Breaking Quantum Limits with Collective Cavity-QED:
Generation of Spin Squeezed States via Quantum
Non-Demolition Measurements
byZilong Chen
B.S., Massachusetts Institute of Technology, 2004
M.S., University of Colorado at Boulder, 2009

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
2013

UMI Number: 3607286

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This thesis entitled:
Breaking Quantum Limits with Collective Cavity-QED: Generation of Spin Squeezed States via
Quantum Non-Demolition Measurements
written by Zilong Chen
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
Chen, Zilong (Ph.D., Physics)
Breaking Quantum Limits with Collective Cavity-QED: Generation of Spin Squeezed States via
Quantum Non-Demolition Measurements
Thesis directed by Prof. James K. Thompson

Large ensembles of uncorrelated atoms are extensively used as precise sensors of time, rotation, and gravity, and for tests of fundamental physics. The quantum nature of the sensors imposesa limit on their ultimate precision. Larger ensembles of N atoms can be used to average the quan√tum noise as 1/ N , a scaling known as the standard quantum limit. However, the ensemble size
may be limited by technical constraints and/or atom-atom collisions – a fundamental distinction
from photon-based sensors. Learning to prepare entangled states of large ensembles with noiseproperties below the standard quantum limit will be key to extending both the precision and/or
bandwidth of atomic sensors. More broadly, the generation and application of entanglement tosolve problems is a core goal of quantum information science being pursued in both atomic andsolid state systems.
In this thesis, we utilize the tools of cavity-QED to prepare entangled spin-squeezed stateswith 3.4(6) dB improvement in spectroscopic sensitivity over the standard quantum limit. The
collective atomic spin is composed of the two-level clock states of 87 Rb confined in a medium
finesse F = 710 optical cavity. We employ cavity-aided quantum non-demolition measurementsof the vacuum Rabi splitting to measure and subtract out the quantum projection noise of thecollective spin state, preparing states with collective atomic spin projection noise 4.9(6) dB below
the projection noise level. The conditionally reduced spin noise combined with the measured
1.5(3) dB reduction in the mean spin length implies a net 3.4(6) dB spectroscopic enhancement orconditional squeezing as defined by the Wineland criterion. Our method does not require single
particle addressability and is applied to a spectroscopically large ensemble of N = 7 × 105 atoms
using two collective population measurements, with the whole squeezing operation taking ∼ 150 µs.

iv
The gain in sensitivity is spectroscopically equivalent to the enhancement obtained had we created> 105 pairs of maximally entangled qubits, demonstrating the power of a top-down approach forentangling large ensembles. The nondemolition probing of atomic populations via the vacuum Rabi
splitting is also of broad interest for non-destructively reading out a wide variety of both atomic
and solid state qubits.

Contents

Chapter
1 Introduction and Motivation

1

1.1

Precision Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Contributions to Superradiant Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82 Spin Squeezing: Overview and Context
2.1

10Collective Spins, Coherent Spin States, and the SQL . . . . . . . . . . . . . . . . . . 10
2.1.1

Collective Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2

Coherent Spin States and the SQL . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

Ramsey Spectroscopy with Coherent Spin States . . . . . . . . . . . . . . . . . . . . 14

2.3

Surpass SQL with Spin Squeezed States . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4

Generation of Spin Squeezed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1

Collisional Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.2

Light-mediated One-Axis Twisting Hamiltonians . . . . . . . . . . . . . . . . 20

2.4.3

Quantum Non-demolition Measurements . . . . . . . . . . . . . . . . . . . . . 23

2.5

Relative advantages and disadvantages to different approaches . . . . . . . . . . . . . 26

2.6

Other work on entanglement-enhanced metrology . . . . . . . . . . . . . . . . . . . . 27

2.7

Outline of our squeezing experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

vi
3 Theory: Probing atoms-cavity system with projection-noise-limited sensitivity

32

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2

Coupled atoms-cavity modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3

3.2.1

Linearized response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.2

Driven and damped dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.3

Full complex field response to probing . . . . . . . . . . . . . . . . . . . . . . 41

3.2.4

Probe vacuum noise and measurement imprecision . . . . . . . . . . . . . . . 43

Quantum-limited signal-to-noise and free space scattering . . . . . . . . . . . . . . . 44
3.3.1

Projection-noise-driven fluctuations of mode frequencies . . . . . . . . . . . . 44

3.3.2

Fundamental measurement imprecision and free space scattering at arbitrary
detuning δc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.3
3.4

at fixed maximum detuning . . . . . . . . . . . . . . . . . . 52
Minimizing mprojsSummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Theory: Probe-induced back-action

54

4.1

Back-action on Jy

4.2

Back-action on |hĴi| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3

Back-action on Jz

4.4

4.6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.1

Simple three-level model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.2

Diffusion of Jz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Measurement limits on Jz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1

4.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Single spin measurement resolution . . . . . . . . . . . . . . . . . . . . . . . . 60

Spectroscopic enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5.1

Spin-Flip Diffusion-Noise-Limited Squeezing/Raman Limit . . . . . . . . . . 62

4.5.2

Decoherence-Limited Squeezing/Cycling Limit . . . . . . . . . . . . . . . . . 62

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

vii
5 Experiment: Spin squeezing via cavity-aided QND measurements

65

5.1

Energy levels and probing scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2

Atom-cavity coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3

Key experimental details for Science Cavity and atomic ensemble . . . . . . . . . . . 68
5.3.1

Science Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3.2

Atomic ensemble properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4

Inhomogeneous coupling: effective coupling 2g and effective atom number N. . . . 71

5.5

Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6

Measuring Jz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.6.1

Single Rabi splitting measurement . . . . . . . . . . . . . . . . . . . . . . . . 73

5.6.2

Measurement uncertainty ∆Jz

5.6.3

Probe-induced Raman scattering . . . . . . . . . . . . . . . . . . . . . . . . . 79

. . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.7

Observation of quantum projection noise in Jz. . . . . . . . . . . . . . . . . . . . . 83

5.8

Conditional spin variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.9

Loss of coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.10 Net spectroscopic enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.11 Back-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6 Advantages of operating in resonant-coupling regime

93

6.1

Fundamental scalings related to projection noise . . . . . . . . . . . . . . . . . . . . 93

6.2

Noise rejection in the resonant limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3

ac-Stark shift cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4

Radial oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.5

Spatial mode matching of atoms-cavity modes . . . . . . . . . . . . . . . . . . . . . . 100

6.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

viii
7 Experiment: Realization of precise rotations

104

7.1

Aligning rotation axis to collective spin vector . . . . . . . . . . . . . . . . . . . . . . 104

7.2

Constraining noise from microwave rotations . . . . . . . . . . . . . . . . . . . . . . 107

7.3

Low phase noise microwave source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3.1

Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.3.2

Low noise, fixed frequency source . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3.3

Agile control of Phase, Frequency and Amplitude . . . . . . . . . . . . . . . . 117

7.3.4

Impact on QND measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8 Theory: Impact Phase Noise on Bloch Vector Rotations

122

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2

Deflection of Bloch Vector due to coherently phase modulated rotation . . . . . . . . 125

8.3

8.4

8.5

8.2.1

System Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.2.2

LO Phase Noise

8.2.3

Bloch Sphere Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.2.4

Small Rotation describing Phase Modulated Rotation . . . . . . . . . . . . . 127

8.2.5

Solution for Deflection Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 128. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.3.1

Physical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.3.2

Response to Coherently Phase Modulated Rotation . . . . . . . . . . . . . . . 131

Noise in Bloch Vector due to Phase Noise in a Single Rotation . . . . . . . . . . . . 132
8.4.1

Covariance Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.4.2

Covariance Noise Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Noise in Bloch Vector due to White Phase Noise From Multiple Rotations . . . . . . 136
8.5.1

Noise Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.5.2

Average Infidelity

8.5.3

Composite π-pulse Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . 138. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

ix
8.5.4
8.6

Spin Echo Pulse Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9 Broader Context, Future Prospects, and Conclusions

144

9.1

Framing our results in broader context . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.2

Spectroscopic gain in the presence of decoherence . . . . . . . . . . . . . . . . . . . . 147

9.3

Fundamental Limits on Squeezing via Probing the 87 Rb cycling transition . . . . . . 149

9.4

Future Experimental Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9.5

9.4.1

Improving Cavity Finesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9.4.2

Single-ended Cavity to Improve Detection Efficiency . . . . . . . . . . . . . . 158

9.4.3

Uniform Coupling of Cavity Mode to Atoms . . . . . . . . . . . . . . . . . . . 159

9.4.4

Removing 17 Hz Vibration Noise Source . . . . . . . . . . . . . . . . . . . . . 159

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Bibliography

162

Appendix
A Coherent Reinitialization for Reduced Recoil Heating

174

B Heterodyne PDH Detection Scheme

177

C AD8015 Photodiode Characteristics

182

D Phase Noise Definitions

185

E Relation between Mean Squared Modulation Amplitude and Single Sideband Phase Noise 187
F Composite and Plain π-pulse Covariance Noise Matrix and Infidelity

188

F.1 CORPSE π-pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
F.2 SCROFULOUS π-pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

x
F.3 BB1 π-pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
F.4 Plain π-pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
F.5 Composite Pulse Static Error Cancellation . . . . . . . . . . . . . . . . . . . . . . . . 191
G Fundamental Limits on Squeezing via Differential Measurement of 87 Rb clock transition

193

xi

TablesTable
3.1

Regimes of cavity probing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1

Science Cavity Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2

Atomic Ensemble Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3

Intracavity Optical Lattice Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.1

Auxillary measurements to constraint noise added by rotations . . . . . . . . . . . . 109

7.2

Estimated added noise from rotations in projection noise measurements . . . . . . . 110

F.1 Commonly used composite π-pulse sequences . . . . . . . . . . . . . . . . . . . . . . 188
F.2 Composite Pulse Static Error Cancellation . . . . . . . . . . . . . . . . . . . . . . . . 192

Figures

Figure1.1

Un-squeezed Coherent Spin State and Spin Squeezed State . . . . . . . . . . . . . .

2

1.2

Essential concepts behind our squeezing experiment . . . . . . . . . . . . . . . . . .

3

2.1

Bloch sphere representation of a Coherent Spin State . . . . . . . . . . . . . . . . . . 13

2.2

Ramsey spectroscopy with coherent spin state, spin squeezed state and maximallyentangled cat state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3

Ramsey Spectroscopy signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4

One-axis twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5

Graphical summary of demonstrated spectroscopic enhancements arising from interparticle entanglement to date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6

Same as Fig. 2.5 on linear dB scale on the vertical axis . . . . . . . . . . . . . . . . . 22

2.7

Nondemolition versus Quantum Nondemolition Measurement . . . . . . . . . . . . . 24

2.8

Conditional Squeezing via QND measurements . . . . . . . . . . . . . . . . . . . . . 25

2.9

Phase array antenna analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10 Energy Levels and the concept of Rabi Splitting. . . . . . . . . . . . . . . . . . . . 29

3.1

Energy level diagram and probing schematic . . . . . . . . . . . . . . . . . . . . . . . 34

3.2

Holstein-Primakoff approximation and linearly coupled-cavities model . . . . . . . . 36

3.3

Avoided Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4

Complex representation of transmitted and reflected probe field . . . . . . . . . . . . 42

xiii
3.5

Dressed cavity linewidth κ0 and projection noise fluctuations ∆ω proj versus cavity
detuning δc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6

Scaling of key quantities Mdproj , Rs , and mproj
with cavity detuning δc . . . . . . . . 49
s

3.7

mproj
versus cavity decay to atomic decay ratio κ/Γ . . . . . . . . . . . . . . . . . . 50s4.1

3 types of Probe-Induced Back-Action . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2

Backaction from Raman spin flips on spin noise variance . . . . . . . . . . . . . . . . 59

4.3

−1
Optimal squeezing ξm,
opt versus spin flip probability p . . . . . . . . . . . . . . . . . 63

5.1

Energy level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2

Salient aspects of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3

Experimental Sequence for a Single Measurement of Jz . . . . . . . . . . . . . . . . . 73

5.4

Heterodyne IQ detection setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.5

Heterodyne IQ detection setup and signals . . . . . . . . . . . . . . . . . . . . . . . . 75

5.6

LabVIEW program Master.vi in operation . . . . . . . . . . . . . . . . . . . . . . . . 77

5.7

Measurement Signal to Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.8

Large fluctuations in fitted resonance frequencies at high missing probe photon number Mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.9

Distortion of signals caused by Raman scattered induced chirping of the resonances . 81

5.10 Linear decrease in Rabi splitting versus missing photons Mm . . . . . . . . . . . . . 82
5.11 Observing Quantum Projection Noise in Jz and the standard quantum limit . . . . . 85
5.12 Correlation between successive QND measurements . . . . . . . . . . . . . . . . . . . 88
5.13 Measured contrast after QND probing . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.14 Measurement back-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1

Probe laser-cavity frequency noise spectrum and Rabi splitting noise versus τm . . . 96

6.2

Cancellation of average ac Stark shifts with simulataneous probing . . . . . . . . . . 97

6.3

Simultaneous Probing effect on Contrast . . . . . . . . . . . . . . . . . . . . . . . . . 98

xiv
6.4

Radial Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.5

Spatial mode-matching of atoms-cavity modes . . . . . . . . . . . . . . . . . . . . . . 102

7.1

Visualization of rotations used to observe projection noise . . . . . . . . . . . . . . . 106

7.2

Aligning rotation axis to collective spin . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.3

Low phase noise microwave synthesis chain . . . . . . . . . . . . . . . . . . . . . . . 113

7.4

Wenzel crystal lock schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.5

Wenzel crysal loop filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.6

High-Q resonant microwave filters diagram

7.7

DDS 500 MHz clock generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.8

Single sideband phase noise of microwave sources . . . . . . . . . . . . . . . . . . . . 118

8.1

Geometrical representation of phase modulated rotations . . . . . . . . . . . . . . . . 127

8.2

Experimental Sequence for measuring the effect of a phase modulated rotation on. . . . . . . . . . . . . . . . . . . . . . . 115

the Bloch vector polar angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.3

Measured evolution of j̃z versus rotation angle ψ . . . . . . . . . . . . . . . . . . . . 131

8.4

Measured phase modulation transfer functions . . . . . . . . . . . . . . . . . . . . . . 134

8.5

Phase noise mapping onto final Bloch vector position . . . . . . . . . . . . . . . . . . 136

8.6

Variance Wzz and infidelity 1 − F for the CORPSE, SCROFULOUS and BB1 composite π-pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9.1

Sampled Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9.2

Probing scheme using the cycling transition in 87 Rb . . . . . . . . . . . . . . . . . . 151

9.3−1
Optimal squeezing ξm,
opt for probing on the cycling transition . . . . . . . . . . . . . 154

9.4

Dressed mode frequency ω− that optimizes squeezing . . . . . . . . . . . . . . . . . . 155

9.5

Free-space-scattering ms that optimizes squeezing . . . . . . . . . . . . . . . . . . . . 156

A.1 Coherent reinitialization scheme to reduce photon recoil heating . . . . . . . . . . . . 175
B.1 Heterodyne PDH setup and signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

xv
B.2 Histogram of successive measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 181
C.1 Transfer Function of AD8015 #2 Photodiode . . . . . . . . . . . . . . . . . . . . . . 183
C.2 Noise properties of AD8015 #2 photodiode. . . . . . . . . . . . . . . . . . . . . . . . 183
C.3 Measured Transimpedance Gain of AD8015 #2 photodiode . . . . . . . . . . . . . . 184−1
87 Rb clock transition . . . . 196
G.1 Optimal squeezing ξm,
opt for differential probing of theG.2 Frequency resolution required for optimal squeezing . . . . . . . . . . . . . . . . . . 197
G.3 Phase resolution required for optimal squeezing . . . . . . . . . . . . . . . . . . . . . 198

Chapter 1

Introduction and Motivation

The idea of using quantum entanglement as a resource to overcome classical and quantum
constraints has become a paradigm in modern physics. Entanglement is a concept that arises naturally when the superposition principle is applied to composite quantum systems. Two quantum
systems are entangled if the total wavefunction for the composite system is in a non-factorizable
superposition of joint-system wavefunctions. This means that the two systems are no longer independent of each other, and possess correlations stronger than classical correlations could create.
Measuring one system necessarily disturbs the other system instantaneously regardless of their
physical separation. This notion of non-locality — that the two sub-systems still remain a single
quantum system no matter how far apart — presented a conundrum for Einstein, Poldosky andRosen [1], who famously called entanglement “spooky action at a distance”.
Whereas Einstein and co-workers viewed entanglement as a problematic feature of quantum
mechanics, others saw entanglement as a fertile land of opportunity on which they sowed the seedsof quantum-enabled strategies that could surpass classical strategies. With constant nurturing
by generations of scientists, these seeds have grown into large healthy trees today — quantum
communication [2], quantum computation [3], quantum simulation [4] and, last but not least,
quantum metrology [5] — with interconnecting branches and cross fertilization resulting in some
utilitarian fruits already. For instance, quantum key distribution was commercialized in 2004,
providing a provably secure method for sharing information [6]. In quantum computation, thecelebrated Shor’s factoring algorithm [7] provides an exponential speed-up over the best classical2
algorithms — a problem with practical applications to cryptography. Quantum simulators ofstrongly correlated entangled many-body systems are within reach of current ultracold atom andtrapped ion technology [8–11]. Quantum metrology aims to achieve measurement precision below
quantum limits using entanglement, and atomic spin squeezing is one way to achieve this. My
contribution in this landscape is a leaf on the spin squeezing branch of the quantum metrology
tree.

(a)

(b)
z
x

Coherent Spin StateySpin Squeezed State

Figure 1.1: (a) A coherent spin state (CSS) features isotropic pointing fluctuations indicated by
the noise blob perpendicular to the mean spin (red arrow). (b) A spin squeezed state possesses
entanglement that reduce fluctuations in one direction. As a consequence of Heisenberg uncertainty
relations, fluctuations in another direction are increased.

This leaf tells the story of using collective cavity quantum nondemolition measurements to
prepare entangled spin squeezed states of atoms that have spectroscopic sensitivities below thestandard quantum limit [12]. As will be explained in more detail in Chapter 2, coherent spinstates are collective spins of N identical but otherwise independent spins. Coherent spin states areused, de facto, in quantum phase estimation — a central problem in quantum metrology. Unlikea classical spin that has no noise in its pointing direction, quantum mechanics imposes pointing
uncertainty in a coherent spin state (CSS), as depicted by the noise blob at the tip of the mean
spin direction in Fig. 1.1(a). The standard quantum limit refers to the CSS angular resolution√∆φSQL = 1/ N due to pointing fluctuations inherent to the CSS. Due to the lack of entanglement
in a CSS, the noise distribution is isotropic to the mean spin direction. In an entangled spinsqueezed state, uncertainty in one direction is reduced at the expense of increasing uncertainty in

3another direction (see Fig. 1.1(b)), as required by Heisenberg uncertainty relations. The squeezed
quadrature provides enhanced angular resolution that can be used to sense a quantum phase below
the standard quantum limit (SQL).

Figure 1.2: Essential concepts behind our squeezing experiment. Our setup consists of an opticalcavity filled with rubidium atoms. The atoms behave like little pieces of glass with state-dependent
refractive index. The phase shift imposed by the atoms on the cavity mode causes the cavity
resonance frequency to shift depending on the collective populations N↑ in spin up and N↓ inspin down. We send a laser through the cavity to determine the frequency shifts, from which we
can determine the population differences or equivalently the collective spin projection Jz . The
information gained is then used to reduce the quantum projection noise below the CSS noise forone of the two components describing the spatial orientation of the collective spin. Because themeasurement is collective in nature, the probing causes little shortening (single atom wavefunction
collapse) of the collective spin, preserving coherence. The net polar angular uncertainty of the
post-measurement state is reduced below that SQL set by the initial CSS composed of identical,
un-entangled atoms. Image credit: Richard Baxley, JILA.

In the experiment described in this thesis, we generate spin squeezed states by directly
measuring the quantum fluctuations in the spin projection Jz of identically prepared CSS, shown
schematically in Fig. 1.2. Our CSS is composed of ∼ 106 elementary spins formed by the twoclock states of 87 Rb. This measurement is implemented by probing the atomic ensemble, preparedin a CSS, with laser light tuned such that quantum fluctuations of the CSS are imposed on the4light phase and/or amplitude. Knowledge of Jz is then obtained by measuring the light. Whenthe measurement imprecision is below that of the CSS quantum noise, we would have reduced theuncertainty along ẑ below the initial CSS uncertainty. However, the measurement process may cause
the collective spin to shorten due to free space scattering of the probe light. Free space scattering
reveals individual spin states, effectively taking individual spins pointing along x̂ and randomly
putting them into spins pointing up (along ẑ) or down (along −ẑ), shortening the collective spin
length along x̂. The shortening would degrade the phase sensitivity as the angular uncertainty is set
by the ratio of the quantum noise in a spin projection perpendicular to the mean spin to the mean
spin length in the small angle limit. In our cavity-aided quantum nondemolition measurement [12],collective information is obtained (largely) without revealing individual spin states, thus preserving
the mean spin length. Using the measurement outcome from the pre-measurement, the quantum
noise can be subtracted from subsequent measurements, effectively creating a spin squeezed statewith improved angular resolution or equivalently phase sensitivity over the SQL.
In this introductory chapter, I will present an overview of precision metrology and provide
motivation for why spin squeezing is important. As mentioned earlier, the fundamental measurement precision achievable with un-entangled ensembles is given by the SQL, which improves with√the ensemble size N as ∆φSQL = 1/ N . However, there may be situations where scaling up N is
non-trivial or not desirable. These situations provide motivation for going beyond the SQL using
entangled states of atoms, a research field known as quantum-enhanced metrology. An importantclass of such states are spin squeezed states, the realization of which forms the core focus of thisthesis. Following a brief description of my contributions to a different research topic, namely superradiant lasers, I will lay out the organization of the thesis. This thesis will focus only on spinsqueezing.

1.1

Precision Metrology
The central goal of quantum metrology is to measure physical quantities very precisely.

Precision is defined by the root mean squared (rms) fluctuation of the output of a quantum standard

5or sensor that would be observed over many independent experimental trials. We will first describe
how information is encoded in a quantum standard/sensor, and then the precision with which theinformation can be extracted.
The basis on which most atomic sensors operate is estimating a quantum phase that evolves
in a dark time using Ramsey spectroscopy1 . This quantum phase φ is encoded as a relative phase in √the superposition |↓i + eiφ |↑i / 2 of the two quantum states |↑i and |↓i employed as the sensor.
The rate at which the quantum phase φ evolves is set by the frequency difference ω↑↓ between
the two states |↑i and |↓i. This frequency can depend on externally applied fields, or conversely,
can be extremely insensitive to the environment in the case of atomic clocks. By measuring thisfrequency very precisely, quantum sensors of exquisite sensitivity can be realized [13]. As the phasecannot be measured directly, it is instead mapped onto populations that can be discriminated via
fluorescence detection or other means. However, even if there is no detection noise, there is stillfundamental quantum noise due to probabilistic collapse of the atoms into spin up or down, much
like N independent coin flips with the probability of heads or tails corresponding to the probabilityof collapsing into spin up or down.
Using large ensembles of N atoms allows many identical experiments to be run in parallel
rather than simply one after another. This is the reason that large ensembles of un-entangled
atoms are extensively used as precise sensors of time [14–18], acceleration, rotation, gravity [19–22],
magnetic fields [23], and for tests of fundamental physics [24–28]. The quantum nature of the sensors
imposes a limit known as the standard quantum limit (SQL) on their ultimate precision. The SQL√
states that the rms uncertainty on the measurement of the quantum phase is ∆φSQL = 1/ N .
Microwave fountain clocks have operated at the SQL since 1999 [29]. More recently, an opticallattice clock was demonstrated that operated only 3 dB above the SQL for ∼ 103 atoms [18]. By
introducing entanglement, it is possible to surpass the SQL, with the fundamental limit allowed byquantum mechanics given by the Heisenberg limit ∆φHL = 1/N [5].
1Ramsey spectroscopy on internal states of atoms is also analogous to Mach-Zehnder interferometry with electrons,
neutrons, atoms, molecules and photons.

6
Entanglement-enhanced precision may be a future key technology in atomic quantum sensors.
This is because the atomic ensemble size may be limited by both technical constraints or atom-atomcollisions that introduce systematic inaccuracies and loss of precision. The atom-atom interactions
present in atomic systems is fundamentally distinct from the vast majority of photon-based sensors.
Due to the lack of direct photon-photon interactions, photon-based sensors can achieve a lower
absolute phase sensitivity by simply using more un-entangled photons. While radiation pressure
shot noise from the photons can degrade the measurement precision [30], such effects are nota limitation unless the optical power is extremely high, and has not been observed until very
recently [31, 32]. This is the reason why squeezed light has only recently found an application in
the field of gravity wave detection such as the Laser Interferometer Gravitational Wave Observatory(LIGO) and the GEO600 detector [33–35], an optical interferometer that performs near fundamental
measurement limits. In contrast, atom-atom collisions already introduce systematic errors thatmust be carefully studied and corrected in optical lattice clocks [36].
Learning to prepare entangled states of large ensembles with noise properties below the SQL
will be key to extending both the precision [37] and/or bandwidth [38] of atomic sensors. Higher
measurement precision in atomic clocks would also allow a clock to average down to its ultimate
accuracy faster. From a fundamental physics perspective, creating large scale quantum entanglement is also very important for answering foundational questions. For instance, the violation ofBell’s inequalities rules out local hidden variable theories of quantum mechanics [39–41]. Tests ofquantum mechanics at macroscopic scales with Schrödinger Cat states [42–44] probe interactions
between quantum mechanics and gravity [45, 46], and modifications of unitary dynamics [47]The goal of this thesis is to realize entangled spin squeezed states using collective cavity
quantum nondemolition measurements.

1.2

Contributions to Superradiant Lasers
Besides spin squeezing, I have also contributed to research on superradiant lasers as part of acollaborative lab effort. Because this thesis will focus on spin squeezing only, I shall briefly describe

7
my contributions to the field of superradiant lasers in this section.
The broad research theme in our lab seeks to exploit collective atomic effects to advanceprecision measurements in ways that single atom physics cannot. Going below the standard quantum limit via collective cavity-aided QND measurements as demonstrated in this thesis has been
one of the major thrusts in our lab. A parallel thrust in our lab is to explore a relatively unexplored regime of the Schawlow-Townes linewidth limit and sensitivity to cavity frequency pullingby studying steady-state superradiant lasers that operate deep in the bad-cavity regime in which
the cavity linewidth κ is much greater than atomic gain medium linewidth Γ.
In Ref. [48], we realized the first quasi-continuous laser that operates with < 1 intra-cavity
photon on average. The system relies on collectively enhanced emission known as superradiance.
Our system relies on Raman transitions between ground hyperfine states of 87 Rb, and serves asa physical analogue to future ultra-narrow lasers based on highly forbidden optical transitions inatoms such as Sr, Yb, Ca or Mg [49]. In a superradiant laser, defined by its operation in the badcavity limit (κ  Γ), the phase flywheel resides in the collective atomic coherence instead of thecavity field, as is the case in a good-cavity optical laser (κ  Γ); the roles of the atoms and photons
are partially swapped compared to a good-cavity laser. The standard Schawlow-Townes linewidth
and the cavity-pulling equations apply only in the good-cavity limit but analogous limits apply toa bad-cavity laser with the roles of the cavity linewidth κ and atomic gain medium linewidth Γinterchanged.
While our superradiant Raman laser does not possess an ultra-narrow linewidth, it is anextremely useful test-bed to explore the physics of superradiant lasers over the parameter spacespanned by the induced spontaneous Raman decay rate Γ from ∼ 2π × (0.3 Hz) to 2π × (100 Hz),thus operating in the regime relevant to some candidate optical clock transitions [50]. The lasingtransition decay rate is tuned via the Raman dressing laser intensity and detuning in our Raman
model system. In contrast, the decay rate of optical clock transitions is fixed by nature.
Building on the fact that the superradiance can be dynamically controlled with the Ramandressing laser, we demonstrated and analyzed a proof-of-concept hybrid active-passive sensor of8collective atomic coherence [51] in an atomic clock configuration. We have also applied the hybrid
sensor concept in a magnetometer configuration demonstrating both wideband and narrowband
sensing of magnetic fields [52].
In our Raman superradiance system, dispersive tuning of the dressed cavity mode with atomic
populations gives rise to rich dynamics, namely cavity feedback mechanisms, that are not present in
the original proposal paper [49]. We have observed and analyzed the amplitude stability of Raman
superradiant lasers using our 87 Rb model system and revealed a cavity feedback mechanism thatcan damp or amplify the superradiant laser output amplitude depending on the relative detuning
between the Raman-dressing laser and the dressed-cavity mode [53].
In summary, our 87 Rb Raman superradiant laser provides a useful model system for studying
how future superradiant lasers might operate, and for understanding potential limitations to theirstability.

1.3

Organization of Thesis
Having described my contributions to superradiant lasers research, I shall devote the rest of

this thesis to spin squeezing.
In Chapter 2, I present an overview of spin squeezing, provide a summary of other spinsqueezing-related results and methods, and develop a high-level description of how we generate spinsqueezed states via collective cavity quantum non-demolition measurements. This then provides anatural transition point to delve into the theoretical details of our method in Chapters 3 and 4 [54].
To achieve spin squeezing, sub-projection noise measurement sensitivity is required. Sub-projection
noise limited measurements are useful not only for squeezing, but also for low noise readout ofa Ramsey interferometer experiment. In Chapter 3, we theoretically analyze projection-noiselimited cavity-aided nondemolition measurements as a function of the cavity detuning from atomic
resonance. The first half of Chapter 4 then discusses the three types of probe-induced back-action
and derives limits on the measurement imprecision of the collective spin projection Jz . Squeezing
requires sufficient preservation of coherence when probing with sub-projection noise measurement

9sensitivity. The second half of Chapter 4 derives limits on the spectroscopic enhancement relative
to the SQL attainable by balancing probe measurement imprecision against probe-induced backaction.
The core of this thesis is in Chapter 5, where we experimentally demonstrate the generationof conditionally spin-squeezed states in a large N = 7 × 105 ensemble via cavity-aided QND measurements [12]. The theory developed in the previous chapters are important for understanding
the experiment. In Chapter 6, we discuss scalings and advantages of probing the coupled atomscavity system in the resonant limit. Experimental data demonstrating some of these advantages
are presented.
Performing microwave rotations that add little noise relative to the SQL for the large ensemble
size used in our squeezing experiment turns out be be a non-trivial undertaking. In Chapter 7, we
provide experimental details for performing precise microwave rotations that enabled the squeezing
results of Chapter 5. Besides relying on suppression or cancellation of static errors, a low phase noise
microwave local oscillator source that adds little noise to the squeezed quadrature is employed [55].
Chapter 8 presents a linear response theory for quantifying the impact of local oscillator phasenoise, or equivalently detuning noise, on rotations of a Bloch vector [56]. Experimental verificationof special cases of the theory are demonstrated.
Finally, in Chapter 9, I consider our experimental squeezing results in the context of recent
squeezing results and also frame our work more broadly as it pertains to the precision measurement
and quantum metrology community. As an outlook to the next generation of QND measurement
squeezing experiments in our lab, I analyze the amount of optimal squeezing attainable by probing
on an optical cycling transition for 87 Rb and show that it could potentially realize much larger
amounts of spectroscopic enhancements that demonstrate Heisenberg-like scaling with ensemble
size [54]. This analysis is relevant to current efforts in the lab that have already yielded quite
promising improvements in squeezing. Finally, I suggest some experimental improvements to thecurrent setup, and finish with a brief summary.

Chapter 2

Spin Squeezing: Overview and Context

This Chapter reviews concepts related to spin squeezing. To begin, I present a convenientmathematical representation of collective spins useful for understanding Ramsey spectroscopy withensemble spins. The standard quantum limit (SQL) set by the projection noise of un-entangled
atoms in a CSS is derived. A discussion on going beyond the SQL with entangled spin-squeezed
states and the Wineland criterion [57, 58] for quantifying spectroscopic enhancement is given. Different methods of generating spin squeezed states and recent experimental results are discussed,
together with considerations of their relative advantages and disadvantages. Finally I give a elementary outline of our experiment that generates conditionally spin squeezed states using collective
cavity QND measurements. Though elementary, the basic description captures the essential physics
and also provides a natural transition to Chapter 3, where I present the theory of quantum projection noise limited measurements via cavity-aided probing of an atoms-cavity system. For a recent
review on the theoretical and experimental aspects of spin squeezing, I point the reader to Ref. [59].

2.1Collective Spins, Coherent Spin States, and the SQLThis section lays the foundations for describing collective spins and coherent spin states thatwill be relied upon throughout the thesis. Notation developed here also carries through to the endof the thesis.

11
2.1.1

Collective Spins

The concept of a collective spin is an elegant visualization and formalism for discussing
Ramsey spectroscopy with an ensemble of N spin-1/2 atoms. A collective spin Ĵ is simply a sum
(i)

(i)

(i)

over its constituent spins ŝ(i) = 21 (σ̂x , σ̂y , σ̂z )
Ĵ =

N
X

ŝ(i) .

(2.1)

i=1

The collective spin Ĵ = Jˆx x̂ + Jˆy ŷ + Jˆz ẑ may be described by its collective spin operators Jˆk given(i)by the sum over the individual spin operators (Pauli matrices) 21 σ̂k , where k = {x, y, z}
Jˆk =

N
X
1
i=1

(i)

σ̂ .
2 k

(2.2)

The mean spin components are then given by expectation values of the collective spin operators as(i)

Jk = hJˆk i. Since the Pauli matrix σ̂z is given by | ↑i ih↑i | − | ↓i ih↓i |, the collective spin operator Jˆz
can also be expressed as the population difference asN̂↑ − N̂↓Jˆz =,2
where N̂↑ =Pi | ↑i ih↑i | and N̂↓ =

(2.3)

P

i | ↓i ih↓i | are the collective atomic population operators forspin up and spin down respectively. Unlike Jˆx and Jˆy , the collective population difference Jˆz is adirectly and easily measurable quantity via fluorescence detection or other methods. Throughout
this thesis, Jˆz will be considered to be the spin projection that can be measured in experiments.
Other spin projections can be accessed by first rotating the collective spin and then measuring thespin projection Jˆz .
The collective spin operators obey the angular momentum commutation relations [Jˆi , Jˆj ] =iijk Jˆk where ijk is the Levi-Civita symbol. The commutation relations lead to a Heisenberg
uncertainty relationship between ∆Jz and ∆Jy given by∆Jz ∆Jy ≥ |Jx |/2 ,

(2.4)

q
where ∆Jz = hJˆz2 i − hJˆz i2 is the standard deviation of the measurement outcomes of the spin
projection Jz for many identical preparations and measurements of the same collective spin state12(and similarly for the spin projection Jy ). I note here that Eq. 2.4 is completely general. Specifically,
it does not rely on any assumption of the orientation of the collective spin. Identical relationships
can be written down by permuting the coordinate labels {x, y, z}, however, the relationship given
by Eq. 2.4 is the relevant case for this thesis in which the mean spin will be oriented along x̂. I shallshow in the next section that the Heisenberg uncertainty relation is the source of uncertainty inquantum phase estimation. This source of uncertainty is fully equivalent to the random projection
noise due to measurement-induced collapse.

2.1.2

Coherent Spin States and the SQL

Coherent spin states (CSS) are the most classical-like quantum states of a symmetric ensembleof N spin-1/2 particles. A CSS is made up of N identical but independent spin-1/2 particles with noquantum correlations/entanglement between the particles. Mathematically, a CSS is constructedby placing all the N particles in the same state|θ, φi =

N
Y
i=1

θ
θ
cos | ↑i i + eiφ sin | ↓i i .
2
2

(2.5)

The physical significance of θ and φ is that the mean spin vector J = hĴi = hθ, φ|Ĵ|θ, φi points
in the direction given by the spherical coordinates (θ, φ) where θ is the polar angle and φ is theazimuthal angle (see Fig. 2.1(c)). The tip of the so-called Bloch vector resides on the surface of asphere of radius J = N/2. Without considering fluctuations or noise in the spin projections, thevector will behave classically.
Quantum mechanics introduces uncertainty in the spin projections as introduced by the
Heisenberg uncertainty relationships of the form Eq. 2.4. Our visualization of the CSS can beextended to include this quantum uncertainty by the addition of an isotropic noise distribution
perpendicular to the mean spin at the tip of the vector as shown in Fig. 2.1(b). The size of the
distribution conveys the magnitude of the rms fluctuations in the spin projections ∆Jy and ∆Jz .
The minimum magnitude and shape of the noise distribution for a CSS can be understoodby realizing that it is a factorizable state. As a result the fluctuation of the total spin vector is(a)N spin-1/2

....

(b)

(c)

z

1/2
y

coherent spin state,
length J = N/2

13

z

θ
x

x

φ

J
y

Figure 2.1: (a) Construction of a coherent spin state (CSS) from N identical but otherwise independent spins. Each spin-1/2 has length 1/2 and rms uncertainty 1/2 in directions orthogonal
to the spin, visualized as an uncertainty
cone. The length adds up coherently as J = N/2 while√
the uncorrelated noise averages to N /2. (b) Bloch sphere representation of a CSS. In this illustration and for the rest of the thesis, I will assume the CSS is prepared along x̂ unless otherwise
noted. The quasi-probability distribution for the CSS noise in Jy and Jz or equivalently the mean
spin polar angle φ and θ respectively (coordinate system defined in (c)) is shown as a noise blob
(red-orange disc) at the√ tip of the collective spin Ĵ (red arrow). The standard quantum limit∆θSQL = ∆φSQL = 1/ N on quantum phase estimation with a CSS, formed by identically√ prepared but otherwise independent atoms, is set by the quantum noise ∆Jz, CSS = ∆Jy, CSS = N /2,
normalized to the mean spin length |hĴi| = N/2 in the limit of small angles. (c) Coordinate systemdefining the collective spin polar angle θ and azimuthal angle φ. The cartesian components of thecollective spin Ĵ in the x, y and z directions are Jx , Jy and Jy .

just given by the incoherent sum of the fluctuations of the N individual spins as illustrated inFig. 2.1(a). To be quantitative, I will assume for concreteness that the CSS is |π/2, 0i because thisis the point of maximum sensitivity in Ramsey spectroscopy, discussed in the next section. The
mean collective spin that represents this CSS is J = N2 x̂ so that Jx = N/2, see Fig. 2.1(b). Each√individual spin in the CSS is in the state |xi = (|↑i+|↓i)/ 2. Using properties of the Pauli matrices,p
an individual spin has an uncertainty in the spin projection ∆sz = hx|σ̂z2 |xi − hx|σ̂z |xi2 = 1/2.
Similarly ∆sy = 1/2. As mentioned earlier, the uncertainties ∆Jz is equal to the incoherent sum√of the uncertainties of each individual spin ∆sz , with the result that ∆Jz, CSS = N /2. Similarly,√∆Jy, CSS = N /2. Note that since ∆Jz, CSS ∆Jy, CSS = |Jx |/2 saturates the Heisenberg uncertainty
relation given in Eq. 2.4, the CSS corresponds to a minimum uncertainty state.
The noise ∆Jz, CSS along Jz is equivalent to quantum projection noise arising from measurementinduced probabilistic collapse of each spin in the ensemble into |↑i or |↓i [58]. Recall that each√spin in the CSS |π/2, 0i is in the state |xi = (|↑i + |↓i)/ 2. In performing a projective population

14
measurement, each spin would collapse into |↑i or |↓i with probability 1/2 each, independent ofwhat the other spins are doing. Therefore, for N spins, the resulting collective spin populationN↑ in |↑i would be N/2 on average. Random and independent collapse of each of the N spins
into | ↑i or | ↓i results in Poissonian fluctuations or equivalently shot noise around the average.
√
It is no surprise then that the rms collective spin population fluctuation ∆N↑ = N /2 contains√
the famous N smoking gun for shot noise fluctuations. The same could be said for the collective spin population N↓ . However, the fluctuations ∆N↑ and ∆N↓ are anti-correlated, that is∆N↑ = −∆N↓ , due to conservation of particle number N↑ + N↓ = N . Thus, the rms fluctuations∆N↑ and ∆N↓ translates into an equivalent rms uncertainty in the collective spin projection Jz via√∆Jz = ∆(N↑ − N↓ )/2 = N /2. This uncertainty ∆Jz is exactly the same as the CSS uncertainty∆Jz, CSS derived earlier, demonstrating the equivalence of ∆Jz, CSS and quantum projection noise.
Information in a quantum sensor is ultimately encoded in a polar angle θ which must be estimated from measurements of Jz or equivalently collective spin populations N↑ and N↓ . Heisenberg
uncertainty relations sets a fundamental limit on our ability to measure the collective spin polar
angle θ or azimuthal angle φ. In the small angle limit, the fluctuations in the spin projection Jz
sets a limit on the uncertainty ∆θSQL given by∆θSQL =

∆Jz, CSS|hĴi|

1
=√ ,
N

(2.6)

a limit known as the standard quantum limit (SQL) for a coherent spin state1 . In this thesis,
we will experimentally demonstrate that quantum nondemolition measurements can be used togenerate entanglement and realize phase estimation uncertainty below the SQL.

2.2

Ramsey Spectroscopy with Coherent Spin States
In this section, I will review the canonical Ramsey spectroscopy that is used in a broad rangeof atomic sensors, and show how the SQL for quantum phase estimation can be surpassed by using
1
The standard quantum limit is not really “standard” in the sense that different communities define an equivalent
SQL for their very specific measurement context, just like we do here in the context of quantum phase estimation
using Ramsey spectroscopy. For example, in the field of opto-mechanics, the SQL for measuring a simple harmonic
oscillator’s position is reached at a measurement strength where the measurement imprecision is equal to the added
noise from measurement back-action [60].

