Microwave Electromechanics: Measuring and Manipulating
the Quantum State of a Macroscopic Mechanical Oscillator
byJennifer Wightman Harlow
A.B., Harvard University, 2007
M.S., University of Colorado, 2010

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This thesis entitled:
Microwave Electromechanics: Measuring and Manipulating the Quantum State of a Macroscopic
Mechanical Oscillator
written by Jennifer Wightman Harlow
has been approved for the Department of Physics

Prof. Konrad W. Lehnert

Prof. Cindy A. Regal

Date

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

iii
Harlow, Jennifer Wightman (Ph.D., Physics)
Microwave Electromechanics: Measuring and Manipulating the Quantum State of a Macroscopic
Mechanical Oscillator
Thesis directed by Prof. Konrad W. Lehnert

In the past several years, the field of optomechanics has progressed from proof-of-principle
experiments to the realization of mechanical oscillators and measurements in the quantum regime.
Mechanical oscillators are of great interest because they can have small dissipation rates, cancouple to many different systems of interest, and are the fundamental elements of ultrasensitive
force detectors. Coupling these mechanical oscillators to microwave or optical fields provides a twofold advantage. Firstly, information about mechanical position can be encoded in the interrogating
field, enabling sensitive readout of the mechanical oscillator. Secondly, the radiation pressure
force of that field can be used to control the state of the mechanical oscillator. Including a highquality microwave or optical cavity enhances both of these effects, as the field strength is resonantly
increased.
The major questions in the field of optomechanics in the last several years have dealt withusing mechanical oscillators for ultrasensitive measurements and as tools for quantum information.
Both of these goals have the prerequisite that we be able to read out the motion of the mechanicaloscillator in a quantum efficient manner. To that end, we developed a nearly shot-noise limited
microwave interferometer capable of measuring mechanical motion with an imprecision below thatat the standard quantum limit. This achievement is not only a critical improvement for the electromechanical experiments we do, but is also an important tool for any experiment that encodes
the information of interest in microwave fields. In order to use mechanical oscillators as tools
for quantum information, the mechanical oscillator must also be cooled into the quantum regime
and fully controllable by the interrogating fields. To this end, we used the radiation pressure of
microwave fields to cool our macroscopic mechanical oscillator to less than one phonon. We also

ivdemonstrated coherent transfer between itinerant microwave states and the mechanical oscillator,even for incident fields with less than one photon of energy.
These accomplishments have set the foundation for further experiments to extend the quantum information abilities of optomechanical systems, couple diverse quantum systems via a mechanical intermediary, and potentially explore the foundations of quantum mechanics at macroscopic
scales.

Dedication

In memory of Ethan.

vi

Acknowledgements

There are many people who deserve thanks for their contributions of one kind or another tomy journey through graduate school.
I’d like to first thank Cindy Regal, James Thompson, Michael Hermele, Scott Bunch, and
Jun Ye for agreeing to be on my PhD defense and/or my Comps III committee and providing
valuable feedback on my work.
I want to thank all of the people who have worked on the Microwave Electromechanics
experiment, either as a Lehnert Lab member or collaborator: Cindy Regal, John Teufel, Tobias
Donner, Mike Demoret, Reed Andrews, Tauno Palomaki, Kat Cicak, and Ray Simmonds. All ofthese people have significantly contributed to the fabrication, data taking, analysis, understanding,
and/or publications contained within this dissertation. In particular, I’d like to thank John forteaching me most of what I know about optomechanics, exhibiting endless patience with all of thestupid questions I have asked over the years, teaching me how to count to 10, and making mememorize h̄.
I additionally want to thank everyone else who has been a member of the Lehnert Lab with
me: Nathan Flowers-Jacobs, Manuel Castellanos-Beltran, Françios Mallet, Scott Hoch, Will Kindel,
Hsiang-Sheng Ku, Adam Reed, Joe Kerckhoff, Michael Schroer, Brad Mitchell, Mehmet Anil, andGerwin Koolstra. We may not share publications or work together directly, but they certainly
contributed in countless ways to the day-to-day success of my individual work and the success of
our experiments as a whole.
Many thanks also go out to the support staff and shops of JILA, without whom we could

vii
not do what we do. In particular, I owe endless thanks to JR as well as everyone in the JILA
instrument shop for always being friendly and helpful beyond my expectations.
I thank the many people have read and given their expert feedback on different sections of
this document: Will Kindel, Brad Mitchell, Adam Reed, Bob Peterson, Joe Britton, Ryan Bowler,
Tom Purdy, Tauno Palomaki, John Teufel, Manuel Castellanos-Beltran, and Justin Bohnet. I
thank Mike Foss-Feig and Dominic Meiser for providing invaluable insights on some of the theory
contained in this dissertation.
I also want to thank the people who have gone above and beyond to support me personallyat some point in the last six years: Craig Hogle, Ethan Townsend, Chris Rosen, Phil Makotyn,
Phoenix Dai, Tobias Donner, John Teufel, Danny Goodman, Rob Barton, Karen Lauffer, Mike
Foss-Feig, Will Kindel, Alejandra Collopy, Teesa Christian, Steve Reaser, Cathy Klauss, Yomay
Shyur, everyone at Wardenburg PHP, and the many other people who have been there for me.
Lastly, I would like to thank Konrad, who fits in every category listed above. Obviously, he
has taught me many things about physics, helped me through many problems both scientific andtechnical, and guided my scientific path through graduate school. However, he has gone above andbeyond the minimum responsibilities of an advisor many times and I would like to acknowledge someof those efforts. I very much appreciate Konrad’s extensive involvement and investment toward
production of high quality talks and publications. Not only has this resulted in many works that I
am proud of, but it has taught me critical professional communication skills that will be relevant
to anything I do in my life. Furthermore, Konrad has read and given feedback on every single
part of this document, in many cases helping to make it more correct, complete, understandable,
and helpful. But most of all, I am particularly thankful for Konrad’s best quality: his willingness
to listen, consider, and adapt to difficult situations and concerns. This quality has helped us toovercome many difficulties and made it possible for me to complete my PhD. Thank you!

Contents

Chapter
1 Introduction to optomechanics
1.1

1

Optomechanical systems as ultrasensitive detectors of motion . . . . . . . . . . . . .

2

1.1.1

Examples of optomechanical systems as ultrasensitive detectors . . . . . . . .

4

1.2

Mechanical control by dynamical radiation pressure forces . . . . . . . . . . . . . . .

9

1.3

Quantum measurement and backaction . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4

1.3.1

Quantum measurement and the associated Heisenberg uncertainty relations . 15

1.3.2

Optomechanics experiments exploring quantum backaction . . . . . . . . . . 17

Macroscopic mechanical oscillators near the ground state. . . . . . . . . . . . . . . 18

1.4.1

Achievements of ground state occupancy of macroscopic mechanical oscillators 19

1.4.2

Quantum effects in macroscopic mechanical oscillators . . . . . . . . . . . . . 20

1.5

Quantum information applications of optomechanics . . . . . . . . . . . . . . . . . . 23

1.6

Tests of quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.7

My contributions to the field of optomechanics . . . . . . . . . . . . . . . . . . . . . 24

2 Theory I: General mathematical formalism
2.1

27

Resonant circuit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1

Simple series RLC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.2

Analogy of parallel RLC circuit to series RLC circuit . . . . . . . . . . . . . . 29

2.1.3

External coupling to measurement . . . . . . . . . . . . . . . . . . . . . . . . 30

ix

2.2

2.3

2.1.4

Voltage response of circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1.5

Energy stored in the circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.6

The scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1.7

Optomechanics introduced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Operator formalism and derivation of equations of motion . . . . . . . . . . . . . . . 37
2.2.1

Canonical quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.2

Operator model including noise and external cavity inputs . . . . . . . . . . . 39

2.2.3

Heisenberg-Langevin equations . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Measurable quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.1

Coherent dressed response of cavity . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.2

Mechanical oscillator spectral density and occupancy . . . . . . . . . . . . . . 44

2.3.3

Output field and voltage spectral densities . . . . . . . . . . . . . . . . . . . . 44

3 Theory II: Specific solutions to the equations of motion
3.1

3.2

46

Single cavity drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1

Input assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.2

Steady state and classical solutions . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.3

Linearized equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.4

Effective mechanical susceptibility . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.5

Phonon spectrum and mechanical occupancy . . . . . . . . . . . . . . . . . . 53

3.1.6

Important limits of the mechanical occupancy . . . . . . . . . . . . . . . . . . 56

3.1.7

Output field spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.8

Limiting cases of the output spectrum . . . . . . . . . . . . . . . . . . . . . . 60

3.1.9

Dressed cavity response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Two cavity drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.1

Input assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.2

Steady state solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

x

3.3

3.2.3

Linearized equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.4

Assumptions used to simplify the 2-drive case . . . . . . . . . . . . . . . . . . 67

3.2.5

Phonon spectrum and mechanical occupancy . . . . . . . . . . . . . . . . . . 69

3.2.6

Important limits of the mechanical occupancy . . . . . . . . . . . . . . . . . . 70

3.2.7

Output field spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2.8

Limiting cases of the output spectrum . . . . . . . . . . . . . . . . . . . . . . 73

3.2.9

Dressed cavity response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Cavity optomechanics in the time-domain . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.1

Simplification of the equations of motion . . . . . . . . . . . . . . . . . . . . . 76

3.3.2

Integrating factor solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3.3

Optimization of efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.4

Pulse shaping examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 Theory III: Interferometric linear measurement

88

4.1

Linear, interferometric measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2

Single quadrature measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3

Two-quadrature measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4

Microwave detection and amplification . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.5

4.4.1

Phase sensitive amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.2

Phase-insensitive amplification . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Quantum limits on detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5.1

Single red-detuned drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5.2

Single blue-detuned drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.5.3

Single on-resonance drive, homodyne detection . . . . . . . . . . . . . . . . . 99

4.5.4

Single on-resonance drive, heterodyne detection . . . . . . . . . . . . . . . . . 102

4.5.5

Generalized single-drive Heisenberg measurement relation . . . . . . . . . . . 102

4.5.6

Double drive scheme, sidebands not overlapping, heterodyne detection . . . . 103

xi
4.5.7

Double drive scheme, sidebands not overlapping, homodyne detection . . . . 104

4.5.8

Double drive scheme, sidebands overlapping . . . . . . . . . . . . . . . . . . . 105

4.5.9

Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5 Device design and fabrication
5.1

5.2

5.3

107

Quantities to optimize by fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1.1

Optomechanical control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.2

Force sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Wire devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.1

Wire device design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.2

Wire device fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.3

Parameters achieved with wire devices . . . . . . . . . . . . . . . . . . . . . . 128

Suspended membrane devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6 Experimental calibrations

135

6.1

JPA gain calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2

Inferring mechanical motion from measured voltage spectral density . . . . . . . . . 136

6.3

Optomechanical coupling calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.4

Amplifier added noise and cable loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7 Experiment I: Quantum efficient microwave measurement

141

7.1

Review of quantum limits on measurement. . . . . . . . . . . . . . . . . . . . . . . 141

7.2

Interferometric measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.3

Device design and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.4

Imprecision below the standard quantum limit . . . . . . . . . . . . . . . . . . . . . 145

7.5

Force sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8 Experiment II: Radiation pressure cooling to the mechanical ground state
8.1

151

Initial realization of radiation pressure cooling . . . . . . . . . . . . . . . . . . . . . . 151

xii

8.2

8.1.1

Review of radiation pressure cooling theory in the weak-coupling limit . . . . 152

8.1.2

Device parameters and interferometric setup . . . . . . . . . . . . . . . . . . 152

8.1.3

Observed change in mechanical susceptibility . . . . . . . . . . . . . . . . . . 155

8.1.4

Cooling to 140 mechanical quanta . . . . . . . . . . . . . . . . . . . . . . . . 156

Cooling to the quantum ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.2.1

Review of radiation pressure cooling theory in the strong-coupling limit . . . 160

8.2.2

Device parameters and interferometric setup . . . . . . . . . . . . . . . . . . 161

8.2.3

Damping and cooling the mechanical oscillator to the quantum ground state 165

9 Experiment III: Two-drive sideband asymmetry

170

9.1

Review of two-drive measurement theory . . . . . . . . . . . . . . . . . . . . . . . . . 171

9.2

Calibration of the two-drive experiment . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.2.1

Calibration of red-drive voltage to photon unit conversion . . . . . . . . . . . 174

9.2.2

Calibration of blue-drive voltage to photon unit conversion . . . . . . . . . . 174

9.3

Interferometric setup and device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

9.4

Measurements of sideband asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 177

10 Experiment IV: Quantum backaction and backaction evasion

181

11 Experiment V: State transfer between coherent itinerant microwave fields and a mechanicaloscillator

187

11.1 Review and details of state transfer theory . . . . . . . . . . . . . . . . . . . . . . . . 189
11.2 Coherent state transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
11.3 Calibration of inferred mechanical state . . . . . . . . . . . . . . . . . . . . . . . . . 195
11.4 Phase-coherent state transfer at the single quanta level . . . . . . . . . . . . . . . . . 196
11.5 State transfer in the strong coupling regime . . . . . . . . . . . . . . . . . . . . . . . 198
12 Conclusions and future directions

201

xiii
Bibliography

203

Appendix
A Symbol definitions

212

B Conventions

214

B.1 Fourier transform conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
B.2 Operator expectation values and spectral densities . . . . . . . . . . . . . . . . . . . 215
B.2.1 Regular Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B.2.2 Windowed Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
B.3 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
C Discussion of different Fano resonance models

219

D Component responses for interferometric detection

221

D.1 Beam splitters and photodetectors as linear detectors . . . . . . . . . . . . . . . . . . 221
D.2 Linear, phase-insensitive amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
D.3 Josephson parametric amplifiers (linear, phase-sensitive amplifiers) . . . . . . . . . . 226
D.3.1 Equation of motion and solution for the JPA . . . . . . . . . . . . . . . . . . 226
D.3.2 JPA direct and intermodulation gains . . . . . . . . . . . . . . . . . . . . . . 229
D.3.3 JPA quadrature amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
E Effective mass

232

F Microwave office simulations of wire capacitance234

F.1 Microwave Office simulations for long wires . . . . . . . . . . . . . . . . . . . . . . . 235
F.1.1 Width dependence for long wires . . . . . . . . . . . . . . . . . . . . . . . . . 235
F.1.2 Length dependence for long wires . . . . . . . . . . . . . . . . . . . . . . . . . 237
F.1.3 Thickness dependence for long wires . . . . . . . . . . . . . . . . . . . . . . . 237

xiv
F.1.4 Final formula for long wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
F.2 Microwave Office simulations for short wires . . . . . . . . . . . . . . . . . . . . . . . 238
F.2.1 Width dependence for short wires . . . . . . . . . . . . . . . . . . . . . . . . 240
F.2.2 Length dependence for short wires . . . . . . . . . . . . . . . . . . . . . . . . 240
F.2.3 Thickness dependence for short wires . . . . . . . . . . . . . . . . . . . . . . . 240
F.2.4 Final formula for short wires . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
F.3 Check of simulation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
G Wire device fabrication recipe244

H Calibration details for two-drive experiments

249

H.1 Calibration of red drive voltage to photon unit conversion . . . . . . . . . . . . . . . 249
H.2 Calibration of blue drive voltage to photon unit conversion . . . . . . . . . . . . . . 253
H.2.1 Direct calibration of blue photon number from the dressed cavity response . . 253
H.2.2 Relative photon number calibrations using high-occupancy spectra or mechanical parametric oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . 258

TablesTable
4.1

Summary of imprecisions and backactions for different cavity drive cases. . . . . . . 106

5.1

Measured device parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

H.1 Red and blue drive powers used in blue calibration example. . . . . . . . . . . . . . . 256

Figures

Figure1.1

Mechanical motion inferred from the phase shift of electromagnetic fields. . . . . . .

3

1.2

Novel optomechanical coupling schemes. . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Schematic of Cavendish’s experiment to determine the density of the earth. . . . . .

5

1.4

Soviet surveillance using optomechanics. . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.5

Optomechanical systems as ultrasensitive gravitational wave detectors. . . . . . . . .

7

1.6

Applications of optomechanical systems for ultrasensitive detection on small lengthscales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.7

Images of radiation pressure in space. . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.8

Nichols radiometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.9

Energy level diagrams for different cavity drive choices. . . . . . . . . . . . . . . . . 13

1.10 Simple scheme for position measurement. . . . . . . . . . . . . . . . . . . . . . . . . 15
1.11 Macroscopic mechanical oscillators in the ground state. . . . . . . . . . . . . . . . . . 19
1.12 Energy level diagram of sideband asymmetry. . . . . . . . . . . . . . . . . . . . . . . 21
1.13 Proposed opto-electromechanical system. . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1

RLC models of a resonant microwave circuit. . . . . . . . . . . . . . . . . . . . . . . 28

2.2

Resonant circuit coupled to measurement ports. . . . . . . . . . . . . . . . . . . . . . 31

2.3

Full circuit model including measurement ports. . . . . . . . . . . . . . . . . . . . . . 32

2.4

T-equivalent circuit representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

xvii
2.5

Operator model of an optomechanical system. . . . . . . . . . . . . . . . . . . . . . . 40

3.1

Regions of cavity nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2

Pulse storage timing diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3

Adiabatic approximation validity on different timescales. . . . . . . . . . . . . . . . . 79

3.4

Optimal transfer pulse shaping results. . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5

Transfer efficiency for different pulse shapes. . . . . . . . . . . . . . . . . . . . . . . . 86

4.1

Interferometric detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2

Model of amplification and added noise. . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3

Mechanical occupancy contributions for various drive schemes. . . . . . . . . . . . . 100

5.1

Power dependence of mechanical temperature in wire devices. . . . . . . . . . . . . . 111

5.2

Measured mechanical quality factors of tensioned aluminum wire devices. . . . . . . 115

5.3

Models for wire capacitance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.4

Illustrative top view of the fabrication process. . . . . . . . . . . . . . . . . . . . . . 122

5.5

Illustrative side view of the fabrication process. . . . . . . . . . . . . . . . . . . . . . 123

5.6

SEM images of the wire suspension process. . . . . . . . . . . . . . . . . . . . . . . . 124

5.7

Images of device sample holders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.8

SEM images of wire actuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.9

Images of visual fabrication concerns. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.10 Images of a device released using the focused ion beam. . . . . . . . . . . . . . . . . 127
5.11 Microwave circuit designs for different wire devices. . . . . . . . . . . . . . . . . . . . 129
5.12 Comparison of device parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.13 Images of a suspended membrane device.

