate, w (x103 s-1)

150

175

200

200
150

wmax

100

wpk

50
0

0

5

10

Decay rate from

15

20

to

, γeg (s-1)

25

30

Figure 5.3: Repumping-induced quenching a, The average number of intracavity photons Mc
versus the repumping rate w for different decay rates γeg . The inferred Bloch vector is shown for
two operating points. b, The repumping rate above which superradiance is quenched wmax (green)
and the repumping rate at peak output power wpk (black) scale linearly with γeg , shown with linear
fits to the data.

responsible for the atom loss and ultimately for the shutoff of the laser. Light-assisted collisions
could be eliminated in the future using higher dimensional trapping lattices.
Ordinary single-particle fluorescence from an atomic ensemble is too weak to be useful as aphase reference, but the emission rate by the collective dipole is increased into a single spatial mode
by approximately N . The predicted peak flux of photons out of the cavity is Ṁpk = RN 2 Cγeg /8,
where R = 1 for the two-level model of Ref. [115]. Typically, R ≈ 0.18 for our multi-level system
due to the necessity of optically repumping the atoms through several intermediate Zeeman states.
We observe the peak output power scaling as N 2 (Fig. 15.1e). For this data, 2γ⊥ /κ ≈ w/κ =3.9(8) × 10−4 to 1.4(3) × 10−3 .
Repumping non-collectively scatters photons transverse to the cavity mode, causing collapseof individual atomic wavefunctions, which are then torqued to rejoin the collective dipole by the
weak intracavity field. If the repumping rate is greater than wmax = N Cγeg , the cavity field

111
10

Intracavity
PhotonsaPhasor Correlationb

All pump light off
only thermal photons in cavity

0

0

Tdark

Time

1.0
0.8
0.6
0.4
0.2
0.0
0

2

4

6

8

10

Dark Time Tdark (ms)

12

14

16

Figure 5.4: Phase coherence maintained with no intracavity photons. a, The dressing andrepumping lasers are shut off for some variable dark time Tdark . The phase of the light relative tothe dressing laser is measured shortly before and after the dark time. b, The phasor correlation
function, defined in Sec. 5.6. At times Tdark < 7 ms, the phases are correlated because the collective
atomic dipole seeds reestablishment of superradiant emission. At longer times, the collective atomic
dipole dephases and superradiant emission restarts with a random phase. A gaussian fit is shownin red line. Error bars ±1 s. d.

cannot replenish the collective dipole relative to the rate of repumping induced collapse [115]. The
predicted peak output power occurs at wpk = wmax /2. The quenching and linear scaling of wmax
and wpk with γeg are shown in Fig. 25.3. The fitted and predicted slopes agree to within the 20%
uncertainty on the predicted values.
To explicitly show that the phase memory of the system resides in the collective dipole, wemeasure the phase of the emitted light just before φ0 and after φ1 a dark period Tdark in which
both the dressing and repumping lasers are shut off (see Fig. 35.4) During this period, any photons
in the cavity exit with exponential time constant 1/κ = 14 ns, rapidly erasing all phase information
stored in the light mode. While the phase φ0 is random from one trial to the next, the phasedifference within a single trial φ1 − φ0 is highly correlated, demonstrating that phase information
is being stored within the collective atomic dipole.

112

b 2.0

0.8

Norm. PSD (x10 -4)

Normalized PSD

a 1.0
0.6
0.4
0.2
0.0

-3 -2 -1 0 1 2 3
Offset Freq. (kHz)

10
10

1.5

Linewidths

Good-Cavity
40000xSchawlow-Townes
Repumping

1.0

3000x

104
103

Doppler
0.5
0-60Magnetic,
0

Offset Freq. (kHz)60

Natural Line
Measured

60x

102

10x

101

1x

100

5

Standard laser or reference cavity
Standard Quantum Limit for 105 atoms

-1

10-2

1

10-3
10-4
10-5

Photon Shot Noise Phase
Uncertainty ∆φ (mrad)

Cavity Pulling P

d

0

c

0.5

0.1

1

10

Decay Rate γeg/2π (Hz)Figure 5.5: Beyond standard optical laser stability. a, The average power spectrum of theheterodyne signal of the emitted light (black circles) at Mc = 0.20(2) intracavity photons with aGaussian fit (red line). b, The average power spectrum of the heterodyne signal of the emitted light(black circles) at Mc = 30.6(5) with a Lorentzian fit that excludes offset frequencies lower than 4kHz. c, The comparison of various characteristic linewidths. Gaussian noise processes are comparedto the measured Gaussian FWHM (dashed), and Lorentzian noise processes were compared to themeasured Lorentzian FWHM (solid). Details of the experimental configurations are included inSec. 5.6. d, As the decay rate γeg decreases, the atomic dipole becomes more isolated from themirrors, as shown by directly measuring cavity frequency pulling P (black circles). Because of
the collectively enhanced emission rate, phase information is still coupled out of the system at a
sufficient rate to achieve an ideal RMS phase noise (red circles) sufficient for spectroscopy below thestandard quantum limit for 105 atoms. The fitted curves are consistent with the predicted scaling
with γeg .

113

5.4

Expt. Measure: Linewidth
In Fig. 45.5a, we present the power spectrum of the emitted light heterodyned with the

dressing laser at Mc = 0.20(2) photons in the cavity mode. The measured Gaussian FWHM, 350(25)Hz, is 4 orders of magnitude below the good-cavity Schawlow-Townes linewidth limit assuming
one intracavity photon. The measured linewidth is also below single particle decoherence rates
arising from inhomogeneous magnetic field broadening, Doppler decoherence, and the repumpingbroadened linewidth w (see Fig. 45.5c). We believe the predicted superradiant linewidth, 2(1) mHz,
is not observed primarily due to dispersive tuning of the cavity mode frequency caused by changes
in atom number between individual experimental trials, an effect present in this Raman system,
but not in proposed active optical clocks, as discussed in Sec. 5.6.
To further probe fundamental noise limits, we note that quantum phase diffusion, as well as
single-particle total excited state decay γe and repumping broadening w, all produce a Lorentzian
contribution to the power spectrum. After removing low frequency spectral components as described in Sec. 5.6, a Lorentzian fit that excludes offset frequencies below 4 kHz yields a FWHM
of 4.5(5) Hz (see Fig. 45.5b). This is still larger than the predicted linewidth, 10(2) mHz, for thisparticular data set. We believe that the linewidth is limited again by atomic population noise as
described above. However, the measured linewidth is less than the total Raman decay linewidthγe = 48(16) Hz (and the repumping linewidth w = 13.6(2.0) kHz), demonstrating that a collective
oscillation can have lower phase noise properties than obtained from a collection of independent
atoms fluorescing in free space, as predicted for C < 1.
In Fig. 45.5d, we demonstrate small cavity pulling P ≈ 10−4 for a range of γeg comparable
to the linewidth of several atoms suited for optical clocks, The pulling P is measured by applying
small changes in the cavity resonance frequency dfcav and measuring the frequency shift of thelight dfγ exiting the cavity. The measured pulling agrees reasonably well with the predicted pulling
P/Ppred = 1.44(22) where Ppred = 2γ⊥ /κ. Assuming the suppression of cavity pulling measured inthis work, current optical cavities stabilized to ≈ 1 Hz are sufficient to reach a linewidth of 1 mHz

114
with the appropriate atom.
To decouple the collective atomic dipole from the environment, one can choose an atom withsmall γeg . However, the emitted light power would also decrease, making the light source too weakto phase stabilize even state-of-the-art narrow lasers. In a 1 kHz bandwidth, we observe a flux ofphotons sufficient to obtain a quantum-limited phase uncertainty ∆φ ≈ 1 mrad or less (Fig. 45.5d
and Sec. 5.6). This phase uncertainty and servo bandwidth would be sufficient to stabilize a bright
1 Hz linewidth laser down to 1 mHz to perform spectroscopy below the standard quantum limit
for ensembles of 105 atoms in a state-of-the-art optical lattice clock.

5.5

Summary and Conclusions
This work explores a novel regime of optical laser physics in which a highly coherent collective

atomic dipole replaces the optical cavity as the high-Q oscillator. Photons play an essential roleby establishing atomic correlations, but a macroscopic number of photons need not occupy the
intracavity mode. This light source, with Mc < 1, N  1, and C  1, operates in the opposite
limit of the single-atom laser [112] with N = 1, Mc > 1 and C  1. The results are particularly
relevant for precision measurement science, since we demonstrate such a source operating in thelikely “no-photon” limit Mc < 1 that may be necessary for superradiant lasers with linewidths < 1
mHz. In addition, we show that as an active optical oscillator, stability in excess of the limits
imposed by single particle decoherence rates can be achieved. Additional experiments are needed
to study photon statistics, atom loss, and possible unforeseen noise that our experiment was unable
to resolve at low offset frequencies.

5.6

Additional Technical Details

5.6.1

Experimental Details

The experimental configuration consists of the cavity (mode waist of 71 µm and length of
1.9 cm) with a quantization axis defined by a 2.7 G magnetic field along the cavity axis. The

115
a

b
F

D1, 795 nm

Repumping

B

i

2'

D2, 780 nm

F 1'

F

2'

2.7 G

e

F

2

F

1

Dressinggmf = 0mf = -2

-1

0

1

2

Figure 5.6: Primary experimental configuration – dressing beam along cavity axis. a,
The physical arrangement of the apparatus. The cavity is vertically oriented, with the quantization
axis defined by the magnetic field. The linearly-polarized D1 dressing beam is in red, and the πpolarized D2 repumping beams are green (F2) and purple (F1). The emitted light, blue, goes into
the cavity mode. It has a linear polarization that is rotated 90◦ from the dressing beam. b, Theenergy level diagram for the D1 and D2 beams. The cavity mode resonance is denoted with a blue
dashed line. The repumping dark state is labeled with a gray circle.

atoms are laser cooled to approximately 40 µK, putting them in the Lamb-Dicke regime along
the cavity axis but not perpendicular to the axis. The cooling is not applied during steady statesuperradiance. We calculate the intracavity photon number via Mc = 2Ṁd /qκ0 , where Ṁd is thedetected photon flux from one end of the symmetric cavity, q ≈ 0.6 is the quantum efficiency fromthe cavity mirror to the heterodyne detector, and κ0 /2π = 5.3(3) MHz is the cavity linewidth due
to mirror transmission alone. The fundamental phase noise due to photon shot noise is calculatedpusing ∆φ = 1/ 4Md (T ) where Md (T ) is the average number of photons measured using homodyne
detection in a time interval T .
Steady state superradiance was primarily observed in an experimental configuration that was
first-order insensitive to Doppler broadening and magnetic field broadening shown in Fig. 5.6. Thisconfiguration was used to measure both linewidths quoted in the text, as well as the linewidthcomparison to the good-cavity Schawlow-Townes, repumping broadening, and natural linewidths.
The 795 nm dressing laser was applied along the cavity axis, and by non-resonantly driving the

a

116

D2, 780 nmbF

D1, 795 nm

Repumping

i

F

2'

F 1'

2'

45

Dressing
B

2.7 G

e

F

2

g

F

1

mf = -2-1

mf = -2

-1

0

1

2

Figure 5.7: Secondary experimental configuration – dressing beam perpendicular tocavity axis. a, The cavity is vertically oriented, with the quantization axis defined by the magnetic
field. The linearly-polarized dressing beam is in red, and the repumping beams are green (F2) andpurple (F1). The circularly-polarized emitted light, blue, goes into the cavity mode. b, The energy
level diagram for the D1 and D2 beams. The cavity mode resonance is denoted with a blue dashedline. The repumping dark state is labeled with a gray circle. The polarization of the repumping
beams is a linear combination of π-polarization (thick lines) and σ − polarization (thin lines).

cavity, we could inject enough intensity to drive the superradiant emission into a resonant cavity
mode. The dressing laser is linearly-polarized and appears as a superposition of σ + and σ − . The
Clebsch-Gordan coefficients on the emission path to |gi ≡ |F = 1, mf = 0i lead to a cancellationof all emission except for linearly-polarized light, rotated 90◦ from the polarization of the dressing
laser. The polarization orthogonality allows the emitted light to be polarization separated fromthe dressing light that also emerges from the cavity. The repumping lasers are π-polarized andtuned such that they create a single state |ei ≡ |F = 2, mf = 0i dark to the repumping. The F1
repumper moves atoms primarily from |F = 1i to |F = 2i, and the F2 repumper pushes population
to |ei, as the Clebsch-Gordan coefficient for the transition |F = 2, mf = 0i → |F = 20 , mf = 00 i iszero.
In this configuration, the nodes and anti-nodes of the standing waves of the dressing laser
and the cavity emission mode do not coincide due to the difference frequency of 6.834 GHz. We

117
D
E D
E
define a spatially averaged decay rate γeg = γeg (z) Ω2ig (z) / Ω2ig (z) weighted by the coupling
to the emission cavity mode Ω2ig (z).
To measure the amplitude and phase of the emitted light, we combined the superradiant emission with a heterodyne reference. The same laser provides the dressing beam and the heterodyne
beam. The schematic and frequencies are described in detail in Fig. 5.8. The beatnote producedon the heterodyne detector is at 50 MHz. The 50 MHz signal is then quadrature demodulatedusing the signal processing chain described in Chapter 2, resulting in the emitted light amplitude
A(t) and phase φe (t) as a function of time.
In order to monitor relative path length fluctuations of the heterodyne detection, we alsomeasure the phase of the Raman dressing beam that passes non-resonantly through the system.
Using the polarization filtering, we allow a small amount of the Raman dressing light to passthrough to the heterodyne detector, which also creates a beatnote on the detector at 84 MHz. Thissignal is separately filtered and quadrature demodulated to provide a path length reference phaseφr (t).
In a similar way, the dressed cavity resonance frequency is probed to measure the total atom
number. A small portion of the heterodyne reference beam is split off to use as a cavity probe(Fig. 5.8). The transmission from this weak beam is detected in heterodyne and the amplitude
quadrature is used to identify the cavity resonance frequency while sweeping the EOM modulation
frequency fm .
Steady-state superradiance was observed in a second experimental configuration that might
be applied for magnetometry, and shown in Figure 5.7. The configuration was first-order sensitive
to magnetic field broadening and Doppler decoherence. The quantization axis was still definedby a magnetic field along the cavity axis. The π-polarized 795 nm dressing laser was applied
perpendicular to the cavity axis, a direction in which the atoms were not in the tightly-confinedLamb-Dicke regime. The state |ei ≡ |F = 2, mf = −2i was the only state dark to the repumping
light. This was achieved by applying two different frequency 780 nm lasers at 45◦ to the cavity
axis, such that they consisted of a linear combination of π and σ − polarization. Light emitted into

118

Figure 5.8: A chart of relevant laser frequencies used in the superradiance experiment. The x-axis
is frequency, but the y-axis is only used to distinguish different beams. The 795nm reference laser(not shown here) starts the frequency chain, stabilized to the atomic reference, as described in
Chapter 2. The second 795 nm laser is then stabilized with respect to the reference 1.184 GHz
detuned from the F = 2 to F = 20 transition, with the transition indicated by the black arrow. Thislaser is split into two paths. The first forms the dressing laser fsp after being shifted 84 MHz lower
in frequency to give the dressing laser detuning ∆ = 1.1 GHz. The other path is ultimately used
for both the heterodyne reference beam fhet and the dressed cavity frequency probe fprobe . Thissecond path is modulated using an EOspace high frequency modulator to produce sidebands at
fm = 6.800 GHz, placing the higher frequency sidebands near the superradiant emission frequency
fγ , which is fixed to the be fsp + fhf , where fhf is the hyperfine splitting frequency. After highfrequency modulation, the probe beam is split from the heterodyne beam, and frequency shifted
lower by 100 MHz. The setup produces 3 frequency separated signals on the heterodyne detectors,
the emitted light at 50 MHz, the dressing laser leakage at 84 MHz, and the cavity probe at 100
MHz. The high frequency modulation fm can be swept using the DDS to scan the probe component
over the cavity resonance, while keeping the probe signal at a constant 100 MHz.

119
the standing wave cavity mode could only be σ + or σ − polarized. This polarization constraint,
combined with cavity resonance frequency conditions, selected |gi ≡ |F = 1, mf = −1i as a uniqueground state for superradiant emission. The dressing laser was tuned 1.1 GHz to the blue of the|F = 2, mf = −2i → |F = 20 , mf = −20 i atomic transition. The cavity mode was tuned 1.1 GHz
to the blue of |F = 1, mf = −1i → |F = 20 , mf = −20 i transition. The laser frequency chain was
similar for this configuration, with small adjustments to account for the slightly different hyperfine
frequency due to Zeeman shifts of the groundstates.

5.6.2

Phasor Correlation

D h −i∆φ
iE
k ik
he
i∆φk
We define a phasor correlation function Cp = Re |heewhere ∆φk = φk1 − φk0−i∆φk i |kk

is the phase difference just before and after the dark time Tdark in a single trial k. In the case thatthe phase measurements have no ambiguities due to measuring modulo 2π, Cp = hcos[φk1 − φk0 −hφk1 − φk0 ik ]ik . This definition is chosen such that if there is no noise in the phase difference, Cp → 1,
while if φ1 is random with respect to φ0 , Cp → 0. The average incorporated 50 to 100 experimental
trials. The phasor correlation function measures the average projection of the final unit length lightphasor eiφ1 onto the initial light phasor eiφ0 . The local oscillator frequency was not equal to theatomic precession frequency in our experiment, so some average relative phase accumulated during
−i∆φk

he
ik
the dark time Tdark . The normalized average difference phasor |he−i∆φk i | accounts for this phasekaccumulation.