15entangled states in Ramsey spectroscopy.
Ramsey spectroscopy with a CSS proceeds as shown in Fig. 2.2(a). The collective spin is firstQ
prepared in the CSS |θ = π, φ = 0i = N
i=1 | ↓i. An initial π/2-pulse about the −ŷ-axis rotates theQ|↑i i+|↓i i√
collective spin up to the x-axis, producing the state |θ = π2 , φ = 0i = N. The rotation is
i=1
2
induced with electromagnetic radiation at frequency ωLO nearly resonant with the spin transition
frequency ω↑↓ . The collective spin is now allowed to evolve for a duration Tevol such that eachconstituent spin acquires a phase φ = (ω↑↓ − ωLO )Tevol in the spin up state relative to the spin
down state in a frame rotating at frequency ωLO . The CSS wavefunction at the end of the evolution
Q|↑i i+eiφ |↓i i√time is given by |θ = π2 , φi = N, corresponding to a precession of the collective spini=12
through an angle φ.
The goal of Ramsey spectroscopy is to measure φ very precisely. Because one cannot measure
φ directly, φ is mapped onto the the collective spin projection Jz = (N↑ −N↓ )/2 which is measurable
via fluorescence detection or other methods. This mapping is performed by a final π/2-pulseQ
π/2−φ
about the x̂-axis and leaves the collective spin in the final state | π2 − φ, 0i = N
i=1 cos
2 | ↑i i +i sin π/2−φ2 | ↓i i up to a global phase. At this stage, the phase information φ has been translated into
the polar angle θ. The collective spin projection of the final state that is measured is Jz = N2 sin φ.
During the measurement process, wavefunction collapse of the individual spins from a superposition
state to one of the eigenstates |↑i or |↓i results in quantum projection noise, sometimes called atom√

shot noise, ∆Jz =N
2 cos φ in the measurement result. See Fig. 2.3(a) for the resulting signal andprojection noise.
The rms uncertainty ∆φ in determining φ is set by ratio of the quantum projection noise∆Jz to the slope of the Ramsey signal dJz /dφ
∆φ =

∆Jz
1
=√ ,
|dJz /dφ|N(2.7)

which is independent of φ. In the presence of technical noise, it is advantageous to operate at
the point of maximum slope φ = 0, π. Note that this derivation shows that the quantum-limited
signal-to-noise (i.e. no additional sources of noise other than quantum projection noise) in Ramsey

(a)

z

y

x

16

Coherent Spin State (identical independent atoms)
Tevol(π/2)-y(π/2)xφ
φ

Coherent Spin State(b)

Spin-Squeezed State (this thesis)

z

x

y

Tevol

(π/2)-y(π/2)xφ
φ

Spin Squeezed State

Figure 2.2: Ramsey spectroscopy with a coherent spin state (a) and a spin squeezed state (b)illustrated in a frame rotating at the LO frequency. In (a) and (b), a π/2 global rotation rotates
the collective spin (red arrow) from pointing along −ẑ to x̂. The fuzzy disc/ellipse at the tip of thecollective spin represents its intrinsic quantum noise. During the evolution time Tevol , the collective
spin freely precesses about the z-axis and accumulates a phase φ = (ω↑↓ − ωLO )Tevol . The phase isthen mapped onto populations, equivalently Jz with a final π/2 rotation about x̂. The observable
Jz is then read out with fluorescence detection or other techniques. Measurement-induced-collapseof the wavefunction results in quantum projection noise ∆Jz . In the illustration, ∆Jz is givenby the projection of the quantum noise distribution onto the z-axis. The goal of this thesis is togenerate entangled spin squeezed states for quantum metrology.

√
spectroscopy with a CSS is exactly the SQL ∆φSQL = 1/ N described earlier.
To go beyond the SQL set by un-entangled atoms, quantum entanglement can be used toeither lower the quantum projection noise ∆Jz or increase the slope dJz /dφ. Spin squeezing, thesubject of this thesis, employs the former strategy at the expense of increased noise in ∆Jy which
is not observed, while maximally entangled cat states employ the latter strategy [61–67].

2.3

Surpass SQL with Spin Squeezed States
In Fig. 2.2(b), I show how the spin squeezed states we generate in our experiment can beused in Ramsey spectroscopy to surpass the SQL. The Ramsey sequence is only sensitive to noisein one quadrature so that the quiet quadrature can be used, while the noisy quadrature is not

(a)
N/2

17

Coherent Spin State

Jz|Slope| = N/2
ΔJz, CSS = �N/2
ΔφSQL = 1/�N

2π

φ

-N/2

(b)
N/2

Jz

Spin-Squeezed State (this thesis)|Slope| = N/2
ΔJz < ΔJz, CSS
Δφ < ΔφSQL

2π

φ

-N/2

Figure 2.3: Ramsey spectroscopy signals with a CSS input (a) and a spin squeezed state input(b). The shaded area represents rms fluctuations in the measured observable Jz that would beobserved if the experiment is repeated many times. (a) Using√ large ensembles of N un-entangled
atoms averages the quantum projection noise as ∆Jz, CSS = N /2, as exemplified by the CSS. The
quantum projection noise ∆Jz, CSS in conjuction
with the slope of the Ramsey signal sets the SQL√
for the phase uncertainty ∆φSQL = 1/ N . (b) The phase sensitivity can be enhanced beyond the
SQL by using a spin squeezed state with reduced projection noise ∆Jz < ∆Jz, CSS .

used. Spin squeezed states reduce the projection noise ∆Jz that appears during the readout, while
maintaining the same signal slope dJz /dφ, therefore enhancing phase sensitivity beyond the SQL,
as illustrated in Fig. 2.3(b).
Having seen how squeezing can improve the spectroscopic sensitivity of Ramsey spectroscopy,
it is important to quantify the amount of spectroscopic enhancement relative to the SQL. Squeezing
has been defined various ways in the literature. The review article [59] provides a comprehensive
list of spin squeezing definitions.
As we are primarily interested in applying squeezing to metrology, we shall take spin squeezing√
2−1 ≡ (∆θto be spectroscopic enhancement ξm
SQL /∆θ) relative to the SQL ∆θSQL = 1/ N set by

18
the CSS, where ∆θ is the polar angle uncertainty of the spin squeezed state. Since we generatesqueezed states with reduced uncertainty in the polar angle in this thesis, we use θ here instead−1 ≥ 1 represents a spectroscopic enhancement in sensitivity thatof φ. Spin squeezing with ξmmust arise from entanglement [68]. Entanglement, while necessary, is not a sufficient condition forspectroscopic enhancement [68].
The above definition for spectroscopic enhancement, synonymous with squeezing in this thesis, is referred to as the Wineland criterion [57, 58] in the literature. The Wineland criterion
accounts for both spin noise reduction and the coherence or Ramsey contrast C = |hĴi|/J of theresulting state, where J = N/2 is the mean spin length corresponding to the fully symmetric
CSS of N spin-1/2 particles. Experimentally, the mean spin length |hĴi| is measured via Ramsey
spectroscopy or Rabi flopping. The measured Jz oscillates with an amplitude set by the mean
spin length |hĴi| as the angle between the mean spin and the rotation axis is varied in Ramsey
spectroscopy or as the pulse area is varied in Rabi flopping. The spectroscopic enhancement canbe expressed as the signal-to-noise ratio in the following form
−1
ξm
=



∆Jz
∆Jz, CSS

−2

C2 .

(2.8)

Accounting for the loss of signal is important because in generating spin squeezed states, the mean
spin length |hĴi| may be reduced below its initial value J = N/2. See Fig. 2.7 for an illustration.
A related quantity frequently encountered in the spin squeezing literature is the spin noise
reductionξn−1 ≡∆Jz∆Jz, CSS

−2
(2.9)

defined by Kitagawa and Ueda [69]. It is sometimes referred to as number squeezing, especially inthe context of Bose-Einstein Condensates. It is possible to have number squeezing according to thedefinition 2.9 and yet have phase estimation sensitivity worse than the SQL.

19

2.4

Generation of Spin Squeezed States
Spin squeezed states in large ensembles have been generated using many diverse approaches.

Generally, the approaches may be divided into two experimental categories as: atom-light interactionbased and atomic collisions-based squeezing. The underlying mechanism that creates squeezing canbe broadly divided into two categories that exploit two different aspects of quantum mechanics:
quantum non-demolition measurements and non-linear Hamiltonians. In the following sections, I
will discuss these approaches, review demonstrated squeezing results, and discuss their relative advantages and disadvantages from the perspective of applying squeezing to metrology. To facilitate
discussion, I will assume the collective atomic spin is initially prepared in a CSS along x̂.

2.4.1

Collisional Interactions

In Bose-Einstein Condensates, atom-atom collisions lead to a nonlinear term in the Hamiltonian HOAT = χJˆz2 [68] (~ = 1 here), referred to as the one-axis twisting (OAT) Hamiltonian by
Kitagawa and Ueda [69]. To explain how squeezing is generated, consider the evolution of a CSS
along x̂ under this Hamiltonian. The Dicke states [70] |J, M i are eigenstates of the OAT Hamil2

tonian with eigenvalue χM 2 , and evolves phase in time as e−iχM t . This implies the precession
frequency of the state |J, M i scales linearly with M as 2χM in a rotating frame whose frequency
is proportional to M . As the CSS is a (weighted) superposition of the Dicke states, the noise
components of the CSS with larger M would precess faster than those with smaller M . Also, noise
components with positive M would precess in the opposite sense compared to noise components
with negative M . This process is depicted pictorially in Fig. 2.4 with the resulting noise distribution sheared such that noise in one direction is lower than the CSS projection noise2 . The shearing
process can be viewed as arising from twisting the Bloch sphere about the z-axis, hence the nameone-axis twisting. A small rotation can be performed to rotate the squeezed quadrature to lie alongŷ or along ẑ.
2

Our semi-classical picture of shearing of the noise distribution breaks down when the shearing wraps aroundthe Bloch sphere, where in fact the system would evolve into a cat state appropriate for Heisenberg-limited spec-

20

z
x

y

Figure 2.4: A CSS evolves into a spin squeezed state under a one-axis twisting Hamiltonian thatcreates a Jz -dependent precession frequency, causing the noise distribution to shear.

−1 ∼ 4 to 8 dB
Recent experiments employing this collision-induced OAT has produced ξmof inferred squeezing after background subtraction in the groups of Oberthaler, Treutlein andSchmiedmayer [72–76] in ensemble sizes of N ∼ 400 to 2000 atoms. It is in fact, not necessary
to utilize Bose-Einstein Condensates, and anti-squeezing has been observed in the non-degenerate
87 Sr fermion optical lattice clock at JILA [77].

Direct collisional interactions causing parametric pair generation into two modes can generate two-mode squeezing, an effect analogous to parametric down conversion in non-linear optics.
The states created are called twin-Fock states and ideally could attain Heisenberg-limited phasesensitivity ∆θHL = 1/N [78]. Because the mean spin length |hĴi| is zero, an observable different
from Jz is required [79]. Parametric two-mode squeezing has been demonstrated in the group of−1 = 2.5(1.0) dB at N = 7×103 . In related setups, parametricKlempt [80] with inferred squeezing ξm

two-mode spin noise reduction has been observed in the groups of Schmiedmayer and Chapman [81,82]. Continuous-variable entanglement with parametric pair generation has also been observed [83].

2.4.2Light-mediated One-Axis Twisting Hamiltonians

Spin squeezed states have been prepared with atom-light interactions that generate effective−1 = 9.6(1.0) dB at 3 × 104 atoms) inlong range interactions on demand (inferred squeezing ξmtroscopy [61, 71].

2

10

Collisional Interactions
Heidelberg - Oberthaler
Basel - Treutlein
Vienna - Schmiedmayer
Hannover - Klempt

Quantum Entangling Gates
NIST - WinelandInnsbruck - Blatt
Osaka - Urabe

-1

10

-2

10

St
an

n
Fu

-3

da

10

Cavity One-Axis
MIT - Vuletic

da

rd

me
nta

-4

10

Qu
a

nt
um

Li

lH

m

it

en

eis

-5

10

QND Conditional Squeezing
Bohr Institute - Polzik
MIT - Vuletic
JILA/NIST - Thompson
ICFO - Mitchellbe

our result

rg
Lim

-6

10

it

Variance of Quantum Phase Estimation [radians ]

21
0

-7

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

Atom or Ion Number
Figure 2.5: Graphical summary of results demonstrating spectroscopic enhancement arising frominter-particle entanglement to date (May 2013). Inferred values based on background subtraction
are plotted. To emphasize the regimes that are technologically relevant for atomic sensors, theabsolute phase uncertainty is plotted versus the ensemble size. Trapped ion optical clocks operatewith < 10 ions [17]. Atom chip sensors (clocks, atom interferometers, magnetometers) operate with102 to 104 atoms [84, 85]. Neutral atom optical lattice clocks operate with 103 to 105 atoms [16,18]. Primary and secondary microwave frequency standards employ 105 to 106 atoms [14, 29]. Free
space atom interferometers and vapor cell magnetometers operate with large ensembles from 105
up to 1012 atoms [19, 20, 23, 24, 86–88]. The ranges provided here are approximate, and certainly
will change as technology evolves.
Nearly maximally entangled states generated in ion traps are depicted in blue, spin-squeezed states
generated by atomic collisions or light-mediated interactions in red, and conditional squeezing via
QND measurements in green.
The absolute phase uncertainty inferred after subtracting out the readout and other technical noise
sources from the observed phase uncertainty is plotted where available. When not available, theobserved phase uncertainty is plotted. The inferred phase uncertainty represents the potential forspectroscopic enhancement when all other noise sources are made negligible. The readout can beseparately optimized from the generation of entanglement or squeezing in principle.

10

Limit

MIT/Vuletic(Cavity One-Axis)

enberg2

Enhancement in Variance of Phase (ΔθSQL/ Δθ) [dB]22

Vienna/Schmiedmayer(Collisions)ntal Heis

8

Heidelberg/Oberthaler(Collisions)

Fundame

6

Basel/Treutlein(Collisions)
NBI/Polzik
Heidelberg/Oberthaler(Free Space QND)
(Collisions)

4

MIT/VuleticBasel/Treutlein(Collisions)

2

H

Quantum Entangling Gates
NIST/WinelandInnsbruck/Blatt
Osaka/Urabean(Cavity QND)

n

(C ov
ol er
lis /K
io le
ns m
)

JILA/Thompson(Cavity QND)

our result
ICFO/Mitchell(Free Space QND)

pt

0
0

10

1

10

2

10

3

10

4

10

Atom or Ion Number

5

10

6

10

7

10

Figure 2.6: Same data as Fig. 2.5, plotted with inferred spectroscopic enhancement in dB on thevertical axis for better visibility. Nearly maximally entangled states generated in ion traps are
depicted in blue, spin-squeezed states generated by atomic collisions or light-mediated interactions
in red, and conditional squeezing via QND measurements in green.

23
a cavity-QED setup [89]. In brief, the approach demonstrated in Ref. [89] begins by imprinting
quantum fluctuations ∆Jz of the collective spin onto probe light which is then filtered by a cavity
response and finally fed-back onto the collective spin. The cavity feedback is tuned such that it
imposes a Jz -dependent ac-Stark shift, i.e. precession frequency, thus implementing an effective
OAT Hamiltonian. Inherent dissipation in this approach due to free space scattering of probe
light, and noise associated with photon number fluctuations in the probe does not result in a truly
coherent evolution. Theoretical proposals on coherent feedback methods to generate OAT havebeen developed for cavities [90, 91] and in free space [92, 93].
Internal state squeezing has been generated in large spin-f atoms using effective OAT Hamiltonians engineered by tensor light shifts [94, 95]. The internal state squeezing generates entanglement between the nuclear spin and the electronic spin without inter-atom entanglement. In−1 ∼ 4 dB of inferred squeezing within the F = 3 Zeeman manifold of 133 Cs was demonRef. [94], ξm−1 = 2.7(8) dB of inferred squeezing within the F = 4
strated with N ∼ 106 atoms. In Ref. [95], ξm

Zeeman manifold of 133 Cs using a macroscopic ensemble of N ∼ 1012 atoms.

2.4.3

Quantum Non-demolition Measurements

A quantum nondemolition (QND) measurement of Jz that also preserves coherence can beused to prepare conditionally spin squeezed states with spectroscopic sensitivity below the SQL [96–98], an approach successfully demonstrated in this thesis. In this thesis, we choose to make a
distinction between nondemolition (ND) measurements and QND measurements, illustrated in
Fig 2.7. In the QND measurement approach, probe light interacts with Jz such that the quantum
fluctuation ∆Jz, CSS is imprinted on the probe. Then a measurement performed on the probe will
project the initial CSS into a post-measurement state consistent with the measurement outcome.
If the measurement imprecision of the probe is sufficient to resolve projection noise fluctuations∆Jz, CSS , then the post-measurement state would have its spin noise reduced below the CSS spinnoise. Provided the coherence C = |hĴi|/J is sufficiently preserved by the measurement, as discussed
earlier in Sect. 2.3, a conditionally spin squeezed state would result. The squeezing is conditional in

24
z
x

y
↑
ND

Fully Collapsed

QND↓

Coherent Spin State
Spin Squeezed State

Figure 2.7: Visualization of a nondemolition (ND) (upper right) and a quantum nondemolition(QND) (lower right) measurement. A CSS for an ensemble of N spin-1/2 atoms prior to measurement is represented as a collective spin (red arrow) of length J = N/2. The quantum noise in
the orientation of the collective spin is visualized as a quasi-probability distribution (red/yellow√
region) perpendicular to the mean spin, with rms opening angle at the SQL ∆θSQL = 1/ N .
A nondemolition measurement in this thesis refers to a measurementof of the spin projection√
Jz = (N↑ − N↓ )/2 with an rms imprecision ∆Jz < ∆Jz, CSS = N /2, and with the majority of theatoms remaining around in the trapping or probing region after the measurement. For example, in
the ND measurement visualized on the right, the measurement imprecision is ∆Jz = 0, but after
the measurement the atoms are described by a product state of atoms in spin up and down due tosingle-atom information gained by the environment via free space scattering of light. We define aquantum nondemolition measurement to satisfy the additional requirement that a sufficiently large
number of atoms remain in a coherent superposition of spin up and down such that the resulting
state, conditioned on the measurement outcome, has a polar angle uncertainty ∆θ < ∆θSQL andcan be used to sense a quantum phase below the SQL of an ensemble of uncorrelated atoms. The
lower right QND example visualizes the conditional state as a product state of a spin-squeezed
state and the atoms that have been collapsed into spin up and down.

the sense that the obtained spin squeezed state is randomly distributed within the range defined by
the initial uncertainty ∆Jz, CSS (see Fig. 2.8). Thus the produced squeezed state is random acrossindependent experimental trials. If the measurement outcome is not used to predict the value of asubsequent measurement of Jz (within the same experimental trial), then no squeezing would beobserved.
QND measurements of large atomic ensembles have recently become precision measurement
tools with the demonstration of QND measurement-based spin-squeezing. In this thesis, cavity-

25

Spin Squeezed
State

Coherent
Spin State
Figure 2.8: Concept of conditional squeezing via QND measurements. A QND measurement of thespin projection Jz of a CSS prepared along x̂ would yield a random measurement outcome within
the range set by the initial CSS uncertainty ∆Jz, CSS (orange region). The QND measurement
projects the state into one that is consistent with the measurement outcome and the measurement
uncertainty. When the measurement uncertainty ∆Jz is lower than the CSS uncertainty and coherence is sufficiently preserved (not shown), the post-measurement state would be a spin squeezed
state with reduced uncertainty in Jz at the expense of increased uncertainty in Jy (green region),conditioned on the measurement outcome.

aided QND measurements of a large ensemble of pseudo-spins composed of the 87 Rb clock transition
projects the ensemble into a conditionally spin squeezed state. The prepared spin squeezed state−1 = 3.4(6) dB over the SQL. At N = 7 × 105 atoms,
has an inferred spectroscopic improvement ξmthis work represents the largest clock ensemble squeezed to date [12]. QND measurement squeezing
has also been demonstrated for clock transitions with atoms in free space in Polzik’s group (inferred−1 = 3.4 dB at N = 1.2 × 105 [99]) and in a cavity in Vuletić’s group (inferred squeezing
squeezing ξm−1 = 3.0 dB at N = 3.3 × 104 [100]). In our work, the cavity mode couples to the atoms resonantly
ξmin the strong collective coupling regime of cavity-QED, generating two new normal modes, called
the vacuum Rabi splitting, that contained equal wavefunction amplitudes in the atom and cavity
components. We then probe both normal modes to project a CSS into a spin squeezed state. In asimilar cavity-QED setup in Vuletić’s group [100], the cavity frequency is tuned such that the cavity
mode couples to the atoms in the far-detuned dispersive regime. In this case, the normal modesof the system remain mostly atom-like and cavity-like, with some dressing of the cavity by the26atoms and vice versa. The cavity-like mode is then probeed to generate conditional spin squeezed−1 = 2.0 dB at N = 7.4 × 105 atoms)
states. Very recently, spin squeezing (inferred squeezing ξmin a spin-1 system demonstrating inter-particle entanglement in a magnetometer configurationfrom Mitchell’s group was reported [101]. Conditional two-mode squeezing of a room temperature−1 = 1.5 dB of observed spectroscopic
vapor of N = 1012 atoms has enabled magnetometry with ξmenhancement and an increased measurement bandwidth [102]. Several experiments [103–106] havereported spin noise reduction without quantifying the loss of signal, so spectroscopic enhancements
are not available for these results. Back-action was observed in Ref. [107]. Spin squeezing of opticallattice clocks using the trap lattice as a QND probe has been proposed [108]. A graphical summaryof these and other squeezing results can be found in Figs. 2.6 and 2.5.

2.5

Relative advantages and disadvantages to different approaches
Spin squeezing based on atom-light interactions offers the advantage that the probe light canbe completely turned off. As an additional bonus, low noise readout techniques that are required
to read out the squeezed state are automatically present in the QND approach. In squeezing
experiments exploiting direct atom-atom interactions, it is not clear if the collisional interactions
required for generating squeezing can be turned off sufficiently. These interactions would cause
systematic mean field shifts, i.e. inaccuracies, and quantum noise redistribution during Ramsey
spectroscopy, and may destroy the squeezing or even make the phase resolution worse than thestandard quantum limit. Conversely, it has been observed that relative number squeezing in atominterferometers can reduce fluctuations in the relative phase shift arising from fluctuating mean
field shifts in the two interferometer paths [76, 109]. The collisional interaction strength χ canbe tuned via magnetic Feshbach resonances or with state-dependent potentials, achieving about afactor of 100 in the on/off ratio in current experiments [73, 74]. Furthermore, inelastic collisions
concomitant with the elastic collisions necessary to generate squeezing would degrade or even limit
the amount of squeezing achievable, as was the case in Ref. [73].
For the particular approach of squeezing on the internal states within an atom, the maximum

27
spectroscopic enhancement permissible is set by the Heisenberg limit corresponding to the numberof internal states, and not by the ensemble size. Since the number of internal states is necessarily
limited by atomic structure, this method is clearly not scalable. However, internal state squeezing
may be combined with inter-atomic squeezing to produce larger amounts of squeezing [110].
Finally, in cavity-based approaches to spin squeezing, either using QND measurements orcavity feedback, the fundamental limit to the amount of squeezing is set by the collective cooperativity of cavity-QED physics, equivalent to optical depth in free space samples. The collective
cooperativity is essentially the single pass optical depth of the sample multiplied by the numberof bounces the light makes in the cavity, which is of the order of the cavity finesse F . Therefore
compared to free space approaches, larger amounts of squeezing are attainable in a cavity at thesame atomic densities. Alternatively, a cavity-based approach can achieve the same amount ofsqueezing as a free space experiment but at atomic densities lowered by order the cavity finesse F .
The reduced atomic density would mitigate density shifts as a source of inaccuracy in the context
of clock applications.

2.6

Other work on entanglement-enhanced metrology
Nearly maximally entangled states of 2 to 14 trapped ions have been generated in the groupsof Wineland, Blatt and Urabe [62–67], with demonstrated observed spectroscopic sensitivities near
the Heisenberg limit ∆θHL = 1/N as shown in Figs. 2.6 and 2.5. These states are typically generated
via phonon-mediated global entangling gates [61] or phonon-mediated stimulated rapid adiabatic
passage [67]. While trapped ion systems have demonstrated near-Heisenberg limited sensitivity, it
may be nontrivial to scale up the method and the ion number to achieve large absolute spectroscopic
gains. Further, it is an open question as to whether such approaches could be applied to muchlarger neutral atom systems that lack long-range Coulomb interactions.

28

Figure 2.9: Phase array analogy to collectively enhanced coupling. A single classical dipole radiates
isotropically with power P0 proportional to the the radiated electric field E 2 (top panel). An arrayof phase synchronized classical dipoles radiate with a factor of N enhancement, a phenomena known
as superradiance, over an array of dipoles with random phases (not shown) into the “end-fire” modes
due to constructive interference in the electric field amplitudes (bottom panel). Consequently, thephase array radiates at single dipole rates (on average) in directions different from the end-firemodes due to destructive interference. The collective enhancement can be viewed as arising fromthe collective coupling of the dipoles to the end-fire modes. The collective coupling causes the arrayto decay into the end-fire modes faster than decaying into all other modes (directions). The same
concept applies to array of quantum atomic dipoles. The phase of a quantum atomic dipole is set
by the relative phase between |↑i and |↓i in a superposition state of |↑i and |↓i. Strong collective
coupling to an array of atoms to a probe mode may be achieved in free space with optically dense
samples along the probing direction. In the strong collective coupling regime, collective information
about the ensemble can be extracted from a detuned probe mode faster than the rate of decoherence
via free-space-scattering into the all directions. The collective coupling can be enhanced by putting
the sample in an optical cavity, the approach pursued in this thesis, so that the probe light interacts
with the same sample multiple times.

2.7

Outline of our squeezing experiment
In this section, I provide an elementary outline that captures the essential physics of how wegenerate spin squeezed states using cavity-aided QND measurements. In brief, we use ∼ 106 87 Rb
laser cooled and trapped inside an optical cavity. The pseudo-spin-1/2 system is comprised of the
microwave clock states |↑i ≡ |F = 2, mF = 0i and |↓i ≡ |F = 1, mF = 0i, see Fig. 2.10(a) for
schematic. The two-level system we utilized is a pseudo-spin-1/2 because the transition frequency
between |↑i and |↓i does not tune with a magnetic field to first order. We prepare a CSS along x̂ by
optical pumping into |↓i followed by a π/2 rotation with resonant microwaves. At this point, each

F’=1,mF=0

5 P1/2Ω↓e2Bare CavityωFrequency cRound Trip Phase modulo 2!

(a)

29
(b)

0

0

F=2,mF=0

5 2S1/2↓

F=1,mF=06.834 GHz

Cavity Transmission

↑

1

(c)

0
0

Cavity Drive Detuning from Bare Cavity Resonance

Figure 2.10: (a) Basic energy level scheme of the experiment. The pseudo-spin-1/2 system comprisesof the microwave clocks states |↑i ≡ |F = 2, mF = 0i and |↓i ≡ |F = 1, mF = 0i. The bare (empty)
cavity frequency is degenerate with the atomic |↑i to |ei ≡ |F 0 = 1, mF = 0i optical transition.
The strong collective coupling of the atomic ensemble to the cavity mode creates a normal mode
splitting, known as the collective vacuum Rabi splitting Ω↑ , which we probe to measure the collective
spin population N↑ . In our setup, only atoms in |↑i couple to the cavity mode. (b) The origin of
the collective Rabi splitting [111] is completely classical and can be understood as follows. A cavity
transmission resonance occurs when the round trip phase shift of the light is an integer multiple of
2π due to constructive interference. In an empty cavity, the round trip phase shift is linear withcavity drive detuning (blue line), resulting in a single transmission resonance at the bare cavity
frequency (and at multiples of the cavity free spectral range). An atomic medium in the cavity
imposes a dispersive phase shift centered on the atomic resonance frequency on top of the free space
propagation phase shift (red curve). In our probing scheme, the bare cavity frequency is degeneratewith the atomic frequency. The two outer zero crossings result in transmission resonances while
the middle zero crossing does not because of atomic absorption of resonant probe light. (c) Cavity
power transmission curves corresponding to (b). The frequency splitting
p between the transmission
resonances is called the Rabi splitting Ω↑ . In our experiment, Ω↑ = N↑ 2g where 2g is the single
atom Rabi frequency.

√
atom is in the same superposition state (|↑i + |↓i)/ 2. The atoms behave like little pieces of glasswith state-dependent refractive index. The phase shift imposed by the atoms on the cavity modecauses the cavity resonance frequency to shift depending on the collective populations N↑ in spin up
and N↓ in spin down. To simplify the discussion, I assume for now that the cavity frequency shift is30
proportional to the collective spin projection Jz = (N↑ − N↓ )/2. By measuring the cavity resonancefrequency precisely3 , we can measure Jz precisely, but in a way that does not reveal which atom
is in spin up or down, therefore preserving coherence. The collective coupling of the ensemble tothe cavity mode, which extracts collective information, over the rate of free space scattering into
all other modes, is the mechanism for preservation of coherence. This collective enhancement in anatoms-cavity system is analogous to the enhancement in emission rate in the “end-fire” directionsof a phased array antenna, illustrated in Fig. 2.9.
We first measure Jz with measurement imprecision below the projection noise limit ∆Jz, CSS ,thus conditionally projecting the initial CSS into a state with reduced spin noise. In order to verify
the spin noise reduction, we show that repeated measurements of the same CSS are correlated below
the projection noise level. In order to claim spectroscopic enhancement, we measure the mean spin
length |hĴi| with a Ramsey sequence and show that the coherence (contrast) is sufficiently preserved
after the first pre-measurement such that the net signal-to-noise of the post-measurement state isimproved beyond the SQL for un-entangled atoms.
Most cavity-aided probing experiments have operated in the dispersive regime. In our experiment, we actually operate in the resonant regime of strong collective coupling cavity-QED.
We access this regime by setting the bare cavity frequency to be resonant with the atomic opticaltransition frequency as shown in Fig. 2.10(a). The resonant exchange of cavity photons and atomic
excitations leads to a normal mode splitting when the coupling rate Ω↑ exceeds the dissipation rates
due to cavity power decay rate κ and atomic optically excited state decay rate Γ. The origin of thenormal mode splitting, also called the collective vacuum Rabi splitting, arises from dispersive phaseshifts imposed by the atomic medium (see Fig. 2.10(b, c), and is therefore completely classical inpnature [111]. In our probing scheme, the Rabi splitting Ω↑ = N↑ 2g depends only on the collective
spin population N↑ and the single atom Rabi frequency 2g set by the atomic optical dipole moment
and the electric field of a single cavity photon. In order to measure Jz , which requires knowledge ofN↑ and N↓ , we first prepare a CSS along x̂ and measure the size of the Rabi splitting to determine3

As is well known, a good way to measure any quantity precisely is to convert it into a frequency.

31N↑ . We then use a microwave π-pulse to phase coherently swap N↑ with N↓ and perform a second
Rabi splitting measurement to determine N↓ .
By operating in the resonant regime instead of the dispersive regime, we hope to reinforce
the idea that the quality of a coherence-preserving QND measurement is not fundamentally linked
to the probe’s large detuning from atomic resonance. Instead, it is the magnitude of the collective
cooperativity parameter N↑ C (equivalent to the optical depth for a free space experiment [38])that sets the fundamental quality of the QND measurement. Here C = (2g)2 /κΓ is the single atom
cooperativity, or equivalently the single atom optical depth. Operating in the resonant regime
also offers advantages in terms of maximizing the absolute size of the cavity frequency fluctuations
caused by projection noise, and technical relative probe-cavity frequency noise immunity.

Chapter 3

Theory: Probing atoms-cavity system with projection-noise-limited sensitivity

3.1

Introduction
Probing the collective spin state of an ensemble of atoms provides a means to reduce heating

via the photon recoil associated with the measurement and provides a robust, scalable route forpreparing highly entangled states with spectroscopic sensitivity below the SQL set by the CSS. The
collective probing relies on obtaining a very large optical depth that can be effectively increased byplacing the ensemble within an optical cavity such that the probe light passes many times throughthe ensemble. In this Chapter, we analyze projection-noise-limited cavity-aided nondemolition
measurements as a function of cavity detuning δc from atomic resonance as well as cavity finesse,
cavity length, cavity linewidth, mode waist, and atom number.
It is well known that significant improvements in readout sensitivity can be achieved by
optically probing ensembles in free space along directions of large resonant optical depth. Thisapproach has been extensively analyzed theoretically [38, 93, 96, 97, 112–119], and experimentallystudied [99, 102, 103, 120–126].
More recently, the technique of free space probing of large optical depth samples has been
extended to using optical cavities to effectively increase the optical depth of the atomic ensemble
by passing the probe light through the same sample many times as shown in Fig. 3.1 [12, 98, 100,
107, 127–130]. In most of these experiments and proposals, the cavity is far-detuned from theoptical transition that was probed. Operating in the resonant regime δc = 0, as demonstrated inour squeezing experiment (Chapter 5), is an exception rather than the norm. In principle, the33cavity detuning δc can be chosen almost arbitrarily. Therefore, a natural question to ask is: How
does cavity detuning affect both the fundamental and technical atomic population measurement
imprecision or the degree of spin squeezing for a given cavity geometry and cavity finesse?
To answer the question posed above, we provide detailed expressions in this chapter for thefundamental scalings for probing an atomic ensemble using an optical cavity that smoothly connects
the resonant regime to the far-detuned regime in this Chapter. We apply our results to estimate theamount of free space scattering, equivalent to photon-recoil heating and potentially wave-function
collapse of the ensemble when the cavity-aided measurement reaches an imprecision at the quantum
projection noise level. The average number of photon recoils per atom sets the degree to which
the measurement can be considered nondemolition. We show that the collective cooperativity
parameter N C plays a crucial role in determining the resolution of cavity-aided nondemolition
measurements. The collective cooperativity parameter N C plays a similar role to the resonant
optical depth of atoms in free space, where N is the number of atoms in the probe volume and C
is the single particle cooperativity parameter [131].
This Chapter is organized as follows. In Section 3.2, we begin with a review of the propertiesof the coupled atoms-cavity system including dissipation. This review also provides definitions andnotation used throughout the thesis. In Section 3.3, we derive the quantum-limited signal-to-noise
ratio for nondemolition measurements of atomic populations considered as a function of the cavity
detuning δc from atomic resonance (see Fig. 3.1a). We identify three different operating regimes, aresonant regime and two detuned regimes separated by a critical detuning δc◦ .

3.2

Coupled atoms-cavity modes
To begin, we briefly review the open coupled atoms-cavity system with the goal of bothproviding a framework for understanding the experimental work and explicitly enumerating theassumptions made to reduce this system to a classical two-mode system [132–134]. The dynamics
of the system under a classical drive and dissipation are then studied with the goal of obtaining
the full complex response of the reflected and transmitted cavity field. Finally, a discussion of the

(a)

e

(b)

δc

Γ

ωc
ωe

34

Cavity QED System

Probe

κ

↓

↑
↓

Photodetectors

Figure 3.1: (a) Relevant energy levels. The pseudo-spin-1/2 system typically comprises two
metastable states |↑i, |↓i, which are utilized in atomic sensors and clocks. The number of atoms N↑
in |↑i modifies the cavity resonance frequency. Initially, we will assume the optically excited state|ei does not shift |↓i because of dipole selection rules or because the coupling is highly non-resonantδc  ωhf , where ωhf is the hyperfine frequency separation between |↑i and |↓i. (b) Transmitted
and/or reflected probe light is monitored to determine the cavity resonance frequencies and hence
the number of atoms in |↑i. An applied NMR-like microwave rotation can swap the populations
between ground states so that a subsequent measurement can also determine the |↓i population
N↓ . The ensemble of atoms are trapped in an intracavity optical lattice (blue-green). The total
cavity power decay rate is κ. Single atom spontaneous decay from |ei into non-cavity modes at
rate Γ leads to photon recoil heating and single-atom wavefunction collapse. The goal then is to
extract collective information from the probe mode more rapidly than the undesired single-atom
photon scattering into free space.

probe signal-to-noise sets the stage for addressing measurement imprecision at the projection noiselevel in Section 3.3.

3.2.1

Linearized response

We consider an ensemble of N atoms with two ground states |↑i and |↓i whose populations
we wish to estimate precisely. The ensemble is confined and collectively coupled to a cavity mode(see Fig. 3.1(b)). Atoms in |↑i can interact with the cavity mode by absorbing a cavity photon andbeing promoted to an optically excited state |ei. On the other hand, atoms in |↓i are assumed to
not interact with the cavity mode because of dipole selection rules, a large energy splitting between
the ground states, or otherwise. A quantum phase may be encoded in the coherence between |↑iand |↓i for realizing an atomic sensor, but is otherwise not important in this section.
Ignoring dissipation, the Tavis-Cummings Hamiltonian that describes the coupled atoms-

35
cavity system is
H = ~δc ĉ† ĉ + ~g σ̂− ĉ† + σ̂+ ĉ .

(3.1)

The Hamiltonian is written in a frame rotating at the atomic transition |↑i → |ei angular frequency
ω↑e . In this Chapter, we assume every atom couples to the cavity mode with the same couplingstrength parameterized by the coupling angular frequency g. Appropriate renormalization of g and
N in the case of inhomogeneous coupling will be given in Chapter 5. The cavity field is describedby the photon annihilation operator ĉ, with cavity photon number given by M̂c = ĉ† ĉ. The cavity
detuning is δc = ωc − ω↑e , where ωc is the empty cavity frequency. The collective raising andP (i)lowering operators (no superscript (i)) σ̂± = i σ̂± are written in terms of the single atom raising
(i)

(i)

and lowering operators σ̂+ = |ei ih↑i | and σ̂− = |↑i ihei |. The atomic populations are given byPPP
the collective projection operators N̂↑ = i |↑i ih↑i |, N̂↓ = i |↓i ih↓i |, and N̂e = i |ei ihei |. For
brevity, we use the following abbreviations: N↑ ≡ hN̂↑ i, N↓ ≡ hN̂↓ i and Ne ≡ hN̂e i.
Although the atoms may in general exist in a superposition of |↑i and |↓i, in the following
analysis we will consider the atoms to be in a definite eigenstate N↑ of the N̂↑ operator. We willthen reintroduce the fluctuations in the operator N̂↑ for atoms in a superposition of |↑i and |↓i inrDE
terms of the rms projection noise about the mean value ∆N↑ =(N̂↑ − hN̂↑ i)2 .
We assume that the system is driven weakly so that the mean number of atoms in the optically
excited state |ei is a small fraction of the total number of atoms in |↑i, such that Ne /N↑  1. In
the weak excitation limit, one can make the Holstein-Primakoff approximation [135] that can bevisualized as replacing the three dimensional vector describing the two-level system composed ofstates |ei and |↑i by its vector projection onto the two dimensional x − y plane (see Fig. 3.2(a)).
When no probe light is applied, the system relaxes into its ground state orientation −ẑ. Whenthe system is weakly excited by probe light, and a small fraction of the atoms are promoted to theoptically excited state |ei, the lowest order effect on the vector is a small angular deflection of thevector away from −ẑ.
In the Holstein-Primakoff approximation, one specifically replaces the atomic raising and

(a)
x

e

z
y

(b) Reflected Port
cr
ci

Incident
Probe

36

Internal Cavity Loss

1

2

L

ct

Transmitted
Port

Cavityc

N g Atoms-Cavity
Coupling

a

Atoms

as
Free Space
Scattering

1 2

N

i

N

as,i

Figure 3.2: (a) Graphical representation of the linearization achieved via the Holstein-Primakoff
approximation. The atomic sub-system |↑i and |ei is described by a vector of length (N↑ + Ne )/2.
In the regime in which the probe light only weakly excites the atoms such that Ne  N↑ , the twolevel system can be described by its projection onto a 2D plane equivalent to that describing a lightfield. (b) With this approximation, the collective atomic mode may be treated as an equivalentcavity mode (lower purple mode) whose coupling to the actual cavity (upper red mode)pis governedby a partially reflecting mirror (center) described by a field coupling rate constant ( N↑ g) that
depends on the number of atoms in state |↑i. The physical mirrors have transmission coefficients√√
described by the field coupling rates κ1 and κ2 , and internal cavity losses are described by√
κL , such that the total power decay rate is κ = κ1 + κ2 + κL . Decay of the√ atoms by emissionof a photon into free space is described by the field transmission coefficient Γ that is the sum of
the field scattering into free space modes by the atoms in |↑i. Because the scattered modes are
distinguishable (expanded view of free space scattering port), it is possible to tell which atoms arein |↑i from the free-space-scattered photons, destroying any coherent superposition between |↑i andanother state |↓i (not shown) that may have been prepared for sensing a quantum phase.

plowering operators with effective creation and annihilation operators as â† ≈ σ̂+ / N↑ and â ≈p
σ̂− / N↑ that satisfy the usual commutation relation [â, â† ] = 1. The number of atoms in |ei is
described by the number operator N̂e = â† â. After substitution in Eq. 3.1, the resulting Hamiltonian describes the coherent coupling of two cavities through a beam-splitter whose transmission
coefficient increases with the number of atoms in |↑ip
H = ~δc ĉ† ĉ + ~ N↑ g âĉ† + â† ĉ .

(3.2)

A visualization of the coupled cavities model is shown in Fig. 3.2(b). The coupled Heisenberg

37
equations of motion for the cavity and atomic operators are
d hĉidt
d hâidtp= −ıδc hĉi − ı N↑ g hâi ,p= −ı N↑ g hĉi .

The eigenfrequencies of the above coupled equations areqδc ± δc2 + Ω2↑
ω± =
,
2

(3.3)

(3.4)

where
Ω↑ ≡

p
N↑ 2g

(3.5)

is the collective vacuum Rabi splitting Ω↑ that sets the difference in the normal mode frequencies
ω+ −ω− at zero detuning δc = 0. The normal mode frequencies in the laboratory frame are ω↑e +ω± .
A useful relation for simplifying expressions involving the mode frequencies is ω+ ω− = −Ω2↑ /4.
Any arbitrary superposition of atomic and cavity amplitudes can be written in terms of complex normal mode amplitudes B± and unit vectors b± as (hĉi , hâi) = e−ıω+ t B+ b+ + e−ıω− t B− b− .
√At resonance δc = 0, the normal mode vectors are b± = (1, ±1)/ 2, and the normal modes beatagainst one another at relative frequency Ω↑ . For example, if at t = 0 there are no atoms in the√
excited state, then B± = 1/ 2. The light in the cavity will Rabi flop (or be converted) into purely
excited-state atoms after a time π/Ω↑ , and then Rabi flop back into light in the cavity after 2π/Ω↑ .
More generally, the unit vectors can be expressed in terms of a mode-mixing angle θmix =arctan(Ω↑ /2ω+ ) as b+ = (cos θmix , sin θmix ), and b− = (− sin θmix , cos θmix ). When δc  Ω↑ ,the b+ normal mode is predominantly cavity-like, while the b− normal mode is predominantly
atom-like. The converse is true when δc  −Ω↑ . This normal mode structure is demonstrated
in Fig. 3.3 with an experimental measurement, described in Chapter 5, of the transmitted probe
power through the cavity as a function of the probe laser detuning δpe from the atomic transition
frequency ω↑e and the cavity detuning δc .

ato
mlike

ca
vit

ylik

e

38

N↑ 2 g

b-

b+

e
-lik
ato
m

ity

cav
re
a
b

bare

v
ca

atom

e

-lik
ity

Figure 3.3: The normal modes of the coupled atoms-cavity system b± exhibit an avoided crossingin theptransmission spectrum at δc = 0, where the size of the collective vacuum Rabi splitting isΩ↑ = N↑ 2g.