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

6.1

Measured voltage spectral densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2

Calibration of interferometer added noise. . . . . . . . . . . . . . . . . . . . . . . . . 140

7.1

Interferometer diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

xviii
7.2

Optomechanical coupling calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.3

Inferred mechanical spectra in displacement and phonon units. . . . . . . . . . . . . 146

7.4

Power-dependence of the imprecision.

7.5

Force sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.1

Optomechanical coupling calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.2

Interferometric measurement setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.3

Optomechanically induced change in mechanical susceptibility. . . . . . . . . . . . . 155

8.4

Calibration of optimal detuning for radiation pressure damping and cooling. . . . . . 157

8.5

Mechanical spectra for several microwave drive powers. . . . . . . . . . . . . . . . . . 158

8.6

Summary of cooling data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.7

Interferometric measurement setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.8Interferometer and device calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.9

Inferred mechanical spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166. . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.10 Results of radiation pressure damping and cooling. . . . . . . . . . . . . . . . . . . . 168
9.1

Interferometric measurement setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.2

Optomechanical coupling calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9.3

Measured sideband asymmetry as a function of red-drive photon number. . . . . . . 178

10.1 Mechanical occupancy contributions near the point where backaction motion becomes dominant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
10.2 Parametric instability in the backaction evasion scheme. . . . . . . . . . . . . . . . . 184
11.1 Sequence of fields for preparation, storage, and retrieval of an itinerant state. . . . . 188
11.2 Coherent state transfer between itinerant photons and mechanical phonons. . . . . . 194
11.3 Calibration of phonon units and apparent mechanical motion. . . . . . . . . . . . . . 196
11.4 Phase-coherent state transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.5 State transfer between the cavity and mechanical oscillator. . . . . . . . . . . . . . . 199

xix
C.1 Fano resonance fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
D.1 Measurement models in the optical regime. . . . . . . . . . . . . . . . . . . . . . . . 222
D.2 Model of the Josephson parametric amplifier. . . . . . . . . . . . . . . . . . . . . . . 227
E.1 Equivalent effective mechanical oscillator. . . . . . . . . . . . . . . . . . . . . . . . . 232
F.1 Geometry for Microwave Office simulations. . . . . . . . . . . . . . . . . . . . . . . . 234
F.2 Width dependence for long wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
F.3 Length dependence for long wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
F.4 Thickness dependence for long wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
F.5 Width dependence for short wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
F.6 Length dependence for short wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
F.7 Thickness dependence for short wires. . . . . . . . . . . . . . . . . . . . . . . . . . . 241
F.8 Check of empirical simulation formulas for long wires. . . . . . . . . . . . . . . . . . 242
F.9 Check of empirical simulation formulas for short wires. . . . . . . . . . . . . . . . . . 242
H.1 Dressed response and spectra with a red-detuned drive only. . . . . . . . . . . . . . . 251
H.2 Extracted parameters from dressed response and spectral data in the presence of asingle red-detuned drive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
H.3 Examples of inconsistent response and spectral features. . . . . . . . . . . . . . . . . 254
H.4 Dressed cavity response in the presence of two drives. . . . . . . . . . . . . . . . . . 256
H.5 Summary of parameters extracted from two-drive dressed response and spectral measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
H.6 Summary of parameters extracted from many different relative calibration measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Chapter 1

Introduction to optomechanics

The field of optomechanics is currently a very popular and fast-progressing area of physics
for a variety of reasons from ultrasensitive measurement applications to quantum information processing to tests of quantum theory at macroscopic scales. The field encompasses a wide rangeof experiments using electromagnetic fields at gigahertz to terahertz frequencies and mechanicaloscillators1 with attogram to kilogram masses. The wide extent of this interest stems from thevery simple yet versatile basis for an optomechanical system; namely, a mechanical oscillator coupled to an electromagnetic field. This coupling provides two main functions. The first is that theelectromagnetic field can be used to interrogate the mechanical oscillator, picking up information
about the mechanical position. The second is that the radiation pressure force of that field on themechanical oscillator can be used to control the mechanical oscillator. In this introductory chapter,
I will focus on these two universal features of optomechanical systems and provide examples ofexperiments that utilize and explore them. Note that the term ‘optomechanics’ is usually used todescribe experiments employing macroscopic mechanical oscillators (collective motional modes ofmany atoms). These experiments are close analogs of cold ion experiments that demonstrate control over the motion of a single trapped ion. Another resource for learning about optomechanicalsystems, the effects they exhibit, and all of the experimental groups investigating these effects is a
recent review paper [1].
1Note that throughout this dissertation, I use the word ‘oscillator’ in the sense of a simple harmonic oscillator, notan electronic oscillator. This use allows more clarity in distinguishing the mechanical element from the electromagnetic
cavity or ‘resonator’. In the few instances where the mechanical element is able to generate its own oscillations, I callthese ‘self-oscillations’.

2

1.1

Optomechanical systems as ultrasensitive detectors of motion
Modern measurement tools frequently employ electromagnetic fields (at either optical orelectrical frequencies), as those fields allow for fast, non-invasive, measurement of systems. Additionally, interferometric techniques enable very sensitive detection of small phase shifts in thesefields due to a system of interest. However, electromagnetic fields are limited in their ability tosensitively read out certain quantities of interest because they cannot image with subwavelength
resolution and they couple poorly to some systems of interest, such as sound waves or gravitational
waves. In contrast, mechanical oscillators are capable of probing very small scales and can universally couple to all forces, and thus many different systems of interest. This makes them very
good intermediaries for measurement, as the mechanical oscillator can interact with the system ofinterest and then be read-out very sensitively by its interactions with an electromagnetic field.
The basic optomechanical (or electromechanical) system consists of a mechanical oscillator
coupled to an optical (or electrical) field. As an example, consider an optical field bouncing off
of a mirror attached to a mechanical oscillator (Figure 1.1(a)). The phase of the reflected wave
is very sensitive to the mechanical displacement; thus, a sensitive readout of the reflected phaseallows sensitive inference of the mechanical position. This interaction, and thus the sensitivity ofreadout, can be enhanced by coupling the mechanical motion to a high-Q electromagnetic resonance,increasing the strength of the interacting field. This situation is shown in Figure 1.1(b), where afixed, partially transparent mirror is placed a distance l away from the mechanically compliant,reflective mirror. These mirrors form a resonant cavity such that interrogating optical fields at
frequencies near the cavity resonance cause both a more intense field within the cavity as wellas an enhanced phase shift of the outgoing field. We often break this down into two steps: themechanical displacement changes the length, and therefore the resonant frequency of the cavity,
and then the resonant frequency change imparts a phase shift onto the outgoing field. This allows
us to extend the ideas from this basic model of cavity optomechanics to any system where theresonant frequency of a resonant structure is coupled to the motion of a mechanical oscillator. In

3

Figure 1.1: Mechanical motion can be inferred from the phase shift of interrogating electromagnetic
fields. (a) An incoming light field (rightgoing, blue) reflects off of a perfect mirror (black), attached
to a mechanical oscillator with mass m and spring constant k (green). The phase of the outgoinglight field (leftgoing, red) depends on the position of the mechanical oscillator. Shown are theoutgoing field reflected from the mirror for two displacements separated by dx (shown as solid ordashed mirrors and fields). (b) Adding a partially reflective mirror (gray) creates a resonant cavityof length l. Again, the steady-state rightgoing (blue) and leftgoing (red) fields are shown for thetwo positions of the mechanical oscillator. For interrogating fields near resonance, the field inside
the cavity is intensified and the phase shift from displacement dx is enhanced. (c) An analogous
system can be made in the microwave frequency regime by replacing the partially reflective mirror
by a coupling capacitor Cc , the perfectly reflecting mirror by a boundary condition (short or open),and the freely propagating optical fields by transmission lines or more generally any resonant LC
circuit with a resonant frequency that is modulated by mechanical motion.

particular, our research focuses on an electromechanical system where mechanical motion changes
the capacitance, and therefore resonant frequency, of an LC resonant circuit, once again impartinga phase shift on the outgoing electrical fields (Figure 1.1(c)).
Many optomechanical experiments rely on either the canonical optical setup of a Fabry-Perot
resonator with a movable mirror [2, 3, 4, 5, 6] or the canonical electrical setup of a mechanicallyvariable capacitance [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. However, these are not the only ways torealize coupling of mechanical motion to the resonant frequency of an electromagnetic resonant
structure. A few examples of optomechanical systems that employ novel coupling mechanisms
are shown in Figure 1.2. Figure 1.2(a) depicts a microtoroid resonator that supports whispering
gallery optical modes around its circumference. Mechanical motion of the toroid can change thiscircumference, thereby changing the resonant frequency for the circular modes [17, 18]. Figure

4

Figure 1.2: Novel optomechanical coupling schemes. Shown for each example are the mechanicalelement (green) and right and left traveling optical modes (red and blue). (a) A microtoroid’s
mechanical motion modifies the circumference for whispering-gallery optical modes. (b) A dielectric
harmonic oscillator dispersively couples to an optical cavity mode. (c) A photonic crystal patternedmechanical beam supports both mechanical and optical modes.

1.2(b) shows a dielectric material confined to a harmonic well and placed in the middle of a rigid
optical cavity. This dielectric material couples to the cavity resonance frequency in analogy todispersive coupling in atomic systems. The dielectric object can be made out of a thin membrane[19, 20, 21], a nanoparticle [22], or even the center of mass motion of an atomic cloud [23, 24, 25].
Figure 1.2(c) shows a mechanical beam that has been patterned with a photonic crystal structure,
allowing photonic and phononic modes to exist and couple in a single structure [26].

1.1.1

Examples of optomechanical systems as ultrasensitive detectors

While the field of optomechanics has recently exploded with many different schemes to optimize sensitive detection and control, the central idea of coupling electromagnetic fields to a mechanical oscillator for ultrasensitive measurement is not new. In fact, Henry Cavendish’s experimentin 1798 to detect the density of the earth could be argued to be optomechanical in nature (seeFigure 1.3 and reference [27]). Cavendish’s experiment measured the angular displacements of two
small masses hung from a torsional pendulum due to the presence of two larger, fixed masses. As
the entire experiment was enclosed in a thick wooden box to isolate it from wind and temperature
gradients, Cavendish read out the displacement of the masses by illuminating the masses with alamp and then observing the result with a telescope. In this way, he used mechanical oscillators toconvert gravitational forces to optically detectable information.

5

Figure 1.3: Schematic of Cavendish’s experiment to determine the density of the earth, adaptedfrom [27]. (a) Two small masses (which I have false colored red) are free to rotate on a torsional
pendulum (yellow). Angular displacements due to their gravitational interaction with two heavier,fixed masses (blue) are read out via light from a lamp (orange) and detected by a telescope (green).
The density of the earth was determined by comparing these measurements with measurementsof the weights of the masses and the torsional force of the pendulum wire. (b) Closer view ofthe optomechanical readout mechanism. I have included an illustration of the incident (blue) andreflected (red) light.

Figure 1.4: Soviet surveillance using optomechanics. (a) Henry Cabot Lodge, Jr., US ambassador
to the United Nations, reveals the Great Seal bug to the UN. Image from [28]. (b) Illustrationof the surveillance device contained within the Great Seal. Image adapted from [29] and included
with permission from Dorling Kindersley.

6Another early optomechanical device was cleverly employed as a surveillance bug by the
USSR during the cold war. In 1946, the United States ambassador in Moscow received a two-footwooden replica of the Great Seal of the United States as a gift from the Soviets (see Figure 1.4 anda much more complete historical description at reference [30]). What the US ambassador did notknow was that the seal was outfitted with a bug designed to allow eavesdropping by Soviets locatedoutside of the building. The bug (illustrated in Figure 1.4(b)) consisted of a diaphragm attached toa resonant cavity. When sound waves hit the diaphragm, they changed the resonant frequency ofthe cavity and also the charge on an attached antenna. Radio waves were reflected off the antennafrom a van parked outside the embassy building. Demodulation of these waves allowed access to
information about sounds in the ambassador’s office. This device was particularly clever because
it was passive and therefore could be entirely contained in the Great Seal with far less chance of
discovery than an active bug. In fact, it took the US six years to eventually find the device.
The Cavendish experiment and the Great Seal bug are surely some of the first uses of optomechanical devices for ultrasensitive measurement. However, neither of these examples reallyrequired sensitivities approaching the fundamental limits of detection. It was instead the search forgravitational waves that brought cavity optomechanics and ultrasensitive interferometric detection
together with the goal of understanding and approaching the ultimate sensitivity. The goal of gravitational wave detection is to observe a propagating curvature of spacetime by determining verysmall changes in lengths. Two main schemes, one electromechanical and one optomechanical, fordetecting such small changes are described and analyzed in detail in a 1978 paper by Braginsky [31].
The first, which had already been experimentally attempted at the time, involved using a Weberantenna or bar. This device was a long cylinder that would support vibrations upon interaction
with a gravitational wave. These vibrations could then be amplified and read out via an electric
circuit (see Figure 1.5(a)). Since the initial proposals, Weber antenna experiments for gravitational wave detection have made vast improvements [7] and are still being carried out by multiple
groups [9, 11]. The second scheme proposed by Braginsky, which at the time was still in the earlystages experimentally, involved using a Michaelson optical interferometer to detect differences in the

7

Figure 1.5: Optomechanical systems as ultrasensitive gravitational wave detectors. (a) Electromagnetic readout scheme for measuring vibrations of a Weber bar due to interaction with a gravitational
wave. The Weber bar is modeled as two masses m on a spring, separated by a distance l and iscoupled to a driven LC circuit. Image from [31]. (b) Interferometric scheme proposed for measuring
differential perpendicular displacements of two masses m. Also from [31]. (c-d) Photograph (from[32]) and schematic (from [2]) of LIGO, demonstrating the realization of large-scale optomechanicalexperiments for detecting gravitational waves. Images included with permission from LIGO andElsevier.

lengths of two perpendicular arms (see Figure 1.5(b)). This idea of an ultrasensitive interferometricdetection of mechanical motion has evolved from its primarily theoretical foundation in 1978 to avery large experimental collaboration, the Laser Interferometer Gravitational Wave Observatory(shown in Figures 1.5(c-d)). Additionally, the long-term interest in optomechanical detection by
the gravitational wave community is responsible for much of the foundational theoretic work uponwhich the field of optomechanics is built.

8
There are also two, more recently conceived, experiments that use optomechanical coupling
for applications requiring extremely sensitive measurement at small length scales. The first, atomic
force microscopy (AFM) is a widely-used technique for imaging surfaces with nanoscale resolution(Figure 1.6(a)). AFM employs a very sharp tip attached to the end of a cantilever (essentially adiving board mechanical oscillator) that can interact with a very small surface area of the sample
of interest. Surface forces on the tip are translated into deflections of the cantilever, which can thenbe read out by an optical field reflected off of the cantilever. The use of the mechanical oscillator
as an intermediary allows spatial measurement resolution on scales far smaller than what could
be achieved by interrogating the sample directly with an optical field. The second ultrasensitivemeasurement application aided by the use of an optomechanical system is magnetic resonance force
microscopy (MRFM). MRFM is similar to AFM, except that the force of interest is not the normal
force between the cantilever tip and the sample, but rather the magnetic dipole force between
nuclear spins in the sample and a nearby magnetic particle (Figure 1.6(b)). MRFM is also different

Figure 1.6: Applications of optomechanical systems for ultrasensitive detection on small lengthscales. (a) Artistic rendering of an AFM. A sharp tip attached to a cantilever (both white) interacts
with the sample surface (blue). The force imparted on the tip causes displacements of the cantilever,which can be read-out by a reflected optical field. Image from [33], and included with permission
from JILA. (b) Artistic rendering of an MRFM experiment. Forces between a magnetic tip andthe nuclear spins in the sample cause displacements of a cantilever, which can again be read-out
with a reflected optical field. MRFM allows a three dimensional image of the sample by using an
external magnetic field to control the position of a resonantly sensitive slice of space (in a similar
scheme to NMR). Image from [5] and included with permission from D. Rugar.

9from AFM in that the obtainable information is not limited to the surface of the sample. It uses
the techniques of nuclear magnetic resonance (NMR) imaging to obtain a three dimensional imageof the sample. So far, the best force sensitivity achieved by an optomechanical system used for
1/2

MRFM was SF F = 0.82 aN/Hz1/2 , as was demonstrated with a 5 kHz mechanical cantilever [34].
The ultimate goal of MRFM is to reach sensitivities good enough to detect single nuclear dipoles
with forces on order 10 fN. This realization would be particularly exciting, as groundbreaking
applications for three dimensional images of nanostructures such as molecules and proteins areabundant. Cold ion systems have also produced exceptionally good force sensitivities, reaching
1/2

SF F = 390 yN/Hz1/2 [35] and are proposed as ultrasensitive detectors of electric and magnetic
fields [36]; however, it may be more difficult to integrate these systems with samples of interest for
applications like MRFM.

1.2

Mechanical control by dynamical radiation pressure forces
At the beginning of this chapter, I emphasized that there were two functions that an optome-

chanical coupling provides. The first, which was introduced in the first section of this chapter, wasultrasensitive measurement of the mechanical position through a phase shift in the interrogating
electromagnetic field. The second function is the ability to control the mechanical oscillator using
the radiation pressure force from the electromagnetic field. The basic idea behind radiation pressure forces is simple to understand. If a continuous electromagnetic wave with power P is perfectlyreflected from a surface (I assume the surface is moving at speed much less than c), then the force
on that surface will be F = dp/dt = (2/c)dE/dt = 2P/c. The idea that we can use this radiationpressure force of photons to control a mechanical oscillator is also easy to understand if you haveever pushed someone on a swing. If you apply a force to the person on the swing once per oscillation
as they are moving away from you, as you typically do when pushing a swing, you amplify theirmotion. If you instead decide to apply a force to the person on the swing once per oscillation as
they are moving toward you, you deamplify their motion. Thus, the phase with which the radiationpressure force acts can cause either damping or amplification of harmonic motion.