5.6.3

Lorentzian Fits

In the measurements of the linewidth of the emitted light, we must account for phase noisefrom relative path length fluctuations. We measure the path length fluctuations in a single trial
k using the measured phase of the Raman dressing beam that non-resonantly passes through thecavity φkr (t). Then we can subtract the path length fluctuations from the phase of the emitted lightfor each 5.5 ms trial is φk (t) = φke (t) − φkr (t).
In this Raman system, the loss of atoms causes a frequency chirp of the cavity resonance that

120
then chirps the superradiant emission frequency through the cavity pulling effect. The frequency
chirp should not be present if an actual highly forbidden optical transition was employed. Because
the frequency chirp is a deterministic effect, not a noise process, we attempt to account for thecontribution to the measured linewidth by calculating an average phase response as a function oftime, averaged over all measured 200 trials hφk (t)ik . This average response was then subtracted
from each measured phase yielding φ0k (t) = φk (t) − hφk (t)ik .
Not only does the dispersive cavity shift chirp the emission frequency in a single trial, changes
in the atom number between trials also shifts the center frequency of the emitted light. We accountfor the fluctuation in the center frequency between trials by subtracting a linear fit from each φ0k (t),resulting in φ00k (t) = φ0k (t) − (m0k t + b0k ). This is equivalent to moving the center frequency of eachtrial to 0 Hz before calculating the average power spectral density.
00

The two-sided power spectrum was calculated for each trial from eiφk (t) using a 4th orderBlackman-Harris windowing function to avoid power spectral leakage of the carrier. The resulting
bin size in the spectrum is 181.55 Hz as a result of using 5 ms of time data over which the frequency
chirp of the laser from atom loss was insignificant. The 4th order Blackman-Harris windowing
function gives a very low noise background at the cost of having a large noise background for the
first 5 bins from the 0 frequency, so the center 10 bins of data should not be included a fit to thefrequency spectrum. The windowing function effectively reduces the close in frequency resolution
to gain a lower noise floor further from the carrier.
Due to technical noise processes, we are interested in the Lorentzian signal from from thecarrier. We performed Lorentzian fits to the resulting average power spectrum, excluding offset
frequencies below 4 kHz to avoid the contributions from these technical noise sources and uncover
the underlying Lorentzian linewidth characteristic of quantum noise. Excluding offset frequencies
greater than 4 kHz did not significantly change the linewidth measurement.

Chapter 6

Linear response theory for a superradiant laser

During the experimental work described in Chapter 5, we observed significant sensitivity of
the laser amplitude to the details of the repumping rate and detuning of the cavity from the laser
emission frequency. Understanding the laser system’s stability is important for guiding the designof future superradiant lasers so they can operate in a stable regime. To study the Raman laser’s
stability, in this chapter we will extend the theoretical model presented in Chapter 4 to allow for
oscillations around the steady state values. Then in Chapter 7, I present our experimental studies
demonstrating the relevance of the linear response model in our superradiant Raman laser.

6.1

Linear Expansion of Uncoupled Equations
For future applications of steady-state superradiant light sources as precision measurement

tools, we are interested in the system’s robustness to external perturbations. As is common in laser
theory [111, 143, 92], here we analyze the system’s linear response to perturbations by considering
small deviations from the steady-state solutions. While some previous theoretical expressions inthis thesis are valid for both the good-cavity and bad-cavity limit, as no assumptions were madeabout the relative magnitudes of κ and γ⊥ , it is convenient now to simplify to two equations for the
dynamics by assuming that the laser is operating deep in the bad-cavity regime, where κ  2γ⊥ .
In this regime, the cavity field adiabatically follows the atomic polarization, providing the physicalmotivation to eliminate the field from Eqns. 4.8-4.11 [115, 97].
The cavity field is eliminated by assuming that the first time derivative of the complex field

122
amplitude C˘ in Eqn. 4.8 is negligible compared to κ2 C. This effectively results in Eqn. 4.12
being the equation for the cavity field. After substituting Eqn. 4.12 into Eqns. 4.9-4.11, we only
concern ourselves with the amplitude responses, simplifying the equations by using Eqn. 4.15 and2 . With these simplifications, the dynamical equations for J˙ , J˙2 , andsubstituting |J− |2 with J⊥z⊥

Ṅ3 areN3 W
CγJ2J˙z = ((Γ3e − W/2)+
(N/2 − Jz )) −2
2
1 + δ002 ⊥(6.1)

2 = −2γ J 2 + 2Cγ J J 2 .
J˙⊥
z ⊥
⊥ ⊥
1 + δ002(6.2)

Ṅ3 = − (Γ3e + W/2) N3 + W (N/2 − Jz )(6.3)

We perform the linear expansion by re-parameterizing the degrees of freedom in terms of
2 (t) = J¯2 (1 + 2j (t)),
fractionally small perturbations about steady-state: Jz (t) = J¯z (1 + jz (t)), J⊥⊥⊥

and N3 (t) = N̄3 (1 + n3 (t)). We also define the response of cavity field through the relationshipp2A(t) ≡ |C(t)|2 = Ā(1 + a(t)). Since |C(t)|2 = 1+δCγ
0 (t)02 J⊥ (t) from Eqn. 4.12, A(t) follows the0atomic polarization, except for the modification from dynamic cavity detuning as will be discussed
below. We analyze the response in the presence of a specific form of external perturbation – themodulation of the repumping rate W (t) = W(1 + w(t)) with w(t) = Re[eiωt ], where  is a real
number much less than 1. The quantities jz (t), j⊥ (t), n3 (t), a(t), and w(t) are unitless fractional
perturbations around the steady-state values that we assume are much less than 1.
We also include, by hand, an inversion-dependent term in the detuning δ00 = δ 0 + αJ¯z jz (t).
The cavity mode’s frequency is tuned by the presence of atoms coupled to the cavity mode. The
tuning is equal but opposite for atoms in the two different quantum states |ei and |gi. The detuning
δ 0 is the steady-state value of the detuning of the dressed cavity from the emitted light frequency.
The variation about this steady-state detuning is governed by the second contribution αJ¯z jz (t).
Effects such as off-resonant dispersive shifts due to coupling to other states can lead to this Jz

123
dependent detuning in real experiments. We derived this dispersive shift of the cavity in Chapter
4.
To linearize the resulting equations, we substitute the expansions around steady-state into
Eqns. 4.11, 6.1, and 6.2. We neglect terms beyond first order in the small quantities jz (t), j⊥ (t),n3 (t), a(t), and w(t). For ease of solving the equations, we treat jz (t), j⊥ (t), n3 (t), and a(t) as
complex numbers where the real part gives the physical value. After eliminating the steady-statepart of the equations, the equations for small signal responses j⊥ (t) and jz (t) can be reduced to
two uncoupled, third order differential equations...
β  ⊥ + ̈⊥ + 2γ0 ̇⊥ + ω02 ⊥ = D⊥ (ω)eiωt

(6.4)

...
β  z + ̈z + 2γ0 ̇z + ω02 z = Dz (ω)eiωt .

(6.5)

We have written the uncoupled differential equations in a form that suggests a driven harmonic
oscillator, with damping rate γ0 , natural frequency ω0 and a drive unique to the ⊥ or z equationD⊥ (ω) or Dz (ω). The drives contain derivatives of the repumping modulation w(t), resulting
in frequency dependence. The third derivative term is a modification to the harmonic oscillator
response from the third level, characterized by the factor β that goes to zero in the two-level
limit (r → ∞). To preserve the readability of the text, we have included the full expressions forthe coefficients as Appendix C. Each of the terms will be discussed subsequently in physically
illuminating limits.
The drive of this harmonic oscillator-like system varies with the modulation frequency andother system parameters. In the case of the two-level model of Ref. [115], with r = ∞, α = 0, andΓR = 0, the drive terms areW(N Γc − 2W − iω)
2

(6.6)

Dz (ω) = (N Γc − W)W + iω).

(6.7)

D⊥ (ω) =124
The modulation-frequency-dependent terms add an extra 90◦ of phase shift at high modulation frequencies to the observed response. Additionally, the cancellation in D⊥ (ω = 0) results
in an insensitivity of the output photon flux to the ground state repumping rate W at Wopt . The
cancellation agrees with the parabolic dependence of M̄˙ c on W , as seen in the steady-state solutions.
The frequency dependent terms in D⊥,z also cause a growing drive magnitude versus ω. Thisis canceled out in the responses j⊥ and jz by the roll-off from the oscillator, keeping the response
finite versus modulation frequency. These characteristic features remain in the response, even as
the complexity of the model increases as additional effects are included.
To proceed, we solve the equations for the complex, steady-state response to a single modulation frequency ω, (e.g. ⊥ (t) = ⊥ eiωt ). The complex response of the cavity field amplitude a(t)results from these solutions,a(t) = ⊥ (t) −δαJ¯z z (t).
1 + δ 02

(6.8)

In contrast to Eqn. 4.12, where |C| depends only on J⊥ , including dispersive cavity tuningfrom the inversion couples the cavity output power to Jz as well.

6.2

Transfer Function Analysis
We analyze the response of the cavity field amplitude to an applied modulation of the re-pumping rates by plotting the amplitude transfer function and the phase transfer function versus


Re[a]
the modulation frequency ω, defined as TA (ω) ≡ |a|/ and Tφ (ω) ≡ arctan Im[a]respectively. We
consider the maximum of the transfer function to define the resonant frequency ωres . The calculated variation in the transfer functions versus various experimental parameters is shown in Figs.
6.1 - 6.5. All results are given as a series of transfer functions varying a single specified systemparameter, with other unspecified parameters set to W = Wopt , r = ∞, δ 0 = 0, and ΓR = 0.
The expressions for the damping γ0 and the natural frequency ω0 guide our understandingof the transfer functions. Holding r = ∞, δ 0 = 0, and ΓR = 0, the damping reduces to γ0 =

125

3.0
W [NC g]
0.1

2.0

AT

0.5
0.75

1.0

0.0
0

T

f

-90

-180
-270
0.0

0.5

1.0

1.5

Modulation Frequency

2.0

w [NCg]Figure 6.1: Output photon flux transfer function for different ground state repumping rates, withr = ∞, δ 0 = 0, α = 0 and ΓR = 0.

6
r
0.1

AT

4

0.5
¥

2

0

T

f

-90

-180
-270
0.0

0.2

0.4

0.6

Modulation Frequency

0.8

1.0

w [NCg]Figure 6.2: Output photon flux transfer function for different repumping ratios r, with W = Wopt ,δ 0 = 0, α = 0 and ΓR = 0.

126
W/2. Physically, the damping enters through the decay of J⊥ at a rate proportional to γ⊥ . Theq√natural frequency ω0 = W(N Cγ − W) = 2J¯⊥ Cγ is set by the steady-state rate of converting
2 C 0 γ, normalized by the steadycollective transverse coherence into atoms in the ground state, J⊥

state transverse coherence J⊥ .
To examine the effect of the steady-state repumping rate W on the response, we plot thetransfer functions TA and Tφ for different values of W in Fig. 6.1. For W < Wopt , we see a narrow
resonance feature in the response(blue curve). The frequency of the resonance increases until
W = Wopt (green curve). Also at W = Wopt , the dc amplitude response TA (ω = 0) = 0, because thedrive D⊥ goes to zero (Eqn. 6.6), consistent with the maximum in Ṁc at Wopt . For W > Wopt , thephase of the response near dc sharply changes sign, as understood from the parabolic response ofṀc versus W; on the W > Wopt side of the parabola, the same change in W produces the oppositechange in the output photon flux compared to the W < Wopt side of the parabola. Meanwhile, thenatural frequency has decreased with the increase in W when W > Wopt . As W approaches Wmax ,the response has essentially become that of a single-pole, low pass filter with an additional π phaseshift.
To examine the effect of population in the third state |3i, we now hold W = Wopt and show
TA and Tφ for different r in Fig. 6.2. The black curve shows the result for r = ∞, which is thetwo-level model of Ref. [115], as no population accumulates in |3i (recall that N̄3 /N̄g = 1/r).
For smaller r, the relaxation oscillations grow, shown by the increasing maximum in TA . This
response is consistent with the reduced damping rate γ0 and increased drive D⊥ seen in the following
expressions.
r
The damping is γ0 = 1+r



N Cγ
4



2

2ω
− N Cγ(1+r). The additional ω dependence, associated withthe repumping delay from atoms spending time in |3i, results from the third derivative term that1
in Eqns. 6.4 and 6.5.
W (1+r)
1+r+2r2
ω2. The term proportional
The complex drive in this limit is D⊥ = iω N 2Cγ (1+r)(1+2r)− 1+r

scales with β =to ω 2 in D⊥ arises from modulating the rate out of the state |3i. Although the ω 2 term in thedamping would introduce a roll off in the transfer function TA with the form 1/ω 2 , the ω 2 frequency

127
4
d', a

3

[NCg]

0
-3

AT

1,10

-3

2

-1,10

-3

1, 2*10-3-1, 2*10

1
0

T

f

-90

-180
-270
0.0

0.5

1.0

Modulation Frequency

1.5

2.0

w [NCg]Figure 6.3: Output photon flux transfer function for different dressed cavity detuning from emitted
N Cγ
light frequency δ, with W = Wopt (δ) = 2(1+δ02 ) , r = ∞, and ΓR = 0. The solid (dashed) lines show−3−3
α = N Cγ × 10 (α = 2N Cγ × 10 ) to demonstrate the effects of increased cavity feedback.

dependence is canceled. The final transfer function maintains a frequency dependence of 1/ω forω  ωres , similar to that of the two-level system.
Next we consider the effect of the dynamically tunable cavity mode. The cavity mode response
can strongly modify the damping of the oscillator and even lead to instabilities in the cavity lightfield, eliminating steady-state solutions. We first consider the damping rate of the two-level model


W
+ΓR
N
W
0
0
0
. The damping(r = ∞) with cavity tuning, γ0 = 2 (1 + h(δ )) where h(δ ) = 2αδ 1+δ02 − Cγ
is modified by a detuning dependent feedback factor h(δ 0 ) that is positive or negative depending
N Cγ
0on the sign of δ 0 . Because W + ΓR < 1+δ
02 to meet superradiant threshold, h(δ ) has the same

sign as δ 0 . Applying negative cavity feedback, when h(δ 0 ) > 0, increases the damping and may
be useful for reducing relaxation oscillations and suppressing the effect of external perturbations.
When h(δ 0 ) < 0, positive feedback decreases γ0 and amplifies the effect of perturbations.
We show the effect of this cavity feedback on the transfer functions in Fig. 6.3 for theconditions r = ∞, W = Wopt (δ 0 ), and ΓR = 0. The red (blue) curves show positive (negative)
feedback, with the black curve serving again as a reference to the model of Ref. [115] with no128

Figure 6.4: Stability plot using γ0 stability condition of Eqn. 6.9 as a function of detuning
δ 0 and the cavity shift parameter α, assuming N = 106 . The stability condition also assumes
W = Wopt (δ 0 , ΓR ). The region of stability is exact for the two-level model (r = ∞), and a good
approximation for all values of r. The blue region shows where the real part of all the poles of the⊥ solution are negative, indicating a damped return to steady-state conditions for a perturbation.
The red region shows where any of the real parts of the poles become positive, making J⊥ unstable,with no steady-state solutions.

129
cavity feedback.
Fig. 6.3 also shows the effect of increasing the cavity shift parameter α. The solid lines
result from α = N Cγ × 10−3 , a cavity shift similar in magnitude to experiments performed in Refs.
[24, 21, 23, 160]. The dashed lines result when α is increased by a factor of two.
With enough positive feedback, the system can become unstable, with any perturbations
exponentially growing instead of damping, which eliminates steady-state solutions. For a driven
harmonic oscillator, the condition for steady-state solutions is γ0 > 0. Again assuming W =N Cγ
Wopt (ΓR , δ 0 ) = 2(1+δ02 ) − ΓR , and remaining in the two level limit (r = ∞) the stability condition

reduces toNαδ 0> −1 .
1 + δ 02

(6.9)

In Fig. 6.4, we plot the stability condition as a red line.
In general, the stability of a linear system can be determined by examining the poles of thesolution. If any pole crosses into the right half of the complex plane, the system is unstable withan oscillating solution that grows exponentially. In the two-level limit (r = ∞), this condition on
the solutions ⊥ and z is mathematically equivalent to the condition on γ0 , Eqn. 6.9. As the levelstructure becomes more complex, e.g. r 6= ∞ or in the full 87 Rb model in Sec. 6.3, we use thepole analysis to examine the regions of stable operation. For the model here, as r changes, the poleanalysis shows that the stability condition in Eqn. 6.9 is no longer exactly correct. However, thechange is small enough that Eqn. 6.9 remains a good approximation of the stability condition for
all values of r.
Finally, in Fig. 6.5 we show the effect of additional decoherence by plotting TA and Tφfor different values of ΓR . Here r = ∞, δ 0 = 0, and α = 0. As a reference, the black curveshows the transfer function with ΓR = 0. For the solid curves, the ground state repumping rate
W = Wopt (ΓR ) is varied with ΓR to remain at the point of maximum output power (Fig. 4.5)
which amounts to holding γ⊥ constant. Thus, as the decoherence increases by increasing the rate

130
1.5

GR [NCg]

0

AT

0.3

1.0

0.6

0.5

0.0
0

T

f

-90

-180
-270
0.0

0.5

1.0

Modulation Frequency

1.5

2.0

w [NCg]Figure 6.5: Output photon flux transfer function for different Rayleigh scattering rates ΓR , with
W = Wopt (ΓR ), r = ∞, α = 0 and δ 0 = 0. The dashed line shows the transfer functions when W is
held to N Cγ/2, not varied to remain at Wopt , and ΓR = 0.3. A dashed red curve is not shown, aswith ΓR = 0.6 and W = N Cγ/2 the maximum repumping rate threshold has been exceeded andthe output photon flux is zero.

of Rayleigh scattering from the ground state, the resonance frequency only moves because W ischanging, as seen in the expression for the natural frequency ω0 = W (N Cγ − 2γ⊥ ). Notice that
additional decoherence does not affect the peak size of the relaxation oscillations. Although thedamping rate decreases because γ0 = W/2, this effect is canceled by the drive decreasing with W
as well, with D⊥ = −iω(W/2) when W = Wopt .
If we hold W constant at N Cγ/2, the resulting transfer function is the dashed line in Fig.
6.5. With W constant, the coherence damping rate γ⊥ varies with ΓR , and the response actually
behaves similar to the case where W is increased (Fig. 6.1) because of the symmetric roles W andΓR have in the natural frequency and the drive.
The main conclusion from our examination of the linear response theory of the three-level,
bad cavity laser is that most conditions for optimizing the output power are compatible with anamplitude stable laser. Operating at the optimum repumping rate in particular suppresses theimpact of low frequency noise on the amplitude stability. However, we also find that because the131cavity detuning δ 0 couples to the population of the laser levels, cavity feedback can act to suppress
perturbations, or cause unstable operation, depending on the sign of δ 0 . A simple relationshipbetween N , δ 0 , and α gives the condition for stable operation at W = Wopt .