3.2.2

Driven and damped dynamics

The Heisenberg equations of motion of Eq. (3.3) can be extended to include cavity damping
and driving, as well as atomic damping via free space scattering of light (i.e., not into the cavitymode) as follows:d hĉidt
d hâidtp√κ= − ıδc +hĉi − ı N↑ g hâi + κ1 ci ,2p
Γ= − hâi − ı N↑ g hĉi .
2

The complex amplitude ci , with units of

(3.6)

p
photons/sec, describes the incident cavity driving field at

frequency ωp in the lab frame. As the above equation is written in a rotating frame at the atomic
frequency ω↑e , the cavity field ci in Eq. 3.6 is ci = |ci |e−iδpe t where δpe = ωp − ω↑e is the drive
detuning from the optically excited state |ei. These equations include the additional non-unitary
damping and drive terms shown schematically in Fig. 3.2(b). The additional terms are derived
using input-output theory [136] and will now be discussed. But first, we note that at large Ω↑ ,the finite damping terms introduce small corrections to ω± of order (Γ − κ)2 /Ω↑ [111] that may in39general be safely neglected in this thesis as we always operate in the well-resolved vacuum Rabi
splitting regime.

3.2.2.1

Cavity damping and input-output fields

As shown in Fig. 3.2(b), the damping of the cavity field at rate κ/2 = (κ1 + κ2 + κL )/2 isset by the mirror power transmission coefficients T1,2 such that κ1,2 = T1,2 × fFSR . The cavity
free spectral range is fFSR = c/2l, with 2l being the round-trip cavity length, and c the speed oflight. The total round-trip scattering and absorption fractional power losses at the mirrors L canbe modelled by an additional beam splitter with field decay rate κL = L × fFSR .
The cavity field is driven by an external incident probe field described by the complex amplitude ci . The reflected and transmitted complex field amplitudes, cr and ct respectively, will be
detected to infer the number of atoms in |↑i. The external field normalizations are chosen such that|ci,r,t |2 is the flux of incident, reflected, and transmitted probe photons in units of photons/second.
The average number of incident, reflected, and transmitted photons Mi,r,t in a measurement timeinterval from t0 to t0 + Tm is then
Z t0 +Tm
Mi,r,t =t0

2

ci,r,t (t00 ) dt00 .

(3.7)

In our experiments, it is convenient to express the number of probe photons coupled into the atomscavity system in terms of the directly measured “missing” photons in the reflected mode comparedto the incident beam Mm ≡ Mi − Mr .
The steady-state reflected and transmitted fields can be found by first solving the coupleddriven Eq. (3.6) for hĉi in steady-state (which oscillates at the cavity drive frequency δpe in therotating frame) and then using the results in the approximate relationships
cr =√κ1 hĉi − ci ,ct =√
κ2 hĉi ,that hold in the limit of a high finesse cavity T1,2 , L  1.

(3.8)

40
3.2.2.2

Atomic damping via free space decay

The atomic damping via scattering of light into free space (i.e. not into the cavity mode)is described by an effective amplitude damping rate Γ/2. To good approximation, the probability
decay rate Γ is simply the single-particle excited state |ei decay rate in free space 1 . The rate of√
scattering into free space is described by the field amplitude as = Γ hâi, normalized such that therate of photons scattered into free space is simply Ṁs = |as |2 .
The above picture of atomic damping can be further refined as shown in Fig. 3.2(b). Whilethe decay of excitation from the cavity mirrors is single-mode in nature, the atoms scatter light into
many free space modes. This multimode scattering can be envisioned by replacing the single decay
process via a single mirror with N↑ independent weak beam splitters, i.e. one for each atom in |↑i. If
the ensemble is optically thin along all directions except the cavity mode, then one can approximate√p that each atom decays into its own bath of states with an amplitude as,i = Γ hâi / N↑ . The
normalization is necessary to impose the fact that if a single photon is emitted into free space,
the probability that it was the ith atom that scattered the photon is reduced as the number ofatoms N↑ increases. The total scattering rate is the incoherent sum of the decay rates, reproducing
the previous decay rate Ṁs = Γ |hâi|2 . However, this refinement importantly emphasizes that themultimode free space scattering leads to in-principle information gain as to which particular atoms
are in |↑i, causing single particle collapse of the atomic wavefunction from a coherent superposition√
into an energy eigenstate, for example (|↑i + |↓i) / 2 → |↑i, thus destroying coherence. In contrast,
the decay of light through the cavity mirrors leads to only collective information as to how many
atoms in total are in spin up and therefore preserves coherence. Thus information gained throughthe cavity will be useful for preparing conditionally spin squeezed states, while the free spacescattering is a competing decoherence mechanism that serves to reduce the attainable degree ofspin squeezing.
1
The approximation that the single-particle decay rate Γ into all modes other than the cavity mode is not modified
by the presence of the cavity holds true in the limit that the cavity subtends a small fraction of the total solid angle
as seen by the atom [137].

41
3.2.2.3

Dressed mode linewidth

The atomic and cavity damping cause the normal modes to decay, resulting in a mode
broadening or narrowing. The approximate Lorentzian full width at half maximum (FWHM)
resonance linewidth given by a weighted average linewidth as

κ0± =

κ+



1+

Ω↑
2ω±



2

Ω↑
2ω±

Γ
2 .

(3.9)

For measuring N↑ via a cavity drive, probing the normal mode that is farthest from atomic resonance, also known as the dressed cavity mode, is most useful because this mode decays predominantly via photons leaking out of the cavity which can be collected with high efficiency. In contrast,
probing the normal mode closest to atomic resonance with a cavity drive results in most of theinformation being carried away by the free-space-scattered photons rather than leaking out throughthe cavity mode. For the sake of brevity, we refer to the further detuned mode’s linewidth andfrequency as simply κ0 and ω such that κ0 = κ0± and ω = ω± when |ω± | ≥ |ω∓ |.

3.2.3

Full complex field response to probing

The reflected and transmitted fields relative to the incident field cr,t /ci = Ir,t + ıQr,t can be
described in the complex plane by the real amplitudes Ir,t and Qr,t . We consider a single incident
cavity drive, or equivalently cavity probe, close to resonance such that the probe frequency ωp isdetuned by δp = (ωp − ω↑e − ω) from the mode resonance frequency. We assume the probe is nearresonance |δp |  ω+ − ω− and that the normal modes are well resolved ω+ − ω−  κ0± so thatinterference effects between normal modes can be ignored. The normalized transmitted electricfield through the cavity is then, to a good approximation, given byA,
1 + (2δp /κ0 )2
A (2δp /κ0 )Qt =,1 + (2δp /κ0 )2It =(3.10)(3.11)

42

(a)
v
u
y

m
w

l

n
j 1
k

x
(b)

1

j

k

FWHM

l

w
u

v

-1

n

m

FWHM
x

0

1

y

Normalized Probe Detuning

Figure 3.4: Transmitted and reflected probe electric fields from driving the atoms-cavity systemthrough the cavity mode. (a) The electric field phasors trace out circles in the I, Q-quadrature plane
as the probe detuning δp from the dressed cavity resonance varies from below to above resonance.
The normalization is chosen such that the reflected electric field goes to 1 when far away fromresonance. The quantum noise of the probe normalized to the incident electric field is represented
as a fuzzy blob with rms diameter 1/|ci |. In this illustration, a symmetric cavity κ1 = κ2 is assumed,so that the circles have the same diameter A. (b) Corresponding power transmission and reflection
signals.

where the unitless amplitude A is given by√2 κ1 κ2A= 2 .
Ω
κ + Γ 2ω↑(3.12)

From Eq. (3.8), the reflected field cr is just the sum of the transmitted field (rescaled for relativeptransmission coefficients) and the promptly reflected field such that Ir = 1 − κ1 /κ2 It , and Qr =p
κ1 /κ2 Qt .
As shown in Fig. 3.4(a), the phasor ct traces out a circle of diameter A in the complex planeas δp varies from  −κ0 to  κ0 . The translation It0 = It − A/2 centers the circle traced out by thephasor It0 + ıQt at the origin. One then sees that the angle with respect to the real axis is givenpby ψt = arctan(Qt /It0 ) = arctan(2δp /κ0 ). Similarly, the translation Ir0 = Ir − (1 − κ1 /κ2 A/2)

43
centers the circle traced out by the phasor Ir0 + ıQr at the origin with the angle with respect to thereal axis ψr = arctan(−Qr /Ir0 ) = ψt . The angles ψr and ψt are the same, but the quantum-limited
estimation of these phases may be different if κ1 6= κ2 .

3.2.4

Probe vacuum noise and measurement imprecision

The size of the quantum vacuum noise that contributes uncertainty to measuring the positionof the phasor is not changed by a linear transformation of coordinates in the complex plane. For
our purposes, the noise can be described as a Gaussian probability distribution with equal anduncorrelated real and imaginary rms fluctuations of magnitude σvac = 1/2. The rms quantum
vacuum uncertainty ∆ψt on the angle ψt is then independent of the average value ψt and is set
only by the average number of detected photons in transmission Md = qd Mt as1∆ψt = √.
2 Md

(3.13)The detection quantum efficiency qd includes any light loss and any excess technical or thermal
noise of the detector relative to vacuum noise. The uncertainty ∆ψt maps onto an uncertainty on
the estimation of δp through ∆δp = |dδp /dψt | ∆ψt = κ0 ∆ψt /2ηd . The detection sensitivity ηd isgiven byηd =1.
1 + (2δp /κ0 )2(3.14)

Probing near resonance δp = 0, one finds ηd = 1. For side-of-fringe probing δp = κ0 /2, one finds
ηd = 1/2. If the probe frequency is linearly and adiabatically scanned from δp  −κ0 to δp  κ0such that the total number of detected photons is fixed to the same Md as in the two previous
scenarios, one finds ηd = 1/2. The optimal readout assumes that as δp is changed, an adaptive
homodyne readout [138] is employed to maximize the measurement sensitivity to small changes inψt . For the spin squeezing experiment presented in Chapter 5, heterodyne detection is employed sothat adaptive detection is not required. However, the effective quantum efficiency qd was reducedby 1/2 as a result of the heterodyne detection.

44
It is straightforward to extend the analysis to a probe signal detected in reflection. However,one must parameterize in terms of the measurable average number of missing photons in thereflection port Mm and the average number of incident photons Mi such that in Eq. (3.13), oneppsubstitutes Md → (κ2 /κ1 )Mi qd (1 ∓ 1 − Mm /Mi )2 when A κ1 /κ2 ≶ 1.

3.3

Quantum-limited signal-to-noise and free space scattering
The measurement of the atomic population N↑ in | ↑i is achieved by precisely measuring

the dressed mode frequency ω+ or ω− or some combination of the two (as was done in our spinsqueezing experiment described in Chapter 5). In essence, the approach used here converts the
problem of measuring an atomic population into a frequency measurement. For atoms in a coherent
superposition of | ↑i and | ↓i, quantum projection noise in the atomic population N↑ causes the
dressed mode frequency to fluctuate.
In this section, we derive the fluctuations on the dressed mode frequency due to quantum
projection noise as a function of cavity detuning δc . We then use the results of Section 3.2 to obtain
the average number of detected probe photons Mdproj and free-space-scattered photons per atom
mproj
s , when the measurement imprecision on the probe field is sufficient to resolve the projection
noise fluctuations of the mode frequency ∆ω proj . The quantity mproj
is the key figure of merit thatscharacterizes the degree to which a measurement is nondemolition. Three limits of cavity probing
are identified, and a summary table of various key quantities in different regimes is presented.

3.3.1

Projection-noise-driven fluctuations of mode frequencies

As stated earlier, the atom number N↑ can be determined by precisely measuring one or both
dressed mode frequencies ω+ , ω− with a cavity probe. The collective enhancement of the Rabipsplitting by N↑ produces a key enhancement of the measurement sensitivity needed to resolve
projection noise. To concretely analyze the measurement imprecision of the probing, we consider ameasurement procedure most relevant to Ramsey spectroscopy as shown in Fig. 2.2(a): we assume
that for each experimental trial, all of the N  1 total atoms are initially prepared in spin down

45
via optical pumping or otherwise. Each atom is then rotated into an equal superposition of spin up
and down, preparing the ensemble in a CSS along x̂. As discussed in Chapter 1, in a CSS preparedalong x̂, the population in spin up and down fluctuate about the average N↑ = N↓ = N/2 with√equal magnitude but perfectly anti-correlated projection noise fluctuations ∆N↑ = −∆N↓ = N /2√
so that ∆Jz, CSS = ∆(N↑ − N↓ )/2 = N /2.
The rms fluctuation ∆ω proj of the individual mode frequencies ω± caused by the projectionnoise-driven fluctuations in N↑ is found by linear expansion as ∆ω proj = |dω± /dN↑ |∆N↑ evaluatedat N↑ = N/2. Making use of Eq. (3.4), one findsΩ↑g∆ω proj = √ q.
2 2 Ω2 + δ 2↑(3.15)

c

Note that ∆ω proj depends on N↑ via the Rabi splitting Ω↑ . The fluctuations of the two mode
frequencies are equal in magnitude but opposite in sign such that the rms differential fluctuationis ∆(ω+ − ω− )proj = 2∆ω proj .
The projection noise variance (∆ω proj )2 decreases as a Lorentzian versus the bare cavity
detuning δc with half width at half maximum (HWHM) Ω↑ . Figure 3.5(b) shows this scaling withdetuning (black, left curve). The technical requirements on the experiment for resolving ∆ω proj
are increased with detuning. Other experimental imprecision and inaccuracies scale relative to themode linewidth κ0 that one must split to the level of ∆ω proj , therefore the ratio ∆ω proj /κ0 is shownin Fig. 3.5(b, left) for three different bare cavity linewidths κ/Γ = 0.01, 1, 100 (blue, red, green).
Note that in the good cavity limit κ/Γ  1 (blue), the experimental requirement on splitting themode line can be somewhat reduced at larger detuning owing to the rapid fall off of κ0 as 1/δc2 in
the approximate region δc /Ω↑ ∈ {1, 10}.

3.3.2

Fundamental measurement imprecision and free space scattering at arbitrary
detuning δc

The resonance frequency ω of the dressed-cavity mode is measured relative to the known
frequency of a coherent (and unsqueezed) laser probe. The rms uncertainty on the probe detuning

46

(a)

100

!' / "

10

!/"
100

1

1
0.1

0.01

0.01

10
10
10
10

0

10

0.01
1

10

-1
-2

100

10

-3
-4

0.01

10
0.1

1

10

0

-1

-2

-3

[ g / ( 2!2 ) ]

proj

10

1

proj

&'

10

&'

/!' [ g / ( " 2!2 ) ]

(b)

100

Bare Cavity Detuning from Atomic Resonance #c / $%Figure 3.5: Theoretical scaling of key quantities with cavity detuning δc expressed in units of thethe collective vacuum Rabi splitting Ω↑ . (a) The dressed cavity linewidth κ0 in units of the atomic
excited state linewidth Γ. In (a) and (b), the scaling for different cavity finesses is shown for κ/Γ =0.01, 1, and 100 (blue, red, and green, respectively). (b, right, black curve) The rms fluctuation
proj
of the dressed cavity mode frequency due to projection noise fluctuations ∆ω+decreases
as 1/δc√above δc /Ω↑ = 1. The normalization is chosen such that one should multiply by g/2 2. (b, left,
red, blue, green curves). The ratio of the projection noise fluctuation of the cavity mode to thedressed cavity linewidth∆ω proj /κ0 is shown normalized such that the plotted values should be√
multiplied by g/(2 2Γ). A large ratio is desirable because technical noise may limit the ability to
split the probed resonance by more than a fractional amount.

47∆δp is equal to the projection noise fluctuation level ∆ω proj at an average detected photon numberof1
Mdproj =2η 0 2κgdδ2
1 + c2Ω↑!
.

(3.16)The passage of light through the cavity also leads to the scattering of Ms = |as |2 Tm probe
photons into free space modes by the atoms in spin up. The ratio of free space scattered to detected
photons Rs = Ms /Md is given by a weighted ratio of the atomic to cavity damping rates using
Eq. 3.9 as
Rs =

1 Γ
qd ηs κ



Ω↑
2ω

2
.

(3.17)

The factor ηs plays an equivalent role to a quantum efficiency and separately accounts for photons
exiting the cavity via an undetected port. In the symmetric cavity example we consider here, only
the transmission port (port 2) is measured, and ηs = κ2 /κ. See Fig. 3.2(b) for an illustration.
The key number of scattered photons into free space normalized to the total number of atoms
N , denoted mproj
s , may then be found from Eqs. (3.17) and (3.16) as
mproj
=
s
=

Rs MdprojN 0 21
κ
4qN↑ C κ(3.18)
1+

δc2
Ω2↑

!

Ω2↑
ω2

,

(3.19)

where C is the single particle cooperativity parameter
(2g)2
,
κΓ

(3.20)

q = qd ηd ηs .

(3.21)

C=
and q is the total effective quantum efficiency

Neglecting any other noise mechanisms, arbitrarily low probe measurement imprecision ∆Jz, imp
relative to the projection noise level ∆Jz, CSS may be attained at large ms relative to mprojs∆Jz, imp∆Jz, CSS

2
=

mproj
s
.
ms

(3.22)

48
The above result refers to the probe measurement imprecision only. However, the total measurement
uncertainty is governed by both the probe measurement imprecision treated in this Chapter, andthe noise added by probe-induced spin flip diffusion of Jz , forthcoming in Chapter 4.
A key result is that the average free space scattered photons per atom ms at a given measurement imprecision ∆Jz, imp saturates to a finite value
ms →4qN↑ C1∆Jz, imp∆Jz, CSS

2 as |δc | → ∞.

(3.23)

How far must one detune to approach this bound? Below we will show that this bound is nearlyreached for detunings greater than the critical detuning δc > δc◦ . The reason for the saturation offree space scattering is because in Eq. (3.19), the ratio of free space to detected photons asymptotically decreases as 1/δc2 , but the required number of detected photons increases asymptotically as δc2 .
Thus the net asymptotic dependence of ms on detuning δc cancels. The nondemolition characterof the measurement is ultimately set, not by the detuning δc , but by the product of the collective
cooperativity parameter and the quantum efficiency qN↑ C. This quantity physically sets the maximum rate at which collective information can be extracted from the ensemble compared to the rate
at which single-particle information is gained by the environment via multimode scattering of lightinto free space. This result is intimately connected to the picture of superradiance in Fig. 2.9. The
single particle forward scattering rate of photons that provide collective information is collectively
enhanced by a factor of N C.
In the good cavity limit κ  Γ, the frequency dependence δc of Eq. (3.19) can be understoodin three regimes: the far-detuned dispersive regime |δc | > δc◦ , the near-detuned dispersive regime|δc | < δc◦ , and the resonant regime δc = 0. The critical cavity detuning δc◦ is written in two equivalentand useful forms asrδc◦ =Γ Ω↑ 1
Γ
√ =
κ 2 q
2

s

N↑ C
.
q

(3.24)

The critical detuning is the cavity detuning at which the dressed cavity linewidth is κ0 = 2κ,possible only in the good cavity limit κ  Γ. Expressions for the number of photons scattered

49

Md

proj

2

2

10

10
10
10
10

10

5
3

10

&/!

10

100

10

1

-1

1

10

0.01

10

-3

100

2

[ ! / (16 q N" g ) ]

(b)

10

10

7

0
-1
-2
-3
-4
-5
-6

Rs [! / ( qd &)][ ! / ( %d g ) ]

(a)

100
0.01

10

ms

proj

2

1

1

0.1
0.01
0.01

0.1

1

10

100

Bare Cavity Detuning from Atomic Resonance #c / $"Figure 3.6: Theoretical scaling of key quantities with cavity detuning δc expressed in units of thethe collective vacuum Rabi splitting Ω↑ . In (a) and (b), the scaling for different cavity finesses isshown giving κ/Γ = 0.01, 1, and 100 (blue, red, and green, respectively). (a, left, blue, red, green )The average number of detected photons needed to resolve the projection noise fluctuations Mdproj
normalized such that the plotted values should be multiplied by Γ2 /(ηd g 2 ). (a, right, black) The
ratio of the number of free-space-scattered photons for every detected photon Rs , normalized suchthat the plotted values should be multiplied by Γ/(qd κ). (b) The crucial average number of scatteredphotons into free space per atom mproj
when the atomic population measurement imprecision issequal to the projection noise fluctuations. The normalization is such that the plotted values should
be multiplied by Γ2 /(16qN↑ g 2 ). In the bad-cavity limit of κ  Γ (green curves), there is little
fundamental advantage to operating away from resonance δc = 0. The technical requirements are
simply increased as a result of detuning. As the finesse of the cavity F is increased, the amountof free space scattering falls roughly as 1/F until the good cavity regime is reached when κ  Γ(blue curves). Here, one must detune by roughly the critical detuning δc◦ in order to realize the fulladvantage of having increased the cavity finesse. Importantly, note that mproj
does not significantlysdecrease above δc◦ owing to cancellation in this regime of the scaling of Rs ∼ 1/δc2 with the scalingof Mdproj ∼ δc2 .

50

10

10
10

ms

2

10

proj

2

[ ! / (16 q N" g ) ]

10

10

6

4

0.5
1
2

2

-4

-100

-10

-2

10

0

-2

0

-0.5 -1$c / %"
100

10

-6

-4

-2

0

2

10
10
10
10
Ratio of Cavity to Atomic Decay Rates # / !

10

4

Figure 3.7: The crucial average number of scattered photons into free space per atom such thatversus the ratio
the detected probe light allows resolution of projection noise fluctuations mprojsof the cavity power to atomic population decay rates κ/Γ. The normalization is chosen such thatthe plotted values should be multiplied by Γ2 /(16qN↑ g 2 ). Here, we assume that κ is varied bychanging the cavity finesse while holding the cavity length fixed (such that g is constant). Eachtrace represents a fixed ratio of the bare cavity detuning to the collective vacuum Rabi splitting
δc /Ω↑ , with values labeled on the traces and blue denoting δc /Ω↑ > 0, green δc /Ω↑ = 0 and dashedyellow δc /Ω↑ < 0 . Here we consider probing the dressed mode at frequency ω+ . At a fixed detuning(and therefore fixed projection-noise-driven frequency fluctuation size) a minimum is reached below
which a better cavity is detrimental due to the dressed cavity linewidth κ0 becoming clamped while
the ratio of free space scattering to detected photon number Rs continues to rise. In the units
above, the locus of the minimum is just 4κ/Γ (red line). Including the normalization, the locusof the minimum reduces to mproj= 1/(qN↑ C). When including the dependence of g 2 , κ ∼ 1/l onscavity length l, the minimum value of mproj
is not changed by just shortening the cavity at fixedsfinesse. Only shortening the cavity length while simultaneously increasing the cavity finesse (suchthat κ is constant) leads to a net fundamental reduction of mproj
s . Finally, the vertical black line
6
indicates the cavity linewidth for a finesse F = 10 cavity, near the highest currently achievable,
for 87 Rb and l = 2 cm (approximately the same as our cavity length).

51
into free space per atom mproj
s , the absolute size of the projection noise fluctuations of the mode
frequency ∆ω proj , and the dressed cavity linewidth κ0 are summarized in these different regimesin Table 3.1. Again, the quantity mproj
is critical for understanding the fundamental limits onsboth probe-induced heating of the sample and potential improvements on measurement sensitivity
beyond the SQL. Note that collective information gained from the cavity results from a forward
scattering process that leaves the momentum state of the atom un-modified and therefore does
not cause recoil heating. The probing-induced free space scattering always cause recoil heating on
average, even if the atoms are tightly confined in the Lamb-Dicke regime in all three dimensions.
Table 3.1: Regimes of cavity probing. The regime name and assumptions used to define the regime
are provided in the first and last columns. The quantity mproj
is the average number of photonssscattered into
free
space
normalized
to
the
total
atom
numberN , required to resolve an rms√
fluctuation N /2 in the spin up population N↑ equal to the projection noise level. The quantity∆ω proj is the rms angular frequency fluctuation of a single coupled atoms-cavity mode ω± due toprojection noise. The quantity κ0 is the dressed cavity power decay linewidth, here taken for themode detuned farthest from atomic resonance. The single-particle cooperativity C, the numberof atoms in spin up N↑ = N/2, the single-particle cavity coupling g, the empty cavity power andatomic population decay rates κ and p
Γ respectively, and the collective vacuum Rabi splitting Ω↑
are related by the following: Ω↑ = N↑ 2g, N↑ C = N↑ (2g)2 /κΓ. The detuning of the empty
cavity resonance frequency from the atomic transition frequency is δc , and the critical detuning at
which κ0 = 2κ is (δc◦ )2 = Ω2↑ Γ/4κ, assuming the good cavity limit κ  Γ. The maximally detunedin the presence of the constraint
regime assumes that the quantity N↑ C is chosen to minimize mprojsthat the cavity detuning cannot be made larger than some maximum value δmax set by technicalconstraints on resolving the projection noise fluctuations or fundamental constraints set by theinternal energy level structure of the atoms being probed (for instance the ground state hyperfine
splitting in 87 Rb).
Regime Name

Resonant
DetunedNear Detuned, Good Cavity

mproj
h s i
1
× 4q

1
N↑ C

Good Cavity, OptimizedκκN↑ C (2δΓ )4
c

4
N↑ C



Γ
2δmax

g
√
q2 2 2N↑ g
2 |δc |

q

2

2

1
2

1+

1 + Γκ

N↑ g
δc2

2



δc = 0Γκ −1δc  Ω ↑2N↑ g
2 |δc |N↑ g Γ
δc2 κ

δc◦  δc  Ω↑ ; Γ  κpκ√g
ΓN↑ g 22 |δc |2δc = δc◦  Ω↑ ; Γ  κ

q2

1
N↑ C

Far Detuned
Maximally Detuned,1+ κ 0 2
4

Critically Detuned, Good Cavity

Assumptions

[×κ]

Γ 2

1
N↑ C

κ0

∆ω proj

g pκ

√

2

Γ

2

1+

N↑ g Γδc2 κ2δc  δc◦ , Ω↑ ; Γ  κ
δc = δmax  Ω↑ ; Γ  κ;
2
N↑ C = 2δmax
Γ

52
3.3.3

Minimizing mproj
at fixed maximum detunings

In some experimental situations, a maximum useful probe detuning |δc | ≤ δmax is set by theenergy structure of the atom. For instance, the 6.8 GHz ground state hyperfine splitting in 87 Rb
imposes δmax ≈ 6.8/2 GHz. One could of course detune farther but the two states being probed
become less distinguishable with respect to the probe light. An optimum value of N↑ C can be foundthat minimizes mproj
when |δc | = δmax . The scaling for this case is shown in the last line of Tables
3.1. Physically, the optimum value of N↑ C is reached (at a fixed detuning) when the dressed cavity
linewidth is related to the bare cavity and atomic linewidths by κ0 /κ = 2Γ/(Γ + κ). In the resonant
limit, δc = 0, one finds an optimum κ0 /κ = 1, while in the detuned limit one finds κ0 /κ ≈ 2, i.e., thedetuned cavity resonance is broadened by a factor of 2 at the optimum. In this same limit, the ratiopof rms fluctuation size to dressed cavity HWHM is given by 2∆ω proj /κ0 = C/8. Larger single
atom cooperativity parameter C reduces the technical requirements on resolving the projection
noise fluctuations of the cavity mode.

3.4

Summary
In this Chapter, we have focussed on the problem of gaining information on the collective

spin projection Jz by probing the coupled atoms-cavity system with an externally applied cavity
drive. We have touched on one aspect of the disturbance imposed by probing the atoms — namely
free-space-scattering, which can cause recoil heating and shortening of the collective spin due tosingle particle wavefunction collapse. We emphasize that the theory presented here is very generaland may be adapted to the many collective cavity-QED setups around the world. The theory canbe adapted with minor modifications taking into account the multi-level structure of real atoms
to predict experimentally relevant quantities. In Chapter 9, we predict the optimum amount of53
spectroscopic enhancements in our 87 Rb system both as an example of how the theory can beadapted for real atoms and also as an outlook on prospects for further squeezing experiments.
In Chapter. 4, we will consider the effect of free-space-scattering shortening of the collective
spin in more detail, as well as two other probe-induced back-actions on the collective spin —fundamental back-action in Jy (anti-squeezing) and probe-induced Raman spin-flip causing Jz
diffusion. We will then derive limits on measurement imprecision limits on Jz and spectroscopic−1 ≡ (∆θ2enhancement in the polar angle sensitivity ξm
SQL /∆θ) in the presence of probe-induced

back-action.

Chapter 4

Theory: Probe-induced back-action

The present Chapter discusses the three types of probe-induced back-action, as shown in
Fig. 4.1, and the limits that the back-action sets on measurement imprecision in Jz as well as
on spectroscopic enhancement or squeezing. In Appendix A, we briefly discuss how coherence
preserving nondemolition measurements at or near the projection noise level can be used to reduce
recoil heating in a full Ramsey spectroscopy cycle to allow neutral atom traps to operate in a duty
cycle regime more like ion traps.
As the collective spin is a 3-dimensional vector, there are three types of back-action associatedwith each direction. For concreteness, we assume the collective spin is initially prepared in a CSS
along x̂, i.e. J = N2 x̂ throughout this Chapter. It is important to note that results in this
Chapter are valid only for CSS prepared in this orientation. The first back-action, ∆Jy , arises
from fundamental Heisenberg uncertainty relations and does not impact squeezing to lowest order.
The second back-action, shortening of the mean spin |hĴi| length, due to probe-induced free spacescattering impacts squeezing by reducing the signal size or contrast C. Finally, the third backaction, diffusion of Jz due to probe-induced Raman spin-flips impacts squeezing through increasing
noise in one’s ability to estimate Jz . The linear scaling of this back-action’s noise variance with theprobe photon number is analogous to radiation pressure shot noise in optomechanical systems [31,32]. The following sections will discuss these back-actions in greater detail.

(a)3 Types of Probe-Induced Back-Action(b)(c)

Fundamental
Heisenberg Uncertainty

Collapse due tofree-space-scattering

55

Diffusion of Jz
due to Raman spin flips(a) Suppress as 1/NC

Suppress as 1/NC(b) Make spin flip
probability small

Figure 4.1: The 3 types of probe-induced back-action associated with measuring Jz . (a) Fundamental Heisenberg uncertainty in the conjugate variable Jy that cannot be engineered away. (b)Shortening of the collective spin due to single atom wavefunction collapse from free space scatteringof probe photons can be suppressed by enhancing the forward scattering rate which scales as thecollective cooperativity N↑ C over the free space scattering rate which stays constant at the single
particle particle rate. (c) Free space scattering events that flip the spin state of the atom causes Jz
to change by ±1 per spin flip. Fluctuations in the number of spin flip events cause Jz to undergo
diffusion, equivalent to a random walk. Raman spin flips can be suppressed by making free spacescattering low with large N↑ C or by choosing a probing scheme that has a low spin flip probability
when undergoing a free space scattering event.

4.1

Back-action on Jy
For a CSS prepared along along x̂, the Heisenberg uncertainty relation Eq. 2.4 implies∆Jz ∆Jy ≥ N/4. In a given Jz measurement scheme, the best possible measurement imprecision ∆Jz, perfect , assuming perfect quantum efficiency in the probe detection process, would result in a fundamental, i.e. minimum, back-action ∆Jy = N/(4∆Jz, perfect ), schematically shownin Fig. 4.1(a). The back-action ∆Jy is also called anti-squeezing. In practice, losses and technical noise limit the effective detection quantum efficiency, resulting in an observed uncertainty

56∆Jz > ∆Jz, perfect . Since the actual measurement uncertainty ∆Jz ≥ ∆Jz, perfect is worse, theHeisenberg uncertainty relation ∆Jy ∆Jz ≥ 1/N is satisfied.
The anti-squeezing ∆Jy does not degrade the polar angle sensitivity ∆θ ≈ ∆Jz /|hĴi| of the
post-measurement state to lowest order. Anti-squeezing ∆Jy can be thought of as spreading thecollective spin out in its azimuthal phase. Provided the anti-squeezing is small compared to thelength of the initial collective spin, i.e. ∆Jy /J  1 where J = N/2, the fractional shortening of
the mean collective spin after the measurement and the impact on the polar angle sensitivity issecond order in ∆Jy /J, and is therefore usually not a problem. In most experimental situations,the direct shortening of the collective spin due to probe-induced free space scattering discussed in
Section 4.2 or the diffusion noise cause by probe-induced Raman spin flips discussed in Section 4.3
practically limits the polar angle sensitivity of the post-measurement state.

4.2

Back-action on |hĴi|
The second type of probe-induced back-action shown in Fig. 4.1(b) is the shortening of thecollective spin Ĵ as a result of probe-induced free space scattering. Free space scattering provides
in-principle information as to which atoms in the ensemble have emitted the photons, leading tosingle atom wavefunction collapse, destroying their superposition states, or equivalently a loss ofcoherence. The shortened length leads in decrease phase sensitivity because the angular resolution∆θ ≈ ∆Jz /|hĴi| is inversely proportional to the mean spin length |hĴi|.
In Section 3.3, we have analyzed the amount of free space scattering due to sending probe
photons into the atoms-cavity system and have shown that the key figure of merit mproj
scalessinversely as the collective cooperativity N↑ C at sufficiently large detuning δc of the cavity resonancefrom the atomic transition frequency. To remind the reader, mproj
is the number of free spaces
scattered photons per atom when the probe measurement imprecision is equal to projection noise
fluctuations of the mode frequency ∆ω proj . Therefore decoherence can be reduced by engineeringsystems with large collective cooperativity N↑ C  1.

57The contrast decays exponentially with the number of scattered photons per atom ms as
C = Ci e−ms ,(4.1)

provided the free space scattering rate for each spin is unchanged by the scattering process. HereCi is the initial contrast before probing. The initial contrast Ci may be reduced below 1 due tobackground decoherence processes such as dephasing due to differential trapping potentials for the|↑i and |↓i states or free space scattering caused by the trap light.
Note that both Rayleigh and Raman scattering lead to a reduction in the mean spin length.
Whereas Raman scattering always leads to single atom wavefunction collapse, in certain cases, freespace Rayleigh scattering does not cause single atom wavefunction collapse [139]. However, thisimmunity to collapse requires indistinguishability in the scattering process with respect to the spin
up and spin down state, which also reduces the information that can be extracted from the probe
mode.

4.3

Back-action on Jz
Raman spin flips caused by free space scattering results in noise added back into the mea-

surement quadrature Jz , causing Jz to diffuse, as shown in Fig. 4.1(c). Measuring with more probe
photons averages down photon shot noise but introduces more Raman spin flips as a result of measuring harder. We will find that this back-action presents limits on the measurement resolution ofJz . We will employ a simple 3-level model to calculate this limit and use that result in Section 4.5
to calculate spectroscopic enhancements relative to the SQL. This simple model will be extended
to include the effect of Raman transitions to other states than |↓i in Chapter 9.

4.3.1

Simple three-level model

In this section, we consider how the free space scattering changes atomic population in thespin up and spin down two-level manifold through Raman or spin flip events. For arriving at theresults presented in this section, only the atomic populations matter, and coherences are irrelevant.

58
A detailed accounting of all possible Raman scattering processes is required to make accurate
predictions for a multi-level atom. To capture the essential physics, we consider the simplest model
here. In this toy model, the only states in the problem are the two-level system |↑i, |↓i and theoptically excited state |ei, as described in Fig. 3.1(b). We assume that a free space scattering event
causes an atom to spin flip from |↑i to |↓i via the intermediate state |ei with probability p. It iseasy to see that the average change in the spin up population per photon scattered into free space
is dN↑ /dMs = −p, and correspondingly dJz /dMs = −p.

4.3.2

Diffusion of Jz

Free space scattering causes Jz to change on average by a certain amount, while the random
nature of the spin flip process leads to a random walk or diffusion of the collective spin projection
Jz . Provided multiple scattering can be neglected, i.e. pms  1, the diffusion process can be
described by the relation∆Jz, ba∆Jz, CSS

2= 4 p ms ,(4.2)

where ∆Jz, ba denotes the back-action noise from Jz diffusion, with the the diffusion constant 4p
set by the spin flip probability p. To remind the reader, ms is average free space scattered photons
per atom induced by the probe.

4.4

Measurement limits on Jz
Given the diffusion in Jz , and the photon shot noise in the probe, we must ask: How well

can one measure Jz or equivalently the atomic population N↑ ? The total measurement uncertainty(∆Jz )2 in the estimate of Jz can be estimated by adding the noise variances from probe measurement
imprecision (∆Jz, imp )2 (Eq. 3.22) and Raman spin-flip-induced diffusion (∆Jz, ba )2 (Eq. (4.2)):
∆Jz∆Jz, CSS

2


=
=

∆Jz, imp∆Jz, CSS

2


+

mproj
s
+ 4 p ms .
ms∆Jz, ba∆Jz, CSS

2
(4.3)
(4.4)

59
Averaging down photon shot noise determines the Jz more and more precisely. Eventually, however,probe-induced diffusion of Jz causes the value of Jz measured at earlier times to become lesscorrelated with the value of Jz measured at later times, adding noise to the estimate of Jz , asshown in Fig. 4.2. We note lowering spin-flip probability p allows one to average down photon shotnoise further to reach a lower measurement imprecision before spin-flip diffusion noise takes over,a situation analogous to back-action-evasion in optomechanical contexts.
0

10

/ms

z

D/ JD( ecnairaV nipS dezilamroN)SSC,zJ

2

p = 1

proj

2ms

-1

10

Photon Shot Noise
proj

ms

/ms
-2

p = 10-2

10Spin Flip Noise
4pms

-3

10

0

10

1

10

2

3

10

10
proj

Fractional Free Space Scattering ms/ms

Figure 4.2: Normalized spin noise variance versus fractional free space scattering ms /mproj
fors
N C = 103 . The measurement variance decreases as 1/ms from averaging down photon shot noise(dotted black line) until noise due to Ramanq spin flipsp(dashed red and black lines) takes over at large
ms . The minimum noise variance is 4 pmproj→ 8p/N C assuming perfect quantum efficiencysand probing in theq far-detuned limit. This minimum is reached when two noise contributions are√
opt
equal at ms = mproj
s /4p → 1/ 8pN C assuming perfect quantum efficiency and probing in the
far-detuned limit. A larger collective cooperativity N C or lower spin flip probability p (analogous
to back-action evasion in optomechanics) allows photon shot noise to be averaged down more before
spin flip noise takes over, as illustrated by the two curves for p/N C = 10−5 (black) and 10−3 (red).
The locus of the minimum variance is 2ms /mproj(green).
s

The optimal uncertainty ∆Jzopt occurs at an optimal scattering mopt
where the noise contris

60
butions from measurement imprecision and diffusion due to Raman spin flips are equal:
1
mopt
,
s = p
8pqN Ch(δc )!2 s∆Jzopt8p=,∆Jz, CSS
qN Ch(δc )

(4.5)

(4.6)

where the detuning dependence has been lumped into
2
κ
ω(δ
)
c
 ,
q
h(δc ) =  0
κ (δc ) δ 2 + Ω2


c

(4.7)

↑

where κ0 (δc ) is the dressed cavity mode linewidth introduced in Eq. (3.9), ω(δc ) is the dressed cavitypmode frequency introduced in Eq. (3.4), and Ω↑ = N↑ 2g is the collective vacuum Rabi splitting.
The parameter h(δc◦ ) = 1/4 at the critical detuning, and h(δc ) → 1 as |δc | → ∞. In the far-detuned2 p∆Jzoptlimit, Raman spin flip noise limits the achievable imprecision to ∆J= 8p/qN C.
z, CSS
4.4.1

Single spin measurement resolution

Single spin resolution ∆Jzopt ≤ 1/2 is required for conditionally preparing states with spectroscopic sensitivity at the Heisenberg limit as well as reading out cat states [140], or Dicke states [78].
Single spin resolution in ensembles of ∼ 100 87 Rb atoms has recently been demonstrated using
cavity-aided nondemolition measurements [130]. We find that in the far-detuned limit and with noqCprior knowledge, single spin resolution is reached for p ≤ 8N
, quantifying how ideal a cycling probetransition needs to be in order to resolve single spins. For qC ∼ 1 and N ∼ 106 , one would needp ≤ 10−7 , which is highly unrealistic for real multi-level alkali-atoms due to off-resonant scatteringfrom other hyperfine states, as will be discussed in Chap. 9 for the case of 87 Rb. Alternatively,
with high single atom cooperativity, qC ≥ 8N p, single spin resolution could be attained withouta cycling transition. For example, with qC ∼ 100 and a worst-case open transition with p = 1/2,
single spin resolution would be reached for N ≤ 25 atoms.

61

4.5

Spectroscopic enhancement
In this section, we derive the fundamental limits to the spectroscopic sensitivity — a key

result of this chapter in which the effects of both Jz back-action and signal reduction back-action in|hĴi| are combined. Spectroscopic sensitivity refers to the ability to resolve the angle through whicha collective spin is rotated. To a first approximation, the polar angle resolution ∆θ ≈ ∆Jz /|hĴi| is
set by the spin noise reduction (∆Jz /∆Jz, CSS )2 , discussed in Section 4.4, and the mean spin length|hĴi|, proportional to the contrast C, that remains after probing, discussed in Section. 4.2.
Angular resolution is improved if the spin noise is reduced by a greater amount than the probeinduced loss of coherence/contrast C. Putting Eq. 4.4 and Eq. 4.1 into Eq. 2.8, the spectroscopicenhancement for a perfect initial contrast Ci = 1 is given by
−1
ξm


=

=

∆Jz
∆Jz, CSS−2∆Jz∆Jz, CSS

−2

Cf

(4.8)

e−2ms .

(4.9)

To relate to our experiment where the initial contrast Ci < 1, the above equation needs tobe modified. An partially decohered reference state with Ci < 1 may be represented as a CSS witha shorter mean spin length |hĴi| = Ci N/2. The shorter mean spin length leads to a higher SQL√√√
in the polar angle resolution ∆θSQL = Ci ∆Jz, CSS /|hĴi| = 1/ Ci N where ∆Jz, CSS = N /2 as
before. As a result, one should normalize the spectroscopic enhancement to the larger SQL set by
the shorter initial collective spin−1ξm−2  2
∆Jz
C
√
=
Ci
Ci ∆Jz, CSS

−2 2
∆Jz
C
=
.
∆Jz, CSS
Ci
(4.10)For the remainder of this Chapter, we shall take Ci = 1.
The optimal spectroscopic enhancement exhibits different scaling with respect to the effective
collective cooperativity qN C in different regimes. If Raman spin-flips dominate over Rayleighp−1
scattering events, the optimal spectroscopic enhancement ξm,
qN C/p is limited by spin-flip
opt ∼

62
diffusion noise as considered in sub-section 4.5.1. If Rayleigh scattering dominates, then the optimal−1
spectroscopic enhancement ξm,
opt ∼ qN C is limited by the reduction of contrast C, discussed in√−1
Section 4.5.2. The change in the scaling of ξm,
N C in the Raman spin-flip limit to N C in
opt from

the cycling transition limit allows far greater amounts of squeezing on a cycling transition. The loss
−1of quantum efficiency q degrades squeezing as ξm,
opt ∼ q on a cycling transition, compared to the−1more forgiving scaling ξm,
opt ∼

√

q in the Raman spin-flip limited regime. However, the Raman

spin-flip limited regime is not really favorable in that the amount of squeezing will typically beworse in the Raman spin-flip regime even at small quantum efficiency q since we typically operatewith N C  1.