10
Ideas about and observations of radiation pressure forces have a long history. Initial theoretic
understanding of radiation pressure and observations of its effect on comets go back to Kepler andMaxwell. The tails of comets (as seen in Figure 1.7(a)), which are made of gas and dust debris fromthe comet nucleus, are observed to always point away from the sun due to forces from solar radiationpressure and solar wind [39]. Along with these observations came the idea of using radiation pressure
forces to propel a spacecraft, known as a solar sail. Until recently, such solar sails were limited
to theory and science fiction; however, in 2010 the Japan Aerospace Exploration Agency (JAXA)
launched and successfully demonstrated solar sailing with the Interplanetary Kite-craft Accelerated
by Radiation of the Sun (IKAROS). In 2011, NASA also launched and demonstrated a solar sailcalled NanoSail-D [40]. To give some sense of the scale of forces involved, the solar radiationpressure force on IKAROS (which is predominantly reflecting and has surface area 200 m2 ) when
it was near Venus (0.73 AU from the sun) is approximately Frp = 3 mN. By contrast, the solar
radiation pressure force on Halley’s comet (which is predominantly absorbing and has diameter 10km) at its closest distance to the sun (0.6 AU) is approximately Frp = 1 kN. However, if we instead
look at the accelerations (to account for the different scales of mass), we would find a = 10 µm/s2
for IKAROS and a = 5 pm/s2 for Halley’s Comet. Forces due to solar wind also effect these objects,

Figure 1.7: Images of radiation pressure in space. (a) Image of Halley’s comet displaying a longtail, from [37]. (b) Image of the IKAROS solar sail from [38].

11sometimes in much more complicated ways [41]. However, the absolute magnitude of the solar windforce on IKAROS, for example, is Fsw = 1 µN, a quantity negligible compared with the radiationpressure force.
The first quantitative measurement of the radiation pressure force was done in 1901 by E. F.
Nichols and G. F. Hull, using a so-called Nichols radiometer [42]. This experiment was similar to
Cavendish’s experiment described earlier in that it used a torsional pendulum to measure the force
and the rotation was read out from outside an enclosure using telescopes aimed at the pendulum.
The apparatus is shown in Figure 1.8 and consists of a hanging wire with two surfaces, one reflecting
and one absorbing. By shining light on these surfaces, the angle of rotation of the pendulum provides
a measure of the differential radiation pressure force on the two surfaces.
More recently, radiation pressure forces have been used to manipulate systems of interest.
Most notably, radiation pressure damping is the mechanism behind atomic cooling, which was

Figure 1.8: Nichols radiometer, adapted from [42]. The inset shows the torsional pendulum thatrotates about the axis defined by a and b, from which hangs one reflecting surface (I false coloredthis red) and one absorbing surface (colored purple). The main figure shows the entire experimental
apparatus viewed from above. The pendulum hangs from the center of a vacuum enclosure. Light(yellow) shines onto the pendulum, exerting a force on both surfaces. The light is aligned and therotation of the pendulum is read out using telescopes (green).

12
first demonstrated in 1978 [43] and is a frequently-used tool of atomic physics. This same radiationpressure force applied to optomechanical experiments leads to all of the same effects and is described
extensively throughout this dissertation.
In order to think about radiation pressure forces in a cavity optomechanical context, we returnto the system model in Figure 1.1(b). The optical resonator is a Fabry-Perot cavity made of one
partially reflecting mirror and one fully reflecting mirror, separated by distance l. This structure
resonates at frequency ωc = πc/l and has a total loss rate κ associated with internal losses as wellas coupling to the itinerant optical modes through the partially reflecting mirror. The amountof energy stored in the cavity can be quantified in terms of number of photons nc = E/(h̄ωc ).
The fully reflecting mirror is mechanically compliant, with mass m and spring constant k. The
presonance frequency of this mechanical oscillator is Ωm = k/m and the loss rate is Γm . Theenergy of the mechanical mode can be quantified in terms of number of phonons nm = E/(h̄Ωm ).
The combined optical cavity and mechanical oscillator can be understood by looking at an energy
level representation of their state (Figure 1.9). The number of mechanical phonons increases fromleft to right, and the number of photons increases by one from the lower to upper row of states.
Any state is separated in energy from the neighboring photon states by h̄ωc and the neighboring
phonon states by h̄Ωm . The linewidth κ of the cavity appears by broadening the levels, determining
the bandwidth for resonant processes. Assuming that Γm is much smaller than all of the other rates
in the problem, it does not play a role in the qualitative understanding of cooling presented here.
Figures 1.9(a,c,e) show a representation of the different frequencies involved in each of threedifferent cavity drive schemes. Each of these images shows the cavity lineshape with the resonant
frequency ωc and linewidth κ, as well as the cavity drive (shown as a tall arrow). The resulting
output fields (called sidebands on the drive) are shown as short arrows, with line thickness indicating
intensity. Figures 1.9(b,d,f) show the accompanying energy level diagrams. The straight arrows
show the cavity drives, while the wavy arrows show photons being emitted from the cavity. Only
processes where a mechanical phonon is added or destroyed are shown. In Figures 1.9(a,b), thecavity drive is applied at the cavity resonance frequency. Two processes are present that add or13

Figure 1.9: Energy level diagrams for different cavity drive choices. For each drive choice, afrequency diagram and energy level diagram are shown. Field intensities are indicated by arrow
thickness. The three drive choices are (a,b) on-resonant, (c,d) blue-detuned, and (e,f) red-detuned.

destroy phonons. First, an on-resonant photon can be converted to a higher frequency photon by
extracting a phonon of energy from the mechanical oscillator: h̄ωc nc + h̄Ωm nm + h̄ωc → h̄ωc nc +h̄Ωm (nm − 1) + h̄(ωc + Ωm ). Alternatively, the on-resonant photon can be converted to a lower
frequency, giving up a phonon to the mechanical oscillator: h̄ωc nc + h̄Ωm nm + h̄ωc → h̄ωc nc +h̄Ωm (nm + 1) + h̄(ωc − Ωm ). These processes are both equally off-resonant and the sidebands they
produce will therefore be equally filtered by the cavity. The result is that, for a large mechanicaloccupation Fock state, no net energy will be added or removed from the mechanical oscillator.
Note that for even a large occupation mechanical state, there will be a significant contribution
from the ground state and the asymmetry of these processes in that case (discussed later in thisintroduction) will exert quantum fluctuations that drive the mechanical oscillator. Figures 1.9(c,d)
show the case of a drive detuned above the cavity resonance frequency (blue-detuned) by themechanical frequency. This drive is now resonant with the process that down-converts the photon
by giving a phonon to the mechanical oscillator. The other process is highly suppressed by thecavity response. Therefore, the blue-detuned drive preferentially gives up energy to the mechanicaloscillator, amplifying its motion. Figures 1.9(e,f) show the case of a drive detuned below the cavity
frequency (red-detuned). In this case, the resonant process is the one that up-converts photons by14extracting energy from the mechanical oscillator. The net effect of this process is thus to cool theoscillator. The cavity linewidth κ and relative positions of the sidebands determine the balance of
the up and down-conversion processes. The preference for processes with sidebands near the cavity
frequency can be attributed to resonant vs. off-resonant energy level transitions, a higher densityof available states near the cavity frequency, or simply suppression of sidebands far off resonance by
the cavity response, but these are all different ways of saying the same thing. For optomechanicalsystems in the resolved sideband regime (Ωm  κ, also sometimes called the ‘good cavity limit’ in
our field), the imbalance between sidebands due to a drive detuned from the cavity frequency by±Ωm is large and the off-resonant sidebands can be ignored. However, for optomechanical systems
not in the resolved sideband regime (Ωm  κ, also sometimes called the ‘bad cavity limit’ in
our field), the imbalance is smaller and the off-resonant process can limit the potential cooling or
amplification.
Cooling the mechanical oscillator into its quantum ground state was a long-standing goalof the optomechanics field, inspired by ground-state cooling of trapped atomic ions [44], as it
is a prerequisite to many interesting experiments involving quantum information and studies offundamental quantum theory. Therefore, nearly every recent cavity optomechanical experimenthas demonstrated radiation pressure damping and cooling to some degree or another [7, 8, 45, 19,
46, 12, 47, 21]. References [48, 49] have used radiation pressure cooling to achieve ground statemechanical occupation, and will be discussed more fully later in this introduction.

1.3

Quantum measurement and backaction
As measurements of mechanical position using optomechanical systems become more andmore accurate, we might wonder what sets the fundamental physical limits on how sensitive ameasurement is possible. The answer is that the quantum properties of light enforce a Heisenberg
principle between the measurement uncertainty of the mechanical position and the perturbation of
the mechanical momentum. I first describe this uncertainty closely following reference [50], which I
find to be a very good introductory reference on quantum measurement. I then take a look at a few

15of the optomechanics experiments that have been able to see the enforcement of the uncertainty
principle on quantum measurement.

1.3.1

Quantum measurement and the associated Heisenberg uncertainty relations

I consider an example measurement of the position of a free mass m where a wave packet
with a single photon of energy (E = h̄ω) is reflected off of the mass (illustrated in Figure 1.10). The
position of the mass can be inferred by measuring the time t it takes for the photon to leave from andreturn to x = 0, such that xm,meas = ct/2. To understand the uncertainty in this measurement, weneed to consider the both the wave-like and particle-like properties of the photon wavepacket. If the
wave packet has duration τ , then the uncertainty in its frequency is ∆ω = 1/τ and the uncertainty
in its position (relative to its mean position xp ) is ∆xp = cτ /2. These relationships combine
to put a Heisenberg limit on the simultaneous knowledge of the displacement xp and momentum
pp = h̄ω/c of the photon: ∆pp ∆xp ≥ h̄/2. This Heisenberg uncertainty between the simultaneous
photon position and momentum knowledge will then lead to an uncertainty in the mass’ position∆xm,meas = c∆t/2 = ∆xp /2 = cτ /4. We could therefore imagine simply making the photon wave
packet shorter and shorter in duration to get a more and more accurate measurement. However,this increase in sensitivity is accompanied by an increasingly strong disturbance back on the mass.
Conservation of momentum pp,i +pm,i = pp,f +pm,f reveals the change in momentum of the mass due

Figure 1.10: Simple scheme for position measurement. A single-photon wavepacket with instantaneous position xp , position uncertainty ∆xp , and momentum pp = h̄ω/c can be reflected off ofa mass m with displacement xm and momentum pm . The mechanical position can be inferred bymeasuring the round trip time for the photon to travel from x = 0, reflect off the mass, and returnto x = 0.

16
to the measurement: pm,f −pm,i = (Ep,i +Ep,f )/c ≈ 2h̄ω/c. Here I have assumed that the speed of themass is much smaller than c. This momentum change of 2h̄ω/c is due to the predictable radiationpressure described in the previous section. However, an extra unknown momentum perturbationarises from the uncertainty in ω, such that ∆pm,perturb = 2h̄/(cτ ). This perturbation will growfor large τ , enforcing a Heisenberg principle on measurement between the measured uncertainty inthe mass’s position and the backaction perturbation on momentum: ∆xm,meas ∆pm,perturb ≥ h̄/2.
Note that this uncertainty principle is not the same thing as the uncertainty principle that would
limit simultaneous knowledge of a mechanical oscillator’s canonically conjugate quantum variables(the mass considered here was described as having a definite displacement xm and position pm , acompletely classical description!) Instead, it is an uncertainty put on quantum measurement as aconsequence of the Heisenberg uncertainty of the photon. Sometimes the backaction is attributed
to the ‘shot noise’ of the photon, but this shot noise is also just the consequence of the photon’s
Heisenberg uncertainty principle. Another thing to note about this interaction is that after themeasurement, the photon frequency uncertainty and the perturbation to the mass’ momentum will
be correlated. Therefore, measurements to demonstrate the presence of a quantum backaction force
on the mass can either focus on directly observing the mass’ extra momentum perturbations oron measuring correlations between perturbations of the mass’ momentum and the measured lightfield.
I will now consider the case where the mass of interest is confined to a harmonic potential, as is
the case for our optomechanics experiments. Now, perturbations of the momentum of the mass will
evolve into perturbations of its position a quarter of an oscillation later. Therefore, many individual
measurements averaged over many oscillations will no longer know about position and momentum
separately, but rather refer to the total motion of the oscillator, which can be quantified by its energy
in terms of phonon number nm = E/(h̄Ωm ) = kx2 /(2h̄Ωm )+p2 /(2mh̄Ωm )). There will be a random
contribution to the inference of mechanical motion due to the uncertainty in the measurements,
which we call the imprecision nimp
m = ∆nm,meas and an uncertainty in the motion due to the extra
backaction fluctuations nba
m = ∆nm,perturb . The quantities xm,meas and pm,perturb can be re-written

172
in terms of phonon number as nimp
= k(∆xmeas )2 /(2h̄Ωm ) and nba
mm = (∆pperturb ) /(2mh̄Ωm ).
ba
The Heisenberg uncertainty product in terms of phonon number is then nimp
m nm ≥ 1/16. The

minimum total uncertainty will occur when the contributions from imprecision and backaction areeach equal to 1/4, a point called the standard quantum limit, which will be discussed in moredetail in the theory sections of this dissertation. Reaching this point requires a perfectly quantum
efficient measurement. For a measurement employing a resonant microwave cavity, it makes sense
to express the imprecision and backaction in terms of a cooperativity C (essentially measurement
strength, proportional to the number of interrogating photons per time) rather than the variable τbaabove for a single photon. This gives nimp
m = 1/(16C) and nm = C.

There are ways to avoid the total uncertainty limit on measurements of both quadraturesof mechanical motion by using clever schemes to access only one quadrature of the motion. Mostof these schemes will not be investigated in this dissertation, but I will address the concept of a
backaction evasion measurement, where the mechanical motion is effectively probed at 2Ωm . This
could be viewed as a train of many wave packets like the one above, spaced in time by (2Ωm )−1 .
Then the backaction perturbations to momentum will always have returned to the momentum
quadrature before the next measurement, enabling a measurement of position better than that at
the standard quantum limit which introduces no extra position fluctuations due to the backaction
force. Quantum backaction evasion has been investigated theoretically in references [51, 52, 53].

1.3.2

Optomechanics experiments exploring quantum backaction

The backaction motion nba
m imparted on a mechanical oscillator due to the interrogating
electromagnetic field will be observable when that motion is comparable to the thermal motionof the oscillator nth
m . Therefore, the important parameter for observing backaction motion in anththexperiment is nba
m /nm = C/nm . There are currently three optomechanics experiments that havebeen able to observe quantum backaction by maximizing this quantity [54, 55, 56].
The first optomechanical experiment able to observe quantum backaction was reference [54],which used the collective motion of an ensemble of trapped atoms as the mechanical oscillator

18
coupled to the optical fields within a Fabry-Perot cavity. This experiment relied on minimizing thethermal motion of the atomic mode by employing simultaneous evaporative cooling of the atomic
cloud, effectively coupling the cloud to a very cold thermal bath. The figure of merit for observing
quantum backaction, C/nthm ∼ 1 was then achievable even for relatively small C. More recently, thisgroup also showed vacuum noise squeezing, another signature of quantum backaction [57].
The second optomechanical experiment to observe quantum backaction [55] employed a thin,megahertz frequency, dielectric membrane as the mechanical oscillator in the center of an optical
Fabry-Perot resonant cavity. Unlike the cold-atom experiment, the difficulty in experiments with
low-frequency mechanical oscillators is that the thermal mechanical motion is very large. In this4case, it started out at nthm = kB T /(h̄Ωm ) = 7 × 10 . Therefore, in order to observe quantum

backaction, this experiment had to achieve very high cooperativity C.
The third optomechanical experiment that observed quantum backaction [56] used a mechanically compliant photonic crystal as the optomechanical system. Rather than looking for excess
mechanical motion due to radiation pressure forces, this experiment showed the presence of quantum radiation pressure by observing squeezing of vacuum noise due to correlations between the
optical field fluctuations (shot noise) and the additional motional fluctuations of the mechanicaloscillator.
In addition to the academic quantum measurement interests behind observing quantum backaction, the quantum limits on measurement may soon become relevant in applications of ultrasensitive measurement. The LIGO collaboration expects for their next generation experiment, advanced
LIGO, to have sensitivity limited by quantum noise fluctuations over most of the frequencies ofinterest [58].

1.4

Macroscopic mechanical oscillators near the ground statePreparing a macroscopic mechanical oscillator in its quantum ground state is a prerequisite formany interesting experiments for applications in quantum information as well as tests of quantum
mechanics at larger and larger scales. These experiments in optomechanical systems follow in19
the steps of the cold atom/ion community, which initially cooled clouds of atoms in 1978 [59, 43]
and reached the ground state of motion in 1995 [44]. Since then, cold ions have been studied
extensively for use in quantum information applications. Optomechanical systems have similarpotential as tools for quantum information but are also capable of coupling to a more diverse setof quantum systems. Additionally, macroscopic mechanical oscillators in the quantum regime areinteresting for studies of the intersection between quantum mechanics and gravity.

1.4.1

Achievements of ground state occupancy of macroscopic mechanical oscillators

The first, and widely attempted, step for optomechanical experiments employing macroscopic
mechanical oscillators was to prepare the mechanical oscillator in its quantum ground state and haveread-out sensitive enough to verify this ground state occupancy. There are now three experimentsthat have demonstrated macroscopic mechanical oscillators with occupancy less than one.
The first experiment [13] to demonstrate ground state cooling of a macroscopic mechanicaloscillator uses a piezoelectric material to form the mechanical oscillator such that mechanical motion
is converted to electrical signals of a microwave circuit (Figure 1.11(a)). This experiment is very

Figure 1.11: Macroscopic mechanical oscillators in the ground state. (a) Piezoelectric mechanical oscillator coupled to a superconducting microwave circuit. Image from [13], included withpermission from Nature and A. N. Cleland. (b) Suspended mechanical membrane also coupled
to a superconducting microwave circuit [48]. (c) Simulated optical (above) and mechanical (below) modes designed to be co-localized in a photonic-phononic crystal structure. Image from [49],included with permission from Nature and O. Painter.

20
different from the two that follow in that it employs a gigahertz frequency mechanical oscillator
which, at the 25 mK temperature of this experiment, began with an occupation of less than 0.07
phonons. Therefore, no radiation pressure cooling needed to be used and the main advance overother systems employing high-frequency mechanical oscillators was the introduction of a coupled
qubit that could sensitively read-out the state of the mechanical oscillator.
The second experiment to achieve ground state cooling [48] was also electromechanical innature and used a suspended mechanical membrane as the upper plate of a resonant LC circuit’s
capacitance (Figure 1.11(b)). This mechanical oscillator has a resonant frequency of 10 MHz andtherefore has an occupancy of 40 phonons even at 20 mK. Radiation pressure cooling was used toreduce the number of phonons from 40 to 0.34. The advantages over other systems of this type thatallowed cooling to the ground state were the realization of superior electromechanical coupling andthe use of a quantum efficient microwave interferometric readout. This details of this device and theinterferometric scheme will be discussed in detail in the experimental sections of this dissertation.
The third experiment to verify mechanical occupancy less than one [49] used a nanobeam
patterned with a periodic structure designed to support localized optical and mechanical modes
that can couple via radiation pressure forces (Figure 1.11(c)). Optical input and output fields
were coupled to this structure via evanescent coupling to a fiber taper. The mechanical resonanceemployed in this experiment was also in the gigahertz frequency range, but as this was an opticalexperiment, it was only cooled to 20 K, where the mechanical occupancy was about 100. This
experiment used radiation pressure to cool the mechanical mode to 0.85 phonons. These threeexperiments represent an exciting first step toward the big goals of optomechanics in both themicrowave and optical regimes.