6.2.1

Bloch Vector Analysis of Response

Relaxation oscillations in a good-cavity laser arise from two coupled degrees of freedom, theintracavity field A and the atomic inversion Jz , responding to perturbations at comparable rates.
Parametric plots of the amplitude and inversion response provide more insight into the nature of
the relaxation oscillations than looking at the laser field amplitude response alone [143]. In the badcavity regime, the cavity-field A adiabatically follows the atomic coherence J⊥ , and the oscillations
arise from a coupling of J⊥ and the inversion Jz . Thus the relevant parametric plot is the 2D
projection of the 3D Bloch vector in the rotating frame of the azimuthal angle. In this section, westudy this response of the Bloch vector to better understand the stability of the bad-cavity laser.
The individual plots of Fig. 6.6 show the trajectory of the Bloch vector for the small signal
response at different applied modulation frequencies ω and different repumping rates W. The
trajectory is calculated using the amplitude and phase quadratures of the responses ⊥ and z
to define the sinusoidal variation of each quadrature with respect to a sinusoidal modulation of
W (t) = W(1 + Re[eiωt ]). The series of plots show the trend in the responses versus the groundstate repumping rate W and modulation frequency ω, with r = ∞, ΓR = 0, and δ 0 = 0. Although
the oscillator characteristics of the two quadratures are identical, they display a differing phase in
their response due to the differences in the drives D⊥ , Dz on the two quadratures.
At high repumping rates W > Wopt and high modulation frequencies ω > ωres , the perturbation modulates the polar angle of the Bloch vector, leaving the length largely unchanged. Near
ωres , the two quadratures have large amplitudes and oscillate close to 90◦ out of phase, leading to
the trajectories that encloses a large area. When ω < ωres and with W near Wopt , the cancellationin the drive term D⊥ leads to almost no amplitude of oscillation in the J⊥ quadrature, making themodulation predominately Jz -like. For α = 0 or δ 0 = 0, this means the cavity field amplitude A132

Increasing modulation frequency ω

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.0
0.0

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.0
0.0

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

Increasing ground state repumping rate W

0.5

Figure 6.6: Response of the 2D Bloch vector to external modulation of the repumping rate. The
steady-state Bloch vector, i.e. J¯⊥ and J¯z from Eqns. 4.18-4.19, is indicated by the blue line, plottedon the axis with units of N , so N/2 is the maximum value. The ellipse is the trajectory of the
Bloch vector responding to the modulation of the repumping rate w(t) = Re[eiωt ], described by
the small signal responses ⊥ and z in Eqns. 6.4 and 6.5. The parameters are  = 0.1, r = 5,
δ 0 = 0, α = 0, and ΓR = 0. The black arrow indicates the direction of the trajectory, starting fromthe blue dot at t = 0. The values of ω are chosen to show ω  ωres , ω ≈ ωres , and ω  ωres .

will also be stabilized, as it is locked to the transverse coherence J⊥ (Eqn. 6.8).
However, dynamic cavity tuning creates a coupling of the inversion to the cavity field as
well, breaking the simple time-independent proportionality of the cavity field amplitude A and theatomic coherence J⊥ , as expected from Eqn. 6.8. Fig. 6.7a show the case of δ 0 < 0. Because of

133

(a)

(b)

(c)

Figure 6.7: Parametric plots of response of the three degrees of freedom Jz , J⊥ , and A, highlighting
effect of cavity frequency tuning on response of atomic coherence and output light field. The blueline represents the steady-state atomic Bloch vector, J¯z and J¯⊥ , from Eqns. 4.18-4.19. The blue
ellipse shows small single response of the Bloch vector to a modulation of the repumping rate W ,given by z and ⊥ from Eqns. 6.4 and 6.5. The parametric response is plotted with units of N ,so N/2 is the maximum value. The red dashed line is the trajectory formed by the response of
the cavity field a (Eqn. 6.8) and z . The cavity field is plotted as a fraction of the average field,
then centered on the steady-state Bloch vector to compare with the atomic response. The arrows
indicate the direction of the response with respect to a modulation W (t) = W(1 + eiωt ). HereN Cγ = 104 s−1 , r = 5, ΓR = 0, W = Wopt (δ 0 ),  = 0.1, and ω = 0.02 N Cγ, chosen to show
the stable J⊥ response. (a) When δ 0 < 0, the cavity feedback can be positive, leading to larger
oscillations compared to the case of no feedback δ 0 = 0 shown in (b). Because of the couplingof Jz to the cavity mode frequency, A is not locked to the J⊥ response, as in (b), but is anticorrelated with Jz . In (c), where δ 0 > 0, the negative feedback reduces the response amplitudes
in all quadratures. The cavity tuning again shifts the cavity amplitude response, but with the
opposite phase relationship due the change in sign of the slope of the Lorentzian, so A follows Jz .
The inset shows a close up of the response.

134

Figure 6.8: Cavity tuning stabilizing the cavity field amplitude. By changing the average repumping
rate W from Wopt to 0.44 × 104 for the same parameters as Fig. 6.7c (N Cγ = 104 s−1 , r = 5,
ΓR = 0,  = 0.1, δ 0 = 1, and ω = 0.02 N Cγ), the response of the Bloch vector (blue ellipse) becomesprimarily perpendicular to the steady state Bloch vector (blue line). Under these conditions, thecavity field response A (red dashed ellipse) has the smallest fractional deviation of the three degrees
of freedom. The inset shows a close up of the response.

135
the coupling to the inversion, the cavity field response has a larger amplitude than J⊥ response inaddition to a phase shift. It is also nearly 180◦ out of phase with the response of the inversion.We
include the case of δ 0 = 0 (Fig. 6.7b) as a reference. The cavity field is locked to the coherence,even for α 6= 0, due to the second order insensitivity in the cavity coupling. For the case of negative
feedback δ 0 > 0, shown in Fig. 6.7c, all the response amplitudes are reduced due to the increased
damping. Notice that the inversion and cavity field are now responding in phase.
Because of the coupling between all three degrees of freedom, it is possible to choose parameters that lead to a stabilization of the cavity field. Operating away from Wopt , the response of
the Bloch vector becomes primarily a modulation of the polar angle as the inversion and coherence
respond 180◦ out of phase. Combined with the cavity tuning, the cavity field is stabilized, as shownin Fig. 6.8, where the parametric plot of A and Jz (dashed red ellipse) shows a response that is
primarily Jz -like. The response of the cavity field has the smallest fractional variation among thethree degrees of freedom.
To conclude our discussion of linear response theory in the three-level model, we point out
that the parametric plot analysis highlights the role that the dispersive cavity frequency tuning
plays in amplifying or suppressing perturbations in both the atomic degrees of freedom and thecavity field. Crucially, frequency stable lasers may need to seek a configuration that suppresses
fluctuations in the Jz degree of freedom to minimize the impact of cavity pulling on the frequencyof the laser. We also see that the dispersive tuning breaks the exact proportionality of the cavity
field and the transverse atomic coherence, restoring an additional degree of freedom that may be
crucial for observing chaotic dynamics in lasers operating deep into the bad-cavity regime [67].

6.3

Linear Response Theory in 87 Rb
To analyze the small signal response about these steady-state solutions analytically, we per-form the analogous linear expansion as was done in Sec. 6.1. We assume the repumping rates are
modulated with W (t) = W(1 + Re[eiωt ]), and assume the resulting modulation of the populations
2 = J¯2 (1 +  (t)). The equations are
and coherence take the form Nλ = N̄λ (1 + nλ (t)) and J⊥
⊥
⊥

136
then linearized by expanding to first order in the small quantities nλ (t), ⊥ (t), and , and thenre-expressed in terms of z (t) = (N̄e ne (t) − N̄g ng (t))/(N̄e − N̄g ) and ⊥ (t).
We solve for the steady-state, complex response amplitude at a single drive frequency z (t) =z (ω)eiωt , ⊥ (t) = ⊥ (ω)eiωt ≈ 2⊥ (ω)eiωt /2 and nλ (t) = nλ (ω)eiωt . The response of the photon0δamplitude flux is a(ω) = ⊥ (ω)− 1+δ
0 dδ(ω), where the detuning response is defined by the populationP
response dδ(ω) = λ (αλ /κ)Nλ nλ (ω) and the αλ are given by elements of the cavity tuning vectorα~ + , given in Appendix B. The predicted normalized fractional amplitude response is TA (ω) = |a|/


Re[a]
and phase response function is Tφ (ω) = arctan Im[a].
Figs. 6.9 - 6.11 contain surface plots showing the light amplitude transfer function TA versus modulation frequency. The third dimension shows how the response changes when a single
parameter W, r, and δ 0 is varied. The lower plots in each figure show the resonant response of
the system, following the frequency of the maximum response ωres and the resonant amplitude
response TA (ωres ). The response functions follow the same general trends as the three-level modelin Sec. 6.1, showing that the simplified model captures the essential physics of our system. The
full model also demonstrates good quantitative agreement with the experimental results as shown
in Ref. [21].
In Fig. 6.9, we show the amplitude transfer function versus the repumping rate W assumingr = 0.71 and δ 0 = 0. The value of r is chosen to reflect the conditions in Ref. [21]. We see theincreasing damping and natural frequency with rising W and the dc response suppression appearing
near Wopt . For W > Wopt , the frequency of the relaxation oscillation moves back towards ω = 0,
as expected from the three-level model. Near W = Wmax , the transfer function no longer has aresonance as it monotonically decreases from its maximum at ω = 0.
We show the effect of the repumping ratio r in Fig. 6.10, where α~ + = ~0, δ 0 = 0 and W = Wopt .
The trends of lower damping and a lower natural frequency as r → 0 are clearly visible, as expectedfrom the three-level model in Sec. 6.1.
We also note that TA has a 1/ω roll-off for ω  ωres , even with higher order derivatives in...
the equations for ⊥ (ω) and z (ω) that function as a low-pass to the response (analogous to the  ⊥137

Figure 6.9: Response for different repumping rates. (a) Light field amplitude response transfer
function TA versus repumping rate W . (b) Resonant response TA (ωmax ) and (c) the resonant
modulation frequency ωmax as a function of repumping rate W . Here N Cγ = 4×105 s−1 , δ 0 = α = 0,
and r = 0.71.

138

Figure 6.10: Response for different repumping ratio. (a) Light field amplitude response transfer
function TA versus repumping ratio r. (b) Resonant response TA (ωmax ) and (c) the resonant
modulation frequency ωmax as a function of repumping ratio r. Here N Cγ = 4 × 105 s−1 , δ 0 = 0
and W = Wopt .

139

Figure 6.11: Response for different detunings. (a) Light field amplitude response transfer function
TA versus detuning δ 0 . The transfer function is not plotted in regions of instability. (b) Resonant
response TA (ωmax ) and (c) the resonant modulation frequency ωmax as a function of δ 0 . The red
shaded regions indicate parameters in which the system is unstable and no steady-state solutions
exist. Here N Cγ = 4 × 105 s−1 , r = 0.71 and W = Wopt . The cavity shift parameter is given byα~ + . These parameter values reflect the conditions of the experimental system in Ref. [21].

140

Figure 6.12: Stability diagram for the full model of a superradiant Raman laser in 87 Rb, plotting
the region any of the real parts of the poles of the ⊥ solution are positive. When any pole becomes
positive, the system is unstable and has no steady-state solutions. The stability regions are shown
versus the detuning of the cavity from the emission frequency δ 0 and the detuning of the bare cavity
frequency from the atomic frequency ∆. The critical contour (bold) marks where the pole changes
sign. The dashed line indicates the detuning ∆ of experimental work (Refs. [24, 21]). For the
calculation we use N Cγ = 10−4 s−1 , W = Wopt , and r = 0.71.

141
term in Eqn. 6.4). However, modulation of the repumping rate out of each hyperfine ground state,
as was done in Ref. [21], puts higher order derivatives in the drive terms as well. The result isa drive that increases with a higher power of the modulation frequency ω, partially balancing thehigher order low-pass filtering. Thus, by modulating the repumping rate out of each hyperfine state,
the amplitude transfer function TA retains 1/ω modulation frequency dependence of the three-level
model in Sec. 6.1.
The response as a function of δ 0 also qualitatively agrees with the simple picture put forward
in Sec. 6.1, as shown in Fig. 6.11. For δ 0 > 0 around δ 0 = 0, we see a lower maximum TA consistent
with heavier damping. When δ 0 < 0, the amplitude of the relaxation oscillations increase as thesystem becomes less damped. But δ 0 continues to decrease, the full model shows a divergence in
TA where the system becomes unstable with no steady-state solutions. In the unstable regime,
we do not plot the transfer function and show a red shaded region in Fig. 6.11b and 6.11c. Thisinstability is consistent with our inability to achieve steady-state superradiance experimentally atdetuning δ 0 < −0.1 [21]. The reduction in ωres with increasing δ 0 is a result of maintaining therepumping W = Wopt , which reduces W at large detunings and affects the natural frequency.
We also use our linear response model to theoretically predict the stability diagram for thefull 87 Rb Raman laser system. We examine the poles of the solution for ⊥ as a function of δ 0 ,the detuning of the dressed cavity resonance frequency from the emission frequency and ∆, thedetuning of the bare cavity resonance frequency from the atomic lasing transition |gi to |ii =|F 0 = 2, mf = ±1i. We plot the regions of stability in Fig. 6.12, which is analogous to Fig. 6.4 in
Sec. 6.1. However, here the physical parameter ∆ controls α~ + , which roughly scales like 1/∆ (weassume ∆ remains large enough such that the system is well described by the dispersive tuning
approximation). Future experiments may benefit from working with larger detuning ∆. Howeverin the standing-wave geometry of Ref. [21], the improved stability would come at the expense ofincreased inhomogeneous ac Stark shifts from the dressing laser. At fixed scattering rate γ, the ac
Stark shift increases linearly with ∆.
Repumping the atoms through multiple grounds states, quantified by the r parameter, has a142

Figure 6.13: Critical stability conditions for variable values of the repumping ratio r. Each line
shows the contour as a function of ∆ and δ 0 separating stable lasing from unstable. The unstableregion is defined as any set of parameters that results in a positive value for the real part of any
pole of the J⊥ response solution. The stability conditions change as a function of the repumping
ratio r. (a) As r increases from 0, the unstable region gets smaller until it reaches some value
between 0.4 and 0.5, after which (b) the unstable region grows to its asymptotic value.

Figure 6.14: The value of ∆ for the critical contour versus r, assuming δ 0 = −1. Lower ∆ indicatesthat more of the parameter space has stable, steady-state solutions. Here we assume N Cγ = 10−4
s−1 and W = Wopt .

143larger impact on the stability digram in this full model than on three level model in Sec. 6.1. To
study the effect repumping through the multiple ground states of 87 Rb has on the stability of thelaser amplitude, we follow the contour of the stability diagram for different values of r, shown in Fig.
6.13. The figure is separated into two parts because the contour does not change monotonically. In
part (a), r is low, indicating much of the population building up outside of the lasing levels, andthe stable region grows in size as the repumping becomes more efficient. However, as r continues togrow, the contour asymptotes to an unstable region about the same size as if r = 0.1. In Fig. 6.14,
we plot the value of ∆ for the critical contour, holding δ 0 = 1, indicating that the largest stableregion occurs when r ≈ 0.45. Here the cavity shift caused by atoms accumulating in the otherhyperfine states acts to partially balance the shift from atoms in the |gi and |ei states, enhancing
the amplitude stability.