4.5.1Spin-Flip Diffusion-Noise-Limited Squeezing/Raman Limit

Here we consider the case where the reduction in the contrast C may be ignored. This isjustified if the optimal scattering moptthat optimizes the measurement imprecision of Jz is small,s 1, equivalently pqN C (κ/Γ)2  1 for probing in the far-detuned limit. In this regime,
mopts
the mean spin length |hĴi| remains approximately N/2, so that the spectroscopic enhancement isprimarily set by the reduction in the spin noise, discussed previously in Section 4.3!−2∆Jzopt−1
ξm, opt ≈,∆Jz, CSS

where∆Jzopt∆Jz, CSS(4.11)

2

has been introduced in Eq. 4.6. In in the far-detuned limit, Raman spin flipp−1
−1
qN C/8p. This result is valid provided ξm,noise limits the achievable squeezing to ξm,
opt .
opt =
0.193/p so that the loss of contrast may be neglected (see Fig. 4.3).

4.5.2

Decoherence-Limited Squeezing/Cycling Limit

Here we consider the case where the reduction of contrast C plays an important role in determining the angular resolution. If the probing is performed on a nominally closed transition where
Raman scattering spin-flips due to probe polarization imperfections and off-resonant scattering arevery improbable, spin-flip diffusion noise is negligible and the only limit to spectroscopic enhance-

63

3

10

Max. Spectroscopic Enhancement-1m, optNC = 1000

Spin Flip Limit2(NC/8p)100.193/pDecoherence Limit

NC/e

1

10

NC = 10
0

10

-5

10

-4

10

-3

10

-2

10

Spin Flip Probability p

-1

10

0

10

−1
Figure 4.3: Optimal squeezing ξm,
opt versus spin flip probability p for N C = 1000 (black curve)
and N C = 10 (red curve) for probing in the far-detuned limit, assuming perfect quantum efficiency
q = 1. Both curves take into account spin flip noise and decoherence. The decoherence-limited−1
squeezing ξm,
N C/e (dotted lines) is approached for p  e2 /8N C, and the spin-flip-limited
opt = p−1
squeezing ξm,
N C/8p (dashed lines) is approached for p  e2 /8N C. The locus of optimal
opt =squeezing at the crossover point p = e2 /8N C is 0.193/p (green line).

ment is the loss of contrast due to free space (Rayleigh) scattering. The probing is performed in thedecoherence-limited or cycling transition regime when the optimal scattering moptthat optimizessthe measurement imprecision of Jz is not small. Formally, this regime is valid when the Raman−1spin-flip limited spectroscopic enhancement calculated from Eq. (4.11) is ξm,
opt ≥ 0.193/p.
−1 = ms e−2ms .
In the decoherence-limited regime, the spectroscopic enhancement is given by ξm
mproj
s
√
opt
An optimum is reached at ms = 1/2, reducing the contrast C by a factor 1/ e ≈ 0.6. The opproj−1
timum squeezing is then ξm,
opt = 1/(2e ms ). In the far-detuned limit, the optimal squeezing−1ξm,
opt → qN C/e. We caution that the simple model presented here is not valid near the Heisen-berg limit because while measurement imprecision below a single spin may be achieved, the effective
spin noise variance must be clamped to 1/4 in order to enforce the Heisenberg limit on spectroscopic
sensitivity.

64

4.6

Conclusions
In conclusion, we have presented detailed expressions for how cavity-aided nondemolition

measurements of atomic populations and spectroscopic enhancement scale with key experimental
parameters: cavity linewidth, collective cooperativity, and Raman transition probabilities in Chapters 3, 4. This analysis will be important in future experiments for reducing photon-recoil heating
and for producing highly entangled states with enhanced spectroscopic sensitivities relative to the
SQL. With the theoretical foundation laid down, we will describe the experimental generationof conditional spin-squeezed states in Chapter 5 – the key results in this thesis — and provide
motivations for probing in the resonant regime δc = 0 in Chapter 6.

Chapter 5

Experiment: Spin squeezing via cavity-aided QND measurements

In this centerpiece chapter, we demonstrate the preparation of conditionally spin squeezed
states of a collective atomic pseudo-spin via cavity-aided QND measurements of the vacuum Rabi
splitting, published in [12]. We infer a 3.4(6) dB improvement in quantum phase estimation relative
to the SQL for a CSS composed of un-entangled atoms. The measured collective spin is composedof the two-level clock states of nearly 106 87 Rb atoms confined inside a medium finesse F = 710
optical cavity. This technique may improve atomic sensor precision and/or bandwidth, and maylead to more precise tests of fundamental physics.

5.1

Energy levels and probing scheme
In this chapter, I will use the abbreviated state notation |F, mF i to denote ground hyperfine

states in the 52 S1/2 manifold of 87 Rb and |F 0 , mF i to denote optically excited hyperfine statesin the 52 P1/2 manifold of 87 Rb. The two-level system whose populations we wish to measure
are the clock states | ↑i = |2, 0i and | ↓i = |1, 0i of 87 Rb with microwave transition frequency
ωhf /2π = 6.834 GHz (see Fig. 5.1).
An ensemble of Ntot ≈ 106 87 Rb atoms are loaded into the TEM00 mode of an optical cavity
whose frequency is made resonant with the |↑i to |ei optical transition with wavelength λ = 795 nm.
Here, the electronic excited state is |ei = |10 , 0i. For π-polarized probe light, the states |↑i and|ei form a nearly ideal two-level cavity-QED system because the dipole matrix element between|↑i and the only other nearby state |20 , 0i is zero. Therefore we would not expect ac-Stark shifts

66
F’=2’,mF=0
5 2P1/2

F’=1’,mF=0

CavityωFrequency c↑

ω− ω +

F=2,mF=0
5 2S1/2

↓

Ω

↓

e

6.834 GHz

F=1,mF=0

Figure 5.1: The clock states |↑i and |↓i form a pseudo spin- 12 system. The population N↑ is measured
by probing near the |↑i to |ei transition, which couples to a degenerate cavity mode, creating acollective vacuum Rabi splitting Ω↑ . A feature of this probing scheme is that the optically excitedstate |F 0 = 20 , mF = 0i (|F 0 = 10 , mF = 0i) does not couple to |↑i (|↓i) due to selection rules,
thus eliminating potential sources of ac-Stark shifts that would otherwise have been caused by
off-resonant cross-couplings if these couplings were dipole-allowed.

(or equivalently level repulsion) from off-resonant coupling of |↑i to |20 , 0i. Similarly, the dipole
matrix element between |↓i and |ei is zero, so we expect no off-resonant shifts of the collective Rabi
splitting from the N↓ atoms in |↓i.
Resonant coupling of the π-polarized cavity mode to the atoms generates a normal modepsplitting called the vacuum Rabi splitting Ω↑ = N↑ 2g. The goal is to produce spin squeezed
states with spectroscopic sensitivity below the SQL by measuring the collective spin projection Jz
or equivalently collective populations N↑ and N↓ precisely via the Rabi splitting without causing(much) decoherence.

5.2

Atom-cavity coupling
The atom-cavity coupling is a key parameter in our experiments. Here, we will provide

details for computing this coupling strength and discuss its significance for our collective cavity-

67

6.834 GHz
Γ = 2π × (5.75 MHz) Microwaves
π-polarized
Probes (795 nm)
Trap (823 nm)

B = 4.8 Gaussκ = 2π × (11.1 MHz)106 87Rb atoms

FilterPhotodiodeFigure 5.2: Salient aspects of the experimental setup. An ensemble of 106 atoms are tightly confinedwithin the TEM00 mode of the cavity using a 1D intra-cavity optical lattice at 823 nm. The cavity
decay and atomic free-space-scattering rates are κ and Γ respectively. A heterodyne interferometer(expanded out in Fig. 5.4) is used to probe the atoms-cavity resonances in both transmission andreflection. The collective nature of the Rabi splitting prevents loss of coherence as atoms remain in
a superposition after the measurement. Microwaves, launched into free space via a dipole antenna,
provide coherent manipulations of the collective spin required for state preparation, constructing ameasurement of Jz , and to spin-echo away probe-induced dephasing.

QED experiment. We begin by computing the peak single atom vacuum Rabi frequency [137] for
an atom at the antinode of the cavity mode and at the cavity waists
2cd2↑e2g0 =~λ◦ Vm= 2π × (607.1(5) kHz) .

(5.1)
(5.2)

The atomic dipole moment is d↑e = 1.465 C·m (Clebsch-Gordon coefficient included) [141], c is the
speed of light, λ is the the wavelength of the |↑i to |ei transition. ◦ is the electric permittivity of
vacuum, and ~ is Planck’s constant. Including the effect of divergence of the mode from waist, themode volume is Vm = 41 πw02 l for the TEM00 mode of a standing wave cavity with mode waist w0
and length l1 . We will later define an effective single atom vacuum Rabi frequency 2g and effective
atom number N to account for the effects of inhomogeneous coupling of the cavity mode to theatoms in Section 5.4. For the discussion in the next paragraph, all that is required to know is that1
For macroscopic cavities l  λ, the effect of the photon field penetration into the mirror multi-layer dielectric
coatings (single layer thickness typically λ/4) may be neglected. The cavity length is simply given by the spacing
between the mirrors. For very short cavities, l ∼ λ, the penetration depth can constitute a major correction to themode volume Vm .

68
2g = 2π × (506 kHz).
The effective single atom cooperativity is given by C = (2g)2 /κΓ where the cavity power
decay rate or FWHM linewidth is κ/2π = 11.1(2) MHz, and the excited state |ei population decayrate is Γ/2π = 5.75 MHz. C is the optical depth presented by a single atom to the cavity mode.
Equivalently, C is the fraction of solid angle subtended by the cavity mode, multiplied by the
number of round trips 2F/π. A single atom in |ei will decay into the cavity mode with rate CΓ
and at rate Γ into free space modes. As C = 4.02(6) × 10−3  1, a single atom hardly feels thepresence of the cavity mode at all.
By using many atoms, the coupling of the atoms to the cavity mode is collectively enhanced
so that the ratio of cavity emission to free space emission is N C. With N C  1, the rate at whichcollective atomic information is extracted from the cavity mode is enhanced relative to the rate at
which single particle information is revealed via free space scattering. In other words, free spacescattering can be engineered away by using large collective cooperativity N C. Information obtained
through the cavity mode preserves coherence or quantum superpositions as cavity emission does notprovide in-principle information about which atom forward scattered into the cavity mode. Free
space scattering into optically thin modes, however, leads to single atom wavefunction collapse
because the environment has in-principle information to figure out which atom has scattered thephoton. In our experiment, coupling to the cavity mode is enhanced by using up to N = 7 × 105 sothat the collective cooperativity parameter during the QND measurements N↑ C ≈ 1400 is large.

5.3

Key experimental details for Science Cavity and atomic ensemble

5.3.1

Science Cavity

We determined the Science Cavity parameters at 795 nm by accurately measuring the cavity
FSR and transverse mode spacings by measuring the frequency beat between two lasers that are
simultaneously resonant with the relevant cavity modes. The parameters for the Science Cavity
are summarized in Table. 5.1.

69
Parameter @ 795 nm
Cavity length l
Mirror radius of curvature Rc
Rayleigh length zR
Free spectral range fFSR
Frequency slitting between TEM01 and TEM00 mode
Frequency splitting between TEM02 and TEM00 mode
Linewidth κ/2π
Linewidth due to mirror transmission only (ATF’s datasheet) κ0 /2π
Linewidth due to absorption and scattering losses κL /2π
Finesse F
Mode waist w0
Mode volume Vm
Antinode cooperativity C0 = (2g0 )2 /κΓ
Effective cooperativity C = (2g)2 /κΓ (includes thermal averaging)
Antinode single atom Rabi frequency 2g0 /2π
Effective single atom Rabi frequency 2g/2π (includes thermal averaging)

1.9149(2) cm
5.000(8) cm
1.968(2) cm
7828(1) MHz
2257(2) MHz
4515(3) MHz
11.1(2) MHz
5.29 MHz5.8(2) MHz
710(10)70.56(4) µm
7.488(8) × 10−5 cm3
5.8(1) × 10−3
4.02(6) × 10−3
607.1(5) kHz
506(8) kHz

Table 5.1: Science Cavity Parameters

5.3.2

Atomic ensemble properties

Key details for the atomic ensemble and the intracavity optical lattice are summarized inTables 5.2 and 5.3, respectively. Here, we describe how these parameters were measured andinferred. We measured a radial temperature of 25(12) µK for the atomic ensemble using a releaseand probe (time of flight) method, where we measure the expansion of the atomic cloud versus
time of flight through the reduction of coupling of the atoms to the cavity mode. To take asnapshot of the dynamics in the lattice at full power, the lattice was switched off on a time scale
much faster than the radial trap frequency. Together with the calculated average lattice depthof U0 /h = 7.4(5) MHz or 360(30) µK, we infer the rms radial extent of the ensemble in the x
and y directions to be xrms = yrms = 10(2) µm. Assuming the axial temperature is in thermal
equilibrium with the radial temperature, the rms amplitude of atomic motion in the axial directionis zrms, pancake = 24(6) nm.
The axial extent of the atomic ensemble was measured by spatially selective population
tomography. To do so, we selectively remove atoms in all but a small sliver of region illuminated70
by a light sheet coming from the side of the cavity. The atomic population N↑ is then measured
as a function of the light sheet position via the vacuum Rabi splitting measurement, described in
Section. 5.6.1. The measured axial distribution of the atomic population is well described by aGaussian with rms width zrms, ensemble = 0.84(9) mm. With this measurement, the center of theatomic ensemble is also confirmed to < 1 mm from the center of the cavity. These length scales
are much smaller than the Rayleigh length of the cavity mode zR = 1.968(2) cm, the characteristic√
length scale over which the cavity mode waist increases from w(0) = w0 to w(zR ) = 2w0 . The
small axial extent and good centering of the atoms will result in only a small reduction in the
coupling of the ensemble to the cavity mode, as we shall see in the next section.
Atomic Ensemble Property
Excited state linewidth Γ/2π
Dipole Matrix Element d↑e
rms axial extent zrms, ensemble
rms radial extent xrms
rms axial oscillation amplitude in each pancake zrms, pancake
Axial Lamb-Dicke parameter k795 zrms, pancake
Offset from center of cavity
Radial Temperature kB Tradial /hPeak atom number per pancake (at center of cavity)

Value
5.7500(56) MHz [141]1.728(1) ea0
or 1.465(1) × 10−29 C · m [141]0.84(9) mm or ∼ 2000 pancakes10(2) µm24(6) nm0.2
< 1 mm25(15) µK∼ 200

Table 5.2: Atomic Ensemble Properties

Lattice Parameter
Wavelength λlatt
Circulating power Platt,circTrap depth or Average ground state ac Stark shift U0 /h
Differential ground state ac Stark shift
Axial frequency faxial
Radial frequency fradial
Cavity mode waist @ 823 nm w0,lattCavity mirror transmission @ 823 nm from ATF datasheet
Probe-Lattice beat length

Value
822.9(6) nm
0.90(7) W
7.4(5) MHz or 360(30) µK
3.1(2) kHz
or -413 ppm of avg ac Stark shift
320(10) kHz
0.82(3) kHz
71.78(4) µm
3615 ppm
13 µm or 32 lattice sites

Table 5.3: Intracavity Optical Lattice Parameters

71

5.4

Inhomogeneous coupling: effective coupling 2g and effective atom number N
The atomic ensemble of Ntot atoms experiences inhomogeneous coupling to the standing-wave probe mode at 795 nm because of the incommensurate 823 nm intra-cavity lattice, with some
atoms essentially uncoupled to the probe mode. Thus the standard quantum limit (SQL) for the
sub-ensemble of probed atoms is larger than that of the total ensemble Ntot . To account for thisinhomogeneous coupling, we define effective atom numbers N↑ , N↓ and N = N↑ + N↓ , and an
effective coupling g. The definitions of the effective atom number N and effective coupling g are
set by the two requirements that the combination correctly yield both the (1) average and the (2)rms fluctuations of the cavity frequency shifts. These two requirements can be expressed by thefollowing two relationships respectively:
tot
DN
X

E

= N↑ (2g)2

(5.3)

E2

= N↑ (2g)4 /2(5.4)

P̂↑,i (2g(~ri ))2

i=1
tot
D N
X

i=1

!2
P̂↑,i (2g(~ri ))2 )

E

−

tot
DN
X

P̂↑,i (2g(~ri ))2

i=1

Here, g(~ri ) is the probe coupling constant for the ith spin at position ~ri , and P̂↑,i = | ↑i ih↑i | is thespin up projection operator for the ith spin. The expectation values are evaluated for a CSS alongx̂, and also include averaging over the thermal distribution of atomic positions ~ri . For the TEM00
x2 +y 2p
w0 − w(z)2mode of the standing wave cavity, 2g (~r) = 2g0 w(z)esin(kz) where w(z) = w0 1 + (z/zR )2 .
Refer to Table. 5.1 for a summary of the Science Cavity properties.
Accounting for both the axial and radial averaging of the probe-atom coupling due to thefinite thermal spread and axial extent of the ensemble, the two conditions Eqs. 5.3, 5.4 yield an
effective coupling g related to the peak coupling g0 by√
2g = 0.963(15) ×

3
× 2g0
2

= 2π × (506(8) kHz)

(5.5)
(5.6)

√

where the factor

3
2 accounts for the inhomogeneous coupling and the factor 0.963(15) represents a

reduction in coupling due to thermal spread in the radial directions and the increasing mode waist

72
size along the axial direction. Similarly, the effective atom number N that goes hand in hand withthe effective single atom vacuum Rabi frequency isN= 0.996(4) ×2
× Ntot3= 0.664(3) × Ntot

(5.7)
(5.8)

where the factor 32 accounts for the inhomogeneous coupling and the factor 0.996(4) arises fromthermal averaging. The uncertainties in 2g and N are dominated by the atom temperature uncertainty. To restate, these definitions are chosen such that the observed vacuum Rabi splitting ispgiven by Ω↑ = N↑ 2g, and the fluctuations due to projection noise fluctuations in N↑ for a CSS√
along x̂ is given by N /2, consistent with the results of Chapter 3.

5.5

Sample preparation
In this brief section, I describe how the sample is initialized prior to the QND measurements.

The temperature measurements described earlier were also performed after this initialization step.
87 Rb atoms are loaded trapped in a magneto-optical trap (MOT) at the middle of the cavity forabout 1 second. The atoms are then loaded and cooled into the one-dimensional intra-cavity lattice
trap at 823 nm and at zero magnetic field. Atom number is controlled by the lattice loading time.
A quantization magnetic field of 4.8 Gauss along the lab physical coordinate system x̂ (see Fig. 5.2)
is turned on and the atoms are optically pumped into |↓i with a π-polarized free space laser beamon F = 1 to F 0 = 10 at 780 nm. Repumper beams at F = 2 to F 0 = 20 at 780 nm recycles atoms
that end up in F = 2 back down to F = 1. As |↓i is dark with respect to both the optical pumping
beam and 3D repumping beams, the atoms eventually accumulate in |↓i. To clear out any atoms in
F = 2, the repumper beams are left on for 100 µs longer after the optical pumping beam is turned
off. At this stage the ensemble ready to be manipulated with microwave rotations and probed withlight through the optical cavity. After the squeezing operation or other experiments are concluded
with the sample, a fresh sample is prepared by repeating the initialization steps.

73

5.6

Measuring Jz
In our probing scheme, we do not measure Jz in single Rabi splitting measurement because

the Rabi splitting Ω↑ only depends on the atomic population N↑ . However, by performing a πpulse to phase coherently swap the population between |↓i and |↑i, and performing a Rabi splitting
measurement again with the results labeled Ω↓ and N↓ , we can construct Jz = (N↑ − N↓ )/2 fromthe measurements. The experimental sequence is illustrated in Fig. 5.3.

Figure 5.3: Experimental Sequence for a single measurement of Jz . The measurement sequence
consists of microwave rotations (dashed line) and QND population measurements (solid line). Each
QND population measurement is implemented by simultaneously probing both modes of the collective vacuum Rabi splitting Ω↑,↓ , discussed in Section 5.6.1. The atomic populations are thenΩ↑,↓ 2
obtained from the measured Rabi splittings Ω↑,↓ via N↑,↓ = 2g
. Finally Jz is constructed fromN↑,↓ via the relation Jz = (N↑ − N↓ )/2.

5.6.1

Single Rabi splitting measurement

In this section, we describe how a single measurement of the vacuum Rabi splitting Ω↑ or Ω↓is performed. We employed heterodyne detection of the probe sidebands, directly IQ demodulated
so that the full amplitude and phase response during a frequency sweep could be used to extract
the mode resonance frequencies, as described in Section 3.2.3.
The probe beam was phase modulated at 150 MHz using a free space electro-optic-modulator(EOM) (Thorlabs EO-PM-NR-C1, Vπ = 180 V) and a high power rf amplifier (Minicircuits LZY-

74

(a)

-55 MHz 150 MHz-80 MHz
AD8015 #5

AOM

AOM

AOM

90T/10R
Phase
Mod

to IQ demodulation
for transmission

106 MHzProbe
Laser
DDS Freq

90T/10R

AD8015 #2

Sweep
Probe Lasert(b)

Heterodyne Reference Beam
201 MHz LO
Upper Probe
Sideband51 MHz LO

from reflection
photodiodeAD8015 #2

to IQ demodulation
for reflection

Probe
Carrier-99 MHz LO
Lower Probe
Sideband

I
Q
I
Q

I
Q

(c)

121 MHz LO
Upper Probe
SidebandADC-31 MHz LO
ADC

from transmission
photodiodeAD8015 #5

Probe
Carrier-179 MHz LO

ADC

Lower Probe
Sideband

I
Q
I
Q

I
Q

ADC

ADC

ADC

Figure 5.4: Experimental schematic of the heterodyne IQ detection system used to probe the vacuum Rabi splitting. (a) Resonant
coupling of the atoms to the cavity mode generates a collectivepvacuum Rabi splitting Ω↑ = N↑ 2g. The trapping optical lattice is not shown here. Both components of the vacuum Rabi splitting are simultaneously probed using the first order phase modulation
sidebands spaced ±150 MHz from the carrier. During a single Ω↑ or N↑ measurement, the probe
laser frequency is linearly swept from high to low optical frequency. Phase coherent modulationand demodulation of the probe signals is accomplished by generating all frequencies with phasecoherent DDSs. An AOM frequency shifts the transmission signals to avoid frequency degeneracy
with the contaminating reflected probe signals on the recycled heterodyne reference beam. (b, c)
The IQ demodulation frequencies were adjusted appropriately to mix the reflection and transmission signals down to DC. The demodulation frequencies are given by the probe component opticalfrequency minus the heterodyne reference beam frequency. In transmission, the carrier signal is
not present when probing atoms because the transmission resonances have shifted to the dressed
atoms-cavity mode frequencies. It is used as a diagnostic for probing the bare cavity.

1+). The phase modulation generates a high frequency and a low frequency probe sideband,
depicted as blue and red colored beams respectively in Fig. 5.4(a), with a modulation index β ∼ 1.5.
For probe sideband power calibration, we measure the fraction of probe power in each first orderprobe sideband to be 0.30(2) of the total probe power. The overall probe power is set by the
55 MHz AOM rf power level, and we typically probe with 1-10 nW total probe power incident at

75

(a)
sweep
resonance
κ’/2π

(b)
Ω ↑ = N↑ 2 g
ωLow Freq Probe

sweep

Probe Carrier

ω+High Freq Probe(c)

Figure 5.5: Simultaneous probing of the vacuum Rabi splitting in transmission and reflection. The
blue and red data points are the individual Rabi splitting signals ω± simultaneously measured bysweeping the phase-modulated probe laser frequency. The resonance frequencies are extracted fromfits (black curves) to the data plotted parametrically versus the probe frequency in the IQ plane (a).
A rotation has been applied to align the circles along the I-quadrature for presentation purposes.
The corresponding power I 2 + Q2 (points – data, curve – from IQ fit results) (b), and phase shiftarctan(Q/I) (points – data, curve – from IQ fit results) for both transmission and reflection (c).
The size of the fitted vacuum Rabi splitting Ω↑ = ω+ − ω− determines the population in |↑i fromN↑ = (Ω↑ /2g)2 . A single scan requires 60 µs. Note: for presentation purposes, only every 5th data
point is plotted.

76
the cavity.
Probing both the high and lower frequency dressed modes simultaneously is advantageous
for probe-induced ac-Stark shift cancellation, to be discussed in Chapter 6. Simultaneous probing
is achieved by adjusting the atom number to produce a Rabi splitting Ω↑ /2π = 300 MHz splittingthat matches the probe sidebands frequency separation. The probe laser optical frequency is swept
at −1184 MHz/ms by linearly ramping a DDS frequency source used to phase lock the probe laser
to a fixed frequency reference laser.
In a single 60 µs measurement of Ω↑ , or equivalently N↑ , the probe laser is swept over a spanof 71 MHz, spending ∼ 7 µs on the dressed modes resonances that have FWHM of ∼ 8.5(3) MHz(see Fig. 5.9). A single measurement of Jz requires two population measurement windows, eachtaking 60 µs, and a 14.5 µs microwave π-pulse to phase coherently swap the populations N↓ andN↑ in between the two measurement windows. The π-pulse plays a crucial role in spin echoing
away inhomogenous probe-induced and lattice-induced ac-Stark shifts that would otherwise leadto severe loss of coherence if left un-cancelled. Chapter 6 will discuss spin-echo in more detail.
To make full use of the available information, both ports of our symmetric cavity are detected.
Nearly photon shot-noise-limited detection of the weak probe (∼nW) is achieved via heterodyne
detection. The reflected probe light is overlapped with a strong heterodyne reference beam on a
90/10 beamsplitter and then coupled into a single mode fiber for good mode matching before sending
to a low noise photodiode. The transmission setup is very similar. The detection path efficiency
is approximated 0.3 in transmission and 0.5 in reflection. The photodiode (Hammamatsu S5973)
internal quantum efficiency is 84%. The relative contribution of the photodiode electronic noise
to the total photocurrent noise at a detected heterodyne power of 1.2 mW reduces net quantum
efficiency by about 1 dB. Transfer function and noise characteristics of the detection photodiodeare made available in Appendix C.
The probe-sidebands and heterodyne beam produce a distinct rf beatnotes that can be frequency separated using rf filters. The beatnote frequencies for the various probe frequency components are given in Fig. 5.4(b, c). Full amplitude and phase information of the probe electric field, or77
equivalently the full in-phase (I) and quadrature (Q) amplitudes are obtained by phase coherently
demodulating the heterodyne signals down to DC.
Phase stabilization of the heterodyne interferometer may be important for getting a goodphase estimate from the IQ fits. However, the measurements are only sensitive to phase drifts ontime scales comparable to a single population measurement. By sweeping across the resonances
quickly, phase stabilization of the interferometer was not crucial for obtaining high quality fits of
the IQ data.
Figure 5.5(a) shows typical IQ signals recorded in both reflection and transmission. The
high (low) frequency probe component sweeps across the resonance ω+ (ω− ), with the resulting
signals shown in blue (red) in Fig. 5.5(b, c, d). The signals are sampled into our data acquisition
program and fitted in real time. See Figure. 5.6 for a screenshot of the data acquisition program.
The resonance frequencies were extracted using least squares fitting in the IQ-quadrature plane,) of the Rabi splitting from the IQ reflection (transmission) fit,(Ωtransyielding estimates Ωrefl↑↑trans ) ≡ Ω exhibited
respectively. Empirically, we find that the weighted average (0.61Ωrefl↑↑ + 0.39Ω↑

the least noise in two repeated measurements of the same Rabi splitting. This ratio reflects therelative quantum efficiencies in reflection and transmission.

Figure 5.6: Screenshot of data acquisition program showing experimental control sequence table(left monitor, middle), raw data (left monitor, top), real-time fitting of the IQ signals in thecomplex plane (right monitor, left top) and accumulation of measurement statistics from different
experimental trials (right monitor, right panel).

78
5.6.2

Measurement uncertainty ∆Jz
10

Probe

m

se ~

2

π
2

Noi

2

1

Tec
hnic
al

Variance[( !" # !" )/2$] [MHz ]

4

M 6

8
6

(0)

4

(1)

8
6

2

Pho
ton

0.1

8
6
4

Projection Noise Level

Sho
t

Noi
se

2

sca

ling

M -1
m

0.01
10

2

3

4

5

6

7 8 9

2

100
(1)

(0)

3

4

5

6

7 8 9

1000

3

Missing Photons Mm in a ( !" # !" ) Measurement [ x 10 ]Figure 5.7: Measurement uncertainty estimated from the noise in the difference of two repeatedmeasurements of the same vacuum Rabi splitting Ω↑ . Inset shows sequence for this measurement.
This measurement is used for calibration because it is insensitive to projection noise. The data
(black dots) versus missing photons Mm (definition of Mm is given in Section 3.2.4) displays photon−1 (black line, fitted to data below 1.8 × 105 missing photons) until technicalshot noise scaling as Mm
induced by frequency chirping of the resonances takes over sharply. An empirical fit to the full
6 . The projection
range of data indicates that the technical noise variance (red line) scales as Mm
noise variance is ∆(Ω↑ − Ω↓ )2CSS /(2π)2 = 0.128 MHz2 (dotted line) and is independent of the atom
number N . Sub-projection noise sensitivity is achieved from Mm ∼ 5 × 104 to 4 × 105 photons. Thisdata was taken at N = 7 × 105 atoms.

The probe photon shot noise along with technical noise contribute an uncertainty ∆Jz to theestimate of the projection Jz . To calibrate this noise, we start with all atoms in |↓i (i.e. θ = π),perform a π/2 rotation to prepare a CSS along x̂. The vacuum Rabi splitting is then measured
(0)

(1)

twice with the results labeled Ω↑ and Ω↑ . Each measurement may fluctuate from one trial to thenext due to total atom number fluctuations, microwave power fluctuations, and quantum projection
noise; however, these sources of noise are common to both measurements within a single trial and

79
(1)

(0)

cancel, at least to lowest order. Statistical error bars of the standard deviation ∆(Ω↑ − Ω↑ )
(1)
(0) p
are estimated via ∆(Ω↑ − Ω↑ )/ 2(ntrials − 1), where ntrials is the number of experimental trials,
typically around 1000 for this data, at each missing probe photon number Mm . The measurement
(1)

(0)

uncertainty data ∆(Ω↑ − Ω↑ ) versus missing probe photon number Mm displayed in Fig. 5.7 was
taken at our squeezed ensemble size of N = 7 × 105 atoms. The measurement uncertainty variance−1 until a technical noise variance that scales as ∼ M 6 takes off
shows photon shot noise scaling Mmm

at around about Mm ∼ 3 × 105 photons.
 (0) 2 !

 (1) 2
The measurement uncertainty is taken to be ∆Jz = ∆

Ω↑
2g

−

Ω↑
2g

/2. For our re-ported squeezing result (to be discussed later), we used an average number of missing probe photons
Mm = 0.96(5)×105 probe photons for a single measurement of Ω↑ , we achieve a probe measurement
imprecision 6.5(5) dB below the projection noise level for N = 7.0(3) × 105 atoms obtained fromthe relative noise between two repeated Ω↑ measurements on the same atomic ensemble. This is
close to the optimal measurement precision achievable in our system.

5.6.3

Probe-induced Raman scattering

This section is aimed at trying to understand the increase in measurement uncertainty as
Mm is increased in Fig. 5.7. We will also make use of results in this section when discussing loss ofcoherence due to free space scattering later in Section 5.9.
Frequency chirping of the dressed modes resonances necessitate sweeping the probe frequency
rather than just parking the probe frequencies on resonance. The chirping occurs because of freespace scattering of the probe photons. The free space scattering causes N↑ to effectively decrease,pthus reducing the size of the Rabi splitting Ω↑ = N↑ 2g, and hence chirping of the dressed mode
resonance frequency with time.
The chirping leads to added noise as photon number is increased above Mm ∼ 3×105 photons
as clearly shown in Fig. 5.7, and the differential line broadening shown in Fig. 5.9. We hypothesizethat the resonance frequencies fluctuate strongly at high photon number due to a positive feedback
between Raman scattering and the vacuum Rabi splitting. The chirping causes more photons to80
be coupled into the cavity, which causes more chirping, which then causes even more photons to becoupled in and so on. This positive feedback could amplify initially small probe power fluctuations
into large fluctuations in the Rabi splitting.

Change in Rabi Splitting(!" # !")/2$ [MHz]
[MHz]

0
-10
-20
-30
-40

π
2

-50
-60
0

100

time
200

300

400

500

600
3

Missing photons Mm per !" measurement [x 10 ]
(1)

(0)

Figure 5.8: We observed a linear decrease in the difference Ω↑ − Ω↑ with missing photon number
Mm below 3 × 105 photons. Above 3 × 105 photons, severe chirping of the resonances discussed inFig. 5.9 results in large fluctuations in the fitted resonance frequencies. The red line is a linear fit
to to low Mm < 2.5 × 105 missing photons to guide the eye.

To quantify the fundamental loss of coherence in Section 5.9 due to probe-induced free spacescattering (technical loss of contrast may result from dephasing, probe-induced or otherwise), we
need to understand the relationship between the directly measured number of missing probe photons
Mm in the reflected port and the number of free space scattered photons Ms which is difficult tomeasure directly. By Eq. (3.12), we find that Ms is related to Mm by the following linear relation
Ms
Mm

=

Γ
,
Γ + κ − κ1

(5.9)

which evaluates to Ms /Mm = 0.405(6) for our system. This relation is valid provided multiple
scattering can be neglected.

(a)

(b)

81
11
Upper Dressed Mode
Lower Dressed Mode
Average

Dressed Mode FWHM [MHz]1.0Power Reflected

0.8

0.6

0.4

5

@ Missing Photons Mm = 2.8 x 10
Higher Dressed Mode
Lower Dressed Mode

0.2

10

9
Theory predicts 8.42 MHz
8

7

0.0
15

20

25

30

35

40

Sweep Time converted to Frequency [MHz]

45

50

0

50

100

150

200

250

300

3

Missing Photons Mm [ x 10 ]Figure 5.9: (a) Distortion of signals caused by Raman scattered induced chirping of the resonances.
The distortion arises from a combination of a linear frequency sweep of the probe laser and asymmetric shrinking of the Rabi splitting about the center of the splitting during the probing.
During probing, the Rabi splitting shrinks symmetrically about bare cavity frequency due to Raman
free space scattering of the probe photons. As the probe laser is swept from higher to lower opticalfrequency, the higher frequency probe sideband would be “chasing after” the higher frequency dressmode of the shrinking Rabi splitting. Conversely, the lower frequency probe sideband would be“running into” the lower frequency dressed mode. This relative chirp causes the linewidth of thehigher (lower) dressed mode to appear higher (lower) than it actually is. For the data shown here,
the upper dressed mode has a fitted apparent linewidth of 10.3 MHz, and the lower dressed mode
has a fitted apparent linewidth of 7.1 MHz. (b) Theory predicts the dressed mode linewidths,
given by the average linewidths of the cavity and atom linewidths to be κ0± = 2π × (8.4(1) MHz)
with the uncertainty dominated by the cavity linewidth κ. Chirping of the resonances lead toan apparent linear increase (decrease) in the linewidths (blue and red data points) of the higher(lower) frequency dressed mode with the number of missing probe photons Mm as shown by thefits (blue and red lines). We average the apparent dressed mode linewidths to provide an estimateof the actual dressed mode linewidth (black data points). The slope of the linear fit to the averageapparent linewidth data (black line) shows a factor of about 9 suppression compared to the fitted
slopes for the upper and lower dressed mode apparent linewidths. The average of the black data
points over the measurement range gives (κ0+ + κ0− )/2 = 2π × (8.5(3) MHz) and agrees with thetheory prediction of 8.4(1) MHz to 1%.

To verify this prediction, we measure the decrease in the Rabi splitting Ω↑ versus the missing
photons Mm , obtaining the readily measurable dΩ↑ /dMm . The Rabi splitting decreases because
atoms that have undergone a Raman free space scattering event can end up in |2, ±1]i and |1, ±1i,with lower coupling strengths to the cavity mode. In fact atoms scattered into |1, ±1i have negligible
coupling to the cavity mode because of the 6.8 GHz detuning from the ground state hyperfine
splitting.

π

(0)

2 0

-20

Ω

(1)

Ω↓

↓

Change in Rabi splitting
[MHz]

82

0

time

Scattering Pulses

-40
-60

0

400

800

1200

Missing Photons M m [10 ]
3

(1)

(0)

Figure 5.10: Change in Rabi splitting (Ω↑ − Ω↑ )/2π due to free space scattering versus missing
Ms
7 √gphotons Mm follows a linear scaling. The slope is predicted to be − 16
Mm . Using the indepenN↑

dently and accurately known values of 2g = 2π × (506(8) kHz) and N↑ = 3.50(15) × 105 , we inferred
Ms /Mm = 0.408(4) from the measured data. This result is in excellent agreement (0.7% level)with an independent theory prediction of 0.405(6). The fit (blue line) is indistinguishable from the(1)(0)
theoretical prediction. The inset shows the experimental sequence used to measure Ω↑ − Ω↑ . A(0)

CSS is prepared using the π/2 pulse (dashed line), and then the first measurement Ω↑ is made.
Next, the system is probed with Mm missing photons by applying a variable number of probe
pulses which induces free space scattering while avoiding nonlinearities at high Mm . Finally, a
(1)
(1)
(0)
second measurement Ω↑ is made to determine the change Ω↑ − Ω↑ .

To connect the experimentally measurable quantity dΩ↑ /dMm to Ms /Mm , we need to calculate dΩ↑ /dMs . Reproducing the results of Section 4.3.1 for the 3-level toy model here, the change
in the atomic population N↑ per free space scattered photon isdN↑= −p ,
dMs(5.10)

where p is the spin flip probability from | ↑i to | ↓i. Accounting for the multi-level structure of
87 Rb, we find the the “spin flip” probability per photon scattered into free space for an actual

physical atom is pactual = 7/24. I put spin flip in quotation marks because in our system, spin
flips from |↑i to |↓i via the intermediate state |ei are actually not optical-dipole-allowed. However,
Raman scattering takes the atoms from |↑i to |2, ±1]i and |1, ±1i which changes the Rabi splitting.
Therefore it is in this sense that the Raman scattering behaves like spin flips. In fact, it is more

83
appropriate to treat these events as loss. Because of the inhomogeneous coupling present in oursystem, the “spin flip” of an actual physical atom is equivalent to “spin flips” of 0.996 × 3/2 ≈ 3/2
effective atoms, therefore the effective “spin flip” probability per photon scattered into free spaceis p = 3/2 × pactual = 7/16. The change in Rabi splitting per free space scattered photon is then
dΩ↑
dMs

g
= −p p
N↑
7 g
,
= − p
16 N↑

(5.11)and the experimentally measurable change in Rabi splitting per missing photon isdΩ↑
7 g Ms=− p.
dMm
16 N↑ Mm

(5.12)

Experimentally, we observed a linear decrease in the Rabi splitting versus the missing photons
Mm as shown in Fig. 5.10. From the slope dΩ↑ /dMm , we inferred Ms /Mm = 0.408(4), in excellent
agreement (0.7% level) with the theory prediction of Ms /Mm = 0.405(6).

5.7

Observation of quantum projection noise in Jz
To ensure that we are observing quantum projection noise as opposed to technical noise, we

check that the observed noise variance in Jz scales linearly with N , and that it also agrees with our
a-priori prediction for the absolute noise level. The ratio of the variances of measured to predicted
projection-noise-driven fluctuations in Ω↑ − Ω↓ is 1.02(6). This level of agreement is possible dueto the very well-defined geometry of the cavity-mode, the trapping intra-cavity lattice and the cold
temperature of the atomic ensemble.
In the resonant coupling limit δc = 0 used in our experiments, the predicted projection noise∆Jz, CSS would produce rms fluctuations in the vacuum Rabi splittings of∆ (Ω↑ − Ω↓ )CSS =√2g= 2π × (358(6) kHz)(5.13)(5.14)

independent of the atom number N . All other QND probing schemes demonstrated to date have

84
been performed in the detuned limit δc  Ω↑ [100], where the rms fluctuations of the dressed cavity√
mode frequency due to projection noise scales as N .
We measure the variance of Ω↑ − Ω↓ , and subtract from it the measured detection noise
(1)

(0)

variance measured from the variance of Ω↑ − Ω↑measured in a separate series of trials (see


(1)
(0) 2
Fig. 5.7). The noise background subtracted variance ∆ Ω↓ − Ω↑is defined as follows


(1)
(0) 2
∆ (Ω↑ − Ω↓ )2bcksub = ∆ (Ω↑ − Ω↓ )2 − ∆ Ω↑ − Ω↑(5.15)

Note that the background subtraction is small at high atom number, but becomes appreciable at
low atom number because the probe photon number must also be reduced to avoid contributionsof Raman-scattering noise to the measurements at small N . Statistical error bars of the standard
p
(1)
(0)
deviation ∆X are estimated via ∆X/ 2(ntrials − 1), where X = Ω↑ − Ω↓ or Ω↑ − Ω↑ here, andntrials is the number of experimental trials, typically between 400 to 600, at each atom number.
The background-free noise data ∆ (Ω↑ − Ω↓ )bcksub , shown in Fig. 5.11(a) agrees with the a√priori prediction (line) for the projection noise level ∆ (Ω↑ − Ω↓ )CSS = 2g/2π = 358(6) kHz to2(6)%. By converting the measured fluctuations in the Rabi splitting to equivalent atom number
variance, we confirmed the predicted projection noise variance (∆Jz, CSS )2 to 2(6)% (see Fig. 5.11(c)(fits not shown)). Projection noise results in a linear dependence of the variance with atom number,whose magnitude is determined using low order polynomial fits. The fitted linear contribution is1.02(6) times (∆Jz, CSS )2 . In this thesis, we conservatively normalize spin noise variances to thepredicted projection noise variance instead of the slightly higher (2% higher) measured projection
noise variance.
Technical noise from the atoms or (homogeneous) microwave rotation noise would exhibit anoise variance in ∆Jz2 scaling as N 2 , illustrated by the orange dashed lines in Fig. 5.11. The ratioof the fitted technical noise variance to the projection noise variance is 21(7)% at our squeezed
ensemble size of N = 7 × 105 atoms. In Chapter 7, we will provide details regarding microwave
rotations added noise.