1.4.2

Quantum effects in macroscopic mechanical oscillators

Cooling a mechanical oscillator to an occupancy less than one hints at the idea that it could
behave quantum mechanically, but does not in itself constitute a measurement of quantum behavior.
In fact, even if the mechanical oscillator behaved completely classically down to zero temperature,21
a measurement of average occupation less than one would be possible. Therefore, measurements of
the intrinsically quantum behavior of such an oscillator are of interest.
Perhaps the simplest indication of quantum behavior is an asymmetry of the measured mechanical sidebands. This effect can be understood by returning to the energy level diagrams usedearlier in this chapter (Figure 1.12). If the optomechanical cavity is driven on resonance and themechanical oscillator is not in its ground state (Figure 1.12(a)), the processes for adding and removing energy from the mechanical oscillator will be equally probable and the measured sidebands
for these two processes will be equal in amplitude. However, if the mechanical oscillator is in its
ground state (Figure 1.12(b)), extracting energy from the mechanical oscillator is forbidden. This
therefore leads to an asymmetry in the sideband amplitudes which is a direct effect of the existenceof a lowest energy state. It turns out (as derived in detail later in this dissertation) that the upper
sideband (associated with removing energy from the mechanical oscillator) will be proportional
to the mechanical occupancy nm , while the lower sideband (associated with adding energy to themechanical oscillator) will be proportional to nm + 1. Derived from the quantum theory, this asymmetry of exactly one comes from the commutation relations of the mechanical field operators. It is,however, important to note that this is still a semiclassical result. If the world behaved completelyclassically down to zero temperature but the light field exhibited classical fluctuations with half

Figure 1.12: Energy level diagram of sideband asymmetry. (a) For most phonon levels, an on resonant cavity drive will excite two equally dominant processes, one each for increasing and decreasing
the phonon occupation. (b) If the mechanical oscillator resides in the ground state of motion, theprocess for removing phonons is not allowed. This creates an imbalance in the two processes andan asymmetry in the associated sidebands.

22a quantum worth of energy, the same result would arise. Nonetheless, seeing the asymmetry ofexactly one suggests quantum behavior and provides a check on inferences of phonon number near
the ground state. This asymmetry was first predicted [60] and demonstrated [61] in the 1980s in
cold ion systems, and was used as proof that the ground state had been reached.
The first experiment with a macroscopic mechanical oscillator to observe the sideband asymmetry [62] was an extension of the third experiment highlighted for reaching the ground state above.
As the mechanical oscillator in this experiment begins with an occupancy of 100 phonons, which
would only present a 1% sideband asymmetry, it was cooled via radiation pressure to 2.8 quanta by
applying a red-detuned cooling drive to one optical mode. They then used a much weaker readout
drive applied either red or blue-detuned to a second optical mode in order to measure each sideband
individually. The readout drive was made weak enough so as not to change the total radiation pressure forces present. This experiment demonstrated clear agreement with the quantum expectations
for the assymmetry.
Perhaps even more impressively, the first experiment to reach ground state cooling [13],which was introduced above, was able to go far past simply observing occupancy or asymmetry
suggestive of quantum behavior to being able to control that quantum behavior. This achievement
was enabled by the very low thermal occupancy of the mechanical oscillator as well as its coupling
to a superconducting qubit, a quantum system capable of very good quantum state preparationand manipulation. They were able to prepare the qubit with a single excitation and transfer
that excitation into the mechanical oscillator, realizing a single-phonon mechanical state. This
experiment demonstrated quantum control of a macroscopic mechanical oscillator unparalleled byother optomechanics experiments to date. However, the mechanical oscillator lifetime was only 6.1
ns, only slightly longer than the state transfer time and much less than typical qubit coherence
times. That the mechanical oscillator is not long-lived compared to these other timescales limits
the future prospects for more complicated quantum manipulation and makes investigation of othertypes of macroscopic mechanical oscillators with longer lifetimes still critical to the longer term
goals of the field.

23

1.5

Quantum information applications of optomechanics
Once a macroscopic mechanical oscillator is in the quantum regime, it has the potential to bea very interesting tool for storage of quantum states as well as an intermediary between otherwise
incompatible quantum systems. The history of using mechanical oscillators to store quantum statesagain goes back to cold ions. Quantum control over the states of these systems was realized in the
1990s [63] and this control has progressed from then to achieve quantum memories with coherence
times on the order of minutes and coherent state manipulation capable of performing qubit logic
gates, reviewed recently in reference [64].
Most of the optomechanics experiments presented so far (with the exception of the Clelandexperiment just discussed) have employed steady-state fields to make continuous measurements ofmechanical position and cool the mechanical motion to its ground state. However, as optomechanical systems move to explore quantum information applications, such as quantum state storage andcoupling between different quantum systems, fast dynamical control of the optomechanical coupling
will be required. In addition to the Cleland experiment, there are a few other demonstrations ofdynamical control of states in optomechanical systems. Reference [18] demonstrated the ability
to transfer an itinerant optical state into the mechanical oscillator, store it there, and recover thestate, albeit all in the classical regime. Reference [65] swapped a signal pulse with an amplitudeof a single quanta back and forth between an optical microtoroid cavity and the mechanical mode.
Lastly, reference [66] demonstrates coherent state transfer between itinerant microwave fields andthe mechanical oscillator at amplitudes of a single photon. This last experiment is discussed in thelast experimental chapter of this dissertation.
In addition to state transfer and storage of quantum electromagnetic signals in a single
optomechanical system, coherent quantum control over mechanical oscillators will open up the
possibility of coupling otherwise incompatible quantum systems. For instance, proposals have been
made for coupling optical and microwave systems via a mechanical oscillator [67, 68] (see Figure1.13).

24

Figure 1.13: Proposed opto-electromechanical system. By coupling the motion of a mechanicaloscillator to both an electrical resonator and an optical resonator, quantum information could betransferred between the GHz and THz frequency regimes.

1.6

Tests of quantum theory
As quantum mechanical effects are observed in increasingly large systems, tests of fundamen-

tal quantum theory are becoming possible. Some of these tests involve investigating increasingly
complicated quantum states [69]. However, the tests that macroscopic mechanical oscillators in thequantum regime are particularly suited for are those of the interaction between quantum mechanics
and gravity. As quantum objects approach larger and larger mass scales, the incompatibility between quantum mechanics and gravitational theory will eventually have to be reconciled. Various
proposals about what measurements might lead to advanced understanding of the intersection between quantum mechanics and gravity have been made (see [70, 71, 72, 73] and references therein).

1.7

My contributions to the field of optomechanics
There are several important ways that my PhD work contributes to the field of optomechanics,

some theoretical and some experimental.
My theoretical contributions were to extend the currently available theory to include effects
that are not treated in other works. The first of these extensions, presented in Chapter 2, was toinclude reactive measurement port components in the model for a microwave circuit in order tocorrectly extract the microwave resonance parameters. This idea had previously been investigatedin reference [74], but the model presented there produces different results, as discussed in Appendix

25
C. The second of these extensions, presented near the end of Chapter 2, was to extend the modeland analysis of an optomechanical system, such as the one presented in [75], to allow excess photon
noise due to the internal environment of the cavity as well as the measurement ports. This noise isoften assumed to be only zero point fluctuations, as the thermal occupancy for a room temperature
optical field or a dilution refrigerator microwave field is in theory zero. However, in Chapters 8.2
and 9 we observe excess cavity photons, which we model as a ‘hot’ internal cavity bath. The
inclusion of excess cavity noise from the measurement ports in the model also makes it possible
to understand the effects of excess generator or laser noise. Including both the Fano and excess
cavity noise effects, I derived, the dressed cavity response and output spectral density for both asingle cavity drive (Chapter 3.1, similar to [75]) and two cavity drives, one optimally red-detuned
and one optimally blue-detuned (Chapter 3.1, to my knowledge this has not been done). Lastly,
I worked through the problem of how to optimally shape the transfer field (which controls thecoupling between the cavity and the mechanical oscillator) in order to achieve maximal transfer,storage, and retrieval efficiency of an arbitrary itinerant microwave field (Chapter 3.3).
I also contributed to the field of optomechanics through many experiments involving electromechanical systems. The first of these ([45], Chapter 8.1) showed our initial attempts at using
the radiation pressure of the microwave field to cool the mechanical oscillator. For this experiment,
I fabricated the device (see Chapter 5), contributed to data taking and analysis, and provided feedback on the manuscript. The second experiment ([76], Chapter 7), demonstrated our developmentof a quantum efficient microwave interferometer. For this experiment, I worked on the device design,
fabricated the device (again see Chapter 5), contributed to data taking and analysis, and helped
substantially with the manuscript. In the next experiment ([48], Chapter 8.2), we demonstrated
achievement of cooling the mechanical oscillator to its quantum ground state. For this experiment,
I mainly contributed to the theoretical framework necessary for analysis, as discussed above. A
substantial part of my experimental time and understanding was devoted to looking for sideband
asymmetry, which is detailed in Chapter 9. For this experiment, I developed the full theoreticalunderstanding for two drive measurements and meticulously examined the complicated nature of

26
the calibrations required to achieve this result. The last experiment ([66], Chapter 11) showed ourachievement of state transfer of coherent itinerant microwave fields to the mechanical oscillator.
For this experiment, I provided experimental support and helped with the manuscript.

Chapter 2Theory I: General mathematical formalism

The first step toward understanding the behavior of any system of interest is to find a set of
equations (namely the equations of motion) that fully describe its evolution. Therefore, I devote
this chapter to a very careful derivation of these equations, starting from a basic resonant circuit
model of an electromechanical system. In Section 2.1, I analyze the microwave circuit used in our
experiment, and then show how coupling a mechanical oscillator to the capacitive degree of freedom
leads to an electromechanical coupling. In Section 2.2, I will then re-express the problem using
operator formalism, which will be generally applicable to all optomechanical systems, regardless of
their physical implementation in the microwave or optical regime.

2.1

Resonant circuit analysis
Our optomechanical system is formed by a mechanical oscillator in a microwave resonant

circuit in such a way that mechanical motion modulates the capacitance, and therefore the resonant
frequency, of the circuit. There are generally many ways to realize resonant circuits in the microwave
regime: they can be made as quarter or half wave coplanar waveguides or as more lumped element
series or parallel RLC circuits. They can also be coupled either inductively or capacitively toa measurement circuit. The goal of this section will be to show that it is generally possible totransform any such complex circuit network, possibly with many resonances, into a simple series
RLC resonator about one of the resonances of the complex network. I will show that this is true
for one specific example, the parallel RLC resonant circuit in series with a coupling capacitor.

28
2.1.1

Simple series RLC circuit

First, I will solve the simple RLC circuit with the foresight that the parallel RLC will be able
to be simplified to a similar form. Here I find expressions for the resonant frequency and impedancelooking into the series RLC circuit (Figure 2.1(a)). I have added 2’s as subscripts to the circuit
elements to distinguish them from similar elements in the parallel RLC circuit in Section 2.1.2.
The complex impedance of the series RLC resonant circuit isZres = jωL2 +1+ R2 ,
jωC2

(2.1)

√
where j = − −1 = −i (see Appendix B.1). At the resonant frequency of a resonant circuit,there will be equal amounts of energy stored in the electric and magnetic components. Thus,
the impedance on resonance must be due only to the resistive part, which is real. Therefore, theresonance frequency is the frequency where the imaginary part of the impedance goes to 0:
Im[Zres ] = ωL2 −

(a)

1
=0
ωC2

when

(b)

L2
R2

1
.
L2 C2

(2.2)

(c)

Cc

C2
Z res

ω = ω0 = √

L

R

V

C

Cc
C3
L2

Z resR2

Figure 2.1: RLC models of a resonant microwave circuit. (a) Series RLC circuit. (b) Parallel RLC
circuit with coupling capacitor. This is the model we generally used for our early devices. (c) Series
RLC circuit with coupling capacitor taken out of C2 . For appropriate relations between the circuit
elements in the parallel and the series cases, the total impedance looking into each of these circuits
from their input node is Zres . The impedance looking into the circuit from the node at voltage Ṽ
is Z̃res .

29
I can expand the impedance to linear order about ω0 : ω = ω0 + ∆ωZres ≈ R2 + jω0 L2 + j∆ωL2 +

2.1.2

1
∆ω
= R2 + 2j∆ωL2 .
−jω0 C2 jω02 C2(2.3)

Analogy of parallel RLC circuit to series RLC circuit

Finding the response of the parallel RLC circuit in series with a coupling capacitor Cc (Figure
2.1(b)) is more analytically complicated than the above series case, so it is helpful to simplify it bymaking an analogy to the series case. The resonator impedance for this circuit is


−j
−j
1 −1
Zres =+ jωC ++ωCc
ωL R
jωLR2 (1 − ω 2 LC)
ω 2 L2 R−j+ 2+=
ωCc R (1 − ω 2 LC)2 + ω 2 L2 R2 (1 − ω 2 LC)2 + ω 2 L2−j
ω 2 L2jωL≈+.
+
ωCc (1 − ω 2 LC) R(1 − ω 2 LC)2

(2.4)
(2.5)
(2.6)

In the last step I have taken the limit of large R  ωL (small internal loss). For ω > (LC)−1/2 ,the second term of the impedance is negative imaginary and therefore acts like a capacitance. This
capacitance adds in parallel with Cc and nothing special happens. However, when ω < (LC)−1/2 ,this term is positive imaginary and therefore acts like an inductance. This inductance can resonate
with Cc at ω = ω0 :Im[Zres ] = −

1
ωL
=0
+
ωCc 1 − ω 2 LCwhen

1ω = ω0 = p.
L(C + Cc )(2.7)

Taylor expanding the impedance about ω0 and keeping only terms to lowest order gives1 :Zres ===1j∆ωL(1 + ω02 LC)
ω02 L2j∆ω++ω02 Cc(1 − ω02 LC)2
R(1 − ω02 LC)2j∆ωL(C + Cc ) j∆ωL(2C + Cc )(C + Cc ) L(C + Cc )
+
+
Cc
Cc2
RCc2
2j∆ωL

(C + Cc )2 L(C + Cc )
+
.
Cc2
RCc2

(2.8)
(2.9)
(2.10)

Below, I will find that adding extra reactive components to the measurement circuit results in a Fano resonanceand modifies the resonant frequency by a small amount. Therefore, I really should have expanded about the newresonant frequency, ωc , instead of ω0 here. This would make extremely minor corrections to the expressions for L2 ,C2 , and R2 depending on the Fano parameters below. However, the important part about this section was that there
was a correspondence between the complex parallel circuit and the simple series RLC circuit. This will allow me touse the series RLC model to find expressions for the scattering matrix and allows extraction of the resonant frequency
and damping rate. It is not important at all that I be able to exactly relate R, L, C, and Cc to these quantities
because in practice R, L, C, and Cc are not known, and we would only ever care to know their approximate values.

30Comparing this impedance to Equation 2.3 and the resonance frequency to Equation 2.2, I can findequivalent series components:
L2 = LC + Cc
Cc

2
,

R2 =

L(C + Cc )
,
RCc2

C2 =

Cc2
.
C + Cc(2.11)

Using reasonable values from one of our devices, L ∼ 10−9 H, C ∼ 10−12 F, Cc ∼ 10−15 F, and R ∼106 Ω, I find the following values for the equivalent series components: L2 ∼ 10−3 H, C2 ∼ 10−18 F,
R2 ∼ 103 Ω.

2.1.3

External coupling to measurement

In order to drive or read out a microwave resonant circuit, we must couple it to externalmeasurement lines. Typically, this involves adding one port (in a reflection geometry) or two ports(in a transmission geometry), each with impedance Z0 . Adding only these ports will result ina purely Lorentzian response of the circuit. However, we sometimes see in experiments that thecircuit has a Fano resonant response rather than a Lorentzian. One example of such a resonancecan be found in Appendix C. A Fano resonance is generally due to the interference between a
Lorentzian resonance and a background standing wave. I attribute this to the presence of reactive
components between the measurement ports and the resonant circuit. These components can set up
an interference between standing waves in the measurement lines and the circuit response, resulting
in a non-Lorentzian resonance at the output of the experiment.
I attribute the extra reactance observed to different effects for the transmission and reflection
geometries. In the transmission geometry, the extra reactance in likely from parasitic inductance of
the wire bonds between our sample and the external microwave lines, as has also been assumed in[74]. I can model this as some frequency-independent reactance BZ0 associated with each line (seeFigure 2.2(a)). In the reflection geometry, wire bonds alone cannot account for the Fano resonance(they would just modify the resonance frequency). However, there can be a Fano shape if there issome power being coupled out of the resonator through a very small capacitor to another invisible‘port’. This has indeed been observed in some of our experiments where resonances on one side of

31
(a)
1

jBZ 0

jBZ 0

(b)
1

2

jZ 0 / J

2

(c)
1

Z res

Z res

jYZ 0

jXZ 0

2

Z res

Figure 2.2: Resonant circuit coupled to measurement ports, including sources of Fano interference
effects. (a) The transmission measurement model includes small reactive components coupling
the resonator to each port. (b) The reflection measurement model includes a very large reactive
component coupling the resonator to a second ‘port’. (c) These two situations can be thought ofas limiting cases of a more general model with reactive components XZ0 and Y Z0 .

a chip are visible from another, seemingly disconnected, port on the other side of the chip. I canmodel this power leakage by coupling the system to a second port through a very large frequencyindependent reactance Z0 /J, where J is small (see Figure 2.2(b)). The two geometries can thus betreated as two limiting cases of the more general geometry in Figure 2.2(c):Case 1:

X=Y =B

Transmission geometry including wire bonds,

Case 2:

X = 0, Y = 1/J

Reflection geometry with weakly coupled invisible ‘port’.

Here, B and J are 0 in the absence of the above effects and can be positive or negative. In our
experiments we typically measure |B| < 1.