Chapter 7

Relaxation oscillations, stability, and cavity feedback in a superradiant Raman
laser

7.1

Introduction
We experimentally study the relaxation oscillations and amplitude stability properties of an

optical laser operating deep into the bad-cavity regime using a laser-cooled 87 Rb Raman laser. By
combining measurements of the laser light field with non-demolition measurements of the atomic
populations, we infer the response of the gain medium represented by a collective atomic Blochvector. The results are qualitatively explained with a simple model. Measurements and theory areextended to include the effect of intermediate repumping states on the closed-loop stability of theoscillator and the role of cavity-feedback on stabilizing or enhancing relaxation oscillations. This
experimental study of the stability of an optical laser operating deep into the bad-cavity regime
will guide future development of superradiant lasers with ultranarrow linewidths.
Optical lasers operating deep in the bad-cavity or superradiant regime, in which the cavity
linewidth κ is much larger than the gain bandwidth γ⊥ , have attracted recent theoretical[115, 33, ?]and experimental [24, 23, 160] interest. The interest has been partially driven by the possibility ofcreating spectrally narrow lasers with linewidths ≤ 1 millihertz and dramatically reduced sensitivity
to the vibrations that limit state of the art narrow lasers and keep them from operating outside thelaboratory environment[78]. These lasers may improve measurements of time[74, 19], gravity[38],and fundamental constants[55, 16] aiding the search for physics beyond the standard model. The
cold-atom Raman superradiant laser utilized here operates deep into the bad-cavity regime (κ/γ⊥ ≈145103  1), making it an important physics test-bed for fundamental and practical explorations of
bad-cavity optical lasers.
In the interest of fundamental science and in light of the potential applications, it is important to understand the impact of external perturbations on lasers operating deep into the badcavity regime. In this Chapter, I present an experimental study of the response to external perturbations of the amplitude, atomic inversion, and atomic polarization of an optical laser operating
deep into the bad-cavity regime. In contrast, experiments have extensively studied the amplitude stability properties of good-cavity lasers (κ  γ⊥ ) (See Ref. [111] and references therein).
Previous experimental work in the extreme bad cavity[24] and crossover regime[97] focused on
the phase properties of the light and atomic medium. Amplitude oscillations, intensity noise,and chaotic instabilities have been observed in gas lasers operating near the cross-over regime(κ/γ⊥ ≤ 10)[30, 154, 70, 161]. Relaxation oscillations of the field have been studied deep into thebad cavity regime using masers[142] in which the radiation wavelength is comparable to the sizeof the gain medium, unlike in the present optical system. Previous theoretical studies of amplitude stability deep in the bad-cavity regime include studies of relaxation oscillations[92], chaotic
instabilities[67], and intensity fluctuations characterized by correlation functions[92, 113].
In good-cavity optical lasers, the atomic polarization (proportional to J⊥ ) can be adiabaticallyeliminated and the relaxation oscillations are associated with the flow of energy back and forth
between the gain inversion (proportional to Jz ) and the cavity field A, where Jz , J⊥ are componentsof the collective Bloch vector J~ describing the atomic gain medium. In contrast, in a bad-cavity
laser, the cavity field can be adiabatically eliminated, and the oscillations are driven by the couplingof J⊥ and Jz . Here, we will measure and infer not only the light field A(t), but also the atomic
degrees of freedom J⊥ (t) and Jz (t) using non-demolition cavity-aided measurements[35, 34] to givethe complete picture of the dynamics of relaxation oscillations in a bad-cavity laser.
We will also consider the effects on the laser’s amplitude stability of non-ideal repumping
through multiple intermediate states. Intermediate repumping states were not included in previous simple theoretical models[115], but are present in most actual realizations. In addition, we(a)Collective
Emission

Repumpbeams(780 nm)

(b)

δ

∆

I

Ωd

ωcav
γ

Dressing
beam(795 nm)

↑

W
Ω1

146

Γ3 Ω 2
ΓII

3↓|A(t)|2[109 s-1]

(c)

Trap
(823nm)Γ

III

Figure 7.1: (a), (b) Physical setup and energy level diagram. The trapping light (orange) and
Raman dressing laser (red, power ∝ Ω2d ) are injected along the cavity axis. The repumping light(purple, green) is applied perpendicular to the cavity axis. The emitted optical laser light (blue)is nearly resonant with the cavity mode (dashed lines) detuned from ωcav by δ. The repumping
is accomplished through a pair of two-photon transitions through intermediate optically excitedstates |IIi and |IIIi with incoherent decay rates Γ. We individually control the two-photon rates
W and Γ3 with the repumping laser powers ∝ Ω21,2 . |3i represents other metastable ground states
besides the laser levels. (c) Example emitted laser photon flux |A(t)|2 versus time showing spiking
and relaxation oscillations at turn-on.

demonstrate that the cavity frequency tuning in response to the distribution of atomic population
among various ground states can be used to suppress or enhance relaxation oscillations in the Raman transition configuration or other configurations with atomic transitions near-detuned from thelasing mode. As evidence, we show stabilization of Jz , J⊥ , and A similar to observations of the
suppression of relaxation oscillations in good-cavity lasers[146]. The cavity frequency tuning mechanism is related to other applications of cavity feedback including the creation of nonlinearities forgenerating spin-squeezed atomic ensembles [104], cavity cooling and amplification in atomic andopto-mechanical systems [86], and the control of instabilities in gravitational wave detectors [79].

147

7.2

Experimental System
Our experimental system consists of a quasi-steady-state Raman laser described in Fig. 7.1

and in Chapter 2. The laser uses N = 1 × 106 to 2 × 106 87 Rb atoms as the gain medium. The
atoms are trapped and laser cooled into the Doppler-insensitive Lamb-Dicke regime (40 µK) in a 1D optical lattice at 823 nm formed by a standing wave in a moderate finesse F ≈ 700 optical cavity
with a cavity power decay rate κ/2π = 11 MHz. The single-atom cavity cooperativity parameteris C = 8 × 10−3  1, and is equivalent to the Purcell factor[149].
Fig. 7.1b shows a simplified energy level diagram of a three level Raman laser system. The
lasing transition is a spontaneous optical Raman transition with single-particle rate γ from |↑i ≡5 2 S1/2 F = 2, mF = 0 to |↓i ≡ 5 2 S1/2 F = 1, mF = 0 . The decay is induced by a 795 nm dressing laser injected into the cavity non-resonantly, and detuned from the |↑i to |Ii ≡ 5 2 P1/2 F 0 = 2transition by ∆/2π = +1.1 GHz. The atoms are incoherently repumped back to |↑i in two steps:
from |↓i to |3i and then from |3i to |↑i, at single-particle rates W and Γ3 respectively. The third
metastable ground state |3i here represents the sum of all other hyperfine ground states in 87 Rb.
The full energy level diagram with details of the dressing and repumping lasers is provided in Sec.
7.4.
We control γ (typical value 60 s−1 ) using the intensity of the dressing laser. We control
the repumping rates W and Γ3 using two 780 nm repumping lasers tuned near resonance withthe 5 2 S1/2 F = 1, 2 → 5 2 P3/2 F 0 = 2 transitions. The repumping intensities are independently
controlled allowing us to set the proportionality factor r ≡ Γ3 /W . In our experiments, W ranges
from 103 s−1 to 105 s−1 , and r ranges from 0.01 to 2. The repumping dominates all other homogenous broadening of the |↑i to |↓i transition such that γ⊥ ≈ W/2. The inhomogenous broadeningof the transition is γin ≈ 103 s−1 . To summarize, the relevant hierarchy of rates characterizing oursystem is κ  γ⊥ ≈ W/2 ∼ N Cγ > γin  γ. The rate N Cγ sets the scale for the single-particle,
collectively-enhanced decay rate from |↑i to |↓i. Coupling to other transverse and longitudinalcavity modes is negligible.

148
The frequency of the superradiantly emitted light ωγ is set by the frequency of the dressing
laser and the hyperfine splitting ωHF /2π = 6.834 GHz. The detuning of light and cavity resonancefrequency is δ = 2(ωcav − ωγ )/κ, normalized to the cavity half linewidth. The single particle

scattering rate from the dressing laser into the cavity mode is Γc (δ) = Cγ/ 1 + δ 2 .
P
The cavity frequency is dispersively tuned by the atomic ensemble ωcav = ωbcav + k αk Nk ,where ωbcav is the bare cavity frequency and αk is the cavity frequency shift for a single atom inthe kth ground Zeeman state resulting from dispersive phase shifts of the intracavity light field.
Since the cavity frequency shift from atoms in the F = 1, 2 hyperfine states are not equal, thecavity frequency can provide a measurement of the atomic populations. We can suddenly switch off
the repumping and dressing lasers to effectively freeze the atomic populations[23]. We then combine
repeated non-demolition cavity frequency measurements like those describe in Chapter 3 and Refs.
P[152, 139, 35, 34] with NMR-like rotations[36] to determine Jz (t) = h 21 N
i=1 (|↑i i h↑i | − |↓i i h↓i |)iand δ(t), described in detail in Sec. 7.4.2. We measure the amplitude of the light field emitted
from the cavity A(t) in heterodyne just prior to freezing the system, along with the measurement
D E
of the cavity frequency detuning δ provides an inferred value of J⊥ = Jˆ− using the relationpPA(t) = Γc (δ(t))J⊥ (t), where Jˆ− = N
i=1 |↓i i h↑i |.
We observe characteristic laser spiking and relaxation oscillation behavior in |A(t)|2 as thelaser turns on and settles to steady state (Fig. 7.1c). To systematically study small amplitudedeviations about the steady-state values J¯z , Ā, and J¯⊥ , we apply a swept sine technique, similar
to Ref. [157]. We apply a simultaneous small amplitude modulation of the repumping rates as
W (t) = W̄ (1 +  Re[eiωt ])) and Γ3 (t) = rW (t). The modulation frequency ω is scanned overfrequencies of order W̄ , such that γ < ω  κ. We then measure and infer the quantities A(t),J⊥ (t), and Jz (t) as described earlier.
To calculate the transfer function of the applied modulation, the measured light field amplitude A(t) exiting the cavity as a function of time is fit to A(t) = Ā(1 + a(ω) cos(ωt + φa (ω))).
The normalized fractional amplitude response transfer function is TA (ω) ≡ a(ω)/ and the phase
response transfer function is Tφ ≡ φa (ω). We also define the modulation frequency that maximizes

149

ω/2π = 0.11 kHz

ω/2π = 2.2 kHz

ω/2π = 8 kHz

2ε = 0.22

ε = 0.11

2ε = 0.22

Jz (106)(a)J⊥(106)

J⊥(106)

J⊥(106)0-π/2-πωres [kHz]

(b)

2

0

0.4

W [Wmax]

0.8

-3π/2

Figure 7.2: (a) Parametric plots of the Bloch vector components Jz (t) and J⊥ (t) over a single
cycle of modulation of the repumping rate W for modulation frequencies below, near, and above
resonance or ω/2π = 0.11, 2.2, 8 kHz, from left to right. The black points are the measured smallsignal deviations about the measured steady-state Bloch vector (red arrow). The blue curve is thepredicted deviation from steady-state given the experimental parameters N = 1.3 × 106 , r = 0.71,
δ̄ = 1, W̄ = 0.35 N Cγ, and N Cγ = 125 × 103 s−1 . The modulation depth  for data below
and above resonance was doubled to make the response more visible. (b) The amplitude (upper)
and phase (lower) response transfer functions TA (ω), Tφ (ω) of the light field for three values of therepumping rate W . The points are measured data, and the lines are zero free-parameter predictionsof the response. (Inset) ωres versus W (points) and a fit to ω0 (line) showing the expected frequency
dependence of the relaxation oscillations on repumping rate.

150
TA (ω) as the resonance frequency ωres .

7.3

Results
We present the measured transfer functions and atomic responses in Figs. 7.2, 7.3, and 7.4,

with theoretical predictions from a full model for 87 Rb for quantitative comparison. To guide theinterpretation of the measurements, we present an analogous 3-level model for the system shown in
Fig. 7.1b that captures qualitative features of the full model, described in detail in Chapters 4 and6. The 3-level model uses semi-classical optical Bloch equations to describe the lasing transition
and the repumping process. Since κ  W, γ, the cavity field can be adiabatically eliminatedfrom the system of equations. Additionally, we have adiabatically eliminated the populations in
the optically excited states |Ii , |IIi , |IIIi, arriving at the steady state solutions for the inversion
J¯z and collective atomic coherence J¯⊥ [115, 114]. The steady state amplitude Ā is maximized at
W = Wpk = 21 N Γc (δ̄) where δ̄ is the steady state cavity detuning.
To predict relaxation oscillations and damping, we do a straightforward expansion about thesteady state values J¯z , J¯⊥ , and N̄3 as Jz (t) ≈ J¯z (1 + Re[z (t)]), J⊥ (t) ≈ J¯⊥ (1 + Re[⊥ (t)]), andN3 ≈ N̄3 (1+Re[n3 (t)]) and ignore terms that are second order in small complex quantities ⊥ , z , n3
and repumping modulation amplitude , as is described in Chapter 6, similar to work analyzing
the small signal response of a laser in Refs. [92, 143]. The coupled quadratures z and ⊥ respondlike the two coupled quadratures of a harmonic oscillator, slightly modified by the presence of theintermediate repumping state |3i. In the limit of ideal repumping (r → ∞) as is considered in Ref.
[115], we can recast the equations as two uncoupled, second order differential equations

̈z,⊥ + 2γ0 ̇z,⊥ + ω02 z,⊥ = Dz,⊥ (ω)eiωt .

(7.1)

When δ̄ = 0, the damping rate γ0 = W̄ /2 is set by the damping of the transverse component⊥ caused by single-particle wave function collapse associated with the repumping. The naturalp√frequency ω0 = W̄ (N Cγ − W̄ ) = 2J¯⊥ Cγ is set by the steady-state rate of converting collective

(a)

151

(b)

Figure 7.3: Effects of finite ratio of repumping rate r (a) Comparison at two values of r ofTA (ω)versus modulation frequency. The points are measured data in good agreement with thezero-free parameter fit (lines). (b) Plot of the resonance frequency ωres (black) and the peak value
TA (ωres ) (red) versus r.

2 Cγ, normalized by the total steady state coherence J¯ .
transverse coherence into inversion J¯⊥⊥

The responses of the two quadratures to the modulation are different because the effective
drives are different with D⊥ (ω) = W̄2 (N Cγ − 2W̄ − iω) and Dz (ω) = (N Cγ − W̄ )(W̄ + iω). Notethat the magnitude and phase of the drives change with the modulation frequency and repumping
rate, even as the modulation depth  remains constant.
We show this driven oscillator response in Fig. 7.2a with the measured and predicted parametric plot of Jz and J⊥ at three different applied modulation frequencies, with repumping nearW̄ = Wpk . Although the characteristic frequencies and rates of the atomic oscillator do notchange, the differing drives lead to a change in the phase relationship between the response of thetwo quadratures. We believe the discrepancy with theory in the center panel of Fig. 7.2a is the
beginning of nonlinearity in the system as it responds beyond the small perturbation regime nearresonance.
In Fig. 7.2b, we focus on the light field’s transfer functions TA (ω) and Tφ (ω). Data for

152

(a)

(b)

Figure 7.4: Evidence of negative and positive cavity feedback. (a) Amplitude transfer functionsof the emitted electric field for three detunings from cavity resonance. The data points are theaverage of 4 experimental trials. The lines are fitted transfer functions with N as a free parameter.
(b) Cavity damping of collective atomic degrees of freedom. The response going from δ̄ = 0.2 (left)
where the γ0 is small to δ̄ = 0.9 (right) where the system is expected to be critically damped. The
red lines are sinusoidal fits to the data (circles). The dashed lines highlight the damping in bothJ⊥ and Jz .

three different average repumping rates W̄ are shown. The data displays the features of the simple
3-level model, namely increased damping with W̄ , ω0 scaling with W̄ (inset), the 270◦ phase shiftof Tφ at high modulation frequencies, the small response near ω = 0 and W̄ = Wpk caused by thecancellation in the drive term D⊥ (ω), and finally the phase reversal of the response near ω = 0
going from below to above Wpk . The data also quantitatively agrees with the displayed theory
calculated for the full model including all 87 Rb levels. We suspect the deviation for W̄ = 0.73N Γc
is a result of a systematic error in measuring the total atom number.

153
In this work, the repumping ratio r 6= ∞, and the theory must be extended to quantitatively
describe the data. The physical effect of finite r is that population builds up in |3i. The ratio of
steady state populations is simply N̄3 /N̄↓ = 1/r. As a result of a non-negligible N̄3 , the naturalqrfrequency is slightly modified as ω0 = 1+r
W̄ (N Cγ − W̄ ). The effective damping in the presence22r(N Cγ−W̄ )rω
of a harmonic drive at frequency ω is γ0 = W̄2 (1+r)(1/2+r) + 2(1+r)(1/2+r) − W̄ (1+r) . The frequency

dependent term results from the additional phase shift introduced into the oscillating system as aresult of time spent in |3i. Despite the frequency-dependent reduction of the damping, as long as
γ0 > 0 near ω = ω0 , the system will remain stable. We can experimentally observe a reduction in
damping as r → 0, shown in Fig. 7.3. From the form of ω0 and γ0 , we expect to see the resonancefrequency sharply decrease, and an increase in the peak relaxation oscillation amplitude, as r → 0,
which we observe in Fig. 7.3b.
To understand the dynamic tuning of the cavity resonance frequency ωcav in response tochanges in the atomic populations, we consider the case r → ∞ and δ̄ 6= 0. We also assume
the cavity frequency is tuned by the atomic inversion Jz as α = α↓ = −α↑ > 0. The dynamic



W̄
N
cavity tuning then modifies the damping rate as γ0 = W̄
−
1
+
2α
δ̄
. The dispersive
2
2
Cγ
1+δ̄
tuning of the cavity frequency can act as either positive or negative feedback on the oscillationsof ⊥,z for δ̄ < 0 and > 0 respectively. As an example, in the case of negative feedback, if theinversion Jz decreases, the cavity tunes away from resonance with the Raman transition, reducing
the superradiant emission from |↑i to |↓i, and allowing the repumping to restore the inversion more
quickly. We observe both positive and negative feedback in the measured transfer function TA (ω)
and the atomic responses Jz (t) and J⊥ (t) as shown in Fig. 7.4.
We have studied the dynamics of the polarization, inversion, and field of an optical laser
operating deep in the bad-cavity regime. We have shown that dispersive cavity frequency tuning
can suppress or enhance relaxation oscillations. Having experimentally validated our model for
optical lasers in the extreme bad-cavity regime, future work can now extend the formalism torealistic models of proposed ultrastable lasers using ultranarrow atomic transitions in atoms such as
Sr and Yb[115]. In the future, it should be possible to directly monitor J⊥ using techniques similar

154
Raman Transition
D1, 795 nm

(a)

(b)

Repumping Transitions
D2, 780 nm

F' 2

I

F' 2

e

F 2

F 2

g

F 1

F 1

mf = -1

01

mf = -2

-1

0

1

2

Figure 7.5: (a) The energy level diagram for the D1 Raman transition used for lasing. The linearlypolarized Raman dressing laser is shown in red and the superradiantly emitted light in blue. The
cavity mode resonance frequency ωcav is denoted with a blue dashed line. With the quantization
axis defined by a 2.7 G magnetic field oriented along the cavity axis, the linearly polarized light is alinear combination of σ+ and σ− polarizations. (b) The energy level diagram for the D2 repumping
beams F2 (green) and F1 (purple). The dark state with respect to the repumping lasers is labeledwith a gray circle and corresponds to |↑i.

to those presented here to monitor Jz . Further studies of the nonlinear dynamics of the extreme
bad-cavity laser system will include investigations of chaos[67] and squeezed light generation[101].