85

[MHz]

(a)

∝ N

Atom Number N [106]
(b)

(c)

-4

2

10

6
5
4
3

2

∝N

1
7
6
5

2

Pr

ed

6
5
4

ict

ed

St
an

da

3

rd

2

Qu

an

tum

Lim

it
3.4 dB spectroscopic enhancement

-6

10

6
5
4

3

6

10

Variance (Jz) [10 ]2Variance (!) [rad ]

4
-5

2

0.1

Pr

7
6
5

ed

ict

ed

d

ar

nd

a
St

m
tu
an

it

m

Li

u

Q

4
3

Prepared Squeezed State

2

3
2

0.01

2

3

4

5

6 7 8 9

0.1

2

3
6

Atom Number N [10 ]45

6 7 8 9

0.01
1

0.01

2

3

4

5 6 7 8 9

0.1

2

3

4

5

6 7 8 91

2

6

Atom Number N [10 ]Figure 5.11: (a) In the resonant limit δc = 0, the rms projection noise fluctuation of the populationN↑ is independent of total atom number N . Shown here are the measured rms fluctuations of
the dressed cavity modes ∆(Ω↑ − Ω↓ )bcksub /2π after subtracting out the noise background inquadrature. The data agrees with the a priori prediction (solid line) for the projection noise level√2g/2π = 358(6) kHz to 2(6)%. The shaded gray area represents the error range of our theory
prediction. At larger atom numbers N > 7 × 105 , additional technical noise was observed. In
contrast, probing in the off-resonant regime results in a cavity mode frequency noise variance thatscales linearly with N , and is always smaller than in the resonant limit [see Eq. (3.15)]. To constrain
potential added noise from the π-pulse rotation that swaps the two atomic populations N↑ andN↓ between the two splitting measurements, we apply the rotation about an axis perpendicular to
the CSS (black points), and about an axis parallel to the CSS (red diamonds). See Chapter 7 forconstraints on microwave rotations added noise.
(b) Same data in (a) converted into fluctuations of the collective spin polar angle showing theobserved variance of θ versus atom number N confirms the predicted SQL. In this Chapter, wedemonstrate the preparation of conditionally spin squeezed states with an inferred spectroscopic
sensitivity 3.4 dB below the SQL at a large ensemble size of N = 7 × 105 atoms (blue point). At
this ensemble size, the SQL is ∆θSQL = 1.2 mrad and the generated squeezed state has an angularresolution of 0.8 mrad.
(c) Same data in (a) converted into variance in Jz , showing the linear scaling of quantum projection
noise with atom number.
In all three plots, the scaling of technical noise with atom number N is schematically illustrated
with an orange dashed curve/line.

86

5.8

Conditional spin variance
We now demonstrate that repeated measurements of Jz are correlated below the projection

noise level ∆Jz, CSS . A first measurement Jz1 estimates Jz to a precision set primarily by the measurement uncertainty ∆Jz , preparing a sub-projection noise state when ∆Jz < ∆Jz, CSS . As shownin Fig. 5.12, quantum projection noise plus added classical and detection noise causes fluctuationsin the measured Jz1 from one trial to the next, but the fluctuations are highly correlated with asecond measurement Jz2 (Fig. 5.12(b)). The degree of correlation between the two measurements
Jz1 and Jz2 is characterized by the correlation coefficient ρ:ρ≡cov(Jz1 , Jz2 ),var(Jz1 )(5.16)

where the covariance is cov(Jz1 Jz2 ) ≡ hJz1 Jz2 i − hJz1 ihJz2 i, and the variance is var(Jz1 ) ≡ hJz1 i2 −hJz1 i2 . The variance var(Jz1 ) is given by the sum of the measurement uncertainty variance (∆Jz )2
from probe measurement imprecision and back-action and state preparation noise variance denoted(∆Jz, prep )2var(Jz1 ) = (∆Jz )2 + (∆Jz, prep )2 .

(5.17)The state preparation noise variance (∆Jz, prep )2 is the sum of the quantum projection noise variance(∆Jz, CSS )2 as well as any classical rotation noise variance from the first π/2-pulse in the CSS
preparation step.
The first measurement Jz1 can be used to predict a second measurement Jz2 with an optimal
estimator ρJz1 , allowing the state preparation noise to be partially canceled in the difference Jz2 −ρJz1 (Fig. 5.12(c)). The estimator ρJz1 is optimal in the sense that it minimizes the variance(∆(Jz2 − ρJz1 ))2 . While the state preparation noise is common-mode to both measurements Jz1 andJz2 , the measurement uncertainty ∆Jz is independent for both measurements. If the measurement
uncertainty ∆Jz is zero, then the two measurements Jz1 and Jz2 would be perfectly correlated, i.e.
ρ → 1. In reality, finite probe measurement uncertainty and uncorrelated microwave rotation noise
degrades the correlation ρ between the two measurements Jz1 and Jz2 .

87
At N = 7.0(3) × 105 atoms and a missing probe photon number of Mm = 1.9(1) × 105 permeasurement of Jz , the observed conditional spin noise reduction relative to the projection noise
limit was∆(Jz2 − ρJz1 )∆Jz, CSS

2= −2.6(3) dB ,(5.18)

where ρ = 0.72(6) is estimated using Eq. 5.16. Subtracting the measurement uncertainty ∆Jzof the second measurement in quadrature gives a conservative estimate of the uncertainty in theinferred conditional spin noise reduction Jz,inferred relative to the projection noise limit



2


∆Jz,inferred 2∆(Jz2 − ρJz1 ) 2∆Jz≡−= −4.9(6) dB .
∆Jz, CSS∆Jz, CSS∆Jz, CSS(5.19)The measurement uncertainty was (∆Jz /∆Jz, CSS )2 = −6.5(5) dB as determined from fluctuationsin the difference of two time adjacent Ω↑ , or equivalently N↑ , measurements in Section. 5.6.2.
A different approach to estimating the measurement uncertainty is to assume each QND
measurement has the same measurement uncertainty, whereas the quantum projection noise (as well
as classical state preparation noise) is common mode to both measurements. Under this assumption,
the difference Jz2 − Jz1 then exhibits twice the amount of measurement uncertainty variance thana single Jz measurement. Using the measured value of (∆(Jz2 − Jz1 )/∆Jz, CSS )2 = −2.1(3) dB anddividing by 2 (i.e. subtracting 3 dB) yields a slightly better conditional spin noise reduction of(∆Jz,inferred /∆Jz, CSS )2 = −5.1(6) dB. Note that this estimate also agrees with the earlier estimateof −4.9(6) dB within error bars. In this thesis, I will use the slightly more conservative inferredconditional spin noise reduction (∆Jz,inferred /∆Jz, CSS )2 = −4.9(6) dB to quantify squeezing.

5.9

Loss of coherence
Reduction of spin noise alone does not allow improved quantum phase estimation unless thelength of the collective spin |hĴi| is sufficiently unchanged. To claim spectroscopic enhancement
or squeezing, we need to measure the coherence remaining after the QND measurement Jz1 andshow that the conditional reduction in spin noise more than makes up for the loss of signal due toprobe-induced decoherence.

(a)

(b)

Measure Quantum Projection Noise

Verify Noise Reduction

(c)

88

Jz1
Jz2 - ρJz1

Figure 5.12: a) Measurement sequence for observing Correlation between successive QND measurements. (b) The first and second QND measurements of Jz exhibit correlated fluctuations thatarise largely due to projection noise. Number of trials ntrials ≈ 570. (c) The degree of correlation ρ
between Jz1 and Jz2 can be used to conditionally reduce the spin noise of the resulting state after
the first Jz1 measurement to -4.9(6) dB below the CSS noise after subtracting out the measurement
uncertainty.

In this section, we quantify the amount of coherence remaining versus the number of missing
probe photons Mm used in a single QND measurement of Jz . The normalized length of the collective
spin is measured by varying the polar angle θ from 0 to 2π and determining the contrast C = |hĴi|/J
from the resulting variation of the population N↑ (see Fig. 5.13).
Having instilled confidence in our theory prediction for the ratio of free space scatteredphotons to missing probe photons via experimental verification in Section 5.6.3, we now use thattheory to predict the fundamental probe-induced loss of coherence/contrast. If each free spacescattered photon leads to the collapse of a single spin, we predict from Eqs. (4.1) and (5.9) that the

89

(a)

(b)

(c)

Missing Photons Mm [x 103]

Figure 5.13: (a) The degree of coherence remaining after the measurement Jz1 is determinedusing the sequence: ( π2 )–(measure N↑ )–(π)–(measure N↓ )–( π2 )–(measure N↑ ). For the coherence
measurements, a large dynamic range in Jz (compared to measuring projection noise fluctuations∆Jz, CSS of a CSS) is required. To obtain a large dynamic range, we scan the probe laser (without
sidebands) across the the full Rabi splitting Ω↑ , and detect the probe carrier in reflection. (b)The sequence is repeated and the final measured value of N↑ is plotted versus the phase of the
final π2 -pulse. With no measurements (empty circles), the background contrast is Ci = 0.97(1).
Probing with 1.8 × 105 missing photons (filled circles), causes a small reduction in contrast despitea measurement sensitivity below the projection noise level. (c) Measured contrast versus missing
probe photon number (solid circles), second order polynomial fit (solid line), and the predicted
contrast loss due to free space scattering alone (dashed line).

fractional reduction in contrast is k1,pred = 6.4(3) × 10−7 per missing probe photon at N = 7 × 105
atoms. This prediction is valid provided multiple scattering may be neglected, i.e. the actual lossof contrast is small.
Fig. 5.13 shows that the measured contrast C versus missing probe photon number Mm is
2 . The fitted value, k−7 per missing
well described by C = Ci − k1 Mm − k2 Mm
1,fit = 5.5(7) × 10

probe photon, agrees with the prediction to 15%, and confirms the fundamental role of free spacescattering as the dominant source of decoherence.

90
The quadratic variation of C, k2 = 1.0(3)×10−12 per (missing probe photon)2 , arises from uncanceled inhomogeneous probe-induced light shifts that result in dephasing of the ensemble. Theselight shifts are largely spin-echoed away with the π-pulse used to measure Jz1 . The uncanceleddephasing arises from radial motion in the trap, which will be discussed in Chapter 6. At fixed
measurement uncertainty relative to the projection noise level, the magnitude of the dephasing
increases linearly with probe detuning, making it easier to reach a scattering-dominated regime inthis work compared to work in a far-detuned dispersive regime [100].
Before the QND measurements of Jz , the measured contrast is Ci = 0.97(1). Using Mm =1.9(1) × 105 missing probe photons in total for the first measurement Jz1 at N = 7.0(3) × 105 atoms
reduces the contrast to Cf = 0.82(2). At this probe photon number, free space scattering of probe
photons is the dominant source of decoherence loss. Loss of coherence from un-canceled ac-Stark
shifts accounts for only 25% of the total loss of coherence.

5.10

Net spectroscopic enhancement
The ability to estimate the mean spin polar angle θ is largely set by the noise in Jz and thesignal size |hĴi|. For a mean spin lying close to the x − y plane, ∆θ ≈ Jz /|hĴi|. Using the Wineland
criterion [58], the directly observed spectroscopic enhancement is given by
−1
ξm


=

∆Jz, CSS∆ (Jz2 − ρJz1 )

2

Cf2
Ci

= 1.1(4) dB
below the SQL. We infer the ability to prepare states with enhanced spectroscopic sensitivities of
−1
ξm


=

∆Jz, CSS∆Jz,inferred

2

Cf2
Ci

= 3.4(6) dB
relative to the SQL for quantum phase estimation for N = 7 × 105 atoms, the largest ensemble sizeon which squeezing on a clock transition has been demonstrated to date. The demonstration ofspin squeezing also implies inter-atomic entanglement in this large ensemble [68].

91
The small correction from Ci normalizes the spectroscopic enhancement relative to the SQL
set by the partially decohered collective spin with shortened mean spin length as discussed in
Section 4.5, Eq. 4.10). Because our initial coherence Ci = 0.97(1) is high, this results in a verysmall correction (increase of 0.1 dB in squeezing) for our reported results2 .

5.11

Back-action
(a)

(c)

25
20

2

Variance [Jz2(ψ) - Jz1(0)] - ΔJz, meas [dB]

(b)

15

16.5

10
5

minimum
uncertainty state0

projection noise level

1.0
1.8

Electronic Noise

3.0

Probe Sweep

3.0

Heterodyne Detection

3.3

Cavity Mirror Losses

4.2

Optical Path and Detector
Quantum Efficiencies

Atomic Absorption

-5
0

45

90

Rotation Angle ψ [deg]135

Figure 5.14: (a) Sequence for measuring back-action. Noise in the back-action quadrature is observed by inserting a rotation through angle ψ between the QND measurements Jz1 and Jz2 . (b)Back-action noise normalized to the projection noise level versus rotation angle ψ. The dashed curveis the calculated response for a minimum uncertainty squeezed state, while the solid curve withthe gray 68% confidence interval is the predicted back-action from the intra-cavity probe vacuum
noise. (c) The ratio of the predicted anti-squeezed quadrature at ψ = π/2 and that of a minimum
uncertainty squeezed state is about 16.5 dB (left column) due to finite quantum and technicalefficiencies. The right column gives the detailed breakdown from various estimated contributions.
These contributions account for an estimated 16.3 dB of the excess back-action.
2

For comparison, in Ref. [100], the initial contrast was Ci = 0.7, resulting in 1.5 dB correction.

92
The un-measured quadrature Jy , or azimuthal angle φB in the limit of slight shorteningof the collective spin, is driven by fluctuating ac-Stark shifts arising from the intra-cavity probe
vacuum noise. Noise in the back-action quadrature is observed by inserting a rotation throughan angle ψ between the QND measurements Jz1 and Jz2 , see Fig. 5.14(a). The measured andpredicted quantum back-action noise levels are 22.3(1) dB and 21.4(1.5) dB relative to projection
noise respectively. The ratio of the predicted anti-squeezed quadrature at ψ = π/2 to that of aminimum uncertainty squeezed state is 16.5 dB due to finite quantum and technical efficiencies
in the probe detection process. We estimate that the compounding effects of finite quantum andtechnical efficiencies accounts for 16.3 dB out of the predicted 16.5 dB excess back-action. Detailed
breakdown of these effects are shown in Fig. 5.14(c).

5.12Conclusions
The key result of my thesis — the experimental demonstration of spectroscopic enhance-

ment and entanglement as a result of collective QND measurements — has been presented in this
Chapter. Specifically we have demonstrated the preparation of spin squeezed states with inferred
spectroscopic enhancement
−1
ξm


=

∆Jz, CSS∆Jz,inferred

2

Cf2
Ci

= 3.4(6) dB
relative to the SQL in a large ensemble of N = 7 × 105 atoms, demonstrating the power of a
top-down approach to generating entangled states for quantum metrology. This result is enabled
by strong collective coupling N↑ C ≈ 1400  1. This proof-of-principle experiment may open theway for QND techniques to enhance the fundamental sensitivity of precision measurements withlarge ensembles of neutral atoms.

Chapter 6

Advantages of operating in resonant-coupling regime

The goal of this Chapter is to provide motivations and analyze unique aspects of probing in
the resonant-coupling limit, δc = 0. In Section 6.1, we review theoretical results of Section 3.3 in thelimit of δc = 0. Section 6.2 and 6.3 details technical advantages and sources of common-mode noise
rejection obtained by working in the resonant limit and simultaneously probing both dressed cavity
modes. Section 6.4 discusses the effects of radial motion of the atoms in the trap on the probing.
Section 6.5 considers the effects of spatial mode mismatch of the probe modes in a standing-wave
cavity on the correlation between the dressed mode frequencies.

6.1

Fundamental scalings related to projection noise
In the resonantly coupled regime δc = 0, the absolute size of the projection noise fluctuations

is maximized (see Fig. 3.5), i.e.,
g
proj
∆ω±
= √ ,
2 2

(6.1)

making it easier to observe projection noise fluctuations in the presence of technical noise in thedetection system. Note that the rms fluctuation is independent of N , as experimentally demonstrated in Fig. 5.11(a). The same is true for the FWHM linewidth which is simply equal to theaverage linewidth [111] due to the equal photonic and atomic contributions to the normal modes:
κ0± = (κ + Γ)/2 .

(6.2)

94
To be able to resolve projection noise, one must detect, on average, a number of probe photons intransmission given by
1
proj
Md±
=
2ηd

 0 2
κ±
.
g

(6.3)

The ratio of free-space-scattered photons to detected probe photons in transmission is
Rs± =

1 Γ
.
qd η s κ(6.4)

Finally, the number of scattered photons into free space for measurement uncertainty at the projection noise level normalized to the total number of atoms N is
mproj
s± =

1
2qN C



Γ 2
1+
.
κ

(6.5)

If the cavity length and mode volume are fixed by experimental constraints, then one is, in principle,
free to minimize Eq. (6.5) by varying the finesse of the cavity mirrors, until a minimum value of= 2/qN C is reached when κ = Γ. The minimization with respect to cavity finesse accounts
mprojs
for the fact that the cooperativity C scales as 1/κ.

6.2

Noise rejection in the resonant limit
Section 6.1 has outlined fundamental quantum limits on detection in the δc = 0 regime. In thissection, we describe how the δc = 0 regime is useful for rejecting several sources of technical noise.
Simultaneous probing of both transmission resonances ω± is advantageous for canceling technical,
but important, laser frequency noise and cavity frequency noise. Importantly, the previous analysis
for the value of mproj
is not modified if Mdproj /2 photons are used to measure each mode frequencysω+ and ω− .
δc2
From Eq. (3.4), we find that the measured differential quantity ω+ − ω− ≈ Ω↑ 1 + 2Ω
is2↑

only quadratically sensitive to the cavity detuning δc . Note that the absolute sensitivity to δc ispdecreasing as 1/ N↑ , a favorable scaling for working with large atom number. As an example,
a typical Ω↑ /2π = 300 MHz and a detuning of δc /2π = 1 MHz causes a change in the measured

95
splitting 100 times smaller than projection noise fluctuation in the collective Rabi splitting ∆Ωproj
=
↑
√
proj
2∆ω±
= g/ 2.
Simultaneous or near-simultaneous measurements of both resonances can strongly suppress
the impact of laser frequency noise. To make these measurements, one can use low phase noise
RF modulation of a single laser with much higher phase noise to generate probe components nearω+ and ω− as was discussed in Section 5.6. The laser carrier frequency noise is common-mode
to the two probes and cancels in the difference ω+ − ω− . The measured power spectral density(PSD) of the probe-cavity relative frequency noise Sν (f ), shown in Fig. 6.1, was measured for oursystem using side-of-fringe transmission. If both the upper and lower transmission resonances aremeasured for time Tm with time separation τm between the measurements, the noise variance inthe measured splitting due to laser noise is!2∆Ωlaser↑
2π

Z ∞Sν (f )T (f )df ,(6.6)

4 sin2 (πf τm ) sin2 (πf Tm ).
(πf Tm )2

(6.7)

=
0

where the transfer function T (f ) is
T (f ) =If the laser possesses a Lorentzian lineshape with FWHM ∆ν in Hz as measured in heterodyne
versus a perfect reference laser, the PSD is constant with value Sν (f ) = ∆ν/π. Integrating Eq. (6.6),
2 for good time overlap τ < T , and S /T for no time overlap/2π)2 = Sν τm /Tm
we find (∆Ωlaser
m
m
ν
m
↑

τm > Tm . Completely unoverlapped measurements would yield an absolute measurement uncerp
tainty of ∆Ωlaser/2π = ∆νBm /π, the geometric mean of the measurement bandwidth Bm = 1/Tm↑
and the laser FWHM linewidth ∆ν. For a typical extended-cavity-diode-laser ∆ν = 100 kHz, andp√ p2.
∆Ωlaser/2π ≈ 180 kHz µs/Tm . For partial time overlap ∆Ωlaser/2π ≈ 180 kHz µs τm /Tm↑↑
In practice, lasers exhibit low frequency 1/f noise, but so long as 1/τm is much larger than
the noise corner frequency, the above description is adequate. The laser noise can be reduced byspreading the Md detected probe photons over a longer measurement period, but at the possible
expense of additional decoherence of the atomic quantum state (arising from non-probe related
mechanisms) and the potential additional contributions from 1/f laser and cavity frequency noise.

τm/Tm = 0.1

Transfer Function T(f)τm/Tm = 1000

[Hz2/Hz]96(a)τm/Tm = 0.05

PSD

FWHM = 50 kHz

Frequency [Hz]

[kHz]

(b)

τm
time

Integrated Noise

Tm

Tm

Time Separation τm [sec]Figure 6.1: (a) Measured probe laser frequency noise spectrum Sν (f ) (solid black curve) and forreference the calculated white frequency noise level for a Lorentzian 50 kHz FWHM linewidth laser(horizontal black line). Also shown are theoretical transfer functions relating laser frequency noise
to Rabi splitting noise for Tm = 20µs and τm /Tm = 0.05, 0.1, 1000 (dotted green, blue, and red
curves, respectively) The inset of (b) defines the measurement timeing. For the case τm /Tm = 1000,
the sin2 (πf Tm )/(πf Tm )2 envelope of the transfer function is shown here. obtained by averaging
over the underlying fast oscillations. (b) RMS noise in the Rabi splitting noise ∆Ωlaser/2π due to↑the measured laser frequency noise versus τm for Tm = 20 µs (solid blue curve) and for Tm = 200 µs(solid red curve). For a white laser frequency noise spectrum of the level shown in (a), the integrated
Rabi splitting noise spectrum versus τm is shown in dashed blue and red curves. Vertical dashedlines indicate the two different Tm used for the transfer function integration. Most of the integrated
noise at larger τm can be attributed to the laser frequency noise bump near 17 kHz.

6.3

ac-Stark shift cancellation
The intra-cavity probe light induces an ac-Stark or light-shift of the |↑i state. At resonance,

simultaneous probing of both modes ω± leads to a cancellation of the shift as the induced shifts

97
have opposite signs. The cancellation arises from symmetrically spaced probe components aboveand below the atomic transition frequency ω↑e . In the case of inhomogeneous coupling of theatoms to the cavity mode, the ac Stark shifts are inhomogeneous and lead to a loss of signal and areduction in the effectiveness of composite pulse sequences used for manipulating the ground statespin manifold. The integrated phase shift caused by passing probe light through the cavity wasmeasured using a microwave spin echo sequence and observing the phase shift of the spin echo
fringe versus the number of photons coupled into the atoms-cavity system. In Fig. 6.2, we show theprobe-induced phase shift if only the ω+ mode or the ω− mode is probed. We then show the highdegree of cancellation of the phase shift, and therefore the average induced ac-Stark shift, whenboth the ω+ and ω− modes are probed simultaneously.

π

|

2 0

π

π|

|

↓ 2 φ

90

2|

π

0

time

πb-

|

2 0

b+

Figure 6.2: The relative quantum phase between states | ↑i and | ↓i is light shifted by passing
probe photons through the atoms-cavity system near only the upper normal mode frequency ω+(blue solid circles) or only near the lower normal mode frequency ω− (red open circles). The shifts
have opposite sign and cancel if probe photons are simultaneously passed through the atoms-cavity
system at both frequencies (purple squares). Black lines are fits to the data.The measurements are
performed using a variable number of photons coupled into the atoms-cavity system Mm during a
microwave spin-echo sequence (inset) employed to remove background dephasing from the trapping
optical lattice.

While simultaneous probing was successful in canceling average ac-Stark shifts, Fig. 6.3 shows

98

Fractional Reduction in Contrast

1.0

0.8

1 x probingπProbe 2π

2 x probingπProbe 2

π

φ
π
2

N

Nf

φ
N

π
2

N

Nf

0.6

0.4

0.2

1 x simultaneous probing
2 x simultaneous probing
1 x non-simultaneous probing
2 x non-simultaneous probing

0.0
4

10

2

3

4

5

6

7 8 9

5

2

10

3

4

5

6

7 8 9

6

10

Total Missing Probe Photons Mm

Figure 6.3: Simultaneous probing effect on contrast. The fractional reduction in contrast (C −Cprobing )/C, where Cprobing (C) is the contrast with (without) probing. The total missing probe
photons Mm is the sum of the number of missing photons used in both timing windows adjacent tothe π-pulse. For the simultaneous probing data, the probes sweep through the 8.4 MHz FHWM of
the resonances at the same time. For non-simultaneous probing, the probes are offset in time within
a single population measurement window of N↑ or N↓ . The red (black) curves are quadratic fits tothe 2(1) time window simultaneous probing data (red solid points (black solid points)) respectively.
This plot shows that probing in both time windows, rather than simultaneous probing, is crucial
for reducing the dephasing. Inhomogeneous ac-Stark shifts from imbalance in upper and lower inthe probe sideband photons coupled into the cavity could explain the observations.

that simultaneous probing was not as effective in canceling out inhomogenous ac-Stark shifts. When
probing only in the second time window in a echo sequence, the fraction loss of contrast is invariably
worse than when probing in both time windows adjacent to the π-pulse. The fractional loss ofcontrast is defined as (C − Cprobing )/C where Cprobing is the contrast remaining after probing with
Mm total missing probe photons in the whole echo sequence. The data shows that there is somebenefit from simultaneous probing when probing in one of the two time window only. However, whenboth time windows are probed, there is little, if any, benefit to probing simultaneously. Instead,
it is crucial to probe in both time windows. This observation could be a result of inhomogeneous

99
ac-Stark shifts imposed by probing both modes; simultaneous or not, is not important. Imbalancein the number of upper and lower probe sideband photons coupled into the cavity would result in
un-cancelled inhomogeneous ac-Stark shifts. Probing in both time windows, in conjunction withthe π-pulse, would reverse the inhomogeneous ac-Stark shifts, leading to re-phasing. However, theπ-pulse alone cannot by itself reverse the inhomogeneous ac-Stark shifts when the probing is applied
in one time window only.

6.4

Radial oscillations
1.0
π
2

φ
π
2

π

Nf
0.8

0.6

0.6

0.4

0.4

0.2

Tevol
Probe

π
2

N

π

0.2

φ
π
2

N

, with probing

0.8

Contrast with no probing

Probe

Fractional Change in Contrast

1.0

Tevol

Nf

0.0

0.0
0.0

0.5

1.0

1.5

2.0

2.5

Spin Echo Time Tevol [ms]Figure 6.4: Radial motion of atoms in the trap manifests as oscillations in the spin echo fringe
contrast. The raw contrast C (red filled squares, right axis) and the contrast with probing Cprobing(red hollow circles, right axis) show small hints of oscillations that are amplified in the fractionalchange in contrast (Cprobing − C)/C (blue filled circles, left axis). The frequency of the oscillations is
expected to be at twice the trap radial frequency. A exponentially damped sinusoidal fit (blue curve)
to the fractional change in contrast yields a fitted radial trap frequency of fradial,fit = 0.92(2) kHz, in
excellent agreement with the radial frequency fradial,pred = 0.91(3) kHz predicted from the measured
lattice power. In this data set, N = 7 × 105 and Mm = 2.6 × 105 in total for both the N↑ , N↓
measurements.

Signatures of radial oscillations of the atomic sample in the trapping lattice can be observed

100
on the spin echo fringe contrast curves, but they are relatively weak, see red color data points in
Fig. 6.4. By comparing the contrast with and without probing, the radial motion is made more
apparent via the fractional reduction in contrast (C − Cprobing )/C (blue data points in Fig. 6.4),
which subtracts out the dominant background dephasing from the trapping lattice. The fractional
reduction in contrast is well described by a damped sinusoid that oscillates at twice the trap radialfrequency of about 1 kHz. While this oscillation in contrast might be interesting, it is not usefulfor our experiment. The key point to take away is that while we could choose a specific spin echo
evolution Tevol time to coincide with a local peak in the contrast curve (which would correspondto a Tevol of ∼ 0.6 ms for our system), the loss of contrast would still significant compared toperforming the probing as fast as possible Tevol → 0. Thus our strategy is go probe as quickly as
possible, completing a complete Jz measurement in ∼ 150 µs.

6.5

Spatial mode matching of atoms-cavity modes
In a standing-wave cavity, the ω± modes arise due to coupling of the cavity mode to atoms

located away from the nodes of the standing-wave of each mode. By probing at two different
frequencies, it is possible that the two standing-wave probe modes effectively couple to two slightly
different sub-ensembles of atoms. Projection noise fluctuations in the two sub-ensembles are not
perfectly correlated such that the fluctuations of ω+ and ω− are also no longer perfectly correlated.
The contribution of projection noise to the difference of the mode frequencies ω+ − ω− would thenbe reduced. In the extreme limit of the nodes of the ω+ mode being aligned with the antinodesof the ω− mode, the two sub-ensembles are completely independent. The projection noise driven√
rms fluctuations in the difference ∆(ω+ − ω− ) would then be 2 lower than for the case of perfectoverlap. Here we calculate this effect and provide numerical results indicating that this effect issmall in our experiment.
In vacuum, the wave vectors for resonant probe light k± = ω± /c have a differential wave
vector given by k+ − k− = Ω↑ /c. However, if the atoms uniformly fill the cavity mode along theaxial or ẑ direction, then the index of refraction of the atoms at the resonant frequencies ω± must be101
0 = n k are identical inside the atomic medium. This is guaranteed
such that two wavevectors k±± ±by the round trip phase being a multiple integer of 2π at resonance.
If, however, the atoms do not uniformly fill the cavity, then the wavevectors of the probe
modes in vacuum and inside the atomic ensemble can be different, leading to coupling to slightly
different atomic sub-ensembles. Using the original condition, one finds the indices of refractionof the atomic ensemble at position z along the cavity axis evaluated for probe frequencies at thenominal resonance frequencies ω± are
n± (z) = 1 ∓Ω↑ 1ρ↑(z) .
4fFSR k±(6.8)

where fFSR = c/2l is the cavity free spectral range, and ρ↑ (z) is the coarse-grained linear probability
distribution function that an atom in |↑i is located at z and normalized such that integrating over
all of space yields 1. The coarse graining is performed over many lattice sites, and quantum
fluctuations are ignored in this expression as they are a higher order effect.
As a first application, we assume the atoms uniformly fill a region of space of length d centeredin the middle of the cavity of length l. In the atomic medium, the spatial phase deviation φ± (z) of
the standing-wave electric fields cos (ωc z/c + φ± ) from the empty cavity spatial phase is given byz Ω↑
φ± (z) = ±
4fFSR



1 1
−
l
d


for |z| ≤

d
.
2

(6.9)

In the limit of uniform filling d = l, the two probe modes are spatially matched, otherwise theyacquire a differential spatial phase φ+ (z) − φ− (z) that grows linearly as z.
In our experiment, the atoms were roughly Gaussian distributed along the cavity axis withstandard deviation zrms, ensemble ≈ 1 mm much less than the cavity length l ≈ 2 cm (see Table 5.2
for exact numbers). In this case, the spatial phase deviation of the standing-wave probe modes
from the empty cavity spatial phase in the atomic medium is given by
!!
Ω↑
z 1
z
φ± (z) = ±− Erf √,
4fFSR l2
2 zrms, ensemble
where Erf(·) is the error function.

(6.10)

(a)

102
atoms

(b)

Figure 6.5: Illustration of the concept of spatial mode-matching of the atoms-cavity modes. (a)In vacuum, the electric field of the higher frequency mode ω+ (blue) has a shorter wavelengththan the lower frequency mode ω− (red), leading to a phase advance of the ω+ mode relative tothe ω− mode as it advances from the left mirror to the left edge of the atomic medium (green).
In the atomic medium, the refractive index n± at frequency ω± is such that the ω+ mode has alonger wavelength than the ω− mode so that the two modes phase up in the middle of the cavity.
For reference, the electric field for the bare cavity mode ωc (dashed black) is shown. The phasemismatch in this illustration is deliberately made large by choosing Ω↑ /2π = fFSR . The length of
the uniformly distributed atomic medium is set to d = l/4, again for illustration. (b) The spatial
phase deviation φ± of the ω± modes from the empty cavity spatial phase. The red(blue) curves
corresponding to φ+ and φ− respectively are used to generate the standing-wave modes in (a).

At a typical Ω↑ = 2π × (300 MHz), the differential probe spatial phase φ+ (z) − φ− (z) is
expected to be only 31(3) mrad at z = zrms, ensemble . Numerical simulations of the phase mismatchand projection noise show that this phase mismatch leads to < 1% reduction in the variance of thedifference ω+ − ω− relative to the variance in the same quantity for perfect spatial overlap of thetwo modes with one another.

6.6

Summary
The previous chapter and this chapter have focussed on probing aspects of our collective

cavity-QED system. The next two chapters will finally focus on the microwave rotation aspects ofsqueezing experiment. In our experiments, a significant portion of our time was spent learning how

103
to perform precise rotations (Chapter 7) and developing a linear response theory for quantifying
the impact of phase noise on the rotations (Chapter 8). Precise rotations (at the 1 mrad level inthe polar angle θ) are necessary so that microwave π-pulse required for a Jz measurement does
not an amount of noise into the measurement larger than the SQL or squeezing we were aiming toobserve.

Chapter 7

Experiment: Realization of precise rotations

To first observe quantum projection noise and then demonstrate conditional squeezing in
Chapter 5, the microwave rotations must add little noise into Jz relative to the projection noiselevel. For our large ensemble of N = 7 × 105 atoms, the projection noise level sets the SQL of∆θSQL = 1.2 mrad. In order to measure below the SQL, the microwave rotations need to add lessthan ∼ 1 mrad of noise to the polar angle θ of the collective spin vector.
Precise rotations are key for the the squeezing experiment in Chapter 5, that is we can consistently rotate the collective spin vector to the same polar angle within a single experimental trial.
Added noise in the azimuthal angle φ due to microwave rotations much less of a concern because
φ is the anti-squeezing quadrature; adding noise into φ does not affect the squeezed quadrature.
Also, the mean spin length |hĴi| ∝ cos ∆φ is only sensitive to second order in ∆φ.
In Section 7.1, we demonstrate a tuning procedure to achieve good alignment of the rotation
axis to the collective spin. In Section 7.2, we present constraints on added-noise from microwave
rotations with the help of auxiliary rotations. Finally, in Section 7.3, we describe a microwave
source with sufficiently low noise to allow spin-squeezing on the 87 Rb clock transition. This source
has been published in Ref. [55].

7.1

Aligning rotation axis to collective spin vector
In order to prevent low frequency amplitude and detuning noise from coupling into themeasurement quadrature Jz , we need to know the rotation axis relative to the collective spin vector

105very well so that we can perform an aligned π-pulse or perpendicular π-pulse back through thesouth-pole as shown in Fig. 7.1. In our experiment, we find it necessary to tune the phase of the
microwave rotation pulse carefully to achieve quiet rotations due to phase shifts that occur during
the time between the CSS preparation to the π-pulse rotation necessary for a Jz measurement.
A source of phase shift is attributed to ac-Stark shifts from off-resonant microwaves. Dueto metallic structures around the atoms, finite extent of the atomic cloud and imperfections in
the microwave dipole antenna, the polarization of the microwave field at the atoms is not purely
π-polarized. Circularly polarized microwaves couple the |F, 0i clock states (F = 1, 2) off-resonantly
to the |F, ±1i states , giving rise to microwave ac-Stark shifts. At the applied quantization field of
4.8 Gauss, the Zeeman splitting is 3.4 MHz. The Rabi frequency for the clock transition is about
40 kHz. Assuming the amplitude of σ± -polarized microwaves are the same as the π-polarized microwaves, the ac-Stark shift from a pair of coupled levels is on the order of 120 Hz1 . When weapply the microwave rotations, the microwave frequency is set to the dressed microwave transition
frequency to avoid detuning errors coupling into the measurements. However, when the microwaves
are turned off, the collective spin precesses at a frequency not dressed by the microwaves, leading to relative phase accumulation between the collective spin and the microwave source. Thisaccumulated phase offset may be measured and tuned away in the procedure described in Fig. 7.2.
It is also important to make sure microwaves are completely turned off during the QND measurements. To do so, we have found it necessary to use two microwave TTL switches (Minicircuits
ZASWA-2-50DR+). With only one TTL switch at the input of the high power microwave amplifier, we measured a Rabi frequency of 48 Hz when the switch is in the “off” state, compared to a
Rabi frequency of 34.5 kHz in the “on” state, corresponding to an isolation of 57 dB, consistent
with network analyzer measurements of the switch isolation. This would cause a residual rotationof the collective spin vector of 300 mrad/ms in the worst case scenario where the rotation axis is
1

While ac-Stark shifts from the σ+ and σ− transitions would cancel due to their opposite signs of detuning fromthe clock transition, we do not have control over the microwave polarization and hence the Rabi frequencies andthe relative magnitudes of these ac-Stark shifts. In the worse case scenario, there is power in only one of the two
σ± -transitions, which would result in the full microwave ac-Stark shift.

106(a) Parallel- Rotation Axis
Composite /2-pulse:

MeasureMeasureMeasureMeasurezxy(b) Perpendicular- Rotation Axis
Plain /2-pulse:zxyFigure 7.1: The two rotation sequences used to observe projection noise mapped out on the Bloch
sphere. The collective spin vector (rotation) is shown in red (blue). To observe projection noisein the difference Ω↑ − Ω↓ , classical rotation noise added by the microwave π-pulse required in the
probing sequence needs to be smaller than the quantum noise we are trying to observe. Static
rotation errors would add an offset but not increase the noise. However, nothing is truly static.
Slowly varying rotation errors would show up as a varying offset, increasing the measured variance.
To mitigate slowly varying drifts in the Rabi frequency, we employed sequences that ensure thez-projection of the mean spin Jz is (a) zero to first order in the fractional Rabi frequency erroror (b) common-mode between the Ω↑ and Ω↓ measurements. Note that the quantum noise ∆Jzremains anti-correlated between the two measurements, as it should be. While slow amplitude
errors are cancelled by these rotations, fast amplitude errors would introduce uncorrelated classicalnoise in Jz . We empirically found that fast amplitude noise is not a limiting factor. To reduce theimpact of phase noise or equivalently detuning added during π-pulse, we used a low phase noise
microwave source, described in Section 7.3, to perform the rotations.

perpendicular to the collective spin vector. For the 150 µs time scale of the squeezing operation,
the collective spin vector would be tipped by 45 mrad, much larger than the SQL of ∼1 mrad
we aimed to observe. While a consistent rotation would not add noise in our measurements, only
a 2% fluctuation in the amount of microwave isolation would add noise equivalent to the SQL.
Therefore we took precautions to install a second microwave TTL switch. We constrain the final
microwave leakage by measuring how much the collective spin vector rotates versus hold time whenboth switches are in the “off” state to result in a Rabi frequency of 0.07(4) Hz, or a rotation rateof 0.5(2) mrad/ms and an angle of 0.07(3) mrad in 150 µs in the worst case. Note this constrain

(a) 1.0(b)

Effect of Phase-offset between Collective Spin and Rotation Axis

Phase-offset accumulated over time from Microwave Detuning

0.5
Change in Polar Angle [rad]Change in Polar Angle [rad]

0.5

107
1.0

0.0

z

-0.5

0.0

-0.5

y

Phase of rotation pulse
90 deg
1.5 deg-1.0

Microwave Frequency - 6834.691 MHz:
2 kHz
1 kHz
0 kHz

-1.0
0

10

20

30

40

50

60

70

0

10

Time Microwaves Applied [us]

20

30

40

50

60

70

Time Microwaves Applied [us]Figure 7.2: (a) Tuning phase offset between CSS and rotation axis. The CSS is prepared along
nominally x̂, with potentially a small offset along the z-axis due to microwave amplitude error(inset, black dot showing the tip of the CSS). A rotation through a variable angle is then applied
using resonant microwaves for a variable amount of time. The phase offset between the rotation
axis (inset, green dot showing tip of rotation axis) and the CSS sets the radius of the circle the tipof the CSS traces on the Bloch sphere (inset, black arrow). The spin projection Jz therefore varies
sinusoidally with the angle rotated through. By fitting the Jz oscillations (black, blue symbols)
to a sinusoid (black, blue curves), the phase offset between the rotation axis and the CSS can be
determined. Deflections more than ∼ 0.5 rad are not observed due to limited dynamic range of themeasurement. (b) Phase offset can also accumulate from microwave detuning during free evolution
time. Same experiment as (a) except that instead of changing the phase of the microwave pulse
performing the rotation, the phase of the pulse is kept fixed here. Here a phase offset is accumulated
from a microwave frequency detuning during a 80 µs free evolution window before the variablerotation was applied. The detuning was implemented by a (phase continuous) frequency offset of
the DDS frequency source that sets the microwave LO frequency. This shows the importance ofaccounting for the effect of frequency detuning during free evolution time causing phase offsets
between the rotation axis and the CSS.

provided by the atoms is consistent with the expected attenuation of 114 dB with two TTL switches.

7.2

Constraining noise from microwave rotations
A classical spin vector can exhibit fluctuations in the measured polar angle θ that arise due

to classical noise introduced during a rotation. In principle, one might mistake such classical fluctuations for the quantum fluctuations due to projection noise. To constrain the possible added-noisein θ due to rotations, we perform a set of auxiliary rotation and measurement sequences (summarized in Table 7.1), each with sensitivity to different types of errors in the rotation process.

108
Rotation-induced noise in θ is distinguished from projection noise by selecting rotations that nominally return all collective spin vectors to their original orientation before each measurement of avacuum Rabi splitting. The chosen rotations constrain the added noise from microwave amplitude
and phase noise, and transition frequency noise (arising from the trapping potential for instance.)
These noise sources are equivalent to fluctuations in the angle of rotation, and the axis of rotation.
For the two rotation sequences used to actually observe projection noise in Section 5.7 (see Table7.2), we estimate that the rotations contribute at most −14(3) dB of added noise in θ relative tothe predicted projection noise level for N = 7 × 105 atoms.
If the rotations are imperfect in length, then noise from the anti-squeezed quadrature can leakinto the measurement quadrature. Estimates of the imperfections in the rotation lengths constrain
this noise leakage to < −8 dB and < −35 dB for the two rotation sequences of Table 7.2.
In order to achieve these low noise rotations, we have built and characterized a custom low
phase noise microwave source, and will be described in Section. 7.3. Long term amplitude stability
was enhanced by installing microwave absorbing material near the atoms.

7.3

Low phase noise microwave source
In this Section, we provide an augmented version of Ref. [55], where we describe and charac-terize a simple, low cost, low phase noise microwave source that operates near 6.800 GHz for agile,coherent manipulation of ensembles of 87 Rb. Low phase noise is achieved by directly multiplying a
low phase noise 100 MHz crystal to 6.8 GHz using a non-linear transmission line and filtering theoutput with custom band-pass filters. The fixed frequency signal is single sideband modulated witha direct digital synthesis frequency source to provide the desired phase, amplitude, and frequency
control. Before modulation, the source has a single sideband phase noise near -140 dBc/Hz in therange of 10 kHz to 1 MHz offset from the carrier frequency and -130 dBc/Hz after modulation.
The resulting source is estimated to contribute added spin-noise variance 16 dB below the quantum
projection noise level during QND measurements of the clock transition in an ensemble 7×105 87 Rb
atoms.