2.1.4

Voltage response of circuit

In an electromechanical experiment, the quantity of interest is the motion of a mechanicaloscillator which varies the capacitance C of a parallel RLC circuit. The only thing that thismechanical oscillator can truly care about is the voltage Ṽ across that capacitor (Figure 2.1(b))(this statement is verified in Appendix C). Therefore, the equivalent series circuit found above is
not completely adequate as a model of the system because it does not have access to the necessary
node. However, I can make a small modification to it by modeling the capacitor C2 as two seriescapacitors: the real capacitor Cc and another capacitor C3 = Cc C2 /(Cc − C2 ) (see Figure 2.1(c)).

32
I1
jXZ 0
Z0

I2

V

jYZ 0

Cc

I3

Z0

V

Vin

Z res1

2

Figure 2.3: Full circuit model including measurement ports. An applied voltage Vin from port 1 ofthe circuit will result in a response of the voltage Ṽ across the capacitor.

The impedance looking into the circuit from the point with voltage Ṽ is thenZ̃res ≈

ω 2 L2jωL+.
(1 − ω 2 LC) R(1 − ω 2 LC)2

(2.12)

Expanding once again about ω0 and keeping terms to leading order givesZ̃res ≈jω0 L
ω02 L2j++ R2 .
=2ω0 Cc(1 − ω0 LC) R(1 − ω02 LC)2

(2.13)

Using the full model of the circuit and measurement in Figure 2.3, the voltage Ṽ can be foundin terms of the driving voltage Vin from port 1 by solving the following equations and doing basic
circuit algebra:
I1 = I2 + I3 ,Ṽ = I3 Z̃res ,V = I3 Zres = I2 Z2 = Vin − I1 Z1 ,(2.14)

where I define Z1 ≡ Z0 (1 + jX) and Z2 ≡ Z0 (1 + jY ). Solving these equations to eliminate I1 , I2 ,and I3 gives−1ṼZ̃res Z2
Z̃res Z2
Z1 Z2==
Zres +,
Vin
Zres (Z1 + Z2 ) + Z1 Z2
Z1 + Z2
Z1 + Z2
κ

Z1 Z2
Z1 Z2
2L2
Zres += R2 + 2j∆ωL2 += 2L2+ j(ω − ωc ) =,Z1 + Z2
Z1 + Z22χc [ω]
Ṽ
Z̃res Z2=χc [ω].
Vin
2L2 (Z1 + Z2 )(2.15)(2.16)(2.17)

The cavity displays a Lorentzian response with resonant frequency ωc = ω0 − Im[Z1 Z2 /(Z1 +Z2 )]/(2L2 ), total loss rate κ = (R2 + Re[Z1 Z2 /(Z1 + Z2 )])/L2 , internal loss rate κ0 = R2 /L2 , and33
external loss rate κext = Re[Z1 Z2 /(Z1 + Z2 )]/L2 = Z0 (|Z1 |2 + |Z2 |2 )/(L2 |Z1 + Z2 |2 ). I can divide up
the external coupling into separate couplings for the left and right ports, κl = Z0 |Z2 |2 /(L2 |Z1 +Z2 |2 )and κr = Z0 |Z1 |2 /(L2 |Z1 + Z2 |2 ). I have written the response in terms of the cavity susceptibilityχc [ω]. The addition of reactive components to the feed line changes the frequency at which Ṽ has
the largest response. Therefore, effects of optomechanical coupling should be centered around ωcrather than ω0 (see Appendix C). Note that, while significant for optomechanical effects, this change
in resonance frequency is not large enough to invalidate the previous assumption that a parallel
RLC circuit can be modeled as a series RLC circuit near resonance (see previous footnote). I also
define a quantity cin ≡ Vin Z̃res Z2 /(2L2 (Z1 + Z2 )) for convenience, such that Ṽ [ω]/cin [ω] = χc [ω].

2.1.5Energy stored in the circuit

Here I find the energy stored in the circuit in terms of the cavity parameters and the input
power sent into port 1 of the network. I assume that the input voltage has the formhiVin [t] = Re V̆in [t] = Re V0 [t]ejωt ,(2.18)

where V0 is complex and I could generalize this to be a sum over many frequencies if there were
multiple inputs. I assume that V0 [t] changes slowly compared to the exponential component:
V˙0  jωV0 . This allows me to write the input power (or available power) as

Pin =

[t]2

Vin
=
4Z0

h
i2
Re V̆in [t]
4Z0=V0 [t]2 e2jωt + V0∗ [t]2 e−2jωt + 2 |V0 [t]|2|V0 [t]|2|V̆in [t]|2≈=, (2.19)
16Z0
8Z0
8Z0

where I have eliminated the quickly oscillating terms. The assumption that V˙0  jωV0 also allows
me to use impedances to go between the scooped voltages and currents, even though I am working
in the time domain. This is illustrated by a calculation of the current through a capacitor with
voltage Vin across it:
h i
h
i
h
i


˘
I[t] = Re I[t]= Re C V̇0 [t]ejωt + jωCV0 [t]ejωt ≈ Re jωCV0 [t]ejωt = Re jωC V̆in [t] . (2.20)

34
The energy stored in the circuit near resonance is thenE[t] = 2=1
L2 I3 [t]2
2



2
2
L2 Ṽ˘ [t]
L2 ˘≈I3 [t] =2
2 Z̃res(2.21)

2
2|V̆in [t]|2 |Z2 |2Ṽ˘ [t]Ṽ˘ [t]= Pin κl.
8L2 |Z1 + Z2 |2 c̆in [t]c̆in [t](2.22)

If the circuit is driven at only one frequency ω, the energy stored in the circuit will be E[t] =Pin κl |χc [ω]|2 .

2.1.6

The scattering matrix

I have calculated how the voltage across the capacitor and the energy stored in the circuit
depend on all of the circuit parameters. However, the quantity actually measured in an experimentis the output microwave field. This is related to the input microwave field through the scatteringmatrix (or S matrix)[77]:[Vout ] = [S][Vin ],Sik =Vi,outVj,in(Vk,in = 0 for k 6= j) .

(2.23)

I could just calculate the S-matrix by brute force. However, there is an equivalence trick that makes
this simpler. This trick uses the impedance matrix:[V ] = [Z][I],Zik =Vi,in + Vi,out
Vi=
Ik
Ik,in + Ik,out(Ik = 0 for k 6= j) .

(2.24)For a 2-port reciprocal network such as the one of interest here, the circuit can be compared to aT-equivalent circuit (Figure 2.4) to identify the elements of the impedance matrix. For our model,
(a)
1

jYZ 0

jXZ 0
Z res

2

(b)
1

Z 22 − Z12

Z11 − Z122

Z12

Figure 2.4: T-equivalent circuit representation. (a) Our circuit of interest. (b) The impedance
matrix for our circuit can be easily found by comparing to the T-equivalent circuit.

35
this gives
Z12 = Zres ,(2.25)

Z11 = jXZ0 + Zres = Zres + Z1 − Z0 ,(2.26)

Z22 = jY Z0 + Zres = Zres + Z2 − Z0 .

(2.27)The S-matrix is then related to the impedance matrix by the expressions [77]S21 =2Z21 Z0
,
∆Z

S11 =

(Z11 − Z0 )(Z22 + Z0 ) − Z12 Z21,∆Z(2.28)∆Z ≡ (Z11 + Z0 )(Z22 + Z0 ) − Z12 Z21 .

(2.29)For our model, the scattering elements are then
S21 [ωp ] ===S11 [ωp ] =2Zres Z0
Zres (Z1 + Z2 ) + Z1 Z2
!
r ∗ ∗
√
Z1 Z2
(Z1∗ + Z2∗ ) Z1 Z2
2Z0
Z0 |Z1 ||Z2 | Ṽ [ωp ]p−
L2 |Z1 + Z2 |2 cin [ωp ](Z1 + Z2 ) Z1∗ Z2∗ Z1∗ + Z2∗ Z1 Z2!
√√Ṽ [ωp ](Z1∗ + Z2∗ ) Z1 Z2
p ∗ ∗ N2 − κl κr,cin [ωp ](Z1 + Z2 ) Z1 Z2(2.31)

Zres (Z1 + Z2 − 2Z0 ) + Z2 (Z1 − 2Z0 )Zres (Z1 + Z2 ) + Z1 Z2(2.33)=

Z2 (Z1∗ + Z2∗ )Z2∗ (Z1 + Z2 )=

Z2 (Z1∗ + Z2∗ )Z2∗ (Z1 + Z2 )Ṽ [ωp ]Z2∗ (Z1 + Z2 − 2Z0 )Z0 |Z2 |2−∗∗Z2 (Z1 + Z2 )L2 |Z1 + Z2 |2 cin [ωp ]!
Ṽ [ωp ]N1 − κl.
cin [ωp ](2.30)(2.32)!
(2.34)(2.35)

In both of these calculations, I have pulled out a factor of magnitude one in front of the expression.
This factor will only add an overall phase offset to the scattering matrix, which we do not careabout. Ignoring these prefactors, both terms of the scattering matrix can be written as one equation
with index i:
Si1 [ωp ] = Ni −

√

κl κi

V [ωp ].
cin [ωp ](2.36)In the past few expressions, Ni is a complex coefficient which stores information about any reactive
components in the feed line and is also the value of the response far off-resonance. The values of36
Ni for our two measurement models are
jB
N1 = 1+jB,
1+jJ
N1 = 1+jJ+2J2,

1,
1 + jB√2J 1 + J 2
N2 =,1 + jJ + 2J 2N2 =for the transmission geometry
for the reflection geometry.

(2.37)(2.38)

If the circuit is driven at only one frequency, the response is that of a Fano resonance,
Si1 [ωp ] = Ni −√κl κi χc [ω] .

(2.39)In the cases of B = 0 or J = 0, the response resumes a Lorentzian form.

2.1.7

Optomechanics introduced

I now consider the effect of the mechanical element, which modulates the capacitor C and thus
the resonance frequency of the circuit. There is a force F [t] = −∂E[t]/∂x, sometimes referred toas the ‘radiation pressure force’ in analogy to the optical world, exerted on the mechanical element
due to the voltage across the capacitor. When the mechanical oscillator moves infinitesimally, thecharge on the capacitor plate stays constant, while the voltage changes. Therefore, I must find thederivative of energy with respect to C with the charge Q fixed:




 2
∂E
∂ Q2
∂ωc −1 ∂ωc ∂Q
F [t] = −
=−
=−
∂x
∂x 2C
∂C
∂x ∂C 2C
2−2(C + Cc )Q
−E[t]G=G 2 ≈= −h̄Gnc [t] .
ωc
2C
ωc

(2.40)(2.41)

Here, G ≡ ∂ωc /∂x is the coupling constant between the cavity and mechanical oscillator andnc = E/(h̄ωc ) is the number of photons in the cavity.
I can Fourier transform the force into the frequency domain (see Appendices B.1 and B.3
about Fourier transform definitions and convolutions):−GL2|Z̃res |2 |Ṽ [t]|2 = −h̄Gβ|Ṽ [t]|2 ,ωcZ ∞F [ω] = −h̄GβṼ [t]Ṽ ∗ [t]e−jωt dt = −h̄Gβ Ṽ [ω] ∗ Ṽ ∗ [−ω].
F [t] =

(2.42)(2.43)−∞

The constants out front are just absorbed into β. This force causes a displacement x[ω] = χ[ω]F [ω]where, for an oscillator with mass m, resonant frequency Ωm , and damping Γm , the susceptibility is37
χ[ω] = [m(Ω2m − ω 2 + jωΓm )]−1 . Note that this susceptibility is related to the mechanical susceptibility χ−1m [ω] = Γm /2 + j(ω − Ωm ), which I will use throughout the majority of this dissertation, by
χ[ω] ≈ (χm [ω] − χ∗m [−ω])/(2jmΩm ). The mechanical motion due to this force then couples to themicrowave frequency such that ωc → ωc + Gx[ω]. This modifies how the voltage inside the cavity
relates to the voltage drive by convolving Ṽ with x:cin [ω] ==κ+ j(ω − ωc ) Ṽ [ω] − jGx[ω] ∗ Ṽ [ω]

2



Ṽ [ω]+ jh̄G2 β χ[ω] Ṽ [ω] ∗ Ṽ ∗ [−ω] ∗ Ṽ [ω].
χc [ω](2.44)(2.45)In theory, this equation should provide a solution for Ṽ , allowing me to find the cavity responseat any frequency, given any arbitrary drive cin [ω]. Indeed, I have used this equation to derive theresponse of the circuit optomechanical system in the presence of both one and two large drives. It is
important to realize this ability because it stresses the fact that the cavity optomechanical response
is a completely classical effect. I will not present this here as the derivation is rather complicatedwith all of the convolutions and is much simpler using the quantized operator formalism that
follows. However, I do want to stress that although I use quantum formalism to derive the dressed
response in the following chapters, this quantity is fully contained in the classical equations of
motion describing an LC resonant circuit coupled to a mechanical oscillator.

2.2

Operator formalism and derivation of equations of motion
So far, I have worked completely in the circuit notation of voltages and currents. However, it

is helpful going forward to change to the canonical quantization representation of these equations,in part for notational simplicity, and in part because I will eventually be interested in quantum
effects. Also note that as soon as I describe the electromechanical system in terms of operators, itwill be completely mathematically equivalent to any optomechanical system in the optical regime.
I will therefore switch at that point from calling the microwave circuit a ‘circuit’ to calling it a‘cavity’.

38
2.2.1

Canonical quantization

The classical series RLC circuit energy can be written in terms of the voltage VC2 across thecapacitor C2 and the current I3 through the inductor L2 (also the current through C2 ) or in termsof the charge Q on the capacitor:
Ecircuit =L2C2
L2 Q̇[t]2 Q[t]2.
I3 [t]2 +VC2 [t]2 =
+
2
2
2
2C2

(2.46)The Lagrangian of this system is then the kinetic part of the energy minus the potential part:
Lcircuit =L2 Q̇[t]2 Q[t]2
−
,
2
2C2

∂Lcircuit= L2 Q̇[t] = φ[t].
∂ Q̇[t](2.47)

The canonical momentum is φ, the magnetic flux through the inductor. The classical Hamiltonianis
Hcircuit =φ[t]2 Q[t]2
+
.
2L2
2C2

(2.48)

To find the quantum Hamiltonian, I can just write these canonically conjugate classical variables
as quantum operators which obey the canonical commutation relation:Ĥcavity =

φ̂2
Q̂2
+
,
2L2 2C2hiQ̂, φ̂ = −jh̄.

(2.49)

Going further, I define lowering and raising operators â and â† :r†â, â

Ĥcavityri −j hhiâ, â† =φ̂, Q̂ = 1,
h̄


h̄
h̄ωc
1
† 2
† 2
†
(â − â ) +(â + â ) = h̄ωc â â +.
= −4
4L2 C2 ωc
2
=

L2 ωc
Q̂ ∓ j2h̄

1φ̂,
2L2 ωc h̄

(2.50)(2.51)

These operators are normalized such that â† â = n̂c , the photon number operator. I also define
input and output raising and lowering operators, such that â†in(out) âin(out) = Pin(out) /(h̄ωc ). I canthen write the cavity response in terms of the operators:
! 
r
√
 √
Ṽˆ
h̄ωc
h̄ωc Z0 |Z2 |L2 |Z1 + Z2 | â
1 â√
= Z̃resâ / Z̃resâin ==√,ĉin
L2
L2 |Z1 + Z2 |κl âin
Z0 |Z2 | âin
Si1 [ωp ] = Ni −√κiâ[ωp ].
âin [ωp ](2.52)(2.53)

39
I can quantize the mechanical oscillator using raising and lowering operators in a very similarmanner to the cavity:†b̂, b̂

r
h
i −j
mωm
1
p̂,
b̂, b̂† =x̂ ∓ j[x̂, p̂] = 1,2h̄2mh̄ωm
h̄
2 x̂2
p̂2
mωm
−h̄ωm
h̄ωm
+
=
(b̂ − b̂† )2 +(b̂ + b̂† )2
2m
2
4
4


1
1
†
†
†
h̄ωm (b̂b̂ + b̂ b̂) = h̄ωm b̂ b̂ +
.
2
2

r
=

Ĥmech ==(2.54)(2.55)(2.56)

An interaction term in the Hamiltonian comes from modifying the cavity resonance frequency inthe cavity Hamiltonian:




1
1
†
†
h̄ωc â â +→ h̄ (ωc + Gx̂) â â +
2
2





1
1
+ h̄g0 b̂ + b̂†â† â += h̄ωc â† â +22=Ĥcavity + Ĥinteraction .

Here, G = dωc /dx, the mechanical zero point motion is xzp =(2.57)(2.58)(2.59)ph̄/2mωm , and g0 = Gxzp is the

single photon coupling rate. The full system Hamiltonian is thenĤsystem = Ĥcavity + Ĥmech + Ĥinteraction







1
1
1
†
†
†
†
= h̄ωc â â +.
+ h̄ωm b̂ b̂ ++ h̄g0 b̂ + b̂â â +
2
2
2
2.2.2

(2.60)(2.61)

Operator model including noise and external cavity inputs

Both the cavity and mechanical oscillator dissipate energy to the external environment. The
cavity can dissipate energy to three different ports: the left measurement port, the right measurement port, and the internal port (this is the energy dissipated in the resistor). The dissipation
rates to each port are κl , κr , and κ0 . The total cavity loss rate is κ = κl + κr + κ0 . The mechanicaloscillator’s dissipation is characterized by rate Γm . The fluctuation dissipation theorem states thatany port with dissipation necessarily has fluctuations associated with it. These fluctuations areminimally quantum noise but in general could be thermal states, so I will model them as such. I
will therefore assign a thermal noise operator ξˆi to each of the ports, where i ∈ {l, r, 0, m} for the left40âin = α in + ξˆ lκrκl
ThermalBath ξ̂0

κ0

ξ̂r
âout

Cavity âg0

ThermalBath ξ̂m

Γm

Mech b̂
Oscillator

Figure 2.5: Operator model of an optomechanical system. The cavity, described by operator â, iscoupled to a mechanical oscillator, described by operator b̂, via single photon coupling rate g0 . The
cavity is coupled to two measurement ports and an internal ‘port’, or thermal reservoir, while themechanical oscillator is only coupled to a thermal reservoir. Each of these ports is described by
an operator ξˆi and has dissipation rate κi (or Γm ). The input fields âin always enter from the left
port. For the transmission geometry, the output field of interest âout is the one at the right port
(as is shown).

feed line port, right feed line port, cavity thermal bath, and mechanical thermal bath, respectively(see Figure 2.5). The expectation values for the bath operators areDEξˆ† [t]ξˆi0 [t0 ] = nth δi,i0 δ[t − t0 ]
i

i

and

D

E
0
ξˆi [t]ξˆi†0 [t0 ] = (nthi + 1)δi,i0 δ[t − t ],(2.62)

where nth
i is the thermal occupancy number of the bath. Note that if at any point in my calculations
I set all of the bath temperatures to zero, I will recover the quantum results. The input field âinwill be defined to be always incident from the left and include both a large classical driving field
αin and the noise operator ξˆl . The output field âout will be defined to always be to the right for thetransmission geometry and always to the left in the reflection geometry. Similar models have beenused to describe an optomechanical system elsewhere, such as in [75], but they often assume thatthth
the thermal occupation of the photon baths is zero (nthl = nr = n0 ). This is not necessarily a bad

choice, as the thermal occupancy for both optical fields at room temperature and microwave fields
at dilution refrigerator temperatures are theoretically negligible. However, in our experiments wesometimes observe excess cavity noise that we choose to model as an elevated temperature of thecavity bath. Allowing all of the noise baths to generally be at nonzero temperature allows us to41
include this effect and other possible ones (such as excess generator/laser noise). The extended
analysis of what happens when these baths are not set to zero is one of the most important
contributions I have made during my PhD.