7.4

Additional Details

7.4.1

Technical Experimental Details

The optical cavity has a mode waist of 71 µm, mirror separation of 1.9 cm, and a finesse of
F = 700. The atoms are trapped by a 1-D intracavity optical lattice at 823 nm and laser cooled toapproximately 40 µK. The sub-wavelength localization of the atoms along the cavity-axis ensuresthat the atoms are in the Lamb-Dicke regime along this direction. However, the atoms are not in
the Lamb-Dicke regime with respect to motion transverse to the cavity axis.
We first note that the energy levels, dressing laser, repumping lasers, and relative frequency
tunings are the same as the primary configuration presented in Chapter 5. We include the details

155
here again for clarity. The |↓i ≡ 5 2 S1/2 F = 1, mF = 0 and |↑i ≡ 5 2 S1/2 F = 2, mF = 0 hyperfine ground states of 87 Rb form the basis of our Raman laser (energy level diagram shown in
Fig. 7.5). The |↑i state is dressed with an optical Raman laser at 795 nm that induces an optical
decay to |↓i, with the cavity tuned to be resonant or near-resonant with the emitted light. The
intensity of the dressing beam and the detuning by ∆/2π = +1.1 GHz from the optically excitedintermediate states |I± i ≡ 5 2 P1/2 F 0 = 2, m0F = ±1 set the single-atom scattering rate into all ofCγ
free space γ. The induced single-atom scattering rate into the cavity mode is Γc = 1+
where
δ̄ 2

δ̄ is the detuning of the cavity resonance frequency from the emitted light in units of the cavity
half-linewidth, and C is the single particle cooperativity parameter of cavity QED[35], equivalent
to a Purcell factor[149]. The branching ratio for decay to |↓i are included in the definition of C.
The quantization axis is set by a 2.7 G magnetic field along the cavity axis. The Raman
dressing laser is injected non-resonantly along the cavity axis (experimental setup shown in Fig.
7.6a). For this quantization axis, the linearly polarized Raman dressing laser is an equal combination of σ + and σ − light. Constructive interference between the two decay paths from |↑i to |↓ithrough the two states |I± i leads to enhancement of light emission with linear polarization rotated90◦ from the polarization of the dressing laser. Conversely, emission of light into the cavity withthe same polarization as the dressing laser is highly suppressed by destructive interference of thetwo decay paths.
Two repumping lasers control the rate out of |↓i and back into |↑i. The laser powers are
independently set using acoustic optic modulators (AOMs) 1 and 2 shown in Fig. 7.6a. Bothrepumpers are π-polarized, applied perpendicular to the cavity, and separately tuned near the
5 2 S1/2 F = 1, 2 to 52 P3/2 , F 0 = 2 transitions. The F1 repumper moves atoms primarily from theground 5 2 S1/2 F = 1, mF states to the ground 5 2 S1/2 F = 2, mF states, and sets the scattering
rate W out of |↓i. The F2 repumper pushes population to |↑i as the dipole matrix element for
the transition |52 S1/2 , F = 2, mf = 0i → |52 P3/2 , F 0 = 2, m0f = 0i is zero. We quantify theF2 repumping rate by calculating the single particle scattering rate for an atom in |52 S1/2 , F =2, mf = 1i state as Γ3 . We define the pump ratio r ≡ ΓW3 to quantify the degree of population

156
buildup outside of the two level manifold |↑i and |↓i, with the ideal repumping case being r → ∞.
Though Γ3 is not perfectly equivalent to Γ3↑ from the three level model, it allows us to define r
in an analogous way. Population will still remain in the states |52 S1/2 , F = 1, mf = ±1i even as
r → ∞, unlike in the ideal three-level model. This why the output power correction factor R does
not asymptote to 1 as r → ∞. Also note that unlike the simple three-level model, here, W includes
Rayleigh scattering back to |↓i, contributing damping of the coherence J⊥ without affecting theinversion Jz .

7.4.2

Response Function Measurement Sequence

In Sec. 7.3, we measure three quantities to characterize the response of the superradiantlaser to small perturbations: the light field amplitude A(t), the magnitude of the coherence J⊥ (t),and the inversion Jz (t). As shown in Fig. 7.6, the perturbation is created by a small fractional
modulation of the F1 and F2 repumping laser power at frequency ω such that W (t) = W̄ (1+ cos ωt)while r(t) remains constant. At time t3 , we shut off the repumping and dressing lasers in less than
100 ns, freezing the atomic populations[23]. The amplitude of the light just before shut off A(t3 )is determined from the IQ-demodulated heterodyne signal shown in Fig. 6a.
We then use methods similar to those in Chapter 3 to non-destructively probe the cavity-mode
to determine the cavity resonance frequency ωcav1 and hence detuning from the emitted light frep
quency δ(t3 ). We then calculate the coherence J⊥ (t3 ) using the relation A(t3 ) = J⊥ (t3 ) Cγ/(1 + δ 2 (t3 )).
The probe is non-destructive in that a small fraction of the atoms are lost or Raman scattered toother states during the measurement.
The inversion Jz (t3 ) is determined by using a microwave π-pulse to swap the populations
between |↑i and |↓i, and then measuring the cavity resonance frequency a second time ωcav2 . Shifts
in the cavity frequency due to atoms in other states are common mode to both measurements,
such that the difference between the two cavity frequency measurements is only proportional to theinversion ωcav1 − ωcav2 = 2(α↑ − α↓ )Jz (t3 ). The frequency tuning of the cavity resonance per atomin each state α↑,↓ is calculated from the known cavity geometry, atomic dipole moments, and bare

Cavity
Probe

DAC

RepumpRatio rI
Q
Loop
Filter

AOM

Raman
Dressing

AOM(a)Local Oscillator
6.834 GHz

1
AOM MasterµWave
Source

2

ADC

W

ADC

Jz , J⊥(b)

157
RepumperDressing Laserπ

Microwave Source
M1

Cavity Frequency
Probe
t1

t2

t3

M2
Time

Figure 7.6: Measurement setup and sequence timing diagram. (a) The physical setup for measuring
the light field amplitude response A(t) and the atomic responses Jz (t), J⊥ (t) to a small modulationof the repumping rate W (t). Optical beams are colored with arrows; RF and microwave signals arein black. (b) Timing diagram for measurements. Superradiance is started at t1 , and the repumping
rate W (t) is modulated (with r constant) starting at time t2 . At t3 , the repumper and dressing
lasers are shut off, freezing the atomic populations. The microwave π-pulse corresponds to a 12 µs
pulse of 6.834 GHz microwaves resonant only with the ground hyperfine state transition |↑i to |↓i.
The pulse completely swaps the populations Ne and Ng of states |↑i and |↓i. The cavity frequency
probe windows M1 and M2 used to determine the dressed cavity frequency ωcav1 and ωcav2 , andfrom which we can determine Jz (t3 ).

cavity detuning from resonance with nearby optical atomic transitions at 795 nm. The process isrepeated for different stopping times t3 in order to sample the modulation period for a set of times
ti . The data Jz (ti ) versus J⊥ (ti ) are shown in several parametric plots in the main text.
In the case of the field transfer functions TA (ω) and Tφ (ω), continuous time traces A(t) areaveraged over several trials at the same modulation frequency and phase to enhance signal to noise.
The average response is then fit to A(t) = As (1 + a(ω) cos(ωt + φA (ω)), with transfer functions
calculated as TA (ω) = a(ω)/ and Tφ (ω) = φA (ω). The measurement is then repeated for different
ω.

7.4.3

Real time cavity probe

For the data presented here, we measured the state population by probing the dressed cavity frequency when superradiant emission had been extinguished by turning off the dressing andrepumping laser. However, leading up to these results, we also developed a method to measure
the dressed cavity frequency in real-time during superradiant emission using the transmission of a
weak probe through the atom-cavity system. Here I will describe the details of this real time cavity

158
probe (RTCP).
The basic idea of the real time probe is to weakly phase modulate the dressing laser at thehyperfine frequency, producing a small component of the light at the dressed cavity frequency.
We can then detect the transmission of this probe to measure the probe sideband detuning fromthe dressed cavity resonance δ = fprobe − fc . The sideband is created using the EOspace phasemodulator in the dressing path as described in Chapter 2. The basic detection scheme is complicatedby presence of the strong off-resonant dressing laser and superradiant emission exiting the cavity
along with the weak probe. The superradiant emission is orthogonally polarized with respect tothe dressing laser, so it is polarization filtered and sent to the heterodyne detection. The dressing
laser and the RTCP are then directed to the APD detector.
There are two difficulties we dealt with in directly detecting the transmission of the weakprobe. The first is that the signal from the off-resonant dressing laser provides a large background,Vbck which could fluctuate due to detailed higher order mode structure of the cavity. The second
difficulty is that the transmission only from a single RTCP does not distinguish the sign of thedetuning δ. To address these issues, we implemented a two stage lock-in detection.
The first stage consisted of switching the modulation that created the probe component on
the dressing laser on and off at 100 kHz. This switching allows us to measure the total average
power transmitted with the Raman dressing laser background subtracted out. The second stage
consisted of switching between two different frequencies for the modulation of the dressing laser
to create the weak probe components separated from the ‘probe center’ by δm = ±20 MHz. We
alternated between the two modulation frequencies fh and fl at a rate of 50 kHz. Each frequency
component produces a signal from the APD, Vh and Vl respectively, that have a Lorentzian shape
as a function of the cavity detuning. By mixing the APD signal with the 50 kHz modulation
frequency, we can extract the difference in the power transmitted at these two frequencies, whichcontains information about the sign of the cavity detuning with respect to the center of the twoprobe components. This assumes that the dynamics of the cavity are stable on timescales of order
20 µs. The slowly varying signal from the APD after mixing with the 50 kHz reference frequency is

159

Figure 7.7: A schematic of the real time cavity probe RF setup. The two ZASWA-2-50DR+
switches provide the modulation of the signals, controlled by the two SRS frequency generators
locked to the atomic clock reference. The signal generated by the light transmitting the cavity isthen demodulated to produce the signals used to determine the dressed cavity detuning.

160
Vdiff = (Vbck1 + Vh )/2 − (Vbck2 + Vl )/2. Notice the background signal largely subtracts out, so Vdiff
is proportional to the difference of two Lorentzians. We filter high frequency components above 25
kHz so we can still detect modulations in the cavity mode on the 7 µs timescale.
The difference in powers provide a dispersive-like signal over a range of 30-40 MHz for thecavity frequency. However, depending on the cavity frequency, the total power transmitted varies
as the sum of two Lorentzians. By separately mixing the APD signal with the 100 kHz reference
frequency and filtering the signal at 50 kHz, we obtain a slowly varying signal Vsum = (−Vbck1 +Vh )/2 + (−Vbck2 + Vl )/2 that is proportional to the average transmitted power. We implemented
a power stabilization feedback loop to keep the total transmitted power constant at a level that
does not interfere with the superradiant emission. The final signal used for measuring the probecavity detuning is the ratio S = Vdiff /Vsum , to normalize out any residual total power variation notmδaccounted for in the feedback loop. We fit S to a signal y0 + (κ/2π)28δ+4(δ
2 +δ 2 ) after sweeping themprobe components over the resonance, which calibrates the unknown scale factors in the system,
allowing us to invert the expression and extract δ for a given S.
The RTCP was useful in first discovering the dynamics of the dressed cavity mode and theimpact of the dressed cavity mode on the amplitude stability of the laser. In particular, it provided
the first evidence of the instability associated with a negative detuning of the dressed cavity mode
from the laser emission frequency, discussed in Chapter 6. This data is shown in Fig. 7.8.

IQ Amplitude of
Detuning fc - f γ [MHz] Emitted Photons [V]

161
0.10
0.08
0.06
0.04
0.02
0.00
Stable

10
5
0
-5
-10

IQ Amplitude of
Detuning fc - f γ [MHz] Emitted Photons [V]

0

10

15

20
Time [s]

25

30

35

40x10

-3

0.16
0.12
0.08
0.04
0.00
Crossover

10
5
0
-5
-10
0

IQ Amplitude of
Detuning fc - f γ [MHz] Emitted Photons [V]

5

5

10

15

20
Time [s]

25

30

35

40x10

35

40x10

-3

0.20
0.10
0.00
Unstable

10
5
0
-5
-10
0

5

10

15

20
Time [s]

25

30

-3

Figure 7.8: Data showing the correlation of amplitude stability of the laser with the cavity detuningfrom the laser emission frequency. The data in particular highlights the transition from stable
operation on when the cavity is positively detuned, but the transition in instability for negative
detuning. Here the IQ amplitude of emitted photons is proportional to the emitted light amplitude,
fc is the dressed cavity, and fγ is the emitted light frequency.

Chapter 8

Active and passive sensing of collective atomic coherence in a superradiant laser

In this chapter, we explore the idea of the superradiant laser acting as a non-demolition
mapping of collective quantum coherence onto a cavity light field. We show theoretically that thefundamental precision of the mapping is near the standard quantum limit on phase estimation for√a coherent spin state, ∆φ = 1/ N , where N is the number of atoms. The associated characteristic
measurement timescale τW ∝ 1/N is collectively enhanced. The non-demolition nature of the measurement is characterized by only 0.5 photon recoils deposited per atom due to optical repumping ina time τW . We experimentally realize conditional Ramsey spectroscopy in our superradiant Raman
laser, compare the results to the predicted precision, and study the mapping in the presence of
decoherence, far from the steady-state conditions previously considered. Finally, we demonstratea hybrid mode of operation in which the laser is repeatedly toggled between active and passive
sensing.

8.1

Introduction
Superradiant lasers have the potential to be the most stable optical frequency references

to date, with broad impacts across science and technology [115, 33]. These frequency references
derive their stability from an ensemble of atoms spontaneously synchronized by cavity-mediatedinteractions, achieving collective coherence times greater than single-particle coherence times [24].
Reliance on inter-particle interactions makes superradiant lasing one of the growing number ofexamples of collective phenomena being explored for enhancing precision measurements [116, 4,

163
139, 104, 35, 62, 130, 42].
The superradiant laser’s defining feature is the storage of the coherence of the laser systemin the gain medium. The coherence is mapped onto the cavity field through superradiant emission(Fig. 8.1a). This conceptual Bloch sphere [51] model of the laser has been the key to understanding
not only the laser spectrum [24], but also the output field stability properties [21] and ability to
use the laser as a sensor of magnetic fields [160]. In a Raman laser configuration, the mapping
can be dynamically controlled through the use of an externally applied dressing laser. The storage
and recovery of phase information in the atomic ensemble is analogous to quantum memories forquantum communication [47, 25, 145, 37, 125, 105].
In this chapter, I present an analysis of the non-demolition mapping of the coherence in aself-synchronized ensemble of atoms onto a cavity field through superradiant emission that relies
on fundamentally collective effects with no single-atom analog. We link the well-known SchawlowTownes laser linewidth limit [136] to the information gained about the atomic system through themapping, which sets the fundamental limit to potential superradiant sensors, and study spontaneoussynchronization in the presence of decoherence far from steady-state conditions. Theoretically, theoutput light provides sufficient information to continuously track the evolving phase of the atomic
coherence φ with a precision within a factor of 2 of the standard quantum limit (SQL) on phaseresolution for a coherent spin state. The coherence readout rate is collectively enhanced by afactor of N compared to single-particle fluorescence readout. Each measurement also prepares
the coherence for the next measurement, while only imparting 1/2 photon recoils per atom in thecharacteristic measurement time.
Our theoretical analysis is compared to experimental data from our proof-of-principle experimental system using 87 Rb (Fig. 8.1b and 8.1c). The results culminate in a hybrid sensor thatcombines active sensing of the collective atomic phase during superradiant emission with passive
phase measurements using Ramsey-like evolution times. We show the sensor can repeat many measurement cycles in a single experimental trial due to the non-demolition nature of the superradiantmapping.

164

8.2

Theory of the Optimal Estimator
Our theory considers an ideal cold atom Raman laser. Examples of cold atom Raman lasers

include Refs. [73, 64, 156, 6]. The ideal laser we consider here operates deep into the bad-cavity,or superradiant, regime [97, 92] where the cavity power decay rate κ greatly exceeds all otherrelevant decay and scattering rates. The laser utilizes N atoms trapped in a one dimension (1D)
optical lattice that is formed inside the optical cavity that also mediates the long-range interactions
between atoms that drive spontaneous synchronization of the atomic dipoles.
The optical Raman dressing laser induces a decay rate γ from |↑i to |↓i (Fig. 8.1c). The
dressing laser is detuned from an optically excited intermediate state |ii, and the rate a single atom
scatters photons from the dressing laser into the resonant cavity mode is Γc = Cγ, where C is the
single-particle cavity cooperativity parameter of cavity quantum electrodynamics (QED)[149]. The
collective cooperativity satisfies the necessary condition for superradiance N C  1. Single-particle
optical repumping proceeds at an optimum repumping rate w ≈ wpk = N Γc /2 [115].
The ensemble of atoms can be represented by a Bloch vector J~ whose azimuthal phase φ(t)evolves in time at a rate set by φ̇(t) = E(t)/~, where E(t) is the instantaneous energy difference
separating |↑i and |↓i, and ~ is the reduced Planck constant. Depending on the sensitivity ofE(t) to environmental conditions, precise measurements of φ(t) correspond to measurements of theenvironment or time[89].
The quantum phase φ(t) is not directly measurable and must be mapped onto an observable
quantity. Bad-cavity active oscillators, such as masers or superradiant lasers, continuously map
the collective phase φ(t) onto the observable phase ψ(t) of an electromagnetic cavity field, because
the rapidly decaying cavity field is slaved to the atomic coherence [21]. Ignoring vacuum noise, thecomplex electric field phasor is given by A(t)eiψ(t) ∝ J⊥ (t)eiφ(t) where J⊥ (t) is the projection of the
Bloch vector onto the x-y plane and A(t) is the amplitude of the electric field phasor. Measuring
the cavity field is equivalent to a continuous non-destructive measurement of the evolving atomic
coherence.