109

Table 7.1: Auxiliary rotation and measurement sequences were used to constrain the magnitudeof the added noise ∆θ in the polar angle of the collective spin vector caused by imperfections in
the rotations. The added noise is expressed in dB relative to the predicted projection noise levelat N = 7 × 105 atoms. The atoms are initially prepared in state |↓i (θ = π). The rotations areparameterized as R [ψ, φ], where ψ specifies the angle of rotation about an axis, while φ specify
the azimuthal of the rotation axis in the x − y plane. The rotation axis is always in the x − yplane nominally. Each measurement N1,2 corresponds to a measurement of the number of atomin |↑i obtained by measuring the vacuum Rabi splitting. The fluctuations in the difference of themeasurements N2 − N1 provides a measure of the fluctuations ∆θ. The rotation noise sources
constrained by each sequence are given in the second column, expressed to leading order in thesmall fluctuating parameters i . The amplitude fluctuations i are re-expressed in terms of either
differential − or common-mode + amplitude fluctuations. The impact of microwave phase noise
and atomic transition noise (for instance, due to power fluctuations in the trapping optical lattice)
can be constrained on both fast and slow time scales.
Auxiliary
Sequence
R [π/2, 0]
measure N1
R [2π, π/2]
measure N2
R [π/2, 0]
measure N1
R [π, π/2]
R [π, −π/2]
measure N2
R [(π/2)(1 + 1 ), 0]
measure N1
R [π (1 + 2 ) , 0]R [π (1 + 3 ) , −π]measure N2
R [(π/2)(1 + 1 ), 0]
measure N1
R [π (1 + 2 ) , 0]R [π (1 + 3 ) , 0]
measure N2∆θfast transition
frequency noise,
microwave phase noise

Measured∆θ2 [dB]−16(6)

slow and fast transition
frequency noise,
microwave phase noise-14(3)π−
− = 3 − 2

−16(3)2π+2 , 3 ≈ +

+4(1)

110

Table 7.2: The added noise from rotations for the two pulse sequences used to observe projection
noise. Two independent rotation sequences were used to observe projection noise–each with different
sensitivities to the noise sources measured with the auxiliary rotations and measurements of Table
7.1. The predicted added noise from each source is expressed in dB relative to the calculated
projection noise for N = 7 × 105 atoms. The phases of the microwave pulses were adjusted suchthat φ1,2,3 < 35 mrad or 2◦ .
Measurement Sequence
R [(π/2) (1 + 2 ) , 0]
R [(π/2) (1 + 1 ) , π/2 + φ2 ]measure N1
R [π (1 + 3 ) , π/2 + φ3 ]measure N2
R [(π/2) (1 + 1 ) , 0]
measure N1
R [π (1 + 2 ) , π + φ2 ]measure N2

Amplitude Noise
∆θ
[dB]
2
2
π + + π+ (φ3 − φ2 ) < −30

Phase Noise[dB]-17(3)

Transition Noise[dB]-20(3)

< −28

< −34

1 , 2 , 3 ≈ +

π 2
2 φ2 + + π−− = 2 − 1

1 ≈ +−14(3)

111
7.3.1

Context

Low phase noise microwave sources are crucial for a broad range of applications, including
radar, communications, navigation, and timing-keeping [142]. For quantum sensors [143, 144],
quantum gates [145, 146], and tests of fundamental physics [147, 148], microwaves are often applied
to perform rotations on pseudo-spins formed from two atomic hyperfine energy levels separated byspin-flip resonant frequencies in the microwave regime.
In many atomic physics applications, the microwave source can be narrowband in its tuning
range, since it is being used to address extremely stable atomic transitions. However, the sourcemust have nimble phase control in order to implement composite pulse sequences [149–151] thatare used to cancel slowly varying amplitude and detuning errors. When resonant microwaves areapplied at the spin-flip frequency, one can view the pseudo-spin as being rotated about a fixed axis
in the x − y plane, with the rotation axis’s azimuthal angle set by the phase of the microwaves.
Composite pulses reduce the impact of amplitude and/or detuning errors for a single rotation aboutan axis by implementing several rotations about several axes with different azimuthal angles. Thus,
it is necessary to be able to quickly change the microwave phase between several values. Conversely,
if the microwave phase randomly fluctuates during the rotation, the orientation of the pseudo-spinat the end of the rotation may fluctuate from one realization to the next, swamping quantum
sources of orientation noise, such as quantum projection noise, that are of fundamental interest [57,58].
Here we present the details of a custom, low phase noise microwave source near 6.8 GHz,
with tuning bandwidth of approximately 40 MHz, where rapid changes of phase, frequency, andamplitude are possible. The 40 MHz tuning range is sufficient to span hyperfine transitions in the
widely used 87 Rb atom, and only minor modifications are needed for operation with other commonlyused alkali-atoms such as 133 Cs, 85 Rb, etc. In the range of 10 to 100 kHz offset frequencies fromthe carrier, the single sideband phase noise of the source is nearly 20 dB lower than a commonlyused YIG oscillator phase locked to the same crystal reference.

112
The offset frequency range 10 to 100 kHz is particularly relevant for our spin-squeezing experiments. In particular, during a crucial π-rotation one finds that only phase noise spectral components
near the Rabi-frequency fR ≈ 40 kHz contribute significant noise into Jz (see Chapter 8), whose
value must be resolved below the projection noise level.
Compared to other microwave frequency sources with the necessary stability and agility,
the source we present is also a cost-efficient option. The total component cost is below $3, 300,
when the Rb atomic reference is excluded. The source achieves lower phase noise than manybroadband commercial microwave synthesizers, such as the Agilent E8257D, which is also roughly
an order of magnitude more expensive. Other narrow-band microwave oscillators, such as cryogenic
sapphire microwave oscillators [152], achieve much lower phase noise, but again at much higher
cost and limited availability. Related low phase noise microwave sources based on a dielectricresonator oscillator (DRO) phase locked with a few 100 kHz bandwidth to a non-linear transmission
line (NLTL) frequency comb have been reported at the 133 Cs 9.2 GHz [153, 154] and the 87 Rb
6.8 GHz [155] hyperfine transition frequencies.

7.3.2

Low noise, fixed frequency source

The microwaves are generated by first multiplying up a low phase noise crystal source togenerate low noise microwaves at 6.800 GHz and then performing single sideband modulation of
the carrier using a direct digital synthesis board as the modulation source (Fig. 7.3).
A low noise 100 MHz crystal oscillator (Wenzel ULN 501-16843) is phase locked to a 10 MHz
Rb clock signal (Stanford Research Systems FS725) for long term frequency stability, see Figs. 7.4
and 7.5. The phase comparison between the 100 MHz crystal and the 10 MHz Rb clock is performedby first mixing with a 110 MHz derived from the 10 MHz Rb clock with a x11 multiplier (Wenzel
LNOM-10-11-13-10-AA-AA), and then mixing the resulting 10 MHz signal with 10 MHz Rb clock
signal. The error signal from the output of the last 10 MHz mixing stage is sent to a loop filter
that feedback on the 100 MHz crystal with servo unity gain frequency of 100 Hz, with schematicshown in Fig. 7.5.

GHz Frequency Comb

113

Single Comb Tooth 6.8 GHz Frequency Filter

Wenzel ULN 100 MHz
Crystal Oscillator ZFDC-10-5

LPN7110-SMT
Nonlinear Transmission LineVLF-6700

ZFHP-2100-S

VLF-6700

PLL
fu =100 Hz

SRS FS725
Rb Atomic Clock

HELA-10D

0.267”

BLP-100 BHP-50
HELA-10D

0.865”

VHF-6010

0.192”
Low Loss Cable
CA400

Stub-Tuned,
Dipole Antenna

ZX60-1402
Filter A

MC

L0607-38 HighMB

LO

Filter B

Q

MA

90

I
IF
HMC496LP3
Single Sideband, Carrier
Suppressed Modulation

SBW-5089(Z)
RFMD Amplifier

AD9959Direct Digital Synthesis

Figure 7.3: Schematic of low phase noise microwave synthesis chain. The source starts with a
100 MHz crystal oscillator phase locked to a 10 MHz atomic lock to provide very low frequency
stability. The 100 MHz signal, after coupling off a monitor output, is amplified and multiplied up
through a NLTL. The 68th harmonic is selected by a custom bandpass filter built from a seriesof commercial filters and homemade microwave resonators. The resulting NLTL source is a single
harmonic that provides the LO input for a single sideband modulator. The I and Q inputs of themodulator are driven by a direct digital synthesis (DDS) board that allows phase coherent control
over the frequency, amplitude, and phase of the microwave signal sent to the high power microwave
amplifier. The microwaves are coupled to the atoms using a printed circuit board dipole antenna.
The signal line on the antenna is in red, and the ground plane, on the opposite side of the board,is in blue. The labels MA , MB and MC indicates points where phase noise was measured.

The 100 MHz signal is amplified to +27 dBm (Minicircuits HELA-10D) to drive a NLTL
(Picosecond Pulse Labs LPN7110-SMT) generating a comb of harmonics out to 20 GHz. For
noiseless multiplication, the timing jitter is preserved, but the phase noise is increased relative tothe source by 20 log10 (n) dB, where n is the harmonic order. We utilize the n=68 harmonic at
6.8 GHz yielding a fundamental phase noise increase of 36.7 dB over the crystal source. Note thatthe frequency comb produced by the NLTL source allows for the flexibility to choose a frequency
relevant for other alkali atoms by choosing a different harmonic to filter.
The harmonic at 6.8 GHz is isolated via a combination of standard filters (Minicircuits VHF6010, VLF-6700, ZFHP-2100-S) for broadband filtering and two high quality factor (Q) microwave
resonators [156] for narrow-band filtering. At 6.8 GHz, the measured insertion losses of the VHF6010, VLF-6700, and ZFHP-2100-S filters are 0.7, 0.8, and 0.3 dB respectively. The insertion losses
are low enough that the thermal noise floor of -174 dBm/Hz at 300 K does not affect the phasenoise of the synthesis chain. The ZFHP-2100-S, nominally a high pass filter with 3 dB point at
2.1 GHz, was used to attenuate high frequency harmonics above 14 GHz. The VHF-6010 high pass

Rb ClockSRS FS725

LNOM-10-11-13-10-AA-AA

LO

x11

110 MHz

10 MHz114

100 MHz
Wenzel Sprinter VLN
501-04517D

Isolation
Amplifier

Mixer
ZAD-1

Wenzel MultiplierRF
100 MHz,IF -2.5 dBm

10 M Hz, 210 MHz

Isolation
Amplifier

PLL fug = 100 Hz

Low Pass Filter
BLP-21.4
10 MHz, -4 dBm

RF

10 MHz LOIF
Mixer
ZRPD-1Isolation=
Amplifier(91 dB isolation)-20 dB
Amplifier ZFL-1000LN+24dB gain, +15V p.s.

Figure 7.4: Wenzel crystal lock schematic.
Loop Filter for Wenzel 100MHz to Rb Clock 10MHz Lock
fug = 100 Hz

Wenzel 100 MHz
converted to
10 MHz, -4 dBm

100 MHz
Wenzel Sprinter VLN
501-04517D

50 kΩ 150 nF

RF
260 Ω

Rb Clock
10 MHz LO-13.70(6) Hz/V
Output range ±4.5V

21 Hz PI Corner

750 Ω

IF

-

2 kΩ

Vtune

LT1028Mixer
ZRPD-1
0.5 V/rad, 1 Vpp
on scope (1 MΩ)

6.8nF 680pF

68pF

110 kHz low pass

+

1 kΩ

150 nF

1.6 kHz low pass

230 kΩ

3.8 nF

180 Hz modulation BW

Figure 7.5: Wenzel crystal loop filter with unity gain BW ∼100 Hz.

3 dB point is at 6.01 GHz, and the VLF-6700 low pass 3 dB point is at 7.6 GHz. At an insertion
loss of IL = 2.8(2) dB, the loaded quality factor is QL ≈ 580, and the loaded half width at half
maximum is ≈ 6 MHz, much smaller than the 100 MHz spacing between the comb harmonics.

The microwave resonators have an unloaded Q0 = QL / 1 − 10−IL/20 ≈ 2100. Two microwave
resonators are used to give strong suppression of the harmonics close to the harmonic at 6.8 GHz.

115
The resonators, illustrated in Fig. 7.6, are constructed from stock copper pipe (inner diameterof 1”, outer diameter of 1.5”), with a lid 0.25” thick made from copper barstock and a base fromstock 0.125” copper sheet. The three pieces are held together by screws that allow for a tight fit
and good conductivity, but still provide the ability for the resonator to be disassembled. We foundno major improvements to the unloaded resonator Q by soldering the pieces together.
The coupling of microwaves into and out of the resonator is provided by two panel mount
SMA connector feedthroughs held to the copper sheet with a lock washer and nut. Trimming theinternal signal pin lengths increased the Q at the expense of increasing the insertion loss. For
example, pins that had been trimmed to be 3 mm above the base plate had an insertion loss of 5.2
dB, but trimming down to 2 mm above the base plate resulted in an insertion loss of 18.9 dB. The
filters used for the final NLTL source were trimmed to be 4 mm above the base plate.
Bottom View

4-40 Brass
Tuning Screw

Sideview

Solder to tune frequency

0.89”
Trimmed to 4mm

0.50”
1.00”

SMA Panelmount feedthroughs

Figure 7.6: Diagram of the high-Q resonant microwave filters. The resonant cavity is constructedfrom a copper pipe, with copper plate used to form the lid and base. The cavity resonance frequency
is tuned with a 4-40 brass screw, secured tightly for good conductivity by a nut on either side of thecenter of the lid, and more finely tuned through a small amount of lead solder attached to the tip of
the screw (see text). The signal is injected and extracted through panel mount SMA feedthroughs,
trimmed to provide 2.8 dB of insertion loss while maintaining a loaded Q of about 580.

Coarse tuning of the resonant frequency was performed by changing the depth of a brassscrew inserted in the center of the lid. We found that good conductivity between the screw andthe lid was one of the most important parameters for obtaining the maximum Q. The Q was also
reduced when using a stainless steel tuning screw in place of the brass screw. After coarse tuning
the resonance frequency to within 20 MHz of the desired 6.800 GHz, two brass nuts were placed

116
on either side of the screw to firmly secure it in place. The final fine-tuning of the center frequency
was performed by attaching a small amount of Sn0.6 Pb0.4 solder to the tip of the tuning screw,
then successively sanding small portions of the solder away until the resonator reached the desired
frequency. Tuning resolution below 2 MHz was possible with this approach.
The center frequency of the two resonators were set to 6.800(2) GHz. An amplifier (Minicircuits ZX60-1402) was placed between the resonators to prevent the formation of a coupledresonance. After filtering, the largest unwanted comb tooth is at 6.7 GHz, −49.5 dB below thedesired 6.8 GHz comb tooth. The resonance frequencies can drift due to temperature changes
with a coefficient of ∼ 150 kHz/K. Mechanical vibrations can modulate the resonance frequency,
giving rise to a phase modulation of the transmitted microwaves to lowest order. To mitigate bothtemperature drifts and vibration-induced phase modulation, we passively isolate the resonators ina foam box.
The single sideband phase noise power spectrum of the 6.8 GHz NLTL source was measured bytransporting the source to the National Institute of Standards and Technology (NIST) at Boulder,
CO, and using a self-homodyne phase noise detector (OEwaves OE8000). Fig. 7.8a shows themeasured noise of our NLTL source at measurement point MA in Fig. 7.3 (after filtering, but
before modulation), a YIG oscillator (Micro Lambda Wireless MLPE-1290) phase locked to thesame ULN crystal reference, and the measurement noise floor of the OE8000 calibrated using a
Poseidon sapphire oscillator at 10 GHz. Our ability to measure phase noise is limited by themeasurement noise floor below ∼ 500 Hz offset frequencies, thus the displayed data is an upperbound for the phase noise below ∼ 500 Hz.
The NLTL source is quieter than the YIG source by ∼ 15 to 20 dB in the offset frequency
range from 10 kHz to 400 kHz. The multiplied-up phase noise of the 100 MHz ULN crystal takenfrom the manufacturer’s datasheet agrees relatively well with the observed data, indicating that
there is little added technical noise at this level in the multiplication, amplification, and filtering
relative to the observed noise level. The 100 MHz ULN phase noise slope changes from 1/f 2 above
1 kHz to 1/f 3 below 1 kHz due to 1/f phase noise from the crystal oscillator feedback amplifier

117
being converted into oscillator phase noise via the Leeson effect [157]. The roll-off of the phasenoise near 5 MHz is believed to arise from the high-Q microwave filters.

7.3.3

Agile control of Phase, Frequency and Amplitude

To generate amplitude, frequency and phase tunable microwaves at the hyperfine clock transition frequency 6.834693 GHz, a single sideband (SSB) modulator (Hittite HMC496LP3), with amanufacturer specified output noise floor of −150 dBm/Hz at 7 GHz, modulates the 6.8 GHz NLTL
source. The SSB modulator is tuned to give −31 dB and −19 dB suppression relative to the +1
sideband for the 0, −1 and −2, +2 sidebands respectively.
The I and Q modulation ports are driven near 34 MHz using 90◦ relative phase signals from
2 of the 4 channels of a direct digital synthesis (DDS) frequency source (Analog Devices AD9959
evaluation board) operating near 34 MHz. The DDS source is phase stabilized to a 500 MHz
reference provided by multiplying the 100 MHz ULN crystal oscillator using a diode pair (Avago
HSMP-3822) and harmonic filtering provided by two bandpass filters (Minicircuits BPF-B503+),
see Fig. 7.7. The DDS SSB phase noise floor specified by the manufacturer is −151 dBc/Hz at
10 kHz offset from the carrier when operated at 40 MHz with a 500 MHz reference signal, but
this specified noise floor rises if the DDS is operated with a lower frequency reference, relying on a
programmable internal PLL to multiply up the reference signal to 500 MHz.
Power Splitter
ZFSC-6-110

500 MHzDDS Clocks
Amplifier
ZKL-1R5+13dBm

100 MHz
Wenzel Sprinter VLN
501-04517D

x5 Multiplier

500 MHz Bandpass
BPF-B503

Amplifier
ZKL-1R5750 MHz Low Pass
BLP-750-26dBm

6 dB

Diode-pair
Avago HSMP-3822

Power Splitter
ZFSC-2-1W-75
400 MHz High Pass
BHP-400
500 MHzDDS Clocks

Power Splitter
ZFSC-6-110

Figure 7.7: DDS 500 MHz clock generation via x5 multiplication of a ULN 100 MHz signal.

Single Sideband Phase Noise [dBc/Hz]

aSingle Sideband Phase Noise [dBc/Hz]

b

118
-60
YIG
NLTL Source
NLTL Source (predicted)
Measurement Noise Floor

-70
-80
-90
-100
-110
-120
-130
-140
-150
-160
-170
-180
1
10

10

2

10

3

10

4

10

5

10

6

10

7

10

8

Offset Frequency from Carrier [Hz]-60
-70

NLTL Source + SSB Modulator
NLTL Source
DDS (Specified)

-80
-90
-100
-110
-120
-130
-140
-150
-160
-170
1
10

10

2

10

3

10

4

10

5

10

6

10

7

10

8

Offset Frequency from Carrier [Hz]Figure 7.8: Single sideband phase noise of several microwave sources all locked to or derived fromthe Wenzel ULN oscillator. The measurement noise floor is in grey. a Comparison of noise at
6.800 GHz for a YIG oscillator (green) and our microwave NLTL source (black) measured atpoint MA in Fig. 7.3. The noise in the NLTL source agrees with the prediction (red circles withdashed line to guide the eye) based on ideal scaling of the noise in the ULN oscillator specified
by the manufacturer. b The modulated signal at 6.834 GHz (orange), measured at point MB
in Fig. 7.3, exhibits higher phase noise than the 6.8 GHz signal from the NLTL source (black).
The manufacturer specified noise of the DDS (blue circles with dashed line) when operated with aninternal PLL of ×5 is plotted to show that the added noise primarily comes from the SSB modulator,
excluding frequencies above 2 MHz where the gain peak from the internal PLL dominates. Spurious
multiples of 60 Hz have been removed for presentation purposes.

We control the DDS source through a custom LabVIEW interface2 that can update thephase, frequency and amplitude in 24, 32, and 28 µs respectively. The tuning resolutions are
0.022◦ , 0.12 Hz, and 0.1% of full scale amplitude respectively. Phase continuous frequency ramps
and amplitude ramps are also possible which enable additional techniques, such as atomic popu2

We thank Hsiang-Sheng Ku for programming the DDS LabVIEW control.

119
lation manipulation using Landau-Zener avoided crossings. The LabVIEW interface utilizes 4-bit
serial communication at 2 MHz, while the DDS is capable of communication at 125 MHz. The
communication rate is presently limited by the speed of the National Instruments DIO card (National Instruments PCIe-6259) used to implement the serial communication. If less flexibility or lessintegration with the data acquisition system is required, the DDS can also be programed through amanufacturer-provided USB interface and software, while still maintaining the ability to modulate
a single parameter between as many as 16 programmable values.
The SSB modulator output is amplified to as much as 8.9 Watts using a high power microwave
amplifier (Microwave Power (now Microsemi-RFIS) L0607-38). Measurements with the OE8000 at
measurement point MC in Fig. 7.3 show that the microwave amplifier adds a negligible amountof phase noise relative to the measured phase noise of the NLTL source derived from the ULN
crystal shown in Fig. 7.8a. The microwave radiation is coupled to the atoms through a low loss
1.5 m cable (CA400) and stub-tuned dipole antenna [158] fabricated directly on a high frequency
laminate circuit board (Rogers RO3035). The critical dimensions for the frequency tuning of theantenna are labeled in Fig. 7.3. The antenna is relatively broadband, with a FWHM of 400 MHz.
After careful alignment of the dipole antenna, we achieved a Rabi frequency of fR ≈ 40 kHz withthe antenna approximately 1.5” or 4 cm from the atoms 3 . When the microwaves are launched
into free space using a dipole antenna, it is possible that the phase noise of the microwave field at
the position of the atoms is degraded due to reflections off of nearby surfaces. One expects thatsuch effects should have negligible impact on Q0 since little mechanical motion is expected near
the microwave Rabi frequency of fR ∼40 kHz.
The SSB modulator adds 5 to 8 dB of phase noise relative to the NLTL source, as seen in
Fig. 7.8b where the measurement is taken at point MB in Fig. 7.3. For frequencies below 2 MHz, we
attribute the added noise to the SSB modulator. This data was taken with the the DDS referenced
3

As a side note, microwave absorbing material (Laird RFLS Single Layer sheets, 0.25 inch thick, measured
attenuation of 10 dB) around the glass cell vacuum chamber was necessary to reduce observed changes in the Rabi
frequency of greater than 10% peak to peak caused by people walking past the optics table. With the shielding, thefluctuations are reduced to ∼1.3% peak-to-peak with people walking around and ∼0.7% with people staying still.

120
to the 100 MHz Wenzel ULN crystal directly, and multiplication to 500 MHz being accomplishedusing the internal DDS PLL multiplier set to ×5. In this configuration, the phase noise of the
DDS specified by the manufacturer, and confirmed by our own measurements, only has a small
contribution at offset frequencies less than 2 MHz where an expected noise peak from the DDS
PLL becomes visible. This noise peak was present for the measurements in Fig. 7.8b, but is notpresent for the final source that has a DDS reference clock of 500 MHz obtained via external ×5
multiplication as described above.

7.3.4

Impact on QND measurements

To achieve conditional spin squeezing using QND measurements of the atomic spin projection
Jz as described in Chapter 5, it was necessary to apply a π-rotation using a microwave source. Using
the YIG PLL microwave source locked directly to the 10 MHz Rb atomic clock for the π-rotation,
we observed added noise in the spin projection Jz that was 18 dB above the quantum projection
noise level for the 7 × 105 atoms. From manufacturer specifications, the multiplied phase noise of
the 10 MHz crystal is -95 dBc/Hz, for the relevant offset frequencies near fR .
Utilizing the NLTL source locked to a slightly noisier 100 MHz crystal oscillator (Wenzel
Sprinter 501-04517D) with the SSB modulation (phase noise measurements taken but not shown),
the predicted added noise noise in Jz due to phase noise during the π-rotation is predicted to be
16 dB below the projection noise level. The prediction is made by simple scaling from the firstmeasurement with the much noisier YIG oscillator. By reducing the added noise from microwave
rotations, we lowered the technical noise floor of the measurement and were able to observe a 5 dB
reduction in spin noise, limited by our probe quantum backaction. By changing the 100 MHz
Sprinter oscillator to the ULN model and improving the added phase noise of the SSB modulator,
future experiments could operate with added microwave phase noise to Jz that is 27 dB below the
projection noise limit for this large ensemble size. Lower microwave phase noise can be exploited toperform composite microwave pulses that would add more noise into Jz than the plain π-rotation
discussed here.

121
In conclusion, we have demonstrated a microwave frequency source with low phase noisethat maintains the flexibility to adjust phase, frequency, and amplitude needed in high precisionexperiments with large atomic ensembles. The source is very cost-effective, providing a phasestability that surpasses many commercial frequency sources for a fraction of the cost. Although
the source is narrowband, the center frequency can be chosen for a particular need or atom that
requires a stable microwave frequency, providing an attractive option for many applications.

Chapter 8Theory: Impact Phase Noise on Bloch Vector Rotations

In Chapter 7, we presented precise rotations in the presence of microwave amplitude, detuning
and phase errors that are static within a single experimental trial but could slowly drift from trial
to trial. We could have used composite π-pulses instead of plain π-pulses to suppress static errors
but we did not. Indeed, composite pulses have been developed to cancel static rotation errors toany degree one desired [159].
In all treatments of composite pulses, these errors are assumed to be static for the durationof the pulse sequence. For our system, this is not a good assumption, even for a single rotation.
Fluctuating phase or equivalent detuning errors, i.e. phase/detuning noise, near the Rabi frequencyΩR during the rotations can add noise to Jz . The phase-noise-induced fluctuations of the collective
spin polar angle prevented us from observing the SQL until we improved the phase noise of our
microwave source, which was presented in Chapter 7. Even with the low phase noise source, added
noise from phase noise can accumulate from many single rotations, potentially offsetting the benefitsof static error cancellation from composite pulses.
In this Chapter, we quantify on the effect of phase noise on a single spin-1/2 or qubit,
conveniently represented by a Bloch vector. Since the phase noise acts identically on each individual
spin making up a CSS, the effect on the polar angle of a single spin in the ensemble is the same as theeffect on the CSS polar angle. We present a general framework for calculating the impact of phasenoise during a rotation on a Bloch vector. The analysis applies to any Bloch vector orientation, andany rotation axis azimuthal angle. The theory is experimentally verified for several special cases.

123
We then extend the analysis include pulse sequences assuming a white phase noise spectrum.
Finally, we calculate the sensitivity of commonly used composite π-pulse sequences: CORPSE,SCROFULOUS, and BB1, used to suppress static amplitude and detuning errors respectively,
and also to spin echo sequences. We expect the formalism presented in this chapter to guide thedevelopment and evaluation of future quantum manipulation protocols. For example, an optical
spectrum analyzer whose noise floor is limited by atomic quantum projection noise has been realized
recently [160]. The material presented in this Chapter has been published in [56].

8.1

Introduction
Atomic quantum sensors and tests of fundamental physics commonly rely on the ability torotate a Bloch vector representing a spin- 12 system or qubit. Besides quantum state manipulation,
rotations can also be used to undo inhomogeneous errors or to reduce other sources of noise, as
is done with spin echo pulses or dynamical decoupling [124, 161–166]. Precise rotations, required
for manipulating collective spin-squeezed states, might also enable dynamical spin-squeezing to the
two-axis squeezing limit [167] using already realized one-axis twisting in cold atoms systems [73,74, 82, 89]. Given that actual rotations are imperfect, it is an open question whether such rotation
protocols can be realized without adding large amounts of additional noise, thus destroying thesqueezing.
Most rotation protocols assume that the phase of the field that rotates the qubit is perfectly
stable, and that imperfections arise only due to slowly varying amplitude errors or detuning errors.
Composite rotations sequences [149–151, 159, 168] and generalizations to shaped pulses [169, 170],
including optimal control theory [171–173], can be used to reduce these errors to essentially arbitraryorder.
In reality, the phase of the qubit-field coupling is never perfectly stable, largely due to phasenoise in the local oscillator (LO) used to generate the field. The LO is typically a radio or microwave
oscillator in nuclear spin, superconducting Josephson junction, quantum dot, neutral atom/ion
Zeeman and hyperfine qubit systems [3]. In the case of highly-forbidden optical transitions [174], the124LO is an ultra-stable laser. Further, qubit transition frequency fluctuations can be straightforwardly
mapped onto an equivalent phase noise of the LO. Such a fluctuation might arise due to noise in
the DC bias current of a superconducting Josephson junction qubit [175], or differential light shifts
for atomic qubits in an optical trap [174]. It is critical for future work beyond proof-of-principle
experiments to develop general tools for analyzing the impact of phase noise on a rotation, both interms of overall fidelity for quantum gates, and quadrature specific noise for manipulating stateswith anisotropic sensitivity to noise, such as spin-squeezed states, and Dicke states [70].
The effect of phase noise on atomic response has been studied in various contexts, for example
on atomic excitation probability [176–179] under continuous drive. In the atomic sensor community,
the impact of phase noise is evaluated for a single quadrature for specific Ramsey sequences [142,180, 181]. In contrast, we present a general framework that may be applied to an arbitrary LO phasenoise spectrum for continuous resonant drive without making assumptions about the orientation of
the Bloch vector. The framework can be extended to arbitrary resonant pulse sequences assuming
white LO phase noise. Furthermore, we fully specify all second order noise moments of the Blochvector including covariances and variances, important for predicting the fidelity of single qubit
gates and manipulations of spin-squeezed or Dicke states. The methodology and tools presented
in this Chapter can help guide the development and evaluation of future quantum control andmeasurement protocols.
This Chapter is organized as follows. In Sec. 8.2, we describe the qubit-field interaction as arotation of a Bloch vector. We show that the net effect of a coherently phase modulated rotation
can be reduced to a small rigid rotation of the Bloch sphere. A description of the experimental
system used to demonstrate the theory is provided in Sec. 8.3. Experimental examples of the
response of the Bloch vector driven by phase modulated rotations are then presented. In Sec. 8.4,
we extend the analysis in Sec. 8.2 assuming linear response to relate the single sideband (SSB)
LO phase noise to Bloch vector noise projections through a covariance transfer matrix for a single
rotation. Experimental realizations for a few special cases are also presented. The Bloch vector noise
projection variances and covariances are captured in the covariance noise matrix. In Sec. 8.5, we125
generalize the single-rotation covariance noise matrix to that for multiple rotations/pulse sequences
and apply the tools to commonly used composite π-pulse sequences: CORPSE, SCROFULOUS,
and BB1 [149–151, 159, 168], and also to spin echo pulse sequences. A simple formula for the
average infidelity of any pulse sequence, appropriate in the context of quantum computing, is alsopresented. Finally, we give a summary and provide an outlook of future work in Sec. 8.6.

8.2

Deflection of Bloch Vector due to coherently phase modulated rotation

8.2.1

System Hamiltonian

The system considered in this Chapter consists of a spin-1/2, or equivalently a qubit, withtransition frequency fa coupled to a resonant classical electromagnetic field with strength characterized by the Rabi frequency fR proportional to the field amplitude. The system’s Hamiltonianin the lab frame is
Hlab = hfa σ̂z + hfR cos (2πfa t + φ(t)) σ̂x ,(8.1)

where h is the Planck constant, φ(t) is the LO phase as a function of time t, and σ̂x , σ̂y , σ̂z are
the Pauli matrices. The first term corresponds to the Hamiltonian of the qubit in the absence ofthe field, and the second term corresponds to the qubit-field interaction that drives Rabi flopping
between the qubit states |↑i and |↓i. By going into a rotating frame at the qubit transition frequency,
and making the rotating wave approximation, the dynamics is described by the Hamiltonian
Hrot =

8.2.2

hfR(cos(φ(t))σ̂x + sin(φ(t)) σ̂y ) .
2

(8.2)

LO Phase Noise

The LO phase φ(t) is random as a function of time, and its statistical properties can be captured by the autocorrelation function hφ(t)φ(t + τ )it in the time domain, or equivalently the power
spectral density of phase fluctuations Sφ (fm ) in the frequency domain, where fm is the frequency
offset from the LO carrier frequency fLO . The power spectral density of phase fluctuations Sφ (fm )is related to the autocorrelation function through the Wiener-Khinchin theorem (see Appendix D).

126
The SSB phase noise L(fm ) = Sφ (fm )/2 of a LO is usually specified in manufacturer datasheets
instead of Sφ (fm ). To facilitate application of our results in an experimental context, the SSB
phase noise L(fm ) is used throughout this Chapter.
The phase noise spectrum of a LO is typically parametrized as a sum of 1/f type phase noise
as follows∞
X
Lk
L(fm ) =
,
k
fm

(8.3)

k=−∞

k . Anwhere the coefficient Lk characterizes the strength of the noise spectrum with dependence 1/fm

important example is white phase noise where there is no frequency dependence. In this case, theonly non-zero coefficient is for k = 0 corresponding to L(fm ) = L◦ , and the autocorrelation function
is given by hφ(t)φ(t + τ )it = L◦ δ(τ ) where δ(τ ) is the delta function. Other frequently encountered2.
phase noise spectra include flicker noise L(fm ) = 1/fm and phase diffusion noise L(fm ) = 1/fm

8.2.3

Bloch Sphere Picture

Since length of the Bloch vector remains the same to second order in the noise, in this Chapter
we normalize the length of the Bloch vector to 1 by re-defining the components of the Bloch vector
Jk ≡ hσ̂k i where k = {x, y, z}. The tip of the Bloch vector thus resides on a Bloch sphere withradius equal to 1 (see Fig. 8.1a). The LO phase φ(t) sets the instantaneous rotation axis aroundwhich the Bloch vector rigidly rotates. For fixed LO phase φ(t) = φR , an initial Bloch vector Ji is
mapped to an ideal final Bloch vector J◦f as
J◦f = R(φR , ψ)Ji ,(8.4)

where R(φR , ψ) performs a counterclockwise rotation through an angle ψ ≥ 0 about an axis in thex − y plane with azimuthal angle φR measured relative to x̂. The rotation angle ψ = 2πfR t is set
by the Rabi frequency fR and the amount of time t the field is applied.

z

(a)

(b)

z

y
x

127

y
x

Figure 8.1: (a) Geometrical representation of the Bloch vector and a coherently phase modulated
rotation on the Bloch sphere. Initial Bloch vector Ji is specified by polar angle θi and azimuthal
angle φi . In the absence of phase modulation, the rotation axis is at a constant angle φR from thex-axis. With phase modulation, the rotation axis oscillates in the x − y plane with amplitude βaround its average position. (b) Effect of a coherently phase modulated rotation. An unmodulated
rotation R(φR , ψ) rotates Ji to J◦f . With phase modulation of the rotation axis, the final Blochvector Jf is deflected slightly by the vector jf that can be regarded as arising from a small rotation
r.

8.2.4

Small Rotation describing Phase Modulated Rotation

Our goal is to generate a general covariance transfer matrix that maps LO phase noise onto
noise projections of the final Bloch vector. To do so, we consider here the response of the Blochvector to a coherent phase modulation of the LO phase in the frequency domain, and generalize to aphase noise process in Sec. 8.4. A sinusoidal modulation of the LO phase φ(t) = φR + β sin(2πfm t +αm ) represents an oscillation of the instantaneous rotation axis in the x − y plane with amplitude
β about its average azimuthal angle φR as illustrated in Fig. 8.1a. The modulation frequency andphase is fm and αm respectively.
A phase modulated rotation causes a deflection of the final vector Jf from its ideal final
orientation without modulation by
jf (φR , ψ, β, fm , αm ) = Jf − J◦f .

(8.5)

To first order in the phase modulation amplitude β  1, the deflection is perpendicular to J◦f . The

128
total effect of the modulation can be described by an additional small rotation r(φR , ψ, β, fm , αm )applied after the ideal rotation such that
Jf = rJ◦f ,(8.6)

where labels on r have been suppressed. The relationships between the vectors Ji , J◦f , Jf , jf , andthe rotations R(φR , ψ), r are depicted in Fig. 8.1b.
Because the dynamics with modulation can be described in terms of infinitesimal rigid rotations, the whole Bloch sphere is rigidly rotated by this additional phase modulation contribution,and r does not depend on Ji . This is confirmed by solving for r analytically. Simple rotations
about ẑ through angle φR relate the small rotation r evaluated at φR = 0 to the small rotation
evaluated at arbitrary φR throughr(φR ) = Rz (φR )r(0)Rz (−φR ) .

(8.7)

Therefore, it is sufficient to determine r for the special case φR = 0 and Ji = x̂. Hereafter, quantities
with a tilde (∼) overhead apply to this special case only. We find


jy −e
jz 
 1 −e
r(0) = Ry (−ejz )Rz (ejy ) = e
,
1
0 
jy



e
jz
0
1

(8.8)

where ejf = (0, ejy , ejz ), and only first order in the small quantities ejy and e
jz are retained.
8.2.5

Solution for Deflection Vector

We now solve for the ejy and ejz that determine the small rotation matrix r defined by Eq. (8.7)
and (8.8). Writing the Heisenberg equations of motion for the Bloch vector components yields tofirst order in the small modulation amplitude
de
j⊥
+ ıe
j⊥ = −β sin (νψ + αm ) ,
dψ

(8.9)

where ej⊥ ≡ ejz + ı ejy , and ν = fm /fR . The rotating wave approximation fm  fa has been made.
Note that this is the equation of motion for the coupled position and momentum of an undamped

129
simple harmonic oscillator with natural resonance frequency fR driven with an externally applied
force at frequency fm . The harmonic oscillator’s displacement and velocity map onto e
jy and −ejzrespectively. Solving for ejy and e
jz using the initial condition ej⊥ (ψ = 0) = 0, we obtain
h
i
β
×
−ν
cos
α
sin
ψ
−
sin
α
cos
ψ
+
sin
(νψ
+
α
)
,
m
m
m
1 − ν2
h
i
β
e
×
ν
cos
α
cos
ψ
−
sin
α
sin
ψ
−
ν
cos
(νψ
+
α
)
.
jz =
m
m
m
1 − ν2ejy =(8.10)(8.11)

The above solutions can be understood as a superposition of the “transient” and steady state
response of a driven harmonic oscillator. The terms proportional to sin ψ and cos ψ in Eq. (8.10)
and (8.11) correspond to the response of the harmonic oscillator at its natural frequency fR . This
response is called the transient response in damped harmonic oscillator systems. Because there is no
damping in this oscillator, the “transient” response does not decay away. The terms proportional
to sin (νψ + αm ) and cos (νψ + αm ) correspond to the steady state response of the oscillator at thedrive frequency fm . At ν = 1, corresponding to the case of driving on resonance, the solutions takeon the following limits
1lim ejy = − (ψ cos (ψ + αm ) − cos αm sin ψ) ,
2
1
lim e
jz = − (ψ sin (ψ + αm ) + sin αm sin ψ) .
ν→1
2

ν→1

(8.12)(8.13)

The amplitude of the response grows roughly linearly with ψ for large ψ  1 as the drive is phasecoherently adding momentum to the oscillator.

8.3

Experimental Verification
To connect the theory described in Sec. 8.2 to an actual physical system, we experimentallydemonstrate the linear response of the Bloch vector to a coherently phase modulated rotation fora few special cases in this section.

130

N↑

Probe
Microwave
Rotation

π/2

N↓

RT

π

Test Rotation

Measure Jz

Optical
Pumping
State Preparation

Time

Figure 8.2: Experimental Sequence for measuring the effect of a phase modulated rotation on
the Bloch vector polar angle. After preparing the initial Bloch vector Ji on the equator throughoptical pumping and a microwave π/2-pulse, a test rotation RT is applied, and the result Jz =(N↑ − N↓ )/(N↑ + N↓ ) is measured probing technique described in Chapter 5. Unlike in Chapter 5
where Jz = (N↑ − N↓ )/2, Jz here has been normalized to the total ensemble size as the length of
the Bloch vector has been normalized to 1 in this Chapter.

8.3.1

Physical Implementation

The experimental system used for these studies has been presented in Chapter 5, and theexperiment schematic can be found in Fig. 5.4. Fig. 8.2 shows the experimental sequence fordemonstrating the impact of rotation phase noise on the Bloch vector polar angle. Instead of
performing the experiment on a single atom many times, we perform the experiment on many
identical but otherwise independent atoms N ∼ 106 , achieving high single shot signal-to-noise in
our quantum-limited readout. Because our measurements are collective in nature, we normalize
the raw measurement results to the ensemble size to obtain single atom Bloch vector information.
All atoms are initially optically pumped into |↓i. Following that, a microwave π/2-pulse rotates
the Bloch vector up to the equator, initializing the system for the experiments. To a very good
approximation, the effects of microwave amplitude and phase inhomogeneity across the atomic
ensemble may be neglected in our experiments. The intrinsic phase noise of the microwave LO
source (Agilent E8257D) is sufficiently low that we could use a small modulation amplitude β inorder to remain in the linear response regime and yet not be affected by phase noise of the LOsource. See Chapter 7 for experimental examples of LO phase noise masking quantum projection
noise.

131
An experiment typically consists of a test rotation RT using phase modulated resonant microwaves coupling the two-level system with Rabi frequency fR = 40.4 kHz. After the rotation iscompleted, the Bloch vector projection Jz = (N↑ −N↓ )/(N↑ +N↓ ) is obtained using the cavity-aided
nondemoliton measurements of the state populations N↑,↓ presented in Chapter 5. The deflection
jz is obtained from the measured Jz and the Jz◦ for the same rotation without phase modulation
using jz = Jz − Jz◦ . For future reference, ejz corresponds to the special case of rotation axis aboutx̂, i.e. φR = 0, and initial Bloch vector Ji = x̂.

8.3.2

(a)

Response to Coherently Phase Modulated Rotation

(b)

0.5

~
jz

~
jz

-0.2
-0.2

0.5

~
jy

0.2

~
jz

~
jy

~
jz

-0.5
-0.5

0.2

/2

/2

Figure 8.3: Examples of the experimentally measured evolution of the Bloch vector along themeasurement axis ẑ versus rotation angle ψ, or equivalently time, for (a) resonant modulation
fm = fR = 40.4 kHz, with β = 0.0125 rad, and (b) non-resonant modulation fm = 1.125fR , with
β = 0.025 rad. The modulation phase αm is 0 for both (a) and (b). Dashed lines joins the data
points (solid circles) to help guide the eye. The system responds with frequency components at fm
and fR , leading to the observed amplitude modulation of the response in (b). Insets show the ideal
theoretical spiral trajectories of the Bloch vector about x̂.