2.2.3

Heisenberg-Langevin equations

I will now show how I can include the thermal baths into the Hamiltonian, resulting in a
Langevin equation of motion (following [78, Chapter 7]). In order to include the thermal bath forthe cavity directly in the Hamiltonian, I need to model it as a set of harmonic oscillators at all
ˆ t] coupled to the cavity via coupling constant µ[ω]:frequencies, represented by operators d[ω,Ĥ = Ĥsystem + Ĥbath + Ĥsys−bath ,


Z ∞
1
†
ˆ
ˆ
Ĥbath [t] = h̄h̄ω d [ω, t]d[ω, t] +
dω,
2
−∞
Z ∞


ˆ t]â† [t] − â[t]dˆ† [ω, t] dω,Ĥsys−bath [t] = −jh̄µ[ω] d[ω,
−∞
h
i
† 0 0
ˆ
ˆ
d[ω, t], d [ω , t ] = δ[ω − ω 0 ]δ[t − t0 ].

(2.63)(2.64)(2.65)(2.66)

ˆ and â are
The Heisenberg equations for the d’sˆ˙ t] =
d[ω,
˙
â[t]
=

i
j hˆ
ˆ t] − µ[ω]â[t],d[ω, t], Ĥ[t] = jω d[ω,
h̄
i jh
i Z ∞
jh
ˆ t]dω.
â[t], Ĥ[t] =â[t], Ĥsystem [t] +µ[ω]d[ω,h̄h̄−∞Solving for the dˆ equations and substituting back in to the â equation, I get
Z t0jω(t−t0 ) ˆˆd[ω, t0 ] − µ[ω]ejω(t−t ) â[t0 ]dt0 ,d[ω, t] = e
t0
h
i Z ∞
j
˙
ˆ t0 ]dωâ[t]=â[t], Ĥsystem [t] +µ[ω]ejω(t−t0 ) d[ω,h̄−∞
Z ∞Z t0−µ2 [ω]ejω(t−t ) â[t0 ]dt0 dω.
−∞(2.67)(2.68)(2.69)(2.70)

t0

At this point, there are a few approximations to make. The first is that µ[ω] is essentially frequencyindependent over the frequency range of interest and that we can identify the dissipation rate
κ = 2πµ2 [ω]. The second is that â goes smoothly to zero at ±∞ and thus
Z tâ[t0 ]δ[t − t0 ]dt0 = â[t]/2.
t0(2.71)

42
The equation for â then becomes
˙
â[t]
=
=

Z t
i rκ Z ∞jhˆ t0 ]dω − κejω(t−t0 ) d[ω,â[t0 ]δ[t − t0 ]dt0â[t], Ĥsystem [t] +
h̄
2π −∞
t0
i √
jh
κ
â[t], Ĥsystem [t] + κâin [t] − â[t],h̄
2(2.72)(2.73)

where I define1âin [t] = √2πZ ∞ˆ t0 ]dω.
ejω(t−t0 ) d[ω,(2.74)−∞

Here, âin has the following commutator and expectation value:hiâin [t], â†in [t0 ]===Z ∞Z ∞hi
10 0
ˆ 0 , t0 ], dˆ† [ω, t0 ] dωdω 0ejω(t−t0 ) e−jω (t −t0 ) d[ω2π −∞ −∞Z ∞Z ∞
1
0 0ejω(t−t0 ) e−jω (t −t0 ) δ[ω − ω 0 ]dωdω 0
2π −∞ −∞Z ∞10ejω(t−t ) dω,
2π −∞= δ[t − t0 ]Z ∞Z ∞
E
D
D
E
1
0 0
ˆ 0 , t0 ] dωdω 0=â†in [t]âin [t0 ]e−jω(t−t0 ) ejω (t −t0 ) dˆ† [ω, t0 ]d[ω2π −∞ −∞Z ∞Z ∞
10 0e−jω(t−t0 ) ejω (t −t0 ) nth [ω]δ[ω − ω 0 ]dωdω 0=
2π −∞ −∞
Z ∞
1
0
= nth
e−jω(t−t ) dω
2π −∞= nth δ[t − t0 ].

(2.75)(2.76)(2.77)(2.78)(2.79)(2.80)(2.81)(2.82)

I assumed that the noise was white in allowing nth to be frequency-independent, as we would
assume for a thermal bath. These are exactly the commutation relation and expectation value thatwere stated above for the different noise bath operators ξˆi . It is also the way to incorporate any
driving field input. The final equation for â˙ (the cavity operator) is called a Langevin equation
˙of motion. There is a very similar equation for b̂ (the mechanical operator). I can therefore write
down the Heisenberg-Langevin equations that correspond to my model:
˙
â[t]
=
˙
b̂[t] =

i κ
X √
√
jh
â, Ĥsystem − â[t] + κl αin [t] +κi ξˆi [t],h̄2i={l,r,0}
i Γ
p
jh
m
b̂, Ĥsystem −b̂[t] + Γm ξˆm [t].
h̄
2(2.83)(2.84)

43
The first term of each expression is just the simple Heisenberg equation of motion for that operator.
The second term of each expression accounts for the total damping of the cavity or mechanical field.
The third term of the â equation accounts for the large classical drive through the left port. The
final terms account for the coupling to all of the different thermal baths. These equations, alongwith the relation between input and output fields (from the bare cavity response found above)
are all that is needed to completely describe the cavity optomechanical system and calculate any
quantity of interest:
κ



X √
√
˙
â[t]
= −
κi ξˆi [t], (2.85)− jωc â[t] + jg0 b̂[t] + b̂† [t] â[t] + κl αin [t] +2i={l,r,0}



 p
Γm
1
˙
b̂[t] = −− jΩm b̂[t] + jg0 â[t]â† [t] ++ Γm ξˆm [t],(2.86)
22√√(2.87)âout,trans [t] = N2 âin [t] + N1 ξˆr [t] − κr â[t] = N2 âin [t] + N1 ξˆr [t] − κl â[t],âout,refl [t] = N1 âin [t] + N2 ξˆr [t] −√κl â[t].

(2.88)

I have replaced the κr with a κl in the expression for âout,trans [t], as κr = κl in the transmission
geometry. This results in the nice consequence that the expressions for âout,trans [t] and âout,refl [t]are exactly the same with N1 ↔ N2 . In the following chapters, I will do all calculations in thetransmission geometry. However, it is easy to recover the reflection geometry result just by switching
the Ni ’s.

2.3

Measurable quantities
In the previous section, I derived coupled equations of motion for the cavity and mechanicaloscillator, as well as a relation of the cavity state and input to the output field. Thus, given any
classical and noise inputs, I should be able to solve for the microwave output in terms of all of those
inputs. However, the goal is to infer the state of the mechanical oscillator from the output microwave
field. So, I will need to solve these equations for the state of the mechanical oscillator and thenwrite the output field in terms of that state. In the following chapter I will make assumptions about
the classical input field (be it at a single frequency or multiple frequencies), use the assumption

44
that that classical field is much larger than any fluctuations in the system to linearize about it, andthen solve the above equations for b̂, â, and âout . What follows in this section are the quantities I
will need to calculate from those fields in order to understand certain measurements of the output
field.

2.3.1

Coherent dressed response of cavity

I have already introduced the scattering matrix (Equation 2.23); however, I have so far only
looked at the response when the excitation is only at one frequency:
Si1 [ωp ] =√ â[ωp ]âout,i [ωp ]= Ni − κl= Ni − κl χc [ω].
âin [ωp ]âin [ωp ](2.89)

However, I can imagine making the same measurement with a probe tone at ωp , but also drive thecavity with another tone at ωd . Measuring the response at ωp now tells me about how the drive
tone effects the cavity and mechanical parameters and their coupling. This is known as the dressed
response (and is once again completely contained withing the classical equations of motion!).

2.3.2

Mechanical oscillator spectral density and occupancy

I can characterize the state of the mechanical oscillator through its spectral density (seeAppendix B.2.2):
DE
Sbb [ω] = b̂† [−ω]b̂[ω] .

(2.90)

This reveals the spectral content of the mechanical oscillator. Integrating over this function gives
the mechanical oscillator’s final phonon occupancy nfm :
nfm =

2.3.3

1
2π

Z ∞
Sbb [ω]dω.

(2.91)−∞

Output field and voltage spectral densities

In order to infer the state of the mechanical oscillator from a measurement, I must also beable to relate the measurement to the input fields. This comes in two parts. I first relate the state45of the mechanical oscillator to the fields âout immediately at the output of the cavity:
D
E
Saout aout [ω] = â†out [−ω]âout [ω] .

(2.92)In the end, I will want to express this output spectral density in terms of nfm in order to infer themechanical state from my measurement. Chapter 3 focuses on finding Saout aout [ω] in terms of nfm
for several different input drive scenarios.
The second piece of the inference from a measurement is to relate Saout aout [ω] to the voltage
spectral density SV V [ω] measured at the output of a homodyne or heterodyne detection. Chapter
4 focuses on relating Saout aout [ω] to SV V [ω].

Chapter 3

Theory II: Specific solutions to the equations of motion

Armed with the equations of motion for an optomechanical system derived in the previous
chapter (Equations 2.85-2.88), I now turn to solving these equations for different choices of the
input drive field. In Section 3.1, I assume a single large microwave drive as the input. In Section
3.2, I assume two large microwave drives, one near optimal red-detuning (ωd − ωc ≈ −Ωm ) and
one near optimal blue-detuning (ωd − ωc ≈ Ωm ). For both the one and two-drive cases, I calculate
the steady-state solutions for the mechanical field, cavity field, and output spectral density andresponse. In Section 3.3, I investigate the equations of motion in the time-domain with pulsed
input tones, rather than in the steady-state regime.

3.1

Single cavity drive
In a majority of the optomechanics experiments done so far, and three out of six of thosepresented in this dissertation, a single input drive has been continuously applied and the ensuingsteady state conditions of the system have been measured. This section focuses on the theoretical
expectations for this case of a single input drive, and discusses the many different limiting casesand effects that are found.

47
3.1.1

Input assumptions

In this section, I will solve Equations 2.85-2.88, given that the input is a single drive at
frequency ωd :αin [t] = αin,0 ejωd t .

(3.1)

I can assume that the cavity field â and the output field âout each have a large component at ωdand smaller components at other frequencies and that the mechanical field operator b̂ also has asteady state value and smaller fluctuating piece:â[t] = α0 ejφd ejωd t + â1 [t],b̂[t] = b0 + b̂1 [t],â† [t] = α0 e−jφd e−jωd t + â†1 [t],b̂† [t] = b∗0 + b̂†1 [t],âout [t] = αout,0 ejωd t + âout,1 [t].

(3.2)

(3.3)

Here, αin,0 and αout,0 are generally complex, but α0 is defined to be real with φd accounting for thephase of the cavity field. The classical drive photon number in the cavity is nd = α02 .
3.1.2

Steady state and classical solutions

To zeroth order (i.e. when â1 , b̂1 , and the ξˆi ’s are zero), solving the equations of motion gives
solutions for each field:
jg0 (1 + 2α02 )α0 ejφd,αin,0 = √,Γm − 2jΩmκl χc [ωd ]√= N2 αin,0 − κl α0 ejφd = (N2 − κl χc [ωd ])αin,0 .

b0 =αout,0

(3.4)
(3.5)

where ω c = ωc + g0 (b0 + b∗0 ) is the new, shifted, cavity resonance frequency. The coherent cavity
photon number and powers are related by
nd = α02 = κl |χc [ωd ]|2 |αin,0 |2 =
Pout
h̄ωd

κl Pin|χc [ωd ]|2 ,
h̄ωd

2
nd N2
= |αout | =− κl ,κl χc [ωd ]

(3.6)

2

(3.7)

Pout = Pin |Ni − κl χc [ωd ]|2 .

(3.8)

48
Although the expressions relating the photon number and powers seem simple, there is ahidden complexity in the fact that ω c depends on nd :
ω c = ωc + g0 (b0 + b∗0 ) = ωc −4Ωm g02 (1 + 2nd )2g02 nd,≈ω−cΓ2m + 4Ω2m
Ωm

4κl Pinnd =h̄ωd κ2 + 4 ωd − ωc + 2g02 nd /Ωm2  .

(3.9)(3.10)

Here, I have assumed a drive of many photons (nd  1) and a high-Q mechanical oscillator(Ωm  Γm ). This results in a cubic equation for nd :
1/2

Ωm
3/2
2 κ3/2 g03



16g06 3 16g04 κz 2n +nd + g02 κ2 (4z 2 + 1)nd − 2κ3 Ωm y
Ω2m d
Ωm


= 0.

(3.11)

Here, z ≡ (ωd − ωc )/κ is the dimensionless drive detuning from the bare cavity resonance frequency,
normalized to the cavity linewidth and y ≡ (2Pin g02 κl )/(h̄ωd Ωm κ3 ) is a dimensionless parameter
proportional to the input power. I have added a prefactor to the cubic equation, which does notchange its solutions, in order to simplify the discriminant below. For some values of detuning andincident power (or z and y), there is only one real solution to this equation and the cavity resonant
frequency will be stable and single-valued. For other values, there are three real solutions, resulting
in a bifurcation. In these regions, the cavity frequency and number of photons will be hysteretic as
power and detuning are changed. The discriminant of the cubic equation for nd is
D = − (4z 2 + 1)2 + 16zy(4z 2 + 9) + 432y 2 .

(3.12)

The cubic equation has three real roots when D > 0 and only one real root when D < 0. Thus, theboundaries between regions with three roots and one root occur when D = 0 at
y± =


1 
−2z(4z 2 + 9) ± (4z 2 − 3)3/2 .
108(3.13)In order for these boundaries to be meaningful, they must be real and positive (since all the√parameters that make up y are real and positive). The boundaries y ± are real as long as |z| ≥ 3/2√
and positive only if z is negative. Therefore, there can be no boundary for z > − 3/2. Since D < 0
at z = 0, this region must have only one root. Looking at the derivatives of the discriminant at the49

Figure 3.1: Regions of cavity nonlinearity as a function of dimensionless drive detuning z = (ωd −ωc )/κ and dimensionless input power y = (2Pin g02 κl )/(h̄ωd Ωm κ3 ). The red lines are the boundaries
y ± between three-solution and one-solution regions. The red shaded area between these lines showswhere there are three solutions. The blue line shows the points where the drive is centered at thecavity frequency. The black line shows the maximum power for which there is only one solution
at all detunings. The black dashed line shows the maximum power for which there is only one
solution for the drive at the cavity frequency. (b) Zoom-in of (a) near the point of interest.

two boundaries reveals the discriminant’s behavior everywhere:3/2∂y D y± = ∓8 4z 2 − 3).

(3.14)√
Therefore, there will be three roots if z ≤ − 3/2 and y − < y < y + (see Figure 3.1 to see thedifferent regions). The highest power for which the cavity is stable for all detunings is y = 3−3/2 .
There are some scenarios (such as the quantum backaction measurement with one drive
discussed later) where I might want the drive to be exactly on resonance (ωd = ω c ). To achievethis, the power and detuning must be related:nd =2κΩm y,g02

z=−

2g02 nd
,
κΩm

z
y=− .
4

(3.15)The highest power for which this point is stable is y = 1/4, which occurs at z = −1, meaning thatthe power applied on resonance is limited: nd ≤ κΩm /(2g02 ). In the remainder of this thesis, I willassume that the power and detuning we apply is such that there is always only one solution for thecavity frequency and number of photons.