165

(a)

AtomicBlochVectorφ(b)

Heterodyne Reference Beam
OpticalRepumping
I

Dressing
Laser

L

EmittedLight

Optical
Lattice

(c)

(d)
ON

OFF

ψ(t) (rad)

i

fhf

DDS

I

R

R

L

I(t) Q(t)90πSuperradiance OFF

ON

0

ON

T

-π
0

1

Time (ms)2

Figure 8.1: (a) Superradiant emission (blue) mapping the collective atomic coherence, represented
by a Bloch vector J~ with phase φ and equatorial projection J⊥ , onto the phasor representing theemitted light field defined by a phase ψ and amplitude A. (b) The experimental setup. 87 Rb atoms
are trapped in a 1D optical lattice (dashed, orange) within an optical cavity. Optical repumping
light is applied perpendicular to the cavity axis. The emitted light is detected in heterodyne, thendemodulated using a direct digital synthesis frequency reference to obtain both field quadratures,I(t) and Q(t), and calculate the light’s amplitude A and phase ψ. (c) An energy level diagram ofa superradiant Raman laser. The Raman dressing laser (red) detuned from an intermediate state|ii induces optical decay at rate γ between ground hyperfine states |↑i and |↓i. The atoms areincoherently repumped back to |↑i at rate w. When the dressing and repumping lasers are off,
the atoms remain in a superposition of |↑i and |↓i, and the quantum phase φ(t) that evolves at
fhf . (d) Example data from a Ramsey-like sequence in a superradiant laser. Superradiant emission
continuously maps φ(t) onto ψ(t). We measure a differential light phase ψ(T ) − ψ(0) over a freeevolution time T .

166
The precision of the φ(t) to ψ(t) mapping is limited by fundamental quantum noise from
Schawlow-Townes phase diffusion of the atomic phase and photon shot noise on the measurement of
the light phase. Measuring ψ(t) longer reduces photon shot noise, but diffusion of the atomic phasemeans later data is less correlated with the initial atomic phase we wish to estimate. Employing
a Kalman filter [82] analysis, we find that the optimal estimate φe (t) of the phase φ(t) is anexponentially weighted average of ψ(t) with a weighting time constantτW = √

1
,
qN Γc

(8.1)

assuming w = wpk . The details are included in Sec. 8.5). Here q is the photon detection efficiency.
A single-pole, low pass filter can be used to implement such a running weighted average.
The mean squared error of the optimal estimator is

σe2 =

D

2

(φe (t) − φ (t))

E

=√

2
.
qN

(8.2)

When q = 1, the error is only a factor of 2 from the SQL on phase variance for a coherent spinstate of N unentangled atoms (∆φSQL )2 = 1/N and is at the SQL, 2(∆φSQL )2 , for the steady-state√ projection during superradiance at the optimum repumping rate wpk . Also, the impact ofJ⊥ = 2N
2√
imperfect detection efficiency (q < 1) is mitigated by the scaling as 1/ q compared to the expected
increase from increased photon shot noise alone, which would scale as 1/q.
√
The measurement is non-destructive in that an atom experiences only 0.5/ q photon recoils
on average during the characteristic measurement time τW . This degree of recoil heating compares
favorably with other sample-preserving [107, 20, 58, 56] and coherence-preserving [4, 139, 35, 34]
probing techniques with precision at the SQL.
This active superradiant mapping provides a continuous measurement of the atomic phase,in contrast to the standard technique for mapping a quantum phase onto the observable difference
in state populations, as is done in passive Ramsey spectroscopy [126] with atomic fountain andoptical lattice clocks, for example [11, 109]. In Ramsey spectroscopy, the atoms accumulate a167
quantum phase φ(t) during free evolution periods after which φ(t) is mapped onto the atomic state
populations. The state population is measured through fluorescence detection, which destroys thecoherence and often destroys the ensemble [10], though the loss of the sample is not fundamental[107, 20, 57, 56, 58, 155, 170, 22]. Passive Ramsey spectroscopy has the benefit of being freefrom perturbations necessarily introduced in the active mapping that disturb the accuracy of themeasurement. By implementing dynamic control of the superradiant mapping, a passive Ramseylike sequence can be realized in a superradiant laser, combining both active and passive sensing
into a single hybrid system[160].
A single cycle of the hybrid sequence (Fig. 8.1d) consists of running steady-state superradiance and estimating the light phase ψ̄(0) just before temporarily shutting off the dressing andrepumping lasers at time t = 0, quenching the superradiant emission. However, the collective Blochvector continues to precess at frequency fhf separating |↑i and |↓i. At time t = T , the repumping and dressing lasers are turned on, and the phase of the light can then be estimated as ψ̄(T )using ψ(t) data at times t > T . The accumulated atomic phase during the time period T is then∆φ(T ) = φ(T ) − φ(0) = ψ̄(T ) − ψ̄(0). The variance on ∆φ(T ) is 2σe2 because of the measurement
necessary to initialize the state at T = 0. The initial measurement is unnecessary in traditionalRamsey measurements with perfect state preparation.
The fundamental limit on the estimation of the phase difference ∆φ(T ) is a factor of 4 larger
than for standard interferometry techniques that use traditional Ramsey spectroscopy at the SQL.
Still, little fundamental precision needs to be sacrificed in future atomic sensors to utilize theproposed mapping. The collective nature of superradiant emission also results in a characteristic
readout rate enhanced by a factor of N compared to the readout rate in the single-atom limit.

8.3

Experimental Demonstration
We have implemented a proof-of-principle atomic sensor using a cold-atom Raman laseron the clock transition of the ground hyperfine states of 87 Rb (|↑i = |F = 2, mF = 0i to |↓i =|F = 1, mF = 0i). For our system, κ/2π = 11 MHz, N = 106 atoms cooled to 40 µK, C = 8 × 10−3 ,168and γ/2π ≈ 1 Hz. The cavity finesse is F = 700. The dressing and repumping lasers can beswitched on and off within 100 ns, much faster than the timescale on which atomic dynamics occur.
We measure the phase ψ(t) and amplitude A(t) of the emitted light via heterodyne detection withrespect to the dressing laser to remove phase noise on the emitted light imposed by phase noiseon the dressing laser. By splitting the resulting rf signal and simultaneously demodulating the twoquadratures, we obtain both A(t) and ψ(t) simultaneously. Example data of the passive sequence
are shown in Fig. 8.1d. We estimate φ(0) (φ(T )) using linear fits to 0.5 ms of ψ(t) data at times
t < 0 (t > T ).
The standard deviation of the light phase difference ψ̄(T ) − ψ̄(0) as T → 0 predicted by the
optimal estimator φe for our system is 12 ± 2 mrad, after accounting for finite quantum efficiency(q = 0.03) and the multi-level structure of 87 Rb. The general expression for the superradiantemission rate,


Rd = R

N
2

2

2w
N



w
1−
N Γc


,

(8.3)

accounts for w 6= wpk and repumping through the multiple levels in the 87 Rb hyperfine structure[24]with the correction factor R. Here N = 9 × 105 atoms, Γc = 0.05 s−1 , R = 0.1, and w = 0.2N Cγ.
The final result of R 6= 1 and w 6= wpk is to reduce Rd by a factor of 0.28. The reduction in theemission rate and the subsequent modification to the phase diffusion increases σe by a factor of 2.4
compared to Eq. 8.2 (see Sec. 8.5).
The observed standard deviation is 70 ± 7 mrad, a factor of 6 above the predicted noise. We
assign the discrepancy to the dispersive tuning of the cavity mode described in Ref. [24], an effectpresent in our system, but not fundamental to the physics of the sensor. Building on this work, wehave also demonstrated sensing of applied phase shifts using a superradiant magnetometer [160].
The analysis of the optimal estimator does not consider the impact of decoherence and dephasing in passive sensors with superradiant readout. The process of superradiant emission continually
maintains the coherence of the ensemble, but when the emission is turned off, the atoms decohere,

(a)Amplitude
A(t) (a.u.)

169

t80
0.8A(0 -)

0

0

Time

T

Amplitude
Ratio

(c)
Std. Dev. ofφe(T) (rad)T+t80
Time to 80% turn on, t80 (ms)A(T+) A(0-)(b)

A(T +)

A(0 -)

0.4
0.2
0Evolution Time T (ms)Figure 8.2: (a) Example time trace of the emitted light amplitude with an evolution time T = 0.4
ms. A(0− ) and A(T + ) are average amplitudes measured in the blue time windows before and after
the evolution time. The characteristic rise time t80 is the time when the amplitude has returnedto 80% of the steady-state value. (b, left) The decay of J⊥ as estimated by (blue squares) the
ratio A(T + )/A(0− ) and (blue line) a fit to loss of contrast c(t) of traditional Ramsey microwave
spectroscopy. (b, right) Measured (red circles) and predicted (line) recovery time t80 . The solid
curve is the prediction accounting for the low-pass filter that was applied to the data. The bandaround the data indicates 1 s. d. on each side of the data point. We attribute the fluctuations
observed at short times to finite measurement precision. Each point is the average of 20 trials. (c)Standard deviation of the phase accumulated in time T as estimated by (diamonds) superradiantmapping and (circle) population mapping. The lines are predictions based on the observed loss ofcontrast c(T ).

170
decreasing J⊥ and A, as illustrated by Fig. 8.1a. In addition to setting the fundamental resolution of the atomic phase, the coherence is vital to the re-establishment of superradiant emission
after the evolution time, so the impact of decoherence must be understood for operation of future
superradiant sensors.
We use the experimental sequence in Fig. 8.2a to study the decay of coherence. The amplitudeof the emitted light field just before turn off, at t = 0, is A(0− ) and just after turn on, at t = T ,the amplitude is A(T + ). When the evolution time T is short, the superradiance promptly returnsto A(0− ), but as the atomic coherence J⊥ decays during the evolution time, A(T + ) decreases
proportionally. The decay of J⊥ in our system is dominated by dephasing caused by inhomogeneous
ac Stark shifts from the optical lattice. The ratio of amplitudes A(T + )/A(0− ) shown in Fig. 8.2b
is well described by a fit to the decay of the contrast fringe c(T ) measured in standard microwave
Ramsey spectroscopy with population readout [96].
The coherence lost during T is eventually restored as the laser returns to steady-state. In
Fig. 8.2b, the time t80 at which A(T + t80 )/A(0− ) = 0.8 is compared to a theoretical prediction
based on the observed atomic contrast c(T ). The prediction is obtained using semi-classical optical
Bloch equations and is described in detail in Sec. 8.6. At short times, little coherence is lost,
and the laser quickly returns to steady-state because the remaining coherence provides a seed forsuperradiance.
Ideally, superradiant readout should last long enough to completely restore the coherence andavoid depletion over multiple measurement cycles. We predict an average recovery time t80 < N 2ΓC ,which is confirmed by our experimental results in Fig. 8.2b. Since the characteristic measurement
time τW > N 1ΓC , each readout will almost fully restore the coherence for the next passive evolution
measurement, as long as at least a small fraction of the coherence remains.
The collective atomic coherence J⊥ (T ) predominantly seeds subsequent superradiance until√(incoh)∼ N /2.
J⊥ (T ) becomes smaller than the rms equatorial projection of the decohered atoms J⊥√
Even for A(T + )/A(0− ) ≈ 0.05, J⊥ (T ) > N /2, as seen by the relatively small t80 (Fig. 8.2b).
(incoh)When J⊥ (T ) < J⊥

, the laser must start an essentially new superradiant emission, resulting

0.5

π

0
S.S.

π
0

0
0

0.1

0.2

0.3

Time (ms)

0.4

0.5

Amplitude (arb.)

ψ(t) (rad)

0
2π
(b)

Weight Function

1

ψ(t) (rad)171(a) 2πFigure 8.3: Demonstration of non-demolition superradiant measurements enabling a hybrid active/passive oscillator for two different duty cycles shown in (a) and (b). The measured light phaseψ(t) is shown when the superradiance is switched on (black). The periods of evolution time, when
the superradiance is switched off, are gray regions (random phase data not shown). The blue andred exponential curves in (a) correspond to the ideal optimal estimate weight functions before andafter an evolution period respectively, calculated with our experimental parameters of N , Γc , andq. (b) The emission amplitude (red) returns to the steady-state value (dashed line) during active
oscillation, reflecting restoration of the coherence J⊥ .

in large fluctuations in t80 . We observe fluctuations of 20 µs, on the same order as the predictedfluctuations in the time to reach the peak intensity of a superradiant pulse after preparation in the
fully excited state [63].
We expect the fundamental phase resolution of the sensor to degrade from decoherence, because the signal, set by J⊥ , decays as c(T ), but the uncertainty from atom shot noise remains
fixed. We predict the measurement noise by adding the background measurement noise in quadrature with the fundamental noise limit for both superradiant and population readout of a Ramsey
sequence, 2σe /c(T ) and σe /c(T ) respectively, shown as the lines in Fig. 8.2c. Experimentally measuring the standard deviation of ψ̄(T ) − ψ̄(0) versus T , we see that even when c(T ) decays to ≈ 5%of the steady-state value at T = 0.5 ms, little phase resolution has been lost. The superradiantmeasurement noise at longer times is not at the decoherence limit due to additional technical noise,
confirmed by the appearance of an equal amount of noise on the standard Ramsey population
readout as well.

172
The culmination of these results is a demonstration of the non-destructive measurement
technique in which we repeat on/off sequences to create a hybrid active/passive phase reference,
shown in Fig. 8.3. The weight functions for the optimal estimate of the phase before and after theevolution periods are shown in Fig. 8.3a. Each measurement both retains a large fraction of theatoms (> 95%, with the loss believed to result primarily from light-assisted atomic collisions) andprepares the atomic coherence for future measurement cycles, as shown by the constant amplitudeof the emission in Fig. 8.3b. The free evolution times would ideally have high accuracy, while
the active oscillation would have much greater measurement bandwidth, providing the strengths ofboth active and passive phase references in a single device.
Furthermore, the duty cycle of the measurements could be adjusted in real time for optimal
overall phase stability and accuracy given knowledge of the environment. Figures 8.3a and 8.3b show
two example experimental trials where only the duty cycle has changed. Real-time adaptation mightallow future hybrid phase references to be employed outside of the laboratory for both scientificand commercial applications [38, 103].

8.4

Conclusion
In some aspects, our technique is similar to recently demonstrated single-atom [20, 57, 56,

58, 155], sample-preserving [107], and non-demolition [4, 139, 35, 34] readout techniques for neutral
atoms where efficient collection of photons allows for detection near the SQL while in principle
imparting only a few photon recoils per atom. Compared to fluorescence detection with a highnumerical aperture lens, here the solid angle of the mode is small (Ω/4π = 8.9 × 10−4 ), providing
low backgrounds and high optical access.
However, our collective measurement is fundamentally different from the above techniques.
Even non-demolition probes cause inevitable decay of the coherence that must be restored via
discrete optical pumping and rotation of the state. In the superradiant readout, the active flow ofcollective information from the cavity also serves to re-prepare the atomic coherence for the next
measurement. The atomic phase continuously evolves across passive and active periods, allowing

173
future frequency sources to be phase-locked rather than frequency locked to an atomic reference,
effectively eliminating aliasing noise in atomic clocks [163].
Looking forward, the ideas studied here might be applied on strictly-forbidden optical atomic
transitions [135], enabling hybrid optical lattice clocks suited for operation outside of carefully
controlled laboratory environments. Though the system demonstrated in this work is of limited
use for precision measurement, it nevertheless points a way forward to developing a different class of
atomic sensors [160] and establishes the theoretical connection between fundamental laser linewidths
and the information gained about the underlying system at the heart of the laser. Our work also
highlights unique phenomena that emerge from collective coupling of many-body systems, a topic
of much recent and future research, e.g. research with quantum dots, superconducting qubits, ions,
nitrogen vacancy centers, and mechanical oscillators [137, 52, 72, 95, 167].

8.5

Derivation of Optimal Estimator
Here we derive the optimal estimator φe (t) of the quantum phase φ(t), and its mean squared

error σe2 , given a measurement record ψ(t) of the phase of the superradiantly emitted optical field.
We use the results from a continuous Kalman filter analysis with uncorrelated process noise andmeasurement noise [81, 82, 169]. Here the measurement noise corresponds to the photon shot noisethat appears in the measurement of the light phase ψ(t) and the process noise corresponds tothe phase diffusion of the collective atomic Bloch vector that sets the Schawlow-Townes frequency
linewidth limit.

8.5.1

Photon Shot Noise

The measured phase of the radiated light is related to the underlying quantum phase φ(t)by, ψ(t) = φ(t) + ∆ψ(t) where the vacuum or photon shot noise adds the noise component∆ψ(t). The noise is Poissonian and described by its lowest order moments as h∆ψ(t)i = 0 andh∆ψ(t) ∆ψ(t + τ )i = δ(τ )Φm . Here, δ(τ ) is the Dirac delta function so that the measurement noise

174
(m)

at different times is uncorrelated, but when τ = 0 it is equal to the constant Φm = 4R1 d . Sφ (ω) is
the two-sided power spectral density (PSD) of phase fluctuations due to measurement noise, whichR∞(m)for photon shot noise Sφ (ω) = −∞ h∆ψ(t) ∆ψ(t + τ )i cos(ωt)dt = Φm [133]. The light phase vari2R
2 = ∞ S (m) (ω) Sinc(ω∆t/2)√
ance for a measurement time window ∆t is σmdω = Φm /∆t = 4R1d ∆t ,−∞ φ
2πwhere Rd is the average rate of detected superradiantly-emitted photons using homodyne detection.
At the optimum superradiant photon emission rate [115, 24] Rd = N 2 Γc /8, obtained at a repumpCγ
ing rate wpk = N Γc /2 where Γc = 1+δ
2 is the single-particle natural decay rate into the cavity

mode, including a finite detuning of the cavity resonance frequency from the emission frequency δin units of the cavity half linewidth κ/2. Taking into account finite quantum efficiency q, we find(m)Sφ (ω) =8.5.2

2.
qRN 2 Γc

(8.4)

Phase Diffusion

In addition to measurement noise, the collective Bloch vector’s quantum phase φ(t) diffuseswith time as a result of quantum noise in the repumping process, the same mechanism that sets
the Schawlow-Townes frequency linewidth limit in a bad-cavity laser or maser [92, 97, 115]. As aresult, values of φ(t) at different times are less correlated with one another as the time separation
grows. Specifically, the two-time phase difference (as measured in an appropriate rotating frame)
averaged over many trials is zero hφ(t + τ ) − φ(t)i = 0, but the variance of the phase difference
grows linearly with the time difference τ as2σD(τ ) = (φ(t + τ ) − φ(t))2 = D2 |τ |(8.5)

The phase diffusion coefficient D for the superradiant source can be derived from the expectation
value of the two-time raising and lowering atomic operator hσ+ (t + τ )σ− (t)i in Ref. [114] and is2

D = Γc



2w
1+
N Γc


(8.6)

175
Assuming operation at the repumping rate wpk one finds D2 = 2Γc . For the Kalman filter analysis
to follow, we utilize the PSD of frequency fluctuations defined as(p)
Sf (ω) =Z ∞DE
φ̇(t + τ )φ̇(t) cos(ωτ ) dτ

(8.7)

−∞
(p)

for the process noise. From Eq. 8.5 and 8.7, one simply finds Sf (ω) = D2 .