The harmonic oscillator-like response to a phase modulated rotation is experimentally demonstrated in Fig. 8.3. The Bloch vector is prepared along x̂, then rotated nominally about x̂ using

132
microwaves whose phase is modulated at a fixed frequency fm . After a variable rotation angle ψ,
the projection e
jz is measured. For phase modulation near resonance fm ≈ fR , the envelope ofthe modulation grows roughly linearly with ψ, whereas away from resonance fm 6= fR , frequency
components at fR and fm beat against one another to create amplitude modulation. The insets ofFig. 8.3 show the ideal theoretical spiral trajectories about x̂ for both cases.

8.4

Noise in Bloch Vector due to Phase Noise in a Single Rotation

8.4.1

Covariance Transfer Matrix

Having obtained and experimentally demonstrated the response of the Bloch vector to acoherently phase modulated rotation in Sec. 8.2, the goal of this section is to define a covariancetransfer matrix that will allow the computation of the variance of the final Bloch vector projection
along any arbitrary axis n̂ due to a randomly phase modulated rotation caused by phase noise in
the LO.
We begin with modeling phase noise at a single discrete frequency f0 by allowing the modulation phase αm and modulation amplitude β to take on random values between realizations of therotation. Statistical results are obtained via ensemble averaging over all possible realizations of thephase modulated rotation. The modulation phase and amplitude are fixed in a single realization
but random from one realization to the next. By drawing the modulation phase αm from a uniform distribution between 0 and 2π, and the modulation amplitude β from a Gaussian distribution
with zero mean and variance hβ 2 i, we model the SSB phase noise L(fm ) = hβ 2 iδ(fm − f0 )/4 (seeAppendix E for details).
We define the covariance transfer matrix as the outer product T (φR , J◦f , ψ, fm ) ≡ 4hjf j|f i/hβ 2 i,
where h·i denotes a statistical average over αm and β. The normalization of T (labels suppressed)
is chosen so that integrating n̂| · T · n̂ over the SSB phase noise L(fm ) of the LO yields the noise
variance h(jf · n̂)2 i of the final Bloch vector projected along a measurement axis n̂. Using thesolutions for e
jy , ejz in Eq. (8.10) and (8.11), we find the covariance transfer matrix Te(ψ, fm ) for the133
special case φR = 0 and Ji = x̂




0 
0 0




Te(ψ, fm ) = 0 Teyy Teyz  ,
0 Tezy Tezz
h
i
2
2
2
Teyy (ψ, fm ) =
×
(cos
ψ
−
cos
(νψ))
+
(ν
sin
ψ
−
sin
(νψ))
,
(1 − ν 2 )2
h
i
2
2
2
2
Tezz (ψ, fm ) =
×
ν
(cos
ψ
−
cos
(νψ))
+
(sin
ψ
−
ν
sin
(νψ))
,
(1 − ν 2 )22
(cos ψ − cos (νψ)) sin ψ .
Teyz (ψ, fm ) = Tezy (ψ, fm ) =1 − ν2(8.14)(8.15)(8.16)(8.17)

The covariance transfer matrix T for arbitrary rotation axis azimuthal angle φR and ideal final
Bloch vector position J◦f = (Jx◦ , Jy◦ , Jz◦ ) is derived using the small rotation r specified in Eq. (8.7)
and (8.8). The full analytic expression for T is cumbersome but can be conveniently obtained
through the transformation
T = D(φR , J◦f )Te(ψ, fm )D(φR , J◦f )| ,
where


0


D(φR , J0f ) = 0


0

−Jy◦

−Jz◦ cos φR

Jx◦−Jz◦ sin φR0Jx◦ cos φR + Jy◦ sin φR(8.18)





.



(8.19)As an example of this general result, we experimentally measure the projection e
jz afterrotating Ji = x̂ about x̂ through different angles ψ = π, 2π and 4π while phase modulating the
microwave source. Integer multiples of π were chosen to minimize sensitivity to intrinsic phase noiseof the microwave source near DC. The transfer function Tezz is obtained from averaging over four
discrete values of the modulation phase αm = {0, π2 π, 3π2 }, while keeping the amplitude β constant,Pe
as Tezz (ψ, fm ) = β12 3n=0 e
jz2 (0, ψ, β, fm , nπ2 ). The measured and theoretical transfer function Tzz
are shown in Fig. 8.4.

134

=

0

~

Transfer Function Tzz( ψ , fm )

10

-2

10

-4

10

=2

0

10

-2

10

-4

10

=4

0

10

-2

10

-4

10

0

1

2

3

4

5

Normalized Frequency [fm/fR]
Figure 8.4: Examples of experimentally measured transfer function Tezz (ψ, fm ) for ψ = π (black filled
circles), 2π (red hollow circles) and 4π (blue filled squares) overlaid on parameter-free theoreticalcurves. The amplitude of the phase modulation was β = 0.125, 0.05, 0.0625 rad for the ψ = π, 2π, 4πtransfer functions respectively. The modulation amplitude β  1 is chosen to keep the response
small to remain in the linear regime, and yet large enough to resolve the nulls at integer multiples
of fR . The transfer function does not depend on β as long as β  1 because both the response ejz
and the modulation amplitude are proportional to β.

8.4.2

Covariance Noise Matrix

In the linear response/small signal limit, the noise in the Bloch vector in some bandwidthis simply the integral of the noise variances due to phase noise at frequency fm over the relevant
frequency bandwidth. Integrating the covariance transfer matrix Te over the SSB phase noise L(fm )of the LO yields the covariance noise matrix, defined for the largest possible bandwidth,
Z ∞Ve (ψ) =Te(ψ, fm )L(fm ) dfm .
0(8.20)

135
The covariance noise matrix transforms to arbitrary φR and J◦f in the same manner as the covariancetransfer matrix in Eq. (8.18) via
V = D(φR , J◦f )Ve (ψ)D(φR , J◦f )| .

(8.21)

Using the covariance noise matrix, the variance in the projection along n̂ may be obtained ash(jf · n̂)2 i = n̂| · V · n̂ .

(8.22)As a useful example of quantifying the noise mapping from the LO onto the Bloch vector,
consider a white noise spectrum L(fm ) = L◦ . Integrating the white phase noise spectrum over thecovariance transfer matrix Te(ψ, fm ) yields],
Ve (ψ) = L◦ NEB(8.23)

where the noise equivalent bandwidth matrix is




0
0
0




] = πfR sgn(ψ) 
2NEB.
0 ψ − 21 sin 2ψ
− sin ψ 0− sin2 ψψ + 12 sin 2ψ

(8.24)

A corollary to Eq. (8.24) is that as |ψ| increases, the covariance transfer matrix Te becomes more
and more sharply peaked around the Rabi frequency fR . Therefore for large |ψ|  1, most of the
Bloch vector noise contribution comes from phase noise near the Rabi frequency. As |ψ| → ∞, thecovariance transfer matrix Te approaches a delta function at fR

0 0 0 Te(ψ, fm ) ∼ πfR |ψ|L◦ δ(fm − fR ) 0 1 0 .




0 0 1

(8.25)The noise mapping for any rotation axis in the x−y plane can be obtained by transforming toarbitrary φR , but keeping J◦f = x̂. We find the non-zero elements are Vzz = Vezz cos2 φR , Vyy = Veyy ,and Vyz = Veyz cos φR . Note that the variance Vzz can be driven to zero by applying the rotation
perpendicular to the Bloch vector as shown in Fig. 8.5(b). Alternately, keeping φR = 0, but letting

~
jy

~
jz
Phase Noise

x

136

(b) 4

z

y

Noise Variance Vzz [arb]

(a)

3

2

1

0
0

90

180

Rotation Axis

270

360

R [deg]Figure 8.5: Noise Mapping. (a) Phase modulation of the LO causes modulation of the rotation
axis (green arrow) about its mean orientation, here along x̂ or φR = 0. The final Bloch sphere ofpoints is deflected by an amount described by a small rigid rotation r(0) = Ry (−ejz )Rz (ejy ). The◦
deflection of a Bloch vector depends on the ideal final vector Jf . If a set of possible deflections forjy2 + ejz2 = const (shown outside spherean ideal final Bloch vector J◦f along x̂ is described by a circle e
for clarity), the same set of deflections for other J◦f are described by ellipses and lines centered at
J◦f . (b) The noise mapping shown in (a) for φR = 0 and for J◦f on the equator can be equivalently
demonstrated by keeping the final vector oriented along J◦f = x̂ and varying the rotation axis
φR . The observed noise variance Vzz (solid circles) of the vector projection along ẑ varies as thepredicted cos2 φR (solid line). The experiment was performed by applying phase modulation at a
discrete frequency fm = 1.125fR to a rotation about x̂ cos φR + ŷ sin φR that nominally rotates the
Bloch vector Ji = x̂ cos 2φR + ŷ sin 2φR through an angle ψ = π to J◦f = x̂. Averaging jz2 over thefour modulation phases αm = {0, π2 , π, 3π2 } simulates phase noise at a single discrete frequency fm .

J◦f = x̂ cos θ +ẑ sin θ, i.e. the ideal final vectors lie in the x − z plane, one finds Vzz,yy = Vezz,yy cos2 θand Vxx = Vezz sin2 θ. This noise mapping is graphically shown in Fig. 8.5(a).

8.5Noise in Bloch Vector due to White Phase Noise From Multiple Rotations

8.5.1

Noise Propagation

The single-rotation covariance noise matrix V allows us to analyze two crucial building blocks
for coherent manipulation of quantum systems – composite pulses, used to suppress static amplitude

137
and detuning errors, and time-separated pulse sequences designed to reduce qubit decoherence, suchas spin echo and dynamical decoupling type sequences. Constituent rotations in composite pulses
are applied in a back-to-back manner, leaving as little time as possible between the rotations. In
contrast, time separation beween rotations in a time-separated pulse sequence may be comparable
or much longer than the time it takes execute a rotation.
Assuming a white phase noise spectrum L(fm ) = L◦ , noise from different rotations become
statistically independent regardless of time separation between rotations. We also assume thatdephasing of the Bloch vector during the time interval between rotations is small, a condition
that is satisfied by time-separated pulse sequences designed to reduce decoherence. This ensures
the opening angle between the Bloch vector and the rotation axis is well defined. Under theseassumptions, the formalism for treating noise from a composite pulse, and noise from a timeseparated pulse sequence are the same.
We now present the noise propagation that gives the multiple-rotation covariance noise matrix
W . The symbol W for the multiple-rotation covariance noise matrix is chosen to differentiate itfrom the single-rotation covariance noise matrix V . A pulse sequence consists of N rotations withthe kth rotation given by
Rk = R(φk , ψk ) .

(8.26)In the absence of noise, the ideal Bloch vector after the kth rotation isJ◦k = Rk · · · R1 Ji .

(8.27)

The added noise from just the kth rotation is
Vk = D(φk , J◦k )Ve (ψk )D(φk , J◦k )| .

(8.28)

By accounting for how the noise from the previous rotation Wk−1 is transformed by subsequentrotations, the total noise after the kth rotation may be computed iteratively using
Wk = Rk Wk−1 Rk| + Vk ,(8.29)

138
with the initial condition W1 = V1 . Finally, the noise variance along any arbitrary projection axisn̂ after the kth rotation is given byh(jk · n̂)2 i = n̂| · Wk · n̂ ,(8.30)

where jk is the noise deflection after the kth rotation.

8.5.2

Average Infidelity

While matrix elements of the covariance noise matrix W depend on pulse sequence specifics
and the initial Bloch vector, we present here a simple formula that evaluates the average quality ofa pulse sequence using only general properties of the pulse sequence. Within the quantum control
and computing community, the state infidelity [182]1 − F = Tr(W )/4(8.31)

is an important measure of the rotation quality. While the state infidelity depends on pulse sequencedetails, the state infidelity averaged over the Bloch sphere of possible initial statesh1 − F i = ΨπfR L◦ /3(8.32)

PN

Thus, adepends only on the total rotation angle Ψ =k=1 ψk and not the rotation axes φk .

pulse sequence with smaller Ψ is preferred over one with larger Ψ if one is mainly concerned withthe average fidelity. In spin echo and dynamical decoupling schemes, suppression of environmentinduced decoherence typically improves with the number of pulses. However, this comes at theexpense of increasing the average phase-noise-induced decoherence. It is, therefore, necessary to
strike a balance between reducing environment-induced decoherence and reducing phase-noiseinduced decoherence.

8.5.3

Composite π-pulse ComparisonsComposite pulses, designed to suppress static amplitude and detuning errors, have beenthoroughly analyzed in the literature with regards to the degree of error cancellation [149–151, 159,

139168]. The influence of phase noise on the Bloch vector through composite pulses, however, hasreceived little attention in literature.
Applying the noise propagation formalism presented in Sec. 8.5.1 to the commonly usedcomposite π-pulse sequences: CORPSE, SCROFULOUS, and BB1 [149–151], which effectively
implement a π-pulse about x̂, we summarize in Fig. 8.6 the final variance Wzz and the state
infidelity 1 − F versus the initial Bloch vector Ji specified by its polar angle θi and azimuthal
angle φi . Expressions for the composite pulse rotation sequences, covariance noise matrices, andstate infidelities are provided in Appendix F. For completeness, we also summarize the static error
cancellation order and static infidelity for these pulses evaluated for an rotation axis parallel andperpendicular to the Bloch vector in Appendix F.
The single quadrature variance Wzz is of interest in metrology applications, particularly
for manipulating spin-squeezed states, as added noise in the squeezed quadrature can potentiallydestroy the squeezing. The state infidelity 1 − F , which includes variances from the two transverse
spin components perpendicular to the ideal final Bloch vector, is particularly pertinent in quantum
control for quantifying the overall quality of the rotations.
It is generally not possible to minimize phase noise sensitivity and optimize static error
cancellation simultaneously. To understand the tradeoffs between phase noise sensitivity and static
error cancellation, we compare and contrast the two for initial Bloch vectors in the x − y plane, i.e.
θi = 0, leaving φi as the only degree of freedom. We use Wzz and 1 − F as the basis for evaluating
sensitivity to phase noise.
A static fractional amplitude error  results in an error ψ in the rotation angle, and a static
detuning error δ = (fLO − fa )/fR , where fLO is the LO frequency, causes the rotation axis to betilted up from the x − y plane by an angle arctan(δ). The degree to which the static errors , δ are
2 (, δ, φ ), and the infidelity
suppressed can be characterized by the static error squared Wzz,st = jz,sti

due to static error 1−Fst = Tr(Wst (, δ, φi ))/4. These definitions, in direct analogy to corresponding
quantities for phase noise, allows meaningful comparison of the phase noise sensitivity and static
error cancellation on the same footing in the next section.

1/2

θi
π

0

SCROFULOUS π
Error Cancellation(b) StaticAmplitude
Detuning

CORPSE π

(a)
5
10
15
20
25
30
35
40

5
10
15
20
25
30
35
40

45

45

0

(d)

5

4
6
8
10

10
15
20
25

12

30

14

35
0.29

(e)

12

7.5

14

14

16

16

18

18

20

(c)

Best
Worst
Same

1

φi /π

10

2

20

140

BB1 π

0.79

(f )
13

8

14
15
16
17
18
19
20

8.5
9
9.5
10
10.5

21

11
0.29

0.79

Figure 8.6: Composite π-pulses. Variance Wzz and infidelity 1 − F of CORPSE π-pulse (a, d),
SCROFULOUS π-pulse (b, e) and BB1 π-pulse (c, f) versus initial Bloch vector orientation (θi , φi ).
θi is measured from the x − y plane and φi is measured from x̂. Contours levels are normalized tofzz (θi = 0, φi = 0) for a simple π-pulse about x̂ is π 2 fR L◦ . Points of best (solid
fR L◦ . For scale, W
circle), worst (hollow circle), and same (hatched circle) order of static amplitude error cancellation,and points of best (solid square), worst (hollow square), and same (hatched square) order of static
detuning error cancellation (offset vertically for clarity) are shown for Bloch vectors with θi = 0.
All plots have the same axes as (a). A quadrant of the Bloch sphere is shown here. The rest of thecontour plot can be generated using reflection symmetry about the x − y and x − z plane. The restof the static error cancellation points can be generated via φi → φi + π.

141
8.5.3.1

CORPSE π-pulse

The CORPSE π-pulse, used to suppress static detuning error, has the best static error cancellation at φi = 0, π where both Wzz,st , 1 − Fst = O(δ 6 ). However, it is also most sensitive to phasenoise at φi = 0, π (see Fig. 8.6(a, d)). At φi = π/2, the impact of phase noise is minimized at theexpense of static error cancellation as both Wzz,st , 1 − Fst = O(δ 4 ) + O(2 ).

8.5.3.2

SCROFULOUS π-pulse

In contrast to the CORPSE π-pulse, sensitivity to phase noise does not vary with φi for the
SCROFULOUS π-pulse designed to suppress static amplitude error. As shown in Fig. 8.6(b, e),
one can simply choose φi to optimize the cancellation of static errors depending on the quantity
Wzz,st or infidelity 1 − Fst to be optimized, and on the dominant source of static errors (amplitude
or detuning) without altering the impact from phase noise.

8.5.3.3

BB1 π-pulse

Finally, the BB1 π-pulse, which compensates for amplitude error with little to no cost to thesensitivity to detuning error, has similar impact from phase noise where the order of amplitude
error cancellation is best (worst) at φi ≈ 0.79π (0.29π) as shown in Fig. 8.6(c, f). Therefore one
may choose to operate at φi ≈ 0.79π where Wzz,st = O(δ 2 ) + O(10 ) and 1 − Fst = O(δ 2 ) + O(8 ).
In fact, the impact from phase noise is slightly lower at φi ≈ 0.79π compared to at φi ≈ 0.29π. On
the other hand, there is a tradeoff between suppressing detuning errors and sensitivity to phasenoise. Detuning error cancellation is best at φi = π/2 as both Wzz,st , 1 − Fst do not scale with δ toany order if  = 0. However, the impact of phase noise is also worst at φi = π/2.
In general, careful evaluation of the relative scalings and contributions of phase noise, static
amplitude and detuning errors is required to optimize the overall fidelity or specific noise quadratures.

142
8.5.4

Spin Echo Pulse Sequences

Spin echo and dynamical decoupling sequences constitute another class of manipulation protocols in quantum control and computing, important for suppressing qubit decoherence, or, forinstance, to undo probe-induced dephasing as was done in our squeezing experiment. We analyze
here spin echo sequences of the form [τ − R(φ1 , π) − 2τ − R(φ2 , π) − 2τ · · · − R(φN , π) − τ ] using
the formalism developed in Sec. 8.5.1 to find the covariance noise matrix W for the sequence.
We consider the following two choices of rotation axes: (a) rotation axis always along x̂, or,(b) alternating between x̂ and −x̂ with the first π-pulse applied along x̂. Note that the widelyused Carr-Purcell [161], Carr-Purcell-Meiboom-Gill [162], and Uhrig Dynamical Decoupling [164]sequences are special cases of (a) corresponding to specific orientations of the Bloch vector withrespect to the rotation axis x̂.
Writing the initial Bloch vector as Ji = (Jxi , Jyi , Jzi ), the covariance noise matrix for bothchoice (a) and (b) reads




i i
i i
i2
1 − Jx sN Jx Jy sN Jx Jz 




i
2
,
WN = N π 2 fR L◦ sN Jxi Jyi
Jx
0 




sN Jxi Jzi
0Jxi 2(8.33)

where sN = (−1)(N +1) . While the noise properties for the two choices are the same, their sensitivity
to static errors are different. Choice (b) offers cancellation of static amplitude error as the Blochvector nominally retraces its path while choice (a) accumulates static amplitude error as the Blochvector keeps rotating about the same axis in the same sense. On the other hand, choice (a) does
not accumulate static detuning error while choice (b) does. In future work, we will explore thepossibility of engineering noise properties of spin echo pulse sequences.

8.6

Summary
In summary, we have developed a general framework for analyzing the mapping of LO phasenoise onto noise projections of a Bloch vector and extend the mapping to pulse sequences for the case143of white LO phase noise. Detuning or transition frequency noise can be handled via the mapping
β → ∆FM /fm where ∆FM is the frequency modulation amplitude. Results for special but importantand illustrative cases are presented, which experimentalists can readily utilize for estimation ordesign. Future directions may include extending the analysis to non-resonant excitation, more
complex spin echo or dynamical decoupling pulse sequences, and account for the effects of 1/f andhigher order phase noise, where time separation between pulses may no longer be ignored, andnoise correlations between pulses play an important role.

Chapter 9

Broader Context, Future Prospects, and Conclusions

In this concluding chapter, we will place the observed squeezing in this work within the context of other experiments and approaches for entanglement-enhanced measurement precision. This
chapter will also explore several future directions to improve the amount of observable squeezing,
including probing on optical cycling transitions and technical upgrades to the experiment. The approach of using cycling transitions is currently being pursued in the laboratory with very promising
initial results.

9.1

Framing our results in broader context
We have utilized the tools of cavity-QED to prepare an entangled ensemble with a 3.4(6)dB improvement in spectroscopic sensitivity over the standard quantum limit (SQL) in the largest
clock-like ensemble N = 7 × 105 to date. Our method does not require single particle addressability
and requires a single fast ∼ 150 µs squeezing operation. The gain in sensitivity is spectroscopicallyequivalent to the enhancement obtained had we created > 105 pairs of maximally entangled qubits,
demonstrating the power of a top-down approach for entangling large ensembles. The probing ofatomic populations via the vacuum Rabi splitting is also of broad interest for non-destructively
reading out a wide variety of both atomic and solid state qubits.
From the perspective of technologies that might impact future physics, the large ensemble
size is a crucial component. Entangled states of cold, neutral atoms are unlikely to impact the
future of quantum sensors and tests of fundamental physics unless the techniques for generating

145
the states are demonstrated to work for the 104 to 107 neutral atom ensembles typically used infrequency standards [14, 16, 18], free space atom interferometers [19, 27, 87, 183], trapped atominterferometers [72, 76, 80, 83] and magnetometers [23, 88, 102].
The approach described in this thesis allows sub-quantum-noise limited readout of a sensor
dMswith dM
× MNm ≈ 0.1 photon recoils/atom (at Mm = 1.9 × 105 photons and N = 7 × 105 atoms),m

producing little heating of the atomic ensemble. Applied to a state-of-the-art optical lattice clock,the non-destructive probing can result in higher clock duty cycle which improves measurement
statistics, and helps to suppress the dominant aliasing of the local oscillator noise [15, 184].
The work presented in this thesis is unique in that we probe the atomic ensemble in theresonant regime of strong collective coupling cavity-QED. By doing so, we hope to counter a commonly held view that the quality of a coherence-preserving QND measurement is fundamentally
linked to the probe’s large detuning from atomic resonance. Instead, it is the magnitude of thecollective cooperativity parameter N↑ C (equivalent to the optical depth for a free space experiment [38]) that sets the fundamental quality of the QND measurement as shown in Chapter. 3.
In the context of free space measurements, detuning from resonance creates little enhancement insensitivity to projection noise fluctuations once the detuned optical depth falls below one. The absolute size of projection noise rms fluctuations in the dispersive regime decreases as 1/δp, f s , whereδp, f s = ωp − ω↑e is the free space probe detuning from the |↑i to |ei transition frequency. In thepresence of a fixed technical noise background, resolving projection noise fluctuations become moredifficult as the probe detuning δp, f s is increased. Using an optical cavity to enhance the cooperativity parameter has the potential to allow similar results to free space experiments, but at atomic
densities lowered by a factor of order the cavity finesse.
The spectroscopic gain demonstrated in this thesis is far from the fundamental Heisenberg
limit which scales as 1/N , indicating that there is a lot of room for improvement. The Heisenberg
limit has been approached by creating nearly maximally entangled states of 2 to 14 ions [62–64, 66,67]. However, the spectroscopic gain relative to the SQL demonstrated in this thesis is comparable
to these experiments, highlighting the fact that neutral ensembles have higher potential for spectro-

146
scopic gains than ion trap experiments because of the large ensemble size. Ensembles of N ≈ 103
atoms have been spin squeezed by exploiting atom-atom collisions within a Bose-Einstein Condensate [72–76, 80–83], however these systems face the significant challenge of managing systematic
errors introduced by the required strong atomic interactions.

Figure 9.1: Schematic of weak sampled measurements. A small fraction  of a large ensemble
N of atoms could be extracted and a strong measurement of the extracted sub-ensemble’s Blochvector polar angle θ is performed. Using this classical sampling strategy, the estimate of the Blochvector polar angle√θ of the initial full ensemble is limited to the SQL for the sub-ensemble to∆θSQL,sample = 1/ N . While the loss of signal in the remaining ensemble can be made small, theuncertainty attainable from measuring sub-ensemble increases concomitantly. In our experiment, = 0.15(2). The sampling strategy would provide an estimate −10 log() = 8.2(7) dB noisier thanthe SQL for the initial full ensemble ∆θSQL and 13.2(9) dB noisier compared to the spin-noisereduction demonstrated in Chapter. 5 using a coherence-preserving QND measurement.

The coherence-preserving nature of the QND measurements presented in this thesis should
also be contrasted with weak, sampled measurements in order to compare this work to fluorescence
detection of an optically thin ensemble. Equivalently, the angle θ of the full ensemble could be
estimated by extracting a fraction  = 1 − Cf /Ci = 0.15(2) of the initially un-decohered atoms, andperforming perfect state detection on this sub-ensemble. The loss of signal would be the same as
observed in our experiment, but the sub-ensemble’s estimate of θ would be G = −10 log10 ()+4.9 =
13.2(9) dB noisier than the 4.9(6) dB sub-projection-noise precision demonstrated in this thesis
using a collective measurement approach. Conceptually, this reduction in noise can be described asarising from a noiseless amplifier of gain G placed before a 15/85 linear atom beam splitter [185].

147
This fundamental gain in precision is enabled by the collective interaction of the atomic ensemble
and the probe mode.

9.2

Spectroscopic gain in the presence of decoherence
Generating spin squeezed states with phase resolution below the SQL takes us one step closer

towards making ever more precise phase and frequency measurements. The natural next step is to
apply the generated squeezed states to an actual measurement. The question of how to harness
the power of entangled states for metrological gain under different decoherence models is currently
a very active field of theoretical research. In this section, I will briefly discuss several situations inwhich decoherence during the phase evolution can reduce or even eliminate the spectroscopic gain
afforded by the intrinsic phase resolution enhancement of the generated squeezed states. The final
step in Ramsey spectroscopy is to map the evolved phase onto a change in populations, and thenread out the populations. To realize the full potential accorded by the squeezed states, it is therefore
necessary to read out the state with sufficiently low noise — a topic already treated in Chapters 3
and 4. More precisely, the readout noise needs to be much lower than the projection noise of thesqueezed state. With low noise readout techniques, recent experiments have demonstrated Ramseyinterferometry beyond the SQL with spin squeezed states over evolution times of .20 ms [73, 75,
76, 186, 187].
In systems whose Ramsey evolution time is limited by uncorrelated single particle decoherence, Ref. [188] has showed that maximally entangled cat states provide no spectroscopic enhancement in frequency measurements over coherent spin states. The reason is because cat states are so
sensitive to atomic decoherence such that the optimal Ramsey evolution time is drastically shortened by a factor of N , completely negating the quantum advantage provided by the N times fasterrate of phase evolution. However, Refs. [188, 189] showed that improvement of frequency measurements from partially entangled states that have a high degree of permutation symmetry, such as
spin squeezed states, can be obtained. Taken together, the gains achievable from modest amountsof spin squeezing and the extreme fragility of maximally entangled states suggest that pursing the148
Heisenberg limit in large ensembles could throw off our scent in the pursuit of higher measurement
precision. Generating macroscopic cat states is a very worthy scientific goal for understanding
quantum superpositions at macroscopic scales, but cat states might not be practical for metrology
with large ensembles.
Besides being extremely sensitive to dephasing [66, 190], maximally entangled states are also
extremely sensitive to losses. The loss of a single atom effectively collapses the wavefunction into
a product state wavefunction into all spin up or all spin down. In the density matrix formalism,
tracing over a single particle results in a completely mixed state. The mixed state can be interpreted
as a classical mixture of all the atoms in spin up or all spin down with probability 50% each. Spinsqueezed states, in contrast, have been shown to be robust against loss [191], uncorrelated single
particle dephasing [192], and to a lesser extent, correlated dephasing [37]. The reason for thisrobustness arises from the permutation symmetry of squeezed state [191]. Unlike a cat state,small losses from a spin squeezed state do not destroy entanglement completely. Losses degrade
the amount of squeezing dependent on the relative levels of losses and squeezing. In analogy tosqueezed light, the degradation of spin squeezing in atomic systems due to losses is analogous tovacuum noise coupling in through a dark port. As an example, for squeezing levels of 3 dB, one
could afford to lose ∼ 50% of the particles before losing the squeezing. Unlike photons, it is easierto trap and hold onto atoms for long periods of times, and one could imagine atomic spin squeezing
might be able to surpass even photonic squeezing one day.
It is important to make a distinction between phase and frequency measurements. Frequency
measurements rely on the accumulation of an evolved phase over time, during which decoherence
can occur. In contrast, a phase measurement is robust because it can be made, in principle, in anarbitrarily short amount of time so that decoherence is frozen in its tracks. As a result decoherence is
not expected to degrade spectroscopic gains for a phase measurement. This does rely on performing
the phase measurement immediately after the squeezed state is prepared. Alternatively, squeezing
may be stored in the θ-quadrature which is insensitive to dephasing in φ while waiting for the phasemeasurement experiment to be readied. When the phase is ready to be measured, a rotation that

149aligns the squeezing along φ activates the phase measurement.
In optical clocks, where coherence times are limited by the laser local oscillator dephasing1(an example of correlated dephasing), the precision of the clock can be enhanced by using squeezed
states of large ensembles, with the optimal Allan variance scaling as N −1/3 relative to the SQL [37].
√−1 'This optimal Allan variance scaling is achieved using moderately squeezed states with ξmN,−1 = N for large N . The favorable scaling with N argues
far from the Heisenberg limit of ξmfor using large ensembles, as is pursued in this thesis, to obtain more potential improvement.
Using an ensemble size of N = 106 as an example, the optimal amount of squeezing would be−1 ' 103 = 30 dB. So for squeezing less than 30 dB, such as our experiment where ξ −1 = 3.4 dB,ξmm

the full amount of squeezing would be reflected in the clock Allan variance improvement.
Instead of making more precise measurements, one can also choose to spend the squeezing
on making more frequent measurements. Increase in measurement bandwidth with squeezed states
can be achieved without loss of sensitivity provided the Ramsey evolution time is shorter than the
single particle decoherence time [38]. Atomic magnetometer bandwidth improvement enabled byentanglement has been demonstrated in Ref. [102]. Bandwidth improvement with nondemolition
measurements have also been demonstrated in Ref. [125].

9.3

Fundamental Limits on Squeezing via Probing the 87 Rb cycling transition
Squeezing via probing an open optical transition, as was the case for the experiment presented

in Chap. 5, is fundamentally limited by Raman spin flips as shown in Chap. 4. As an outlook on
future experiments, we consider a situation in which Raman spin flips are greatly reduced, namely
probing on a cycling transition [130, 193]. Using the framework developed in Chapters 3 and 4,
we show that larger amounts of squeezing are possible in this configuration than with probing on
an open transition. The key results are that as N C is increased, the spectroscopic enhancement−1 ∼ N C, as expected in the cycling limit, until it enters a region of saturation of thescales as ξm
1

Nature provides atoms with such long coherence times that the best ultra-narrow linewidth lasers cannot yet
match.

150
spectroscopic enhancement set by the ratio ωhf /Γ of the hyperfine splitting to the excited state
decay linewidth as a result of the Raman spin flips caused off-resonant scattering off other states,
i.e. the notion of a cycling transition is an approximate concept.
As a concrete example of how a cycling transition can be used to enhance probing, we consider
the cycling transition in 87 Rb, |↑i ≡ |F = 2, mF = 2i to |ei ≡ |F = 30 , mF = 3i at wavelength
780 nm. The spin down state is chosen as |↓i ≡ |F = 1, mF = 1i. For the following, the probing
scheme with relevant energy levels, dipole matrix elements, decay branching ratios, dressed mode
frequencies, and probe laser detunings are shown and defined in Fig. 9.2. Here we will extend theprevious models of the precision of the estimation of Jz and the loss of signal due to wavefunction
collapse to capture the essential physics for this system. Key results are that there exists a regionof saturation, or universal spectroscopic enhancement, set only by atomic properties and in which
varying atom number and cavity finesse can have little impact. The asymmetry in the cavity
coupling to |↑i and |↓i allows this saturation region to be surpassed at large values of N C but the√−1 ∼scaling then proceeds as ξm
N C.
We must first consider what limits the rate of Raman scattering processes that can leadto diffusion of the spin projection Jz . The probe polarization can be set to pure σ + to better
than 10−4 , so that Raman scattering from |↑i is suppressed to at least this level or greater. The
more fundamental Raman scattering limitation arises from the finite hyperfine splitting ωhf =2π × (6834 MHz). Specifically, atoms in |↓i can non-resonantly Raman scatter probe photons from|e0 i ≡ |F = 20 , mF = 2i.
In the following discussion, the quantity ms is importantly defined as the average numberof probe photons (normalized to the total atom number) Rayleigh scattered into free space byatoms in |↑i. All other scattering processes will be scaled from this quantity using the quantities
defined in Fig. 9.2. The key parameters for rescaling are the ratio of the dipole matrix elements√r = M↓e0 /M↑e = 1 2, the decay branching ratio B3e0 = 1/6 from |e0 i to |3i ≡ |F = 2, mF = 1i, andthe detunings of the probe light δe and δe0 = δe − ωhf − ωehf from resonance with the transitions|↑i → |ei and |↓i → |e0 i respectively. Because ωhf  ωehf , we neglect the excited state hyperfine

mF = 0

+1

+2

151

+3

F’ = 3F’ = 2F=2

F=1Figure 9.2: Probing scheme using the cycling transition in 87 Rb. The ground hyperfine states F = 2
and F = 1 are split by ωhf = 2π × (6834 MHz). The values of mF are labeled across the top. The
relevant F 0 = 3 and F 0 = 2 excited D2 transitions at wavelength 780 nm are also shown with theexcited state splitting ωehf = 2π × (267 MHz). The pseudo spin-1/2 system is here composed of|↑i = |F = 2, mF = 2i and |↓i = |F = 1, mF = 1i. Ideally, the σ + -polarized probing laser couples
only |↑i to the optically excited state |ei with dipole matrix element M↑e at a frequency detuningδe that is approximately equal to the dressed-cavity mode frequency ω− . In this illustration, δe
is negative. By dipole selection rules, |ei can only decay back to |↑i. However, the same probe
laser also couples |↓i to the single excited state |e0 i with dipole matrix element M↓e0 and larger√
detuning from resonance δe0 = δe − ωhf + ωehf . The ratio of matrix elements is M↓e0 /M↑e = 1/ 2.
Finally, the state |e0 i decays to states |↓i, |↑i, and |3i with fractional branching ratios B↓e0 = 1/2,
B↑e0 = 1/3, and B3e0 = 1/6 respectively.

splittings so that δe0 ≈ δe − ωhf .
The rms imprecision in the estimate of Jz relative to the projection noise level can be approximately modeled as∆Jz∆Jz, CSS

2= 4p↑ ms + p3 ms +me proj
s
.
ms

(9.1)

Any small gains in the imprecision of the estimate of Jz from state preparation or prior knowledge
has been neglected. Starting in order of physical significance, the first and second terms arise fromdiffusion of Jz caused by Raman transitions from |↓i → |↑i and |3i with effective probabilities p↑

152
and p3 given approximately byp↑,3 = B↑,3 e0 r

2



δe
δe0

2
.

(9.2)

In this simple treatment, Raman decays to |3i are treated as loss, as reflected in the smaller
numerical pre-factor in front of the second term of Eq. (9.1).
The third term in Eq. (9.1) is modified to reflect that both states can interact with theprobe at large detunings such that the dressed cavity mode frequency is less sensitive to quantum
projection noise in Jz , and thus more probe photons must be used to resolve Jz at the projection
noise level, i.e.,me proj
=
s

mproj
s
.
Rray

(9.3)

Here, mproj
is defined by Eqs. (3.19), (3.9), and (3.4). Indistinguishability is accounted for by
s


δe 2
Rray = 1 − r
.
δe0

(9.4)

Note that Rray ≤ 1 with an asymptotic value of Rray → (1 − r)2 at large detunings.
There are two effects that are neglected in Eq. (9.3) by first assuming they are small, and thenverifying this to be the case after the calculations. First, in applying the dressed cavity linewidth
result for κ0 from Eq. (3.9), we assume that the cavity mode is negligibly further broadened by atoms
in state |↓i. By estimating the additional broadening evaluated at the optimal cavity detuning
and average number of scattered photons, we find that the optimal spectroscopic enhancements
calculated in Fig. 9.3 are reduced by < 0.3 dB due to the neglected mode broadening. Second, we
assume that the dressed mode frequency ω− calculated from Eq. (3.4) is only modified by a small
fraction by atoms in state |↓i. Again, this assumption is verified to be the case at the optimal
cavity detuning and the average number of scattered photons, with the exception of the case whereN > 108 and cavity finesse F = 100, as shown in Fig. 9.3 where several dB of deviations are possible
due to this effect.

153
Next, we consider how collapse due to free space scattering reduces the coherence C, specificallyC = e−ms (Rray +Rram ) ,(9.5)

where the partial cancellation of wavefunction collapse due to indistinguishable Rayleigh scattering
off of both |↑i and |↓i (see [139]) is accounted for by Rray .
The term Rram accounts for Raman scattering from |↓i to |3i.
 2
δe
2
Rram = B3e0 r
.
δe0

(9.6)

As before, we assume that Raman scattering to state |3i is equivalent to atom loss. Note also that
Rram ≤ B3e0 r2 .
Equations (9.1) - (9.6) are used to numerically estimate the optimal spectroscopic enhance−1
ment ξm,
opt shown versus atom number N in Fig. 9.3 for a range of technologically reasonable cavity

finesses assuming our cavity geometry and perfect quantum efficiency q = 1. The optimization isdone with respect to both ms and the dressed cavity mode frequency ω− (tuned by changing thebare cavity frequency). The mode frequency ω− at the optimum is shown in Fig. 9.4. The loss of
signal due to wavefunction collapse and scattering to |3i at the optimum is shown in Fig. 9.5.
−1
At low atom number, the spectroscopic enhancement scales as ξm,
opt ∼ qN C. At high atom√
−1
qN C. There is an intermediate region
number, the spectroscopic enhancement scales as ξm,
opt ∼of atom number for which the spectroscopic enhancement is relatively flat versus atom number with−1ξm,
opt ∼ ωhf /Γ. The physical origin of this plateau arises from the form of the critical detuning of

Eq. (3.24).
In the good cavity limit, the scattering necessary to reach projection noise level sensitivitypmproj
falls as 1/δc4 for δc < δc◦ , making it beneficial to operate with |ω− | ∼ δc ≥ Γ N C/q.
s
However, the Raman transition probabilities p↑,3 continue to grow quadratically with the detuning,
while detuning farther no longer rapidly reduces mprojs .
Assuming the critical detuning δc◦ is optimal for the reasons above, and in the limit of |ω− | <ωhf , the Raman transition probabilities scale as p↑,3 ∼ (N C/q)(Γ/ωhf )2 , while mproj∼ 1/qN C.
s

-1
,

154

−1
Figure 9.3: Theoretical optimal spectroscopic enhancement ξm,
opt (solid curves) in dB relative to87
the SQL versus total Rb atom number N for our current cavity geometry and probing near theD2 cycling transition as shown in Fig. 9.2. The optimization is performed for a range of technicallyrealizable cavity finesses F = 102 , 103 , 104 , 105 , and 106 (blue, cyan, green, yellow, and red curves
respectively) under the assumption of perfect quantum efficiency q = 1. Note the unphysical−1
regime below the Heisenberg limit (disallowed grey region). A region of saturation of ξm,
opt versus−1
N occurs near ξm, opt ≈ ωhf /Γ shown by the labeled horizontal dashed line. The physical origin
for the saturation region arises from competition between the scaling of the off-resonance Raman
scattering probabilities and the dressed cavity mode broadening, as described in the√ text. The
points of 3 dB deviation from N C scaling at low atom number (solid circles) and N C scaling−1
at large atom number (open circles) occur when the solid curves cross ξm,
opt = 27 dB and 33 dB,√
respectively. The approximate transition from q to q scaling occurs when the solid curves cross
−1
3/4 .
ξm,
opt = 28 dB, the point at which the scaling is approximately q

Optimizing the total noise in our estimate of Jz using Eq. (9.1) with respect to ms reproduces the−13
observed plateau value ξm,
opt ∼ ωhf /Γ ∼ 10 . The plateau region is exited at low atom number

when loss of signal, described by Eq. (9.5), dominates the reduction in spectroscopic enhancement,
as illustrated by the loss of signal due to wavefunction collapse shown in Fig. 9.4. At high atom
number, the plateau region is exited when the optimum mode frequency becomes large comparedto the hyperfine splitting |ω− | > ωhf , as shown in Fig. 9.5.
Importantly, this analysis shows that there is a range in which increasing either finesse or

155
fh

4

w / -w gninuteD edoM desserD lamitpO

10

Cavity Finesse
6

10

3

10

5

10
2

10

4

10

3

1

10

10

2

10

0

10
-1

10
-2

10
-3

10
0

10

1

10

2

10

3

10

4

10

5

10

6

10

Atom Number

N

7

10

8

10

9

10

10

10

−1 forFigure 9.4: The dressed mode frequency ω− that optimizes the spectroscopic enhancement ξm2
3
4
5
the conditions described in Fig. 9.3, where again the cavity finesses F = 10 , 10 , 10 , 10 , and 106
correspond to blue, cyan, green, yellow, and red curves, respectively. The open and closed circles
indicate the locations of the 3 dB points in Fig. 9.3. The mode frequency is normalized to thehyperfine splitting ωhf /2π = 6.834 GHz. Note that ω− ≈ Γ at ω− /ωhf ≈ 10−3 . Comparing√ to−1
Fig. 9.3, one finds that the transition to spectroscopic enhancement scaling as ξm, opt ∝ qN C
occurs near ω− ∼ 1.8 × ωhf . Above this point, the spin flip probabilities p↑,3 change little with ω− .

atom number can have little effect on the optimal spectroscopic enhancement achieved. Also, notethat the value of the plateau does not depend on the cavity geometry, and therefore represents
a universal value that depends only on the atomic properties. One can also show that the rangeof atom number spanned by the plateau and the position of the first corner scales roughly as
N ∼ w02 /F , but does not depend on the cavity length l. Finally, for atom numbers below 103 , itappears possible to both prepare and readout states near the Heisenberg limit using this approachand technologically feasible cavity finesses. Single atom measurement resolution for N . 100 wasrecently demonstrated using the approach described here [130].
Looking ahead at how the squeezed state might be of use, we note that the squeezing is
generated on a magnetic field sensitive transition desirable for enhancing magnetometer precision

156
0

10

Cavity Finesse
-1

10

2

10
3

10
4

10

-2

sm lamitpO

10

5

10
6

10
-3

10

-4

10

-5

10
0

10

1

10

2

10

3

10

4

10

5

10

6

10

Atom Number

N

7

10

8

10

9

10

10

10

−1 shown in Fig. 9.3.
Figure 9.5: The scattering ms that optimizes the spectroscopic enhancement ξm
2
3
4
5
6
Again, the cavity finesses F = 10 , 10 , 10 , 10 , and 10 correspond to the solid blue, cyan, green,
yellow, and red curves, respectively. The open and closed circles indicate the locations of the
3 dB points in Fig. 9.3. Comparing to Fig. 9.3, one finds that the transition to spectroscopic−1enhancement scaling as ξm,
opt ∝ qN C occurs when ms ≈ 0.25, indicating that at larger N , far
off-resonance Raman scattering begins to create significant diffusion of Jz that limits the precisionof the estimation of Jz as discussed in the main text.

and/or bandwidth. Looking from a different perspective, the squeezing could be generated using afavorable probing scheme such as the one presented here, and then mapped onto the clock states
for precision time keeping via microwave and rf Rabi rotations or rapid adiabatic passage. Ramsey
spectroscopy can then be performed on the squeezed clock states, and for low noise readout, theoutput of the Ramsey interferometer could be mapped back onto the same states squeezing was
originally generated on. During the readout, the probe measurement strength can then be increased
to provide very high resolution, performing essentially a non-demolition measurement as describedin Fig. 2.7. Destroying coherence during the readout step does not affect the precision of theRamsey interferometer.