50
3.1.3

Linearized equations of motion

Substituting the derived steady-state values into the equations of motion gives the linearized
equations of motion:
κ



X √
â˙ 1 [t] = −κi ξˆi [t],(3.16)− jω c â1 [t] + jgej(ωd t+φd ) b̂1 [t] + b̂†1 [t] +2i=l,r,0



 p
Γm
˙
b̂1 [t] = −− jΩm b̂1 [t] + jg e−j(ωd t+φd ) â1 [t] + ej(ωd t+φd ) â†1 [t] + Γm ξˆm [t],(3.17)
2√âout,1 [t] = N2 ξˆl [t] + N1 ξˆr [t] − κl â1 [t],(3.18)√
where g ≡ g0 nd is the photon enhanced coupling rate. Multiplying the â1 and b̂1 equations by
e−jωt and integrating gives the linearized equations of motion and their conjugates in the frequency
domain:â1 [ω]χc [ω]= jgejφd (b̂1 [ω − ωd ] + b̂†1 [ω − ωd ]) +X √κi ξˆi [ω],(3.19)

i=l,r,0â†1 [−ω]χ∗c [ω]= −jge−jφd (b̂†1 [−ω + ωd ] + b̂1 [−ω + ωd ]) +b̂1 [w]χm [ω] p

= jg ejφd â†1 [ω − ωd ] + e−jφd â1 [ω + ωd ] + Γm ξˆm [ω],(3.21)b̂†1 [−w]χ∗m [ω] p†= −jg e−jφd â1 [−ω + ωd ] + ejφd â†1 [−ω − ωd ] + Γm ξˆm[−ω],(3.22)

where χc [ω] ≡X √κi ξˆi† [−ω],(3.20)

i=l,r,0κ

2 + j(ω − ω c )−1and χm [ω] ≡

 Γm
2

−1
are the cavity and mechanical+ j(ω − Ωm )susceptibilities, respectively.
These four equations can be written in terms of only four variables (â[ω + ωd ], â† [ω − ωd ],b̂[ω], and b̂† [ω]) and can thus be solved exactly, resulting in the following solutions:â1 [ω] =χc [ω]
2∗
1 + g (χc [ω] − χc [−ω + 2ωd ]) (χm [ω − ωd ] − χ∗m [−ω + ωd ])
×




 X √
1 − g 2 χ∗c [−ω + 2ωd ] (χm [ω − ωd ] − χ∗m [−ω + ωd ])κi ξˆi [ω](3.23)

i=l,r,0−g 2 e2jφd χ∗c [−ω + 2ωd ] (χm [ω − ωd ] − χ∗m [−ω + ωd ])X √κi ξˆi† [ω − 2ωd ]

i=l,r,0

+jge

jφd

p

Γm



o

†
χm [ω − ωd ]ξˆm [ω − ωd ] + χ∗m [−ω + ωd ]ξˆm[ω − ωd ],51â†1 [ω] =χ∗c [−ω]
1 + g 2 (χc [ω + 2ωd ] − χ∗c [−ω]) (χm [ω + ωd ] − χ∗m [−ω − ωd ])X √× g 2 e−2jφd χc [ω + 2ωd ] (χm [ω + ωd ] − χ∗m [−ω − ωd ])κi ξˆi [ω + 2ωd ] (3.24)
i=l,r,0X √ †κi ξˆi [ω]+ 1 + g 2 χc [ω + 2ωd ] (χm [ω + ωd ] − χ∗m [−ω − ωd ])
i=l,r,0op †−jge−jφd Γm χm [ω + ωd ]ξˆm [ω + ωd ] + χ∗m [−ω − ωd ]ξˆm[ω + ωd ] ,b̂1 [ω] =χm [ω]∗
2
1 + g (χc [ω + ωd ] − χc [−ω + ωd ]) (χm [ω] − χ∗m [−ω])×npΓm 1 − g 2 χ∗m [−ω] (χc [ω + ωd ] − χ∗c [−ω + ωd ]) ξˆm [ω](3.25)p †+ Γm −g 2 χ∗m [−ω] (χc [ω + ωd ] − χ∗c [−ω + ωd ]) ξˆm[ω]+jge−jφd χc [ω + ωd ]X √κi ξˆi [ω + ωd ]
i=l,r,0X √+jgejφd χ∗c [−ω + ωd ]κi ξˆi† [ω − ωd ],

i=l,r,0b̂†1 [ω] =χ∗m [−ω]2∗
1 + g (χc [ω + ωd ] − χc [−ω + ωd ]) (χm [ω] − χ∗m [−ω])×npΓm g 2 χm [ω] (χc [ω + ωd ] − χ∗c [−ω + ωd ]) ξˆm [ω](3.26)p †+ Γm 1 + g 2 χm [ω] (χc [ω + ωd ] − χ∗c [−ω + ωd ]) ξˆm[ω]X √κi ξˆi [ω + ωd ]−jge−jφd χc [ω + ωd ]

i=l,r,0−jgejφd χ∗c [−ω + ωd ]X √i=l,r,0κi ξˆi† [ω − ωd ]




.



Here, the daggered equations contain the same information as the undaggered. However, I haveincluded them here because it is helpful to see how to convert between the operators and their
daggers (for any operator (Â[ω])† = Â† [−ω]). Later in this chapter I may omit daggered equations
as they are easily found from the undaggered ones.

52
3.1.4

Effective mechanical susceptibility

In the absence of optomechanical coupling (g = 0), I can write the position of the mechanicaloscillator coupled to it’s thermal bath asx̂[ω] = xzp (b̂[ω] + b̂† [ω]) = xzpp†Γm (χm [ω]ξˆm [ω] + χ∗m [−ω]ξˆm[ω]).

(3.27)

When I re-introduce coupling to the cavity, but keep the cavity isolated from the cavity baths, themechanical position is√†
xzp Γm (χm [ω]ξˆm [ω] + χ∗m [−ω]ξˆm[ω]).
x̂[ω] =
2∗
1 + g (χc [ω + ωd ] − χc [−ω + ωd ]) (χm [ω] − χ∗m [−ω])(3.28)

Here, mechanical fluctuations lead to a change in the cavity field, which exerts a force back on themechanical oscillator. This built in feedback effect leads to a new, effective mechanical susceptibility
for the mechanical oscillator, whose form is familiar from control theory1 [79, Chapter 11]:χm,eff [ω] =χm [ω].
1 + g 2 (χc [ω + ωd ] − χ∗c [−ω + ωd ]) (χm [ω] − χ∗m [−ω])(3.29)In the weak-coupling limit, g  κ, the effective mechanical susceptibility can be approximatedas a Lorentzian by evaluating the χc ’s at ω = Ωm . This is a good approximation because the χc ’s
are fairly constant over the frequency range of interest (near the mechanical resonance frequency).
I also make the approximation that χ∗m [−ω] near ω ≈ Ωm is negligible (when Ωm  Γm ). Writing
the effective susceptibility as a function of an effective total frequency and total linewidth, I find1
χm,eff [ω]

≈

1
Γtot
+ g 2 (χc [Ωm + ωd ] − χ∗c [−Ωm + ωd ]) ≡+ j(ω − Ωtot ),χm [ω]
2(3.31)

1

Some authors like to emphasize the relationship to Dyson’s equation rather than control theory and define aquantity called self-energy to relate the effective mechanical susceptibility to the bare susceptibility:1
1≈ jg 2 (χc [ω + ωd ] − χ∗c [−ω + ωd ]) .
(3.30)
Σ[ω] = j−χm,eff [ω]χm [ω]
This expression is easily found from Equation 3.29 by making the approximation that χ∗m [−ω] is negligible near
ω = Ωm (which is valid if Ωm  Γm ).

53
where the effective total and optomechanically induced mechanical damping and resonance frequency areΓtot = Γm + Γopt ,Ωtot = Ωm + Ωopt ,(3.32)

Γopt = 2g 2 Re [χc [Ωm + ωd ] − χ∗c [−Ωm + ωd ]]


κ
κ
2
= 4g
−
,
κ2 + 4(∆ + Ωm )2 κ2 + 4(∆ − Ωm )2Ωopt = −g 2 Im [χc [Ωm + ωd ] − χ∗c [−Ωm + ωd ]]∆ + Ωm∆ − Ωm2= 4g+.
κ2 + 4(∆ + Ωm )2 κ2 + 4(∆ − Ωm )2(3.33)(3.34)(3.35)(3.36)

Here, ∆ is the detuning of the drive away from the cavity resonance frequency: ∆ = ωd − ω c .
The χc ’s above really should have been evaluated at Ωtot rather than Ωm . However, in the weakcoupling limit, the change in resonance frequency is fractionally very small, so evaluating at Ωmwill not make much difference.

3.1.5

Phonon spectrum and mechanical occupancy

I can re-write Equations 3.25-3.26 for b̂ and b̂† with coefficients to simplify the subsequent
algebra:†b̂1 [ω] = s1 [ω]ξˆm[ω] + s2 [ω]ξˆm [ω]rr
X κi †X κi+s3 [ω]ξˆ [ω − ωd ] + s4 [ω]ξˆi [ω + ωd ],
κ i
κ

i
†
∗
†
ˆ
b̂1 [−ω] = s1 [ω]ξm [−ω] + s∗2 [ω]ξˆm[−ω]+s∗3 [ω]Xir(3.37)

i(3.38)X
κi ˆξi [−ω + ωd ] + s∗4 [ω]
κ
i

r

κi ˆ†
ξ [−ω − ωd ].
κ i

54
As I explained in Section 2.3.2, the phonon spectrum is given byEb̂† [−ω]b̂[ω]
D
E
D
E
†
†
= |s1 [ω]|2 ξˆm [−ω]ξˆm[ω] + |s2 [ω]|2 ξˆm[−ω]ξˆm [ω]E
X √κi κj D2
+ |s3 [ω]|ξˆi [−ω + ωd ]ξˆj† [ω − ωd ]
κ
i,j
√
E
X κi κj D †+ |s4 [ω]|2ξˆi [−ω − ωd ]ξˆj [ω + ωd ]
κ
i,j


2 th
= |s1 [ω]|2 nthm + 1 + |s2 [ω]| nmX κi 
X κi
2+ |s3 [ω]|2nthnthi + 1 + |s4 [ω]|
κ
κ i
i
i




2
2 th
2
th
th
= |s1 [ω]| nm + 1 + |s2 [ω]| nm + |s3 [ω]| nc + 1 + |s4 [ω]|2 nthc ,Sbb [ω] =Dth−1
where nthm is the thermal mechanical occupancy and nc ≡ κ(3.39)(3.40)(3.41)(3.42)

thi=l,r,0 κi ni is the total cavity

P

thermal occupancy. Substituting in the coefficients and simplifying, I find the mechanical spectrum:Sbb [ω] =Γm |χm [ω]|2 |χm,eff [−ω] − χm [−ω]|2  thn+1m|χm [ω] − χ∗m [−ω]|2++Γm |χm [ω]|2 |χm,eff [ω] − χ∗m [−ω]|2|χm [ω] − χ∗m [−ω]|2
x2zp
h̄

2

2 SF F [−ω] |χm,eff [ω]|



nth
c +1

nth
m



+

(3.43)
x2zp
h̄

2 th
2 SF F [ω] |χm,eff [ω]| nc .

Here, SF F is the the radiation pressure force spectrum,
SF F [ω] ≡h̄2 g 2 κ|χc [ω + ωd ]|2 .
x2zp(3.44)

This result is very similar to that in reference [75], except that they write the answer in terms of
the self energy (see Footnote 1) and there are a several minor differences in notation between theirwork in mine. Perhaps the most important (and least obvious) difference is that they define theRspectral density as Scc [ω] = dteiωt ĉ† [t]ĉ[0] = ĉ† [ω]ĉ[−ω] . This is my definition with ω → −ω.
This will not affect the final mechanical occupancy, because that involves integrating the spectral
density over all negative and positive frequencies, but it does mean that their spectral density ends
up being centered about ω = −Ωm rather than ω = Ωm .

55
My expression for the spectrum is peaked near ω = Ωm . Therefore, I can simplify by making
the very good approximation that χm [−ω] = χm,eff [−ω] = 0 near ω ≈ Ωm (valid in the limitΩm  Γm ):Sbb [ω] =nthm Γm +

x2zp 
2h̄th
SF F [−ω](nthc + 1) + SF F [ω]nc!
|χm,eff [ω]|2 .

(3.45)

This result is fully general for any coupling regime. I can specialize to the weak-coupling regime by
substituting in the effective mechanical susceptibility and making the approximation that SF F [ω] isrelatively constant over the frequency range of interest (near the mechanical resonance frequency).
If this is true, then I can simply evaluate it at Ωm (not Ωtot ). This is reasonable because SF F [ω] is afunction of only the χc ’s, which change on frequencies of order κ  Γm . Making this approximation,
SF F [ω] ≈ SF F [Ωm ] =h̄2 (1 + n0m )Γopt,
x2zp(3.46)h̄2 n0m Γopt,
x2zp

(3.47)SF F [−ω] ≈ SF F [−Ωm ] =n0m ≡κ2 + 4(∆ + Ωm )2
SF F [−Ωm ]=−.
SF F [Ωm ] − SF F [−Ωm ]16∆Ωm(3.48)

Here, n0m is the mechanical occupancy in the absence of both mechanical damping and cavity
thermal photons (Γm = 0, nthc = 0). For a red-detuned drive at frequency ωd ≈ ωc − Ωm , this factor
is n0m = κ2 /(16Ω2m ). It therefore depends on how far in the resolved sideband regime the system
is and is zero in the far resolved sideband limit (Ωm  κ). Conversely, for a blue-detuned driveat frequency ωd ≈ ωc + Ωm , this factor is n0m = −1 − κ2 /(16Ω2m ), which is nonzero even in the far
resolved sideband regime.
The mechanical spectrum in the weak-coupling regime is




2
th
0
th 0
Sbb [ω] = nthm Γm + nc + nm + 2nc nm Γopt |χm,eff [ω]| ,(3.49)

which has a Lorentzian profile of width Γtot , centered about Ωtot . The final mechanical occupancy

56
in the weak coupling regime is
nfm

Z ∞
1
=
Sbb [ω]dω
2π −∞



 1 Z ∞
1
th
th
0
th 0
=
nm Γm + nc + nm + 2nc nm Γoptdω
22π −∞ (Γtot /2) + (ω − Ωm )2

th
0
th 0
nth
m Γm + nc + nm + 2nc nm Γopt=.
Γm + Γopt(3.50)(3.51)(3.52)The total motion of the oscillator (including zero point motion) can be divided up into
a component originating from the thermal phonon bath and a component originating from thevarious cavity baths:
nfm + 1/2 =Γopt (1 + 2n0m )(nthΓm (nthΓm (nthc + 1/2)
m + 1/2)
m + 1/2)
+
=
+ nba
m.
Γm + Γopt
Γm + Γopt
Γm + Γopt(3.53)

The first of these components is a weighted mechanical bath occupancy. By modifying Γopt , thecoupling of the mechanical oscillator to the mechanical bath can be either diluted or strengthened.
The second component is called the backaction occupation nba
m , as it is the motion of the mechanicaloscillator caused by the radiation pressure of the photon field. This expression, converted to a force
spectral density, gives the same result (up to a factor of two due to the definition of spectralqba[Ω] in reference [80].
densities) at that found for SFF

3.1.6

Important limits of the mechanical occupancy

Here, I will look at various interesting limits of Equation 3.52, the final mechanical occupancy
in the weak coupling regime. For an off-resonant drive (∆ = ωd − ω c 6= 0), Γopt will be nonzero andthere will be a net dynamical radiation pressure on the mechanical oscillator. Turning the drive
power up sufficiently such that |Γopt |  Γm nth
m , the final occupancy will be


2
2
0
th 0
th
th 4(Ωm + ∆) + κ.
nfm = nth
+
n
+
2n
n
=
n
−
1
+
2n
c
m
c m
c
c
16Ωm ∆The detunings that minimize and maximize the final mechanical occupancies are
s
s
κ2
κ2
∆optimal,min = −Ωm 1 +
and
∆
=
Ω
1
+
.
m
optimal,max
4Ω2m
4Ω2m

(3.54)(3.55)

57
In the far resolved sideband limit (Ωm  κ), these detunings are just ±Ωm , while in the far
unresolved sideband limit (Ωm  κ), they are ±κ/2. Using these detunings, I will now go back to
the original expression for nfm , as (even ignoring technical experimental reasons) there are limits on
the largest drive powers that can be applied usefully. In the amplifying case (∆ > 0), increasing thedrive power will decrease the total mechanical linewidth. The mechanical oscillator can no longer
be stable once its linewidth becomes negative. Thus, the largest |Γopt | can be is Γm . As |Γopt | goesto Γm , the total linewidth goes to zero and the occupancy is amplified until it diverges. In thecooling case (∆ < 0), Γopt is positive and the total mechanical linewidth grows with power. The
mechanical oscillator will only remain an underdamped oscillator while Qtot = Ωtot /Γtot > 1/2.
Thus, the largest Γtot can be is 2Ωtot and the minimum possible final occupancy (assuming an
initially high Q oscillator, Qm = Ωm /Γm  1) is
nfm,min =

nth
1
m
+
2Qm 2

s


1 + 2nth
c



!
κ2
−1 .
1+
4Ω2m

(3.56)In order for the mechanical occupancy to reach the ground state (nfm,min < 1), the system musthave a sufficiently large initial mechanical quality factor such that nth
m < 2Qm , it must be in the√
resolved sideband limit (4 2Ωm > κ), and the cavity thermal occupancy must be less than one(nthc < 1). If the cavity and mechanical oscillator are coupled to baths at the same temperature,
this last requirement can be written another way: nthm < ωc /Ωm . However, in experiments, weobserve excess cavity occupancy that indicates that the thermal temperature of the cavity is higher
than that of the mechanics. Thus, the requirement that nth
c < 1 is the more fundamental one. The
last requirement is technical rather than fundamental - that the incident power can be turned up
sufficiently high that Γopt  nthm Γm .
In the far resolved sideband limit applicable to our most recent experiments on cooling andcontrol of the mechanical oscillator, the final occupancies given an optimally red and optimally58
blue-detuned drive are
nfm =

th
nth
m Γm + nc Γopt
Γm + Γopt

nfm =

th
nth
m Γm − (nc + 1)ΓoptΓm + Γopt

for ωd = ωc − Ωm ,for ωd = ωc + Ωm .