8.5.3

Optimal Phase Estimation with Kalman Filter

The Kalman filter [81, 82, 169] is designed to provide an optimal estimate φe (t) of the phaseDE
φ(t) that minimizes the mean squared error in the estimate σe2 = (φe (t) − φ(t))2 . The Kalmanfilter assumes the state model and noise sources are well known, and the that the process noise andmeasurement noise are uncorrelated, an assumption we verify by extending the theoretical work ofRef. [92] to the spectrum of phase fluctuations in a homodyne measurement.
For this simple case, the optimal Kalman filter takes the form of a single pole low-pass filter.
In the time domain, this is equivalent to an exponential weighting of the measurement record
characterized by the exponential time constant τW . The time constant is the inverse of the Kalman
gain K, which is calculated in steady state by a ratio of the noise power spectral densities


τW =

(m)

Sφ (ω)

1/21=  (p)
K
S (ω)
f

=√

1
,
qN Γc

(8.8)

assuming w = wpk . The optimal estimate is then the exponentially weighted average φe (t) =Rt
10 −(t−t0 )/τW dt0 .
τW −∞ ψ (t ) e
The mean squared error in the optimal estimate is given by the geometric mean of the noise spectral
densities


1/2
2
(p)
(m)
=√
σe2 = Sf (ω)Sφ (ω)
qN

(8.9)

Here we simply considered portions of the measurement record ψ(t) at times t ≤ t◦ , as this is
the only information actually available were the superradiance to be shut off at time t◦ as part of a

176
Ramsey-like measurement. Conversely, an estimator of the phase just after superradiance is turnedback on φe (t◦ + T ) will only include the measurement record at times t ≥ t◦ + T . The symmetryof the two noise processes with respect to time reversal makes it sufficient to consider only the first
case.

8.5.4

Modifications for Experimental Conditions

To obtain a prediction for σe2 observed here, we must also consider the modifications to
(m)

(p)

Sφ (ω) and Sf (ω) due to imperfections in our experiment. The general superradiant emission
(m)

(p)

rate, Eq. 8.3, clearly modifies Sφ (ω) in Eq. 8.4. However, Sf (ω), given by the phase diffusion(p)of the Schawlow-Townes limit, is also modified. As Sf (ω) corresponds to the full width at half
maximum (FWHM) of the laser linewidth [114], we estimate the general form of the linewidth using
Ref. [97]∆fFWHM =

1 w2
ninv
4π Rd

(8.10)

N(p)↑
to calculate the modified Sf (ω). Here, ninv = N↑ −N
is an inversion factor that depends on w and↓R through N↑ and N↓ , the steady-state populations of |↑i and |↓i respectively. After simplification,(p)

Sf (ω) reduces to(p)
Sf (ω) = Γc

8.6

3
1
+
N
Γ
2 R( w c − 1)!
.

(8.11)

Prediction of Time to Return to Steady-state Superradiant Emission
To predict the time to restore the superradiant emission to steady state, characterized by

t80 in the main text, we begin with a model based on semi-classical optical Bloch equations forthe collective Bloch vector [114]. The cavity mode is adiabatically eliminated from the set of
equations, justified by the cavity damping rate exceeding all other relevant rates in the system. We
extend the model to include the full ground hyperfine structure of 87 Rb, which requires repumping

177
lasers on both the |F = 1i → |F 0 = 2i transition and the |F = 2i → |F 0 = 2i transition to return
atoms from |↓i to |↑i. We also account for the Raman transition in our system by adiabaticallyeliminating the intermediate excited state and defining a two-photon atom cavity coupling g2 as
described in Ref. [24]. The model is similar to the cold atom laser in Ref. [156], except that inthis work, the cavity decay rate κ is much greater than the atomic coherence decay rate γ⊥ ≈ w/2.
Compared to the cold atom lasers in Ref. [73, 64], the atomic decoherence rate is much smaller
here, because the atoms are confined in a far detuned optical lattice instead of a MOT. Also, here
the lasing occurs via a Raman transition between the magnetic field insensitive ground hyperfine
states |↑i = |F = 2, mF = 0i and |↓i = |F = 1, mF = 0i, as opposed to different Zeeman states.
To find the characteristic rise time t80 (T ) as a function of the evolution time T , we first solve
the system of differential equations under steady-state conditions with the values of N , Γc , w, andR calculated from the measured shift of the cavity mode and laser powers in our experiment. We
then assume that during the dark evolution time all populations remain at their original steadystate values, but that the collective coherence is reduced with respect to the original steady-state
coherence J¯⊥ as J⊥ (T ) = c(T )J¯⊥ , where c(T ) is the fractional reduction in the Ramsey contrast
fringe measured using traditional microwave spectroscopy and population readout shown in Fig.
3b in the main text. The loss of contrast c(T ) is consistent with dephasing caused by differential
ac Stark shifts experienced by the trapped atoms [96]. To model the behavior once the couplingis restored at time T , we use the steady-state populations and modified coherence J⊥ (T ) as initialconditions to numerically integrate the equations and extract a predicted value of t80 .

Chapter 9

Summary and Conclusion

9.1

Summary of Results
Throughout this thesis, we have leveraged the additional degree of freedom present in theatom-atom interactions of a large atomic ensemble to generate useful collective states for enhancing
precision measurements. Though my work has taken the form of two separate projects over thecourse of my thesis, both projects are connected by the use of an optical cavity mode as the medium
through which the atoms interact.
In Chapter 3, I presented coherence preserving, non-demolition measurements of 4.8 × 105
87 Rb atoms using their collective coupling to the optical cavity.

The combination of the highmeasurement precision, more than 12 dB below the projection noise limit, and the reduced measurement back-action enabled by probing on an atomic cycling transition resulted in an ensemble
with directly observed phase resolution 10.2(6) dB better in variance than the SQL. By probing
the atoms on the F = 2 to F = 30 cycling transition, nearly eliminating the Raman spin flip noisethat limited our previous attempts to generate spin squeezed states to an equivalent squeezing of
1.1(4) dB [35].
Our measurements of spin flips indicate that they are no longer the limitation in our 87 Rb
system, and we have uncovered previously ignored forms of measurement back-action, such as
photon recoil heating and opto-mechanical motion.
The other collective phenomena investigated in this thesis is steady-state, collective emission
in the superradiant Raman laser. In Chapter 4, I gave a semi-classical model for the superradiant179
laser including important extensions beyond the model presented in Refs. [115, 33]. Specifically, I
included an additional state for the atoms to be repumped through, additional decoherence fromthe repumping process, and a detuning of the cavity from the emission frequency. Additionally, I
motivated the use of the Raman system as a good model for future optical superradiant lasers by
doing a explicit adiabatic elimination of the optically excited state in the Raman system, showing
it can be reduced to an effective two level system.
The extensions presented in Chapter 4 were key for understanding the experimental resultsin Chapter 5, where we experimentally realized the first steady-state Raman laser operating deep inthe superradiant regime. Using the ensemble of laser cooled and trapped atoms as the gain medium
for the laser, we could operate the Raman laser with a ratio of 2γ⊥ /κ = 5 × 10−5 . As a result, weobserved steady-state lasing and linewidth narrowing even with < 1 photon in the optical cavity on
average. We also confirmed the prediction that the sensitivity of the laser emission frequency to thecavity frequency should be reduced by 2γ⊥ /κ, a crucial feature of superradiant lasers for getting
ultrastable optical frequency sources out of stable laboratory environments. Although we wereunable to observe the predicted quantum-limited linewidth, we were able to observe spectral featuresindicating a linewidth below that set by the single particle excited state decay rate, indicating thatthe collective coherence is driving the coherence of the laser.
Further work in understanding the stability of the laser lead to the results in Chapter 7, where
we experimentally investigated the sensitivity of the superradiant Raman laser to fluctuationsin the repumping rate by modulating the repumping beam power. We showed that the laser
amplitude stability is sensitive how much of the atomic population remains in ground and excitedstates where the coherence is stored, which we associated with the repumping efficiency. We
also investigated a cavity feed-back mechanism by which fluctuations in the ground and excitedstate populations cause a shift in the cavity mode, and therefore modify the collective coupling.
We experimentally demonstrated that this cavity feedback could act to both stabilize the output
amplitude or destabilize the system. Finally, using the non-demolition measurement techniques
similar to those described in Chapter 3, we were able to gain a unique view of the state of the180laser, going beyond just observing the output light and observing the inversion and polarization of
the gain medium as well. The using theory laid out in Chapter 6, we could understand all theseexperimental results in the context of a simple linear response theory. In the future, this linear
response theory will be useful for designing superradiant lasers with robust amplitude stability andmay help us understand the impact of quantum noise in the repumping process on the phase noise
spectrum of a superradiant laser.
Finally, in Chapter 8, I presented a new conceptual view of the superradiant Raman laser,
treating it not only as a stable optical frequency source but as a non-demolition mapping of aquantum phase onto the cavity light field phase. This concept of a superradiant laser naturally
leads to the question “What is the fundamental precision of the non-demolition mapping?” We
theoretically show in this chapter that the fundamental precision is set by the SQL for phaseestimation of the coherent spin state, linking the quantum limited linewidth of the laser to atomic
projection noise. Combining this superradiant mapping with the ability to dynamically switch on
and off the superradiant emission with the Raman dressing laser, we experimentally explore theidea of a hybrid atomic sensor that combines periods of high bandwidth readout of the atomic
phase with high accuracy phase measurements using a Ramsey-like sequence. We also observed the
decay and re-establishment of atomic coherence during the mapping. Unlike other non-demolition
measurements, the superradiant mapping actively establishes coherence during the measurement.

9.2

Looking Forward
The concluding message from this thesis is that the future looks promising for using thesecollective states as tools for precision measurements. As this thesis has shown, effective interactions
generated by collective coupling to an optical cavity is particularly well-suited for implementationin precision measurement experiments.
In the case of generating squeezed states, our work demonstrated a large spectroscopic enhancement in an ensemble of cold atoms trapped in an optical lattice, a system very applicable toone of the premier precision measurement devices today, the optical lattice atomic clock. The ex-

181
perimental demonstration of more than an order of magnitude observed enhancement is a true leap
forward for quantum measurement science, indicating that entanglement-enhanced spectroscopy
has real relevance for future work. Though our technique requires the use of a cycling transition forprobing the atoms, current optical clock candidates like Sr or Yb have a readily available, nearly
cycling transition in the 1 S0 to 1 P1 transition. The factor of 10 enhancement in spectroscopic sensitivity represents a real resource for these atomic sensors, in that it is currency that can be spent
either improving the precision, accuracy and/or bandwidth. One example would be using a factorof 10 less atoms to achieve the same phase sensitivity while reducing collisional perturbations. In
that case, entanglement-enhanced spectroscopic sensitivity is used to reduce a systematic error
instead of improving the sensor’s precision.
The order of magnitude improvement observed in our experiments is not the fundamental
limit possible for our collective pre-measurement scheme. As discussed in Chapter 3, one of the
primary limitations is that the effective probe detection efficiency is only ∼ 8%. With a few
modifications to the apparatus, it should be possible to approach a factor of 100 spectroscopicenhancement in essentially the same system. To address the poor detection efficiency, in the futureclean mirrors should be installed and periodically heated to keep rubidium from spoiling theirfinesse. Furthermore, an single sided cavity (i.e. a cavity with asymmetric transmission properties)
could be used to implement a high efficiency homodyne detection in reflection where path lengths
can be better stabilized. To realize a factor of 100 in spectroscopic enhancement, we also neednarrower linewidth probe lasers, which could be obtained by locking to a stable optical cavity
reference. Going further, the sources of measurement induced back-action uncovered in our work
will need to be further investigated and understood. For example, opto-mechanical forces associatedwith the probing appeared significant in our work, and exploring adiabatic turn on and off of theprobe may be useful in reducing the impact of the atomic motion on the collective measurement.
Precise non-demolition measurements are also useful in implementing quantum feedback
schemes. One possibility going forward is that these precise, non-destructive measurements canbe used after a Ramsey sequence to reposition a squeezed state along the equator of the BlochFourier Frequency [kHz]

100

-100

-200

-200

-100

0

100

ω-	

200

ω+	

0

ω+	

0

-100

Fourier Frequency [kHz]

200

100

100

Magnetic Field Zeeman Shift [kHz]

0

-200

-100

200

Magnetic Field Zeeman Shift [kHz]

-200

182

B	
  

200

A	
  

Figure 9.1: Frequency spectrums (horizontal axis) of the superradiantly emitted light, shown as
a function of an applied magnetic field (vertical axis) that tunes the lasing transition frequency.
Warmer colors indicate more power at a particular frequency. With the atomic ensembles pumping
to magnetic field sensitive hyperfine states, as in the secondary configuration discussed in Chapter
5, Sec. 5.6, the laser emission frequency can be controlled with the magnetic field. (A) In thecase where all atoms are optically pumped to the same state F = 2, mF = +2 (see Fig 9.2a), thelaser emission frequency follows nearly linearly with the change in the transition frequency. (B)When two ensembles are pumped to the opposite Zeeman states F = 2, mF = +2 and mF = −2,
we can observe two distinct superradiant emission frequencies, each associated with one ensemble.
As the magnetic field is tuned to bring the laser emission frequencies closer to degeneracy, theensembles begin to synchronize and eventually lock up, even though their transition frequencies
may not exactly the same.

sphere, thus allowing the atomic sensor to circumvent the limitations to spectroscopic enhancement imposed by the curvature of the Bloch sphere [3]. Future work in our system could explore
the feasibility of these quantum feedback techniques.
Our work also continues to investigate other collective states with interesting applications foratomic sensors. One example is experiments first conceived during the spin squeezing experimentsof Chapter 3 in which state rotation noise could be mitigated by applying controlled dephasing to
the ensemble. This work was further developed during the writing of this thesis by colleages in
the lab. This use of controlled dephasing is also related to recent theoretical work proposing theenhancement of evolution times in Ramsey spectroscopy using dephased states [121].
Our work with a steady-state superradiant Raman laser investigated an interesting and unusual regime of laser physics, but also serves as an important test bed for future superradiant lightA	
  mf = 0 1

B	
  

183

2

Figure 9.2: Results from a demonstration of a superradiant magnetometer [160]. (A) By operating
the superradiant Raman laser in the secondary configuration described in in Chapter 5, Sec. 5.6, thelaser emission frequency is sensitive to a magnetic field. (B) We can continuously readout the effectsof the oscillating magnetic field applied in the lab during superradiant emission by observing thephase of the emitted light (red). However, for high frequency modulations, a large phase deviation
has no time to accumulate, reducing the signal and thus the sensitivity of the magnetometer. By
turning off the superradiant emission (blue region), and applying microwave π pulses that coincidewith the know frequency of magnetic field oscillations, the phase accumulated by the atoms canbuild up. Then at a later time, the superradiance can be restored and magnetic field can be inferredfrom the difference in the light phase before and after the turn off. The ‘lock-in’ detection scheme
can theoretically recover the DC sensitivity of the magnetometer.

sources. This thesis presents a promising path forward as we were able to confirm a number ofkey predictions relevant for future superradiant lasers, such as steady state collective emission andreduced cavity pulling. Our work with the model system has also informed the design of future superradiant lasers. Cavity pulling can still have a significant impact on the laser emission frequency,
as seen by the limitation to the observed linewidth in our system. Additionally, the loss of atoms
from the trap in our system highlights the need to replenish the ensemble. Schemes with moving
optical lattices or continuous atom lasers [147] could be used in the future for reloading the trap.
A truly continuous light source will be useful in observing the narrow predicted quantum-limited
linewidth that we were unable to observe.
Future work continues in our model system aimed at understanding the role of synchronization in the superradiant laser. Partially motivated by the question of mode competition on multiple

184
transitions in a future superradiant laser, we have developed a scheme in which two spatially separated ensembles act as independent superradiant lasers in the same optical cavity. By independently
controlling the lasing frequencies of each ensemble, we can tune the system from a single synchronized ensemble into two independent ensembles, as shown in Fig. 9.1. Recent theoretical work [166]has also investigated this transition, finding interesting connections to phase transitions in open
quantum systems. The flexibility and tunability of the superradiant Raman system may provide agood experimental system to investigate aspects of quantum phase transitions in open systems likean optical cavity. Already we have made technical improvements to the experimental apparatus torealize this goal, implementing spatially separated dressing beams to individually control the lasing
frequencies.
Our work investigating the dynamic superradiant mapping of atomic coherence onto thecavity field also offers another avenue of future study, as it broadens the idea of what superradiantemission can be harnessed to accomplish. This work emphasizes the idea that the coherence of thelaser is stored in the atomic gain medium. This represents yet another new set of possibilities for use
in precision measurements because now essentially the laser coherence can be manipulated with all
the tools that have been developed for controlling atomic states. In particular, we demonstrated thisidea by creating a magnetometer with a superradiant readout out that operate both in a continuous,
wideband readout mode or high sensitivity, lock-in detection configuration [160]. The basic idea is
illustrated in Fig. 9.2. Ultimately, this work points toward a future in which superradiant lasers
can act as both an absolute atomic reference and a frequency source with good short term stability
as the detected cavity field is directly phase locked to the atomic reference.