157

9.4

Future Experimental Upgrades
In this section, I consider several improvements in the experimental setup that could be

implemented in the next generation Science Cavity.

9.4.1

Improving Cavity Finesse

Having spent a lot of time with the current experimental apparatus whose crucial element— the Science Cavity — has been sealed up in UHV since summer 2008, I can point to many
imperfections that have developed since its assembly in a pristine condition. My chief concern
is with absorption/scattering losses on the mirrors, costing us a ∼50% loss in detection quantum
efficiency. This loss impacts our ability to prepare measurement-induced spin squeezed states withmore than 3 dB below the SQL. The squeezing we have generated is not yet limited by Raman spin
flip diffusion, but by technical noise. Therefore, at fixed probe photon number and sufficiently far
away from the technical noise that behaves almost like a vertical wall, the measurement imprecision
would scale inversely to the detection quantum efficiency, just like in the cycling limit. Thus a lossof ∼50% would result in a reduction of ∼ 3 dB of spectroscopic enhancement. I will also mention
that the UHV 45◦ folding mirror presents a 8% loss in detection efficiency in the cavity reflection
path. The cause of this loss is unknown, although we hypothesize it could be due to rubidium
coating the folding mirror due to its proximity to the rubidium reservoir source.
When the cavity was first assembled, its measured linewidth κ2008 was in good agreementwith the predicted value κ0 /2π = 5.29 MHz from ATF’s mirror transmission measurements. Overtime, we noticed a gradual increase in linewidth of ∼1 MHz that can be recovered using 405 nm
blue laser diodes to remove adsorbed rubidium on the mirrors surfaces via light-induced desorption.
A catastrophic high voltage discharge near the silver epoxy connection between the bottom mirror
PZT and the kapton wires in summer 2009 coated the mirrors with material that the blue laser
light cannot remove, leading to an almost doubling of the linewidth to κ2009 /2π = 9.2 MHz. Since
then, the linewidth has gradually crept up to a stable value of κ/2π = 11.1(2) MHz. After the

158discharge, we measured a finite resistance between the PZT wires that is not stable, indicative of
an intermittent connection or flaky short between the wires. We believe this is due to the kapton
coating on the wires soaking up rubidium atoms, leading to a short where the supply and return
voltage wires crossed. The short also limits our ability to scan the Science Cavity to a fraction of
a FSR currently, although that can be mitigated with laser heating of the PZTs.
For the next generation Science Cavity, we could go up a factor of 10x to 100x in finesse
to F ∼ 104 to 105 to increase the single particle cooperativity, a key figure-of-merit in cavityQED experiments. While a higher finesse cavity is more sensitive to losses, we are optimistic thatpreventive measures can be taken to to avoid degradation of the cavity finesse resulting in reductionin quantum efficiency. Careful assembly of the kapton wires such that they do not cross each otheror touch the stainless steel vacuum chamber would prevent shorts. Using kapton dipped wires
would also help. We should drive each PZT independently with its own UHV vacuum feedthroughso that in case one goes bad, we have a fallback. A 2D MOT source can be installed to deliver ahigh flux and collimated beam of cold atoms to the cavity instead of enveloping the whole cavity
with rubidium vapor as is currently the case. Heater coils can be installed around the mirrors todrive off rubidium vapor on the mirror surfaces, an approach successfully used in Ref. [194].

9.4.2

Single-ended Cavity to Improve Detection Efficiency

To simplify the detection, a single-ended cavity with the transmissive end 10x to 100x moretransmissive than the highly reflective end would be highly desirable so that there is nominally only
one detection port (unless we find a good reason to detect the cavity transmission also). We can also
perform homodyne detection of the cavity-probe sideband, winning back the 3 dB of signal-to-noise
lost in heterodyne detection. We can do so by stabilizing the phase of the interferometer in reflection(described in Chap. 5) with a strong phase-reference sideband (∼ µW) detected in heterodyne witha very strong reference beam (∼mW to be shot noise limited) that is frequency degenerate withthe weak cavity-probe sideband (∼nW). Using the detected phase-reference sideband phase as theerror signal, feedback can be applied to the phase of the probe laser beam that carriers both the159phase-reference and cavity-probe sidebands. By choosing the carrier phase value to lock to, we canperform homodyne detection of the cavity-probe sideband in the phase quadrature.

9.4.3

Uniform Coupling of Cavity Mode to Atoms

To generate spin squeezed states that can be launched into free space for interferometry suchas in a fountain clock or in an atom interferometer requires uniform coupling of the cavity mode to
the atoms. There are two ways to obtain uniform coupling — ring cavity or commensurate lattice.
In a ring cavity, the cavity mode is a running wave, unlike the standing wave in a Fabry-Perotcavity. By trapping atoms in a small region near the center of the cavity, fairly uniform coupling
can be achieved. In a ring cavity, however, the two polarization modes of the cavity typicallyhave different finesses and free spectral ranges due to unequal s- and p-reflection coefficients (bothamplitude and phase). At high finesse, orthogonal polarization modes may be split, impacting one’s
ability to address both polarization modes simultaneously. In a commensurate lattice, one utilizes
a lattice at 2x the probe wavelength such that the antinode of the lattice lines up with every otherantinode of the probe mode. Since the atoms are tightly confined in each pancake, good uniform
coupling may be achieved. To be consistent with mirror boundary conditions, the commensuratelattice and probe mode starts out in phase in the middle of the cavity and acquires a relative phaseslip of λ/2 by the time they reach the mirrors. Provided the axial extent of the atomic ensemble ismuch shorter than the cavity length, so the atoms do not see the phase slip, uniform coupling canbe achieved.

9.4.4

Removing 17 Hz Vibration Noise Source

An annoying vibration at 17 Hz of the Science Cavity relative to the optics table causes themode-matching to the cavity to change on timescales of 60 ms. Up and down motion of ∼ 2λ795
peak-to-peak imposes phase noise in the reflected light, which we removed using carrier phasestabilization in Chapter 5. Another effect is the changing of the coupling to the cavity mode due
to sideways motion of the cavity at 17 Hz. The effect is mitigated by optimizing the coupling to

160
the cavity mode in the transverse directions so that any offset in the transverse plane affects thetransmitted lattice and probe power only to second order. We find a 1−2% peak-to-peak fluctuationin the reflected probe power when mode matching is optimized. The source of the 17 Hz is the ACunit attached to the ceiling of the lab below our lab. Thus our lab floor as well as the non-floating
optics table is vibrating at 17 Hz. The cavity, being vibrationally isolated, stays relatively fixedwith respect to the local inertial frame defined by the earth. We could float our optical table ormodify the vibration isolation. We can replace the viton rubber balls used in the first and lowest
frequency stage of vibration isolation with steel ball bearings to transmit the 17 Hz to the cavity
while still leaving the second and third stage for high frequency vibration isolation, so there is nodifferential motion between the optics table and the cavity at low frequencies. While the 17 Hz canaffect the cavity resonance frequency via differential acceleration of the top and bottom mirrors,
the feedback loop has plenty of gain at the low frequency of 17 Hz compared to the unity gain
frequency of ∼2 kHz to squash that noise down.

9.5

Summary and Conclusions
We have demonstrated the preparation of conditionally spin squeezed states with 3.4(6) dB

spectroscopic enhancement beyond the standard quantum limit. The degree of squeezing achievedin our experiment is currently limited by technical noise associated with the frequency chirping of
6 has an extremely
the Rabi splitting resonances. This technical noise variance, scaling as ∼ Mm

unfavorable scaling with the probe photon number Mm .
In the future, our method can be extended to achieve greater violations of the SQL or
equivalently obtaining larger amounts of squeezing by reducing the chirping of the resonances via
probing on an optical cycling transition. Squeezing via probing on an optical cycling transition
with the pseudo spin-1/2 system comprised of the stretched hyperfine ground states of 87 Rb is apromising path to achieving more squeezing. By probing on an optical cycling transition, Raman
spin-flips that would fundamentally limit squeezing on the clock transition in the scheme presented
in this thesis would be greatly suppressed. We also expect that technical noise associated with the161frequency chirping of the dressed cavity mode resonance due to probe-induced Raman scattering
would be greatly reduced, improving our measurement precision.
Furthermore, many experimental aspects, such as cavity finesse and length, can be easily
improved. Running wave cavities or commensurate optical lattices can be employed to create
squeezed states appropriate for launching ensembles into free space for matter wave interferometry.
It will also be interesting to understand and explore how this system might be applied to probe
and manipulate quantum many-body entangled states of large atomic ensembles [195].

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Appendix A

Coherent Reinitialization for Reduced Recoil Heating

In this Appendix, we consider whether coherence-preserving QND measurements can also be
used to reduce the total photon-recoil heating in a Ramsey spectroscopy measurement cycle. A low
number of photon recoils imparted in a complete experimental cycle, not just the readout, would
allow neutral atom traps to operate more like ion traps, with their advantages of high duty cycle
and high repetition rates. We use the canonical example of Ramsey spectroscopy to show how low
photon recoil in a complete experimental cycle can be achieved. Imagine reading out the collective
spin projection Jz at the end of a Ramsey sequence with a QND measurement at or below the
projection noise limit. If one were to repeat the experimental cycle by optically repumping all
the atoms back into | ↓i, each atom in the ensemble would experience ≈ s/2 photons worth ofrecoil heating, where s is the average number of photons scattered by an atom to get from | ↑ito |↓i during the repumping process. In Fig. A.1, we show how recoil heating can be reduced bycoherently reinitializing most of the atoms into |↓i using a microwave π/2 pulse before turning on
the repumping.
The loss of contrast  due to dephasing during the Ramsey evolution time is assumed tobe small for the results presented in this section to be valid. Dephasing causes the off-diagonal
elements of the density matrix representing the state of the ensemble to decay, resulting in a mixed
state. The mixed state can be regarded as a tensor product of a pure state representing a smaller
ensemble of size N (1 − ) atoms in a coherent superposition of |↑i and |↓i [Fig. A.1(a)], with purestates representing the decohered atoms with N/2 atoms collapsed in |↑i and |↓i on average. The

BIBLIOGRAPHY

175(a) Atomic spin state after QND measurement:zxy∆Jy, QBA
atoms

atoms

atoms

(b) After π/2-rotation about y:atoms∆Jz = (∆Jy, QBA)2/N(c) After repumping:

Figure A.1: Coherent reinitialization scheme to reduce photon recoil heating in a complete experimental cycle. (a) Dephasing during the Ramsey evolution can be described as a shrinking of thecoherent part of the collective spin from length N/2 to length N (1−)/2, where   1 characterizes
the amount of dephasing. The dephased atoms can be represented as if they had collapsed into|↑i and |↓i with N/2 atoms in each state. Performing a QND readout with ms = Ms /N  1
results in further shrinkage of the coherent collective spin and additional collapse of the atoms. (b)Performing a microwave π/2-pulse about ŷ rotates the coherent collective spin to the south poleand the collapsed atoms onto the equator of the Bloch sphere. (c) Applying optical repumping
beams reinitializes the ensemble to the south pole, with the photon recoil heating arising fromrepumping the atoms in superposition states into |↓i. The contribution arising from the quantum
back-action of repumping back into a CSS is negligible provided the spectroscopic enhancement issmall compared to the Heisenberg limit.

QND readout causes an additional (1 − p)Ms atoms to collapse into |↑i and pMs atoms to collapse
into |↓i, provided ms = Ms /N  1. Following a microwave π/2 pulse that rotates most of theatoms into | ↓i, we find that approximately s(ms + )/2 photon recoils per atom is required toBIBLIOGRAPHY

176

reinitialize the entire ensemble into |↓i. The contribution from quantum back-action into Jy is oforder sms /N mproj
and may be safely ignored for small spectroscopic enhancements.
s
In summary, we have shown that it is possible to impart only s(ms + )/2 optical photon
recoils per atom through the coherent reinitialization scheme, as opposed to the much larger s/2
recoils per atom without coherent reinitialization. Note the reason why the scheme works is because
we have traded optical photon recoils with microwave photon recoils, which impart recoil energy∼ 8 orders of magnitude smaller than that of optical photons. If an optical transition separated|↑i from |↓i, such as in optical lattice clocks, the scheme presented would not reduce recoil heating
substantially.

Appendix B

Heterodyne PDH Detection Scheme

This appendix describes a heterodyne Pound-Drever-Hall (PDH) detection method in our
early attempts at squeezing. Initial attempts using this method provided nearly projection-noiselimited resolution. We chose not to optimize the sensitivity as we realized the IQ detection method,
described in Chap. 5 would give fundamentally 3 dB higher signal-to-noise for a symmetric cavity
because both transmission and reflection are detected for the IQ demodulation method while information in transmission is not used in the heterodyne PDH method. It is then simple to see thatboth methods fundamentally have the same signal-to-noise when applied to a single-ended cavity(and driving the more transmissive mirror) as all the information comes out from the reflection
port only. However, this technique can be relatively simple to implement and produces a dispersive error signal in real time (without computer processing) appropriate for real-time feedback.
A technical advantage of this detection scheme over the heterodyne IQ-demodulation scheme (inreflection) presented in Chap. 5 is that the heterodyne PDH method is insensitive to optical path
length fluctuations because the phase reference sidebands and the probe sidebands travel along thesame exact optical path.
Here I briefly describe the experiment since the concepts are already laid down in Chapter. 5,
and instead focus on the methodology more. We prepare a CSS along x̂ using optical pumpingp
all the atoms into |↓i followed by a microwave π/2-pulse. A vacuum Rabi splitting Ω↑ = N↑ 2g
develops from the strong resonant coupling between the atoms and the cavity mode. The goal
then is to measure N↑ very precisely by measuring size of the vacuum Rabi splitting precisely. The

BIBLIOGRAPHY

(a)

(b)
e

Phase
Mod

Phase
Mod

106 -54151

15.3 MHzF’=1,mf=0

5 2P1/2

ωc795 nm

↑

ADC

V(t)2

ADC

V(t)

φ1 φ2

(c)
Ω↑

ProbesF=2,mf=0

5 2S1/2↓

90R/10T BS

AOMHeterodyne Reference

AOM

Probe
Laser

178
V(t)2

6.8 GHz

F=1,mf=0

Figure B.1: (a) Experimental setup of the heterodyne PDH scheme used to probe the vacuum
Rabi splitting. Both components of the vacuum Rabi splitting are simultaneously probed using
the first order phase modulation sidebands generated by the first electro-optic modulator (EOM)and spaced ±151 MHz from the carrier. Additional ±15.3 MHz PDH phase reference sidebands
are imposed by the second EOM. (b) Relevant energy level structure of 87 Rb. The bare cavity ismade resonant with the |↑i to |ei transition, and π-polarized probe light is used. (c) Simultaneous
probing of the vacuum Rabi splitting. The blue and red points are the individual PDH error
signals simultaneously measured by sweeping the phase-modulated probe laser frequency acrossthe higher and lower frequency modes of the vacuum Rabi splitting respectively. The blue and redsolid lines are curve fits to the data using an analytic model of the PDH error signal that includes
contributions up to the 5th order sidebands.

vacuum Rabi splitting Ω↑ is measured by sweeping the probe laser frequency and detecting thelight reflected from the cavity input mirror. The resonance signals are extracted using a modified
PDH detection scheme, which I call heterodyne PDH, that employs optical heterodyne detection
to achieve near photon-shot-noise-limited detection at probe powers of < 100 pW to several nW

BIBLIOGRAPHY

179(see Fig. B.1). To reduce sensitivity to laser and cavity noise, we measure both resonances ω+ andω− simultaneously as discussed in Sec. 6.2. The two probe components are generated by phasemodulating the probe light to create sidebands at ±151 MHz from the carrier, chosen to matchthe vacuum Rabi splitting of 302 MHz. The PDH phase reference sidebands at ±15.3 MHz arethen imposed around the probe components by a second phase modulator. To avoid degeneracyof the two probe components in the rf spectrum generated at the photodiode, the probe carrier isblue-shifted 52 MHz with respect to the heterodyne reference beam derived from the same laser as
the probe.
In principle, a single 52 MHz AOM on the probe or heterodyne reference beam would suffice.
However, we observed spurious signals when using a single 52 MHz AOM that is due to rf pick-up
on the probe laser, generating undesired sidebands at our signal frequency. To mitigate pickup, weused doubly shielded BNC cables on all high power rf (∼ 30 dBm) signals in our lab. Because weuse a heterodyne beam power of ∼1 mW and probe powers at the nW level, spurious modulationof the heterodyne beam at the 10−5 to 10−6 level would produce heterodyne beatnote powers of
the same magnitude as the actual probe signal. From our experience, spurious modulation at thislevel can occur if attention is not paid to shielding rf signals carefully. The problem was finallyeliminated by using two AOMs, one that blueshifts by 106 MHz and another that redshifts by 54
MHz, resulting in a net blueshift of the probe beam by 52 MHz. Any spurious sidebands at 52
MHz would now be second order in the pickup. The probe power is controlled via the rf drive
power delivered to the 54 MHz AOM. I note here that these AOMs were tuned from its nominal
operating frequency of 80 MHz to 106 MHz and 54 MHz via tuning the LC-resonant circuit inside
the AOM. No significant loss of diffraction efficiency was observed. The 106 MHz AOM crystal 1
has a chip near the transducer, which we believe is the cause of its long thermal time constant – aslow relaxation of the beam deflection when the rf power is pulsed fully-on.
The light reflected from the cavity is overlapped with the heterodyne light on a 90/10 beam
1

This AOM is current being used in the 823 nm Lattice Laser setup to provide additional isolation from theTransfer Cavity, and fiber tip reflections.

BIBLIOGRAPHY

180

splitter. Both beams are coupled into a single mode polarization-maintaining (PM) fiber to ensurehigh spatial mode matching. The heterodyne laser power delivered to the photodiode is typically 0.8
to 1.2 mW. Working at low heterodyne powers provides additional immunity to classical intensity
noise, as well as increased probe detection efficiency. The photodetector consists of a Hamamatsu
S5973 photodiode, ac-coupled to an Analog Devices AD8015 240 MHz transimpedance amplifier
described in Appendix C.
The phase shift of the probe carrier E0 relative to the phase reference sidebands E±1 may begenerated via the PDH method, which relies on a square law detector to generate the cross terms
E0 E+1 + E0 E−1 resulting in a rf beatnote at the sideband modulation frequency. Demodulating
the beatnote into the quadrature component yields the phase shift of the carrier. For good SNR,
the carrier photon shot noise needs to dominate over the photodiode electronic noise. Normallythis is not a problem when probing an empty cavity; one can simply increase the carrier power.
The situation is different when probing atoms in a cavity because atoms can be saturated at highintensities, collapsing the Rabi splitting we wish to probe. Furthermore, to minimize decoherence
caused by free space scattering, we could not probe with high powers. The standard methods
to achieve (close to) shot noise limited detection is to use a avalanche photodiode (APD) or homodyne/heterodyne detection. We opted for heterodyne detection. Because the heterodyne rf
beatnote ELO (E−1 + E0 + E+1 ) does not contain the desired cross terms between the carrier andthe sideband electric fields, we generated the cross terms by electronically squaring the rf beatnote.
The two probe components are separated in the heterodyne rf spectrum using high andlow pass filters, and notch filters at 99 MHz and 203 MHz. Each time-domain signal is squared
using an Analog Devices AD835 250 MHz voltage multiplier. This step essentially mimics thesquare law detection of the photodiode that occurs in the usual PDH detection scheme. The
resulting 15.3 MHz signals are then demodulated to dc. Figure B.1(c) shows the resulting nearly
simultaneously measured PDH error signals when the probe laser frequency is swept across a
vacuum Rabi splitting. Linear fits to the central 15 µs of each probe component sweeping throughresonance were used to extract the vacuum Rabi splitting Ω↑ .

BIBLIOGRAPHY

181

50

Number of Events

40

30

20

10

0
-8

-6

-4

-2

0

2

4

6

8

Difference in vacuum Rabi splitting (re-centered to 0) norm. to Projection Noise
(1)

(0)

(0)

(1)

Figure B.2: Histogram of the difference Ω↑ − Ω↑ between two successive measurements Ω↑ , Ω↑of the Rabi splitting showing nearly projection-noise-limited measurement precision. A small
horizontal
offset has been applied to center the distribution about zero. The rms fluctuation(1)(0)∆ Ω↑ − Ω↑is 2.1(1) times the projection noise level in standard deviation for an ensemble of
N = 7.1(3) × 105 effective atoms. Taking the measurement precision of both measurements to be
the same, we infer the precision of a single Rabi splitting is 3.5(4) dB above the quantum projection
noise level.

Using this technique at nano-Watt probe powers, we were able to measure the collective spin
at a precision of 3.5(4) dB above the quantum projection noise level for an ensemble of 7.1(3) × 105
effective atoms (see Fig. B.2). At this measurement precision, we calculate ms = 0.16 photons/atomworth of free space scattering, directly proportional to photon recoil heating and loss of contrast
as described in Sec. 6.1. Simple extrapolation implies that with ≈ 2.2× more probe photons, themeasurement precision would reach the projection noise level while causing ≈ 0.35 photon/atomworth of free space scattering.

Appendix C

AD8015 Photodiode Characteristics

In this Appendix, I include characterizations of the photodiode used in squeezing experimentdescribed in Chap. 5. We used a Hamamatsu S5973 photodiode (1 GHz intrinsic bandwidth) whose
uncoated glass cover has been removed to recover about 8% in quantum efficiency. Its sensitivity
without glass is 0.54 A/W at 795 nm corresponding to internal photodiode quantum efficiency of84%.
The photocurrent generated at the photodiode is converted to electric voltage with a 240 MHz
transimpedance amplifier (Analog Devices AD8015). The transfer function, noise and transimpedance
gain of our AD8015 #2 photodiode used in reflection detection are shown in Figs. C.1, C.2, C.3
respectively. The AD8015 #5 photodiode used in transmission detection has similar characteristics
as the AD8015 #2 photodiode.

BIBLIOGRAPHY

183

Transfer Function Magnitude relative to 30 MHz [dB]2
AD8015 #2 Photodiode Transfer Function
0

-2

-4

-6

-8

-10
0

50

100

150
200
Frequency [MHz]

250

300

350

Figure C.1: Transfer Function of AD8015 #2 Photodiode10

AD8015 #2 Photodiode Dark Noise

8

300

6

200

4

100

2

0

0
0

50

100

150
200
Frequency [MHz]

250

300

1/2

400

Equivalent Input Noise Current [pA/Hz ]Optical Power to generate Input Noise Current [uW]

500

350

Figure C.2: Noise properties of AD8015 #2 photodiode. Shot-noise-limited detection may beachieved with 0.2 mW and 0.35 mW optical power at detection frequencies of 100 MHz and 200 MHz
respectively.

BIBLIOGRAPHY

184

8200
AD8015 #2 Photodiode Transimpedance Gain in Ohms

Transimpedance Gain [Ohms]

8100
8000
7900
7800
7700
7600
7500
7400
0

20

40

60

80

100

120

140

160

180 200x106

Frequency [Hz]Figure C.3: Measured Transimpedance Gain of AD8015 #2 photodiode. While the AD8015 chip is
spec for 10 kOhm transimpedance gain (nominal) in the data sheet, we measured a transimpedance
gain of 9 kOhm near DC (at 100 kHz).

Appendix D

Phase Noise Definitions

The power spectral density of phase fluctuations Sφ (fm ) of an oscillator is the mean squared
phase fluctuations (∆φ(fm ))2 at frequency offset fm from the carrier in a 1 Hz measurement bandwidthSφ (fm ) ≡ ∆φ(fm )2 /Hz .

(D.1)

The phase noise Sφ (fm ) has units of rad2 /Hz, and includes contributions from both upper andlower noise sidebands at ±fm .
One can measure Sφ (fm ) by mixing the oscillator-under-test with a reference oscillator at thesame frequency, and with much lower phase noise. The phase of the reference oscillator is chosen
so that the mixer output v(t) is proportional to the relative phase difference φ(t) between theoscillator-under-test and the reference oscillator. The power spectral density Sφ (fm ) is computedfrom the autocorrelation function of φ(t) via the Wiener-Khinchin theorem as
Z ∞Sφ (fm ) = 2

hφ(t)φ(t + τ )it e−i2πfm τ dτ ,

(D.2)

−∞

where fm lies in the range (0, ∞).
The definition of the single sideband (SSB) phase noise L(fm ) is1
L(fm ) ≡ Sφ (fm ) .
2

(D.3)

The units for L(fm ) are rad2 /Hz. It is also commonly expressed in the form 10 log10 L(fm ), which
has units of dB below the carrier in a 1 Hz bandwidth (dBc/Hz). The mean squared phase

BIBLIOGRAPHY

186

fluctuations observed in a Fourier frequency range from fl to fh is given by2

Z fh
L(fm ) dfm .

(∆φ) = 2

(D.4)

fl

For an oscillator whose amplitude noise is much lower than its phase noise, the SSB phasenoise L(fm ) is equivalent to the ratio of the power of a noise sideband PSSB (fm ) at frequency
offset fm in a 1 Hz measurement bandwidth to the power in the carrier Pcar as measured on aradio/microwave/optical frequency spectrum analyzer
L(fm ) =PSSB (fm )
.
Pcar

(D.5)

Appendix E

Relation between Mean Squared Modulation Amplitude and Single Sideband
Phase NoiseIn this short note, we establish the connection between the mean squared phase modulation
amplitude hβ 2 i, used to normalize the covariance noise matrix T in Sec. 8.4.1, and the SSB phasenoise L(fm ). We model the phase modulation φ(t) = β sin(2πf0 t + αm ), at fixed modulation
frequency f0 , as being drawn from a random distribution of αm and β. The modulation phase αmis uniformly distributed from 0 to 2π, and the modulation amplitude β is Gaussian distributedwith zero mean and variance hβ 2 i. The time-averaged squared phase modulation should be further
averaged over the distribution for αm and β to yield the statistical phase fluctuations as
hhφ(t)2 it iαm ,β = hφ(t)2 iαm ,β =hβ 2 i,2(E.1)

where we made use of the fact that averaging over time t has the same effect as averaging overphase αm . Using Eq. D.1, D.3 and E.1, we obtain the relation between the SSB phase noise L(fm )and mean squared modulation amplitude hβ 2 i as
L(fm ) =hβ 2 iδ(fm − f0 ) .
4

(E.2)

Appendix F

Composite and Plain π-pulse Covariance Noise Matrix and Infidelity

Table F.1: Rotation sequence for commonly used composite π-pulses.
Pulse Name
Plain πCORPSE π
SCROFULOUS π
BB1 π

Pulse Sequence π  R0◦ [π]
 7π 
◦◦R0 3 R180◦ 5πR
033R60◦ [π] R300◦ [π] R60◦ [π]R0◦ [π] R104.5◦ [π] R313.4◦ [2π] R104.5◦ [π]The covariance noise matrix W and infidelity 1−F for the CORPSE π-pulse, SCROFULOUS
π-pulse, BB1 π-pulse and a single π-pulse, which all effectively implement a π-pulse about x̂, aregiven below. The results assume a white LO phase noise spectrum L(fm ) = L◦ . Here we specify
the initial Bloch vector Ji by its polar angle θi , measured from the x − y plane, and its azimuthal
angle φi measured from x̂ so that Ji = (cos θi cos φi , cos θi sin φi , sin θi ).

BIBLIOGRAPHY

F.1

189

CORPSE π-pulse
7π
For the CORPSE π-pulse sequence R 0, π3 R π, 5πR0,(time ordering right to left),3
3

used to suppress static detuning error, we have:



√ 
√ 
1
Wxx = πfR L◦ 13π − 3 3 cos2 θi sin2 φi + 13π + 3 3 sin2 θi
3

√ 
1
Wyy = πfR L◦ 13π − 3 3 cos2 θi cos2 φi
3

√ 
1
Wzz = πfR L◦ 13π + 3 3 cos2 θi cos2 φi
3

√ 
1
Wxy = Wyx = πfR L◦ 13π − 3 3 cos2 θi sin 2φi6Wyz = Wyz = 0(F.1b)(F.1c)(F.1d)(F.1e)√ 
1
Wxz = Wzx = πfR L◦ 13π + 3 3 sin 2θi cos φi
6



√
√ 
1
1 − F = πfR L◦ 13π − 3 3 cos 2θi + 13π + 3 3 cos2 θi cos2 φi
12


F.2

(F.1a)(F.1f)(F.1g)SCROFULOUS π-pulse
For the SCROFULOUS π-pulse sequence R

π
3,π



R

5π
3 ,π



R

π
3,π



(time ordering right toleft), used to suppress static amplitude error, we have:3
Wxx = π 2 fR L◦ 2 cos2 θi sin2 φi + sin2 θi23Wyy = π 2 fR L◦ 2 cos2 θi cos2 φi + sin2 θi23Wzz = π 2 fR L◦ cos2 θi23
Wxy = Wyx = π 2 fR L◦ cos2 θi sin 2φi23Wyz = Wzy = − π 2 fR L◦ sin 2θi sin φi
43Wxz = Wzx = π 2 fR L◦ sin 2θi cos φi
4
3
1 − F = π 2 fR L◦ (5 + cos 2θi )16(F.2a)(F.2b)(F.2c)(F.2d)(F.2e)(F.2f)(F.2g)

BIBLIOGRAPHY

F.3

190

BB1 π-pulse

The BB1 π-pulse sequence R (0, π) R (φR , π) R (3φR , 2π) R (φR , π) (time ordering right toleft), where φR = arccos − 14 ≈ 104.5◦ , is used to compensate for amplitude error with little tono cost in the sensitivty to detuning error. The covariance noise matrix W and infidelity 1 − F for
the BB1 π-pulse are:5
Wxx = π 2 fR L◦ 4 cos2 θi sin2 φi + sin2 θi45
Wyy = π 2 fR L◦ 4 cos2 θi cos2 φi + 3 sin2 θi
45Wzz = π 2 fR L◦ cos2 θi (2 − cos 2φi )45
Wxy = Wyx = π 2 fR L◦ cos2 θi sin 2φi215Wyz = Wzy = − π 2 fR L◦ sin 2θi sin φi
8
5 2
Wxz = Wzx = π fR L◦ sin 2θi cos φi
8
5 2
1 − F = π fR L◦ 4 + 2 cos2 θi − cos2 θi cos 2φi
16

F.4(F.3a)(F.3b)(F.3c)(F.3d)(F.3e)(F.3f)(F.3g)

Plain π-pulse
Finally, we provide expressions for the covariance noise matrix and infidelity for a plain π-pulse R (0, π) as a useful benchmark to compare against composite π-pulses. The noise covariancematrix W and infidelity 1 − F for a plain π-pulse are:Wxx = π 2 fR L◦ 1 − cos2 θi cos2 φi(F.4a)

Wyy = π 2 fR L◦ cos2 θi cos2 φi(F.4b)

Wzz = π 2 fR L◦ cos2 θi cos2 φi(F.4c)

1
Wxy = Wyx = π 2 fR L◦ cos2 θi sin 2φi2(F.4d)

Wyz = Wyz = 0(F.4e)

1
Wxz = Wzx = π 2 fR L◦ sin 2θi cos φi21
1 − F = π 2 fR L◦ 1 + cos2 θi cos2 φi
4(F.4f)(F.4g)

BIBLIOGRAPHY

F.5

191

Composite Pulse Static Error Cancellation
Table. F.2 shows the lowest order static error cancellation for commonly used composite π-pulses, and also the plain π-pulse as a benchmark for comparison. The following parameters areused: static fractional amplitude error as  = ∆fR /fR and the static fractional detuning error of
the local oscillator as δ = (fLO − fa )/fa .

O(L2◦ )3π 2
2 fR L ◦3π 2
2 fR L ◦5π 2
4 fR L ◦15π 2
4 fR L◦CORPSE ⊥SCROFULOUS k
SCROFULOUS ⊥
BB1 k
BB1 ⊥

CORPSE k

Plain kPlain ⊥

Phase Noise Sensitivity(∆θ)2
π 2 fR L ◦O(L2◦ )√
1
3 π 3 3 + 13π fR L◦

Pulse Name√√
−1 − 25π
δ 3 − 5πδ
3
3
√

π
2
3 − 2 δ − π
√ 5 51
32 3π  √− 4δ√ 3
3 3
6 3δ + 3πδ 2  − 21 3π 2 δ2 − π 4√1
3 3
32 15π  − 2δ 3 3
15πδ 415
5π 
216 + 8 πδ  − 32Static Error∆θstatic−2δ
2− πδ2 − π

4 44δ 2 + 3π6427δ 4
9 2 2 23π 4 4
4 −√8 π δ  + 64 6 61δ 2 − 32
15π 3 δ3 + 15π
4096
25π 2 δ 6
25 2 2 225π 6 6
+ 16 π δ  + 4096
64

Static Infidelity∆J2 /4 = 1 − F
δ2
π2 δ2
1 2
π 2 ε2 16 + 4 π δε√+ 4 
2
1
5π
37π
1
2 2 2
√
δ 6 − 12
−10 3 − 37π πδ 4  + 37
4 + 4 3 + 48
12 π δ 

√
√  2
3π
3
1
π 2 2π24
4 − 4 + 16 δ + 4 π π − 2 3 δ  + 4

Table F.2: Comparison of strengths and weaknesses of composite pulse sequences for performing a π-pulse with the rotation axis parallelor perpendicular to the Bloch vector.

BIBLIOGRAPHY
192

Appendix G
Fundamental Limits on Squeezing via Differential Measurement of 87 Rb clocktransition

Using the framework developed in Chapters 3 and 4, we now analyze the fundamental limits
on squeezing on a clock transition comprised of two hyperfine ground states, taking into accountRaman spin flips and decoherence.
From Sec. 3.3, we have shown that the measurement resolution at fixed free space scattering
improves with cavity detuning δc , but ultimately saturates to a value set by the collective cooperativity parameter N C. For squeezing on a clock transition comprised of two hyperfine ground states,
the maximal detuning is approximately half of the ground state hyperfine splitting δc = ωhf /2. We
find that the squeezing is fundamentally limit by the ratio ωhf /Γ of the hyperfine splitting to theexcited state decay linewidth.
To be specific, we analyze the measurement scheme demonstrated by the MIT group [100]in which they used the same pseudospin states presented in this thesis, namely the clock states of
87 Rb |↑i ≡ |52 S

2
1/2 , F = 2, mF = 0i and |↓i ≡ |5 S1/2 , F = 1, mF = 0i.

For differential probing of

the hyperfine clock transition, the bare cavity frequency is tuned to the average of the two groundstates to optically excited state transitions near 780 nm. This tuning ensures that an atom in |↓ishifts the dressed cavity resonance frequency by an equal, but opposite amount as an atom in |↑i.
The excited state hyperfine splitting ∼ 500 MHz is much less than the ground hyperfine splitting
and is taken to be zero for the following analysis.
The problem is analyzed by extending the linearized two-mode model of Eq. (3.6) to a lin-

BIBLIOGRAPHY

194

earized three-mode model in which the atomic operator â is generalized to operators â↓ and â↑to yield three coupled differential equations, along with the same input-output relations given by
Eq. (3.8):
d hĉidt
d hâ↑ idtd hâ↓ i
dt

 √
p
p
1
N↑ hâ↑ i − N↓ hâ↓ i + κ1 ci ,= − κ hĉi − ıg2p1
= − (Γ + ı ωhf ) hâ↑ i − ı N↑ g hĉi ,2p1= − (Γ − ı ωhf ) hâ↓ i − ı N↓ g hĉi .
2

(G.1)

The equations are now written in a rotating frame at the bare cavity resonance frequency that ischosen such that the two optical atomic transitions are detuned by ±ωhf /2. The rate of scattering√
into free space is described by the two field amplitudes as,↑,↓ = Γ hâ↑,↓ i and normalized such thatthe rate of photons scattered into free space is simply Ṁs = |as,↑ |2 + |as,↓ |2 .
From the coupled set of Eq. (G.1), we find that the rms phase shift of the transmitted lightfield caused by the rms projection noise level fluctuation in the population difference is:
∆φ

proj

√
=


NC

Γ
ωhf



1
2
1 + N CΓ2 /ωhf


.

(G.2)

p
This expression assumes the damping rates are small κ, Γ  ωhf , N/2 2g. The phase shift initially√√climbs with increasing atom number as N , but saturates to a maximum value ∆φproj = C/2 at√a critical atom number given by N C = (ωhf /Γ)2 , after which the phase shift decreases as 1/ N .
The physical interpretation for this decrease is that above the critical atom number, the dressedcavity mode linewidth κ0 rapidly starts to broaden with increasing atom number. The number offree-space-scattered photons required to resolve the projection noise level phase shift of the probe
is
1
mproj
=
s
4qN C


1 + NC

Γ
ωhf

2 !
.

(G.3)

The diffusion of the difference between the estimate of Jz and the actual value of Jz is driven
by Raman transitions that move atoms from |↑i to |F = 1i or |↓i to |F = 2i. Raman transitions
between states of the same F (i.e., ∆F = 0) lead to loss of coherence, but do not change the couplingof the atom to the cavity mode in the limit where the excited state hyperfine splitting is neglected,

BIBLIOGRAPHY

195

as we do here. Hyperfine changing transitions ∆F 6= 0 cause the detuning to change sign, but notmagnitude, making such a process equivalent to a spin flip. Accounting for transition branchingratios, we find that to a good approximation, we can apply Eq. (4.4), with an effective spin flip
probability p = 1/6. Assuming the loss of coherence is small, then the optimal spectroscopicenhancement with respect to average probe photon number is
√

−1
ξm,
opt =

6qN C
2 .
1 + 4N CΓ2 /ωhf

(G.4)

√
−1
At small N , the spectroscopic enhancement scales as 6qN C, reaching a peak value of ξm,
opt =p
3q/8 ωhf /Γ at a value N C = 41 (ωhf /Γ)2 , slightly before the maximum phase shift is reached. At
p−1
3q/8N C(ωhf /Γ)2 .
larger N C, the spectroscopic enhancement scales as ξm,
opt =Taking the quantum efficiency to be q = 1, the maximum spectroscopic enhancement for
87 Rb is quite large at 28 dB. The exact details of the full measurement sequence (i.e., whether

rotations such as π-pulses are used to cancel sources of technical noise) are needed to construct an
optimal estimator of Jz , but, at best, a 3 dB further improvement may result.
Because C does not depend on the cavity length, the optimum N for peak spectroscopicenhancement scales as (w02 /F )(ωhf /Γ)2 , where w0 is the cavity mode waist, and F is the cavity
finesse. More fundamentally, no change in the cavity geometry (w0 and l) or finesse F changes the
maximum obtainable enhancement in spectroscopic sensitivity. This enhancement is determinedsolely by the atomic properties. Figure. G.1 shows the spectroscopic enhancement versus atom
number for a range of technologically feasible cavity finesses.
Resolving very small phase deviations or small frequency shifts imposes technical challenges
that are modified with cavity geometry or finesse, as shown by the probe frequency (Fig. G.2) andprobe phase shift (Fig. G.3) resolutions required to obtain the spectroscopic sensitivities shown in
Fig. G.1. All three figures assume the same cavity geometry as ours, l = 1.91 cm and w0 = 71 µm.
Finally, we note that Fig. G.2 shows the technical requirement on probe frequency resolution ismore relaxed above the optimum N compared to achieving the same spectroscopic enhancement
at a value below the optimum N .

196

-1
,

BIBLIOGRAPHY

Figure G.1: The fundamental optimum spectroscopic enhancement for differential probing of the
87 Rb clock transition, as performed in Ref. [100]. The calculations are performed for our current
cavity geometry and assume a net quantum efficiency of q = 1. The purple, light blue, green,orange, and red curves correspond to cavity finesses of F = 102 , 103 , 104 , 105 , and 106 , respectively.
All these finesses are experimentally feasible. The atom number N at which the spectroscopicenhancement is maximized scales with the cavity mode waist w0 and cavity finesse F as w02 /F .

BIBLIOGRAPHY

197

6

10

]zH[ noituloseR ycneuqerF eborP

5

10

4

10

Cavity Finesse
2

10

3

10

3

10
4

10

2

10

5

10
6

10

1

10
0

10

1

10

2

10

3

10

4

10

5

10

6

10

Atom Number

N

7

10

8

10

9

10

10

10

Figure G.2: Frequency resolution with which the relative frequency of the probe and dressed cavity
mode must be measured to obtain the spectroscopic enhancements shown in (and under the same
conditions as) Fig. G.1 . The purple, light blue, green, orange, and red curves correspond to cavity
finesses of F = 102 , 103 , 104 , 105 , and 106 , respectively.

BIBLIOGRAPHY

198

-1

10

]dar[ noituloseR tfihS esahP eborP

Cavity Finesse
6

10
-2

10

5

10

4

10

-3

10
3

10

2

10

-4

10

-5

10
0

10

1

10

2

10

3

10

4

10

5

10

6

10

Atom Number

N

7

10

8

10

9

10

10

10

Figure G.3: Phase resolution with which the transmitted probe light must be measured to obtainthe spectroscopic enhancements shown in (and under the same conditions as) Fig. G.1. The purple,
light blue, green, orange, and red curves correspond to cavity finesses of F = 102 , 103 , 104 , 105 , and
106 , respectively.