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Appendix A

Details of Raman laser model

A.1

Adiabatic Elimination of the Optically Excited State
Here we explicitly derive the adiabatic elimination of an intermediate, optically excited stateof a cold atom Raman laser described in Sec. 4.2 and Fig. 4.6. The result is a system of equations
describing the laser, Eqns. 4.26-4.28.
The Louivillian L[ρ̂] = Lc [ρ̂] + Lik [ρ̂] + Lge [ρ̂] includes the cavity decay term Lc [ρ̂], thespontaneous emission terms from state |iiNΓ X X (q) (q)(q) (q)(q)(q)
Lik [ρ̂] = −(σ̂ik σ̂ki ρ̂ + ρ̂σ̂ik σ̂ki − 2σ̂ki ρ̂σ̂ik ),
2

(A.1)

q=1 k=e,g

and an incoherent repumping term that looks like spontaneous decay from |gi to |eiN

W X (q) (q)(q) (q)(q)(q)
Lge [ρ̂] = −(σ̂ge σ̂eg ρ̂ + ρ̂σ̂ge
σ̂eg − 2σ̂eg
ρ̂σ̂ge
).
2

(A.2)

q=1

As in Sec. 4.1, we assume we are able to factorize the expectation values hĉσ̂ie i = Cσie , hĉσ̂ii i = Cσii ,hĉσ̂gg i = Cσgg , hĉσ̂eg i = Cσeg , hĉσ̂ig i = Cσig , and hĉσ̂ie i = Cσie .
Applying these assumptions to the master equation results in the equations of motion

198

C˙ = − (κ/2 + iωc )C − igJgiΩdJ˙ge = − (W/2 + iωeg )Jge − i eiωd t Jgi + igCJie
2


Γ
+
W
J˙gi = −+ iωig Jgi
2Ωd− i e−iωd t Jge + igC(Ni − Ng )

2
Γ+W
˙
Jei = −+ iωie Jei2Ωd+ i e−iωd t (Ni − Ne ) − igCJeg
2
1
J˙z = (Ṅe − Ṅg ) = W (N/2 − Jz )2Ωd− i (σei eiωd t − Jie e−iωd t ) + ig(C ∗ Jgi − CJig ).
2
Here we identity the relevant transverse atomic decay rate γ⊥ = W/2.

(A.3)
(A.4)

(A.5)

(A.6)

(A.7)

The equation for J˙z assumes only a negligible fraction of the atomic ensemble resides in |ii,
an assumption we justify shortly. For convenience, we go into a rotating frame (often called thenatural frame [27]) defined by the transformation of variables:C˜ = Ceiωc t(A.8)

J˜ge = Jge eiωeg t(A.9)

J˜gi = Jgi ei(ωc +δ0 /2)t

(A.10)

J˜ei = Jei ei(ωd −δ0 /2)t

(A.11)(A.12)

where δ0 is the two-photon detuning δ0 = ωd − ωc + ωeg . With these substitutions, Eqns. A.3 - A.7
becomeC˜˙ = −(κ/2)C˜ − ig J˜gi e−iδ0 t/2(A.13)ΩdJ˜˙ge = −γ⊥ J˜ge − i eiδ0 t/2 J˜gi + ig C˜J˜ie eiδ0 t/2
2(A.14)

Γ+W ˜ΩdJ˜˙gi =(i∆ −)Jgi − i e−iδ0 t/2 J˜ge
2
2

199
(A.15)

˜ iδ0 t/2 (Ni − Ng )+ ig CeΓ+W ˜ΩdJ˜˙ei =(i∆ −)Jei + i e−iδ0 t/2 (Ni − Ne )2
2

(A.16)

˜ iδ0 t/2 J˜eg
− ig CeΩdJ˙z =W (N/2 − Jz ) − i (J˜ei eiδ0 t/2 − J˜ie e−iδ0 t/2 )2(A.17)+ ig(C˜∗ J˜gi e−iδ0 t/2 − C˜J˜ig eiδ0 t/2 )Here ∆ = ∆d + (δ0 /2) is also equivalent to the average detuning of the Raman dressing laser ∆dand the cavity mode ∆c from their respective optical atomic transitions.
To reduce these equations to those of an effective two-level system coupled to a cavity field, we
assume that we can adiabatically eliminate the collective coherences J˜gi , J˜ei and that the populationof the intermediate state is small Ni  Ng , Ne . These assumptions are justified due to large
detuning ∆  Γ, γ⊥ , δ0 . The adiabatic elimination of the coherence proceeds as follows [27]: byexamining the form of the equations for J˜˙gi and J˜˙ei , we expect that each one can be written as
the sum of a term rapidly oscillating at frequency ∆, and a term varying on the timescale of the
population dynamics, much more slowly than 1/∆. By averaging over a timescale long comparedto the rapid oscillation, but short compared to the population dynamics, we essentially perform
a coarse graining approximation and are left with only slowly varying terms. The derivatives ofthese coarse-grained collective amplitudes are negligible. Here we consider small fluctuations aboutsteady-state values at frequency ω, so the approximation will be valid when ∆  ω. Thus, to agood approximation for the cases considered here, the derivatives J˜˙gi , J˜˙ei can be set to zero. We
then solve for J˜gi and J˜ei asgΩd −iδ0 t/2 ˜˜ geJge + eiδ0 t/2 CNJ˜gi ≈2∆∆Ωd iδ0 t/2g
J˜ei ≈e
Ne + e−iδ0 t/2 C˜J˜eg
2∆∆(A.18)(A.19)

200
where we have approximated i∆ + Γ+W≈ i∆. After including these simplifications and transform2
ing back to the original frame, we arrive at Eqns. 4.26-4.28 in the main text.

A.2

Steady-state Emission Frequency of the Raman Laser
We find the steady-state cold atom Raman laser frequency from Sec. 4.2 by assuming the

˘ −iωγ t and Jeg = J˘eg e−i(ωγ −ωd )t . Substituting in Eqns.
laser is oscillating at frequency ωγ , so C = Ce
4.26 - 4.28 gives˙C˘ = (−κ/2 − iδ)C˘ − ig2 J˘ge

(A.20)˙J˘ge = (−γ⊥ − i(ωeg − ωac + ωd − ωγ )) J˘ge + i2g2 Jz C˘(A.21)

J˙z = W (N/2 − Jz ) + ig2 (C˘∗ J˘ge − C˘J˘eg )(A.22)

where δ is the detuning of the emission frequency from the dressed cavity mode

δ = ωD − ωγ .

(A.23)The steady-state emission frequency ωγ is constrained by the condition that Jz must be real, andfollowing the procedure in Sec. 4.1, we arrive at the the laser oscillation frequency

ωγ =

2γ⊥
κ
ωD +
(ωeg + ωd − ωac ) .
2γ⊥ + κ2γ⊥ + κ(A.24)

Note the insensitivity of the oscillation frequency to changes in the cavity frequency in the badcavity limit where κ  W = 2γ⊥ .

Appendix B
Detailed Equations for 87 Rb Model

B.1

Repumping Scheme
We begin the description of our model for a cold atom laser in 87 Rb with the details of

the repumping process. The equations to describe the repumping are arrived at after adiabatic
elimination of the optically excited states 52 P3/2 , F 0 = 2, mF through which the Raman transitions
for repumping are driven. However, unlike in the Sec. 4.2, the scattered photons lack a resonantcavity mode, so the scattering is presumed to be primarily into free space (i.e. non-cavity) modes.
The relevant set of Rabi frequencies describing the resonant repumper laser coupling the
ground state 52 S1/2 , F, mF state to an optically excited state 52 P3/2 , F 0 = 2, m0F is given by the
dipole matrix element between the states asΩF,mF ,2,m0F =~ F 52 S1/2 , F, mF
52 P3/2 , 2, mF d~ · E

/~

(B.1)

where d~ is the atomic dipole moment operator and the electric field of the two repumping lasers~ 1 and E~ 2 . If an atom is in the excited state 52 P3/2 , F 0 = 2, m0 then it spontaneously decays
are E
F
to the ground state 52 S1/2 , F, mF with fractional probability given by the branching ratio

BF,mF ,F 0 ,m0F = |hF mf |F 0 1 m0f pi|2 ,where p labels the polarization of the emitted light (σ + = −1, π = 0, σ − = +1 and
1.

(B.2)
P

F,mF BF,mF ,F 0 ,m0F =202
The repumping rate W 0 in our model is calculated as the resonant, unsaturated scattering
rate

W0 =

Ω21,0,2,0(1 − B1,0,2,0 )
2ΓD2

(B.3)

where ΓD2 is the D2 excited state decay rate ΓD2 /2π = 6.07 MHz. Note that the rate W 0 is thescattering rate out of the ground state, and does not include Rayleigh scattering into free spaceΓR =Ω21,0,2,0
2Γ B1,0,2,0 which causes the scattering atom to collapse into |gi due the optically thinnature of the atomic ensemble along nearly all directions other than the cavity mode. Since ΓRscales with

Ω21,0,2,02Γ

as does W 0 , we will group both together into a single rate W =Ω21,0,2,0
2ΓD2 , anddistinguish the two rates with branching ratios in our equations for the population equations.

B.2

Reduced Optical Bloch Equations
2
Including the coherent dynamics of the effective two-level system adds the coherence J⊥

driving the population from |ei to |gi as was seen in Sec. 4.2, Eqn. 4.35

Ṅe,g = ∓CγJ2 .
1 + δ 02 ⊥(B.4)

Here δ 0 is the detuning of the dressed cavity mode from the emitted light frequency, normalized by
κ/2, as in Eqn. A.23. In the subsequent equations, we neglect the single particle scattering from|ei to |gi at rate γ as it is much less than the collective emission rate.
With these terms, we can write the reduced optical Bloch equations for the the ground state
populations as

2

X
dNF,mFW= 2
dtΩ1,0,2,0 0

0

F
X




BF,mF ,2,m0F − δF,F 0 , δmF ,m0F Ω2F 0 ,m0 ,2,m0 NF 0 ,m0F

(B.5)

Cγ
J 2 (δF,1 , δmF ,0 − δF,2 , δmF ,0 )1 + δ 02 ⊥

(B.6)

F

F =1 m0F =−F 0

+

F

203
where δF,F 0 is the Kronecker delta function. The sums have been reduced using the assumptionthat the repumping light is π-polarized.
We also must include the equation for the coherence, driven by the population inversion

2 = −W J 2 +
J˙⊥
⊥

Cγ
2
(N2,0 − N1,0 )J⊥
1 + δ 02

(B.7)

which is analogous to Eqn. 4.34 in Sec. 4.2, except that here W is the sum of the ground state
repumping rate and the Rayleigh scattering rate, where W in Eqn. 4.34 contains only the groundstate repumping rate.
Including the equation for the coherence, along with the terms for the effects of collective
emission, the resulting equations evaluated for 87 Rb can be compactly expressed asd~~
V = (Mcav + MW + MrW ) V
dt

(B.8)

with the column vector given by N2,2  N2,1 






 Ne





 N2,−1 






~
V =  N2,−2 






 N1,1 




 Ng







 N1,−1 




2
J⊥

(B.9)

where Ne = N2,0 and Ng = N1,0 are specially labeled to indicate their importance as the lasinglevels.
The matrices governing the time evolution are separated into the evolution governed by thecollective coupling to the cavity

204




 0


 0



 0


 0

Cγ 

Mcav =
02
1+δ  0


 0


 0



 0


0

0

0

0 0 0

0

0

0

0

0 0 0

0

0

0

0

0 0 0

0

0

0

0

0 0 0

0

0

0

0

0 0 0

0

0

0

0

0 0 0

0

0

0

0

0 0 0

0

0

0

0

0 0 0

0

0

2 0 0 0 −J 2 0
0 J⊥
⊥

0 


0 



−1 


0 



,
0 



0 


1 



0 


0

(B.10)

the repumping of atoms in F = 1



 0


 0



 0


 0



MW = W  0



 0


 0



 0


0
and the repumping of atoms in F = 2

0 0 0 01
8

0

0

0 0 0 0

1
16

1
4

0

0 0 0 0

3
16

0

3
16

0 0 0 0

0

1
4

1
16

0 0 0 0

0

0

1
8

9
0 0 0 0 − 16

1
12

0

0 0 0 0

3
16

− 32

3
16

0 0 0 0

0

1
12

9
− 16

0 0 0 0

0

0

0

0 


0 



0 


0 



,
0 



0 


0 



0 


−1

(B.11)

205



1
6

0

0

0

2
3

− 11
12

0

0

0

0

1
4

0

1
4

0

0

0

0 − 11
12

0

0

0

1
6

− 83

2

1
4

0

0

0

0

1
4

0

1
4

0

0

0

0

1
4

2

0

0

0

0

0

8
 −3














MrW = rW 














2
3

0 0 0 0 
0 0 0 0 

0 0 0 0 
0 0 0 0 
.
0 0 0 0 

0 0 0 0 
0 0 0 0 

0 0 0 0 
0 0 0 0(B.12)The repumping rates induced by the F2 repumper are parameterized by repumping ratio r asr=

Ω22,1,2,1Ω21,0,2,0.

(B.13)The normalized detuning δ 0 of the dressed cavity resonant frequency with the emitted lightfrequency ωγ carries implicit dependence on the populations NF,mF as derived in Sec. 4.2

~ − ωγ /κ .
δ 0 = 2 ωc + α~ ·V(B.14)The elements of the single-atom cavity tuning vector α~ comes from the cavity dressing as derived
from Eqn. 4.32. There we see α~ is set by the detuning of the cavity frequency and the detuningfrom the atomic transition frequencies ωF,F 0 between the ground 52 S1/2 , F states and optically
excited states 52 P1/2 , F 0 as

αF,mF =

2X
(2gF,m ,F 0 ,m +p )2
F

F 0 =1F

4(ωbcav − ωF,F 0 )(B.15)

where p = ±1 for the σ ± polarized cavity mode, and the single-particle vacuum Rabi frequencies
are evaluated for each transition. For the quantization axis along the cavity axis, the σ + and σ −polarizations modes will shift in frequency by different amounts specified by two vectors α~ ± . However, the symmetry of the atomic population equations ensure that the populations are symmetric206
such that NF,mF = NF,−mF . Thus, only the shift of the σ + cavity mode needs to be considered.
We use the dressing laser detuning ∆ = +1.1 GHz as was present in Refs. [24, 21]. The resulting
cavity tuning vector is




0






 0.0776 






0.1515 0.222 






α
~ + = 2π(33.8 Hz)  0.289  .






1.679








1






 0.440 




0

(B.16)~ /dt = 0 and
We find the steady-state solutions to the system of equations by setting dV~ . From the results, we can for the expressions for J¯2 , J¯z , and N̄other in the main
solving for V⊥
text.

Appendix C

Full Expressions for the Three-level Model Linear Response Theory

The full expressions for the coefficients in the three-level response equations Eqns. 6.4 and
6.5:

r
γ0 =
(2N Γc − 2γ⊥ ) + 2W r − ΓR2ζ+ h(δ)(4γ⊥ + W (1 + 2r))−r(1 + 2r)W
ω02 =
ζ

(C.1)



(2γ⊥ − N Γc ) − 2γ⊥ h(δ)

β=

(C.2)

(1 + 2r)Wζ

where the denominator factor ζ = 2rh(δ) + (1 + r)(1 + 2r) and h(δ) = 2αδ

(C.3)


N
⊥
− 2γ
Cγ
1+δ 2



.

The drive terms areD⊥,z (ω) = D0,⊥,z + iωD1,⊥,z − ω 2 D2,⊥,z(C.4)

where the coefficients are−r(1 + 2r)W
(W + 2γ⊥ − N Γc − ΓR h(δ))
2ζ

(C.5)

W (1 + r)(1 + 2r) + 2r (2γ⊥ − N Γc − ΓR h(δ)) /2
2ζ

(C.6)

D0,⊥ =

D1,⊥ = −208

D2,⊥ = −

W
2γ⊥








D0,z =

D1,z =

W
2γ⊥

C.1

(C.7)

r(1 + 2r)W (N Γc − 2γ⊥ )ζD2,z =

1 + 2r2ζ

r(3 + 2r)(N Γc − 2γ⊥ )
ζ

−2r
2γ⊥



N Γc − 2γ⊥
ζ

(C.8)

(C.9)

(C.10)

Interesting Limiting Cases
Perfect repumping, on resonance:

ω02 = W (N Cγ − 2γ⊥ )(C.11)

γ0 = W/2(C.12)

β=0(C.13)D⊥ (ω) =
Dz (ω) =W(N Cγ − W − 2γ⊥ − iω)
2

W
2γ⊥

(C.14)

(N Cγ − 2γ⊥ )(W + iω)(C.15)

Perfect repumping, with detuning:ω02 = W (N Γc − 2γ⊥ )

2γ⊥
1 + 2αδCγ(C.16)

209

γ0 =D⊥ (ω) =W(1 + h(δ))
2(C.17)

β=0(C.18)W(N Γc − W − 2γ⊥ − ΓR h(δ) − iω)2Dz (ω) =W
2γ⊥(C.19)

(N Γc − 2γ⊥ )(W + iω)(C.20)

Imperfect repumping, on resonance:

ω02 =


γ0 =



r
1+r


W (N Γc − 2γ⊥ )(C.21)(N Cγ + (r + 1/2)W − 2γ⊥ )(C.22)1
W (1 + r)(C.23)

ω2
2(1 + r)2riω(N Cγ − 2γ⊥ ) − W+
2 (1 + r)(1 + 2r)rW+(N Cγ − 2γ⊥ − W )2(1 + r)(C.24)r(1 + r)(1 + 2r)β=D⊥ (ω) =2r(N Cγ − 2γ⊥ )Dz (ω) = −ω 22γ⊥ (1 + r)(1 + 2r)
r(3 + 2r)(N Cγ − 2γ⊥ )
W
+ iω
2γ⊥
(1 + r)(1 + 2r)W+(N Cγ − 2γ⊥ )2γ⊥(C.25)

Appendix D

Heterodyne PhotodiodeThis circuit diagram contains the details of the photodiode circuit that provided most of thedata in the thesis.

211

Figure D.1: A schematic of the transimpedance amplifier circuit for the Hamamatsu S5973 photodiode used as the Heterodyne detector in this thesis. Crucial notes about the circuit are noted intext on the schematic. The most important features for a photodiode with a flat gain response up
to 200 MHz was robust connections to the ground plane, even adding additional ground connections
near the AD8015 transimpedance amplifier. The noise properties of the circuit were also improvedby making the capacitance up to the input of the AD8015 low with short, in-air connections fromthe S5973 output pin to the input of the AD8015. The circuit layout was designed by Terry Brown
in the JILA Electronics Shop.