(3.57)(3.58)

The case of an on-resonance drive is much simpler than the detuned drive cases because Γopt is zero(although n0m blows up so it is critical to use n0m Γopt = 4g 2 κ/(κ2 + 4Ω2m )). The final occupancy isbathnfm = nthm + nm = nm +4g 2 κ(1 + 2nthc ).
2Γm (κ + 4Ω2m )(3.59)

There is no net dynamic radiation pressure at this point, leaving the mechanical occupancy approximately at its initial value. However, there is an additional backaction heating term whichcorresponds to the extra motion induced in the mechanical oscillator by fluctuations of the input
drive field (even when the cavity thermal occupancy is 0). Note that the quantum backaction seen
with a single, on-resonance drive is maximized in the unresolved sideband regime (when Ωm  κ).
One last interesting case is that of strong coupling for a red-detuned drive in the far resolvedsideband case. If I go back to Equation 3.42 and set the far-off resonance terms to zero (χm [−ω] =χc [−ω + ωd ] = 0), I find the following equation for the mechanical spectrum:

2 th
2
2
nth
m Γm + g nc κ|χc [ω + ωd ]| |χm [ω]|Sbb [ω] =.
|1 + g 2 χm [ω]χc [ω + ωd ]|2(3.60)The denominator of this equation has roots at ω = ω ± + j(κ + Γm )/4 where ω ± = Ωm ±p
g 2 − (κ − Γm )2 /16. For large couplings, the spectrum looks like two peaks of width (Γm + κ)/2p
with splitting 2 g 2 − (κ − Γm )2 /16 ≈ 2g. These are normal modes of the cavity and mechanicaloscillator. The expression for the mechanical spectrum can then be re-written as
2
2 + 4g 2 nth κ
nthm Γm κ + 4(ω − Ωm )c
.
Sbb [ω] = Γm +κ( 2 )2 + 4(ω − ω − )2 ( Γm2+κ )2 + 4(ω − ω + )2

(3.61)

Integrating this equation, the final occupancy is
nfm

=
≈

1
2π

Z ∞
Sbb [ω]dω =−∞2
th(4g 2 + κ(κ + Γm ))Γm nth
m + 4g κnc(κ + Γm )(4g 2 + κΓm )2
th(4g 2 + κ2 )Γm nth
m + 4g κnc.
κ(4g 2 + κΓm )(3.62)(3.63)

59
3.1.7

Output field spectrum

I can substitute Equations 3.23-3.24 (re-written in terms of coefficients to simply the subsequent algebra) into Equation 2.88 for the output field to find âout,1 and â†out,1 (remember that thisis for the transmission geometry; swap N1 ↔ N2 for the reflection geometry):†âout,1 [ω] = N2 ξˆl [ω] + N1 ξˆr [ω] + r1 [ω]ξˆm[ω − ωd ] + r2 [ω]ξˆm [ω − ωd ]rr
X κi
X κi †
ˆ+r3 [ω]ξ [ω − 2ωd ] + r4 [ω]ξˆi [ω],κ iκâ†out,1 [−ω]

=

i
i
∗ ˆ†
∗ ˆ†
∗
†
N2 ξl [−ω] + N1 ξr [−ω] + r1 [ω]ξˆm [−ω + ωd ] + r2∗ [ω]ξˆm[−ω + ωd ]+r3∗ [ω]

X
i

r

X
κi ˆ
ξi [−ω + 2ωd ] + r4∗ [ω]κir(3.64)(3.65)κi ˆ†
ξ [−ω].
κ i

The spectral density of the output field operators isDEâ†out,1 [−ω]âout,1 [ω](3.66)
DE
D
E††= |r1 [ω]|2 ξˆm [−ω + ωd ]ξˆm[ω − ωd ] + |r2 [ω]|2 ξˆm[−ω + ωd ]ξˆm [ω − ωd ]√E
X κi κj D
ξˆi [−ω + 2ωd ]ξˆj† [ω − 2ωd ]+ |r3 [ω]|2
κ
i,j
√
E
X κi κj D †[+ |r4 [ω]|2ξˆi [−ω]ξˆj ω](3.67)
κ
i,j
D
E
D
E
+ |N2 |2 ξˆl† [−ω]ξˆl [ω] + |N1 |2 ξˆr† [−ω]ξˆr [ω]rrEEX κi D †X κi D †∗∗ˆˆξi [−ω]ξl [ω] + N2 r4 [ω]ξˆl [−ω]ξˆi [ω]+N2 r4 [ω]
κ
κ
i
i
r D
r D
E
E
X κi †
X κi
∗
∗
†
ˆ
ˆ
ˆ
ˆ
+N1 r4 [ω]ξ [−ω]ξr [ω] + N1 r4 [ω]ξ [−ω]ξi [ω]
κ i
κ r
i
i




2 th
2
th
= |r1 [ω]|2 nth
+
1
+
|r
[ω]|
n
+
|r
[ω]|
n
+
1
+ |r4 [ω]|2 nth
2
3
m
m
c
c
r 
r

κl
κl
+ N2∗ r4 [ω]nth(3.68)+ |N2 |2 + N2 r4∗ [ω]
l
κ
κ
r
r 

κl
κl
2
∗
∗
+ |N1 | + N1 r4 [ω]+ N1 r4 [ω]
nth
r .
κ
κ

Saout aout [ω] =60−1Remember that nth
c =κ

th
th
th
th
i κi ni for i = {l, r, 0} and that if nl or nr is nonzero, nc has a lowerPbound on its possible values. Re-written in terms of the final occupancy, the output spectrum isΓtot fΓtot fSaout aout [ω] = |r1 [ω]|2(nm + 1) + |r2 [ω]|2
n
Γm
Γm m


0
0
2 nm Γopt
2
2 (1 + nm )Γopt− |r2 [ω]|(nth
+ |r3 [ω]| − |r1 [ω]|c + 1)ΓmΓm

(1 + n0m )Γopt
n0 Γopt+ |r4 [ω]|2 − |r2 [ω]|2− |r1 [ω]|2 m
nth
c
Γm
Γm
r
r 

κl
κl
+ |N2 |2 + N2 r4∗ [ω]nth+ N2∗ r4 [ω]
l
κ
κ
r
r 

κl
κl
+ |N1 |2 + N1 r4∗ [ω]nth+ N1∗ r4 [ω]r .
κκ(3.69)

Re-writing the different coefficients in terms of effective mechanical susceptibility gives|r1 [ω]|2 = g 2 κl Γm |χc [ω]|2 |χm,eff [−ω + ωd ]|2 ,(3.70)|r2 [ω]|2 = g 2 κl Γm |χc [ω]|2 |χm,eff [ω − ωd ]|2 ,(3.71)
2|r3 [ω]|2 = g 4 κκl |χc [ω]|2 |χc [−ω + 2ωd ]|2 χm,eff [ω − ωd ] − χ∗m,eff [−ω + ωd ] ,(3.72)

2|r4 [ω]|2 = κκl |χc [ω]|2 1 − g 2 χc [ω] χm,eff [ω − ωd ] − χ∗m,eff [−ω + ωd ] ,(3.73)

r

3.1.8

κl
(Ni r4∗ [ω] + Ni∗ r4 [ω])κ= −2κl Re Ni∗ χc [ω] 1 − g 2 χc [ω] χm,eff [ω − ωd ] − χ∗m,eff [−ω + ωd ] . (3.74)

Limiting cases of the output spectrum

I assume from here on that the feed line has a purely real impedance: N2 = 1, N1 = 0.
These results are now valid for either the transmission or reflection geometry. I also assume thatmeasurements made with a detuned drive are done with a system in the far resolved sideband
regime. Measurements with an on-resonance drive are still general for either sideband regime.
If the cavity is driven with a close to optimally red-detuned drive at ωr ≈ ω c − Ωm , thesideband enhanced by the cavity is the upper sideband, so I will set χm,eff [−ω + ωr ] = 0 andχc [−ω + 2ωr ] = 0. Not making the weak-coupling limit assumptions, the output spectrum is

th
2
th
th
κl |χc [ω]|2 g 2 Γm (nthm − nl )|χm [ω − ωd ]| + κ(nc − nl )thSaout aout [ω] = nl +. (3.75)|1 + g 2 χc [ω]χm [ω − ωd ]|2

61
To get this expression, I used the identities χm [ω] + χ∗m [ω] = Γm |χm [ω]|2 and χc [ω] + χ∗c [ω] =κ |χc [ω]|2 . In the weak coupling limit, I can assume that n0m = 0 and Γopt = 4gr2 /κ. I will also
assume that the drive is very close to perfectly detuned. This allows me to only keep the χc [ω]’s
not found with more narrowly varying features. I will set the others to their approximate value
near cavity resonance χc [ωc ] = 2/κ. The output spectrum in the weak coupling limit is
4g 2 κl Γtot  fththth
Saout aout [ω] = nth
nm − 2nth|χm,eff [ω − ωr ]|2 ,(3.76)|χc [ω]|2 + r 2
c + nll + κκl nc − nlκwhich exhibits a constant background due to extra input fluctuations, a wide spectral component
filtered by the cavity susceptibility due to extra thermal cavity fluctuations, and a narrow featurefiltered by the mechanical susceptibility due to the final occupancy of the mechanical oscillator.
If the cavity is instead driven with a blue-detuned drive at ωb ≈ ω c + Ωm , the sideband
of interest is the lower sideband and I set χm [ω − ωb ] = 0 and χc [−ω + 2ωb ] = 0. I once again
assume weak coupling and a close to perfectly detuned drive, but in this case, that requires setting
n0m = −1 and Γopt = −4gb2 /κ. The output spectrum is


th
th
|χc [ω]|2
+
κκ
n
−
n
Saout aout [ω] = nth
l
c
l
l

4g 2 κl Γtot  fth+ b 2
nm + 1 + 2nth−n|χm,eff [−ω + ωb ]|2 .
clκ

(3.77)

It is important to notice how the presence of thermal cavity occupancy nthc and extra input noisenthl affects the red and blue-detuned cases. In the former (latter), cavity occupancy decreases(increases) the size of the sideband, while input noise increases (decreases) the size of the sideband.
th
It is thus extremely important to independently measure nthc and nl in order to correctly infer themechanical occupancy from a measurement of the output fields.
The last case I’ll consider is if the cavity drive is applied on-resonance at ω c , no longer making
any assumption about the relative values of κ and Ωm . For this case, all of the χ’s are important,
but the expression can be simplified a bit by the fact that χc [−ω + 2ω c ] = χ∗c [ω] and the two

62
sidebands are far separated in frequency compared to their widths. The output spectrum is


th
th
Saout aout [ω] ≈ nth
+
κκ
n
−
n
|χc [ω]|2(3.78)
l
c
l
l




4g 2 κl Γm  f
th
th
2
f
th
2
+
n
−
2n
+
n
κ
+
4
n
−
n
Ω
|χm [ω − ω c ]|2
m
c
m
m
l
l
2
2
2
(κ + 4Ωm )
4g 2 κl Γm  f
th
th
2
f
th
2
+
n
+
1
+
2n
−
n
κ
+
4
n
+
1
+
n
Ω
|χm [−ω + ω c ]|2 .
m
c
m
m
l
l
2
2
2
(κ + 4Ωm )To get this expression, I used the approximations
χc [ω]2 χm [ω − ω c ] + χ∗c [ω]2 χ∗m [ω − ω c ] ≈χc [ω]χm [ω − ω c ] + χ∗c [ω]χ∗m [ω − ω c ] ≈χc [ω]2 χ∗m [−ω + ω c ] + χ∗c [ω]2 χm [−ω + ω c ] ≈χc [ω]χ∗m [−ω + ω c ] + χ∗c [ω]χm [−ω + ω c ] ≈Γm 2
κ − 4Ωm |χc [ω]|4 |χm [ω − ω c ]|2 , (3.79)
4κΓm|χc [ω]|2 |χm [ω − ω c ]|2 ,(3.80)2
Γm 2
κ − 4Ωm |χc [ω]|4 |χm [−ω + ω c ]|2 , (3.81)
4κΓm|χc [ω]|2 |χm [−ω + ω c ]|2 .
(3.82)
2

The first two approximations are only good for ω ≈ ω c + Ωm , while the second two are only goodthfor ω ≈ ω c − Ωm . It is interesting to see how nthc and nl affect the on-resonance drive measurement

differently in the resolved or unresolved sideband limits. In the resolved sideband limit, cavity
noise is irrelevant, as the sidebands are not located near the cavity frequency, whereas input noiseinterferes with the mechanical occupancy with the opposite sign as it did in the red and bluedetuned drive cases. In the unresolved sideband limit, the extra noises add with the same sign as
for the red and blue-detuned drive cases.
In all drive cases, if the extra noises are zero, the upper sideband is proportional to nfm , while
the lower sideband is proportional to nfm + 1. This is generally referred to as ‘sideband asymmetry’.
This asymmetry indicates the quantum nature of the effective mechanical oscillator. When viewed
in terms of the backaction mechanical occupancy introduced above, the asymmetry can also beviewed as a consequence of the photon field shot noise (as is discussed in [81]).

3.1.9

Dressed cavity response

In this section, I will calculate the response of the cavity to a small probe tone at ωp , in thepresence of a strong drive tone at ωd . The amplitude of the probe tone is small enough that it

63should not change the cavity or mechanical parameters. If I were to start from Equations 2.852.88, I would use the input âin = αin,d ejωd t + αin,p ejωp t , set all of the noise operators to zero, andthen linearize around the driving field. However, I have already done a lot of this work in theprevious sections, and I can just use the solution for â1 (Equation 3.23), making the replacementsξˆl [ω] → αin,p δ[ω − ωp ] and ξˆr,0,m = 0:â1 [ω] =χc [ω] 1 − g 2 χ∗c [−ω + 2ωd ] (χm [ω − ωd ] − χ∗m [−ω + ωd ]) √κl αin,p δ[0]. (3.83)
1 + g 2 (χc [ω] − χ∗c [−ω + 2ωd ]) (χm [ω − ωd ] − χ∗m [−ω + ωd ])Substituting this into the expression for the dressed cavity response (Equation 2.89), I find (remember that i = 2 for the transmission geometry and i = 1 for the reflection geometry) that thisquantity is
Si1 [ωp ] ===√ â1 [ωp ]αout,p= Ni − κlαin,pαin,p δ[0]χc [ωp ] 1 − g 2 χ∗c [−ωp + 2ωd ] (χm [ωp − ωd ] − χ∗m [−ωp + ωd ])N2 − κl
1 + g 2 (χc [ω] − χ∗c [−ωp + 2ωd ]) (χm [ωp − ωd ] − χ∗m [−ωp + ωd ])
N2 − κl χc [ωp ] + g 2 κl χ2c [ωp ] χm,eff [ωp − ωd ] − χ∗m,eff [−ωp + ωd ] .

(3.84)(3.85)(3.86)The second boxed expression makes it clear that in the absence of drive photons (g = 0), the cavity
resumes its bare response, and that for large drives, there are extra features shaped like the totalmechanical susceptibility.
This dressed response is zero on resonance for an optimally red-detuned drive in the resolved
sideband limit when 4g 2 /Γm = 2κl − κ. This is the point where the impedance between itinerant
photons in the microwave drive is matched to the impedance of the circuit. In this case, thosephotons are turned into mechanical phonons at the same rate that phonons are dissipated from themechanical oscillator, and so they are completely absorbed. Note that this point is only possible
for an over-coupled circuit in the reflection geometry (κl > κ/2).

3.2

Two cavity drives
Two very interesting effects appeared in the previous section for the case of an on-resonancedrive: sideband asymmetry of the measurement (see Equation 3.79) and quantum backaction of

64
photon fluctuations on the mechanical oscillator (see Equation 3.59). If the system is far in the resolved sideband regime, an on-resonant drive results in extremely attenuated sidebands. Therefore,
I might like to apply a red or blue-detuned drive in order to enhance these sidebands. However, for
the quantum backaction measurement, this will result in a very large dynamic radiation pressure
force, dwarfing the quantum backaction. I can get around this fact by applying two drive tones,
one red-detuned and one blue-detuned, such that the net dynamic radiation pressure force is zero,
once again allowing access to the quantum backaction piece.
A similar difficulty and solution arise for the sideband asymmetry measurement. The asymmetry is most visible when the mechanical oscillator has very few phonons, so I would like to apply a
red-detuned drive to cool the oscillator to low occupancy. Unfortunately, this will even further suppress the lower sideband. However, adding a second, blue-detuned, drive of smaller amplitude than
the red-detuned one will allow measurement of its lower sideband while still cooling the oscillator.
Thus, the ideal strategy for seeing both quantum backaction and sideband asymmetry in the
far resolved sideband regime is to apply two drives, one red-detuned and one blue-detuned. In
this section I will follow a similar procedure to the previous section for one drive, only this time
I will include a second drive. Therefore, some of the discussion and definitions of the quantities
I calculate will not be repeated. This situation will require more assumptions than the one-drivecase, namely that the drives are close to optimally red and blue-detuned and that the system is in
the far sideband resolved regime.

3.2.1

Input assumptions

In this section, I will once again solve Equations 2.85-2.88, this time given that there are two
inputs, one approximately red-detuned at ωr ≈ ωc − Ωm and one approximately blue-detuned atωb ≈ ωc + Ωm :αin [t] = αin,r ejωr t + αin,b ejωb t .

(3.87)

65
I can assume that â and âout have large components at ωr and ωb and smaller components at otherfrequencies and that b̂ also has a steady state value and smaller fluctuating piece:â[t]= αr ejφr ejωr t + αb ejφb ejωb t + â1 [t],â† [t] = αr e−jφr e−jωr t + αb e−jφb e−jωb t + â†1 [t],b̂[t] = b0 [t] + b̂1 [t],(3.88)b̂† [t] = b∗0 [t] + b̂†1 [t],âout [t] = αout,r ejωr t + αout,b ejωb t + âout,1 [t].

(3.89)

Here, αin,0 and αout,0 are generally complex, but αr and αb are real and are related to the photon
numbers by nr = αr2 and nb = αb2 . I apologize for the confusing notation between the red drive
photon number nr and the right port thermal photon occupancy nthr . However, the superscript andcontext should always distinguish the two symbols.

3.2.2

Steady state solution

To zero-ith order (i.e. when â1 , b̂1 , and the ξˆi ’s are zero), solving the equations of motion
gives
b0 [t] =αin,r =

jg0 1 + 2 αr2 + αb2 + 2αr αb cos [(ωb − ωr )t + φb − φr ]Γm − 2jΩm
jφr
αb ejφbαr e,αin,b = √,√κl χc [ωr ]κl χc [ωb ]αout,r = (N2 − κl χc [ωr ])αin,r ,

αout,b = (N2 − κl χc [ωb ])αin,b ,,(3.90)(3.91)(3.92)

where ω c = ωc + g0 (b0 [t] + b∗0 [t]) is the new, shifted, cavity resonance frequency. The number ofdrive photons in the cavity and the output power from each drive are the same as if they were used
independently.

66
3.2.3

Linearized equations of motion

Plugging the derived steady-state values into the equations of motion gives the linearized
equations of motion:
κ




â˙ 1 [t] = −− jω c â1 [t] + j ej(ωr t+φr ) gr + ej(ωr t+φb ) gb b̂1 [t] + b̂†1 [t]2X √κi ξˆi [t],+(3.93)

i=l,r,0˙b̂1 [t] = −






Γm
− jΩm b̂1 [t] + j e−j(ωr t+φr ) gr + e−j(ωb t+φb ) gb â1 [t]2p+j ej(ωr t+φr ) gr + ej(ωb t+φb ) gb â†1 [t] + Γm ξˆm [t],(3.94)√√
where gr ≡ g0 nr and gb ≡ g0 nb are the photon enhanced coupling rates. Multiplying theseequations by e−jωt and integrating gives the linearized equations of motion and their conjugates in
the frequency domain:â1 [ω]χc [ω]= jgr ejφr (b̂1 [ω − ωr ] + b̂†1 [ω − ωr ])(3.95)+jgb ejφb (b̂1 [ω − ωb ] + b̂†1 [ω − ωb ]) +X √κi ξˆi [ω],i=l,r,0â†1 [−ω]χ∗c [ω]= −jgr e−jφr (b̂†1 [−ω + ωr ] + b̂1 [−ω + ωr ])−jgb e−jφb (b̂†1 [−ω + ωb ] + b̂1 [−ω + ωb ]) +(3.96)X √κi ξˆi† [−ω],i=l,r,0b̂1 [ω]χm [ω]b̂†1 [−ω]χ∗m [ω]
ejφr â†1 [ω − ωr ] + e−jφr â1 [ω + ωr ] p+jgb ejφb â†1 [ω − ωb ] + e−jφb â1 [ω + ωb ] + Γm ξˆm [ω],(3.97)

= −jgr e−jφr â1 [−ω + ωr ] + ejφr â†1 [−ω − ωr ]

 p
†
[−ω],
−jgb e−jφb â1 [−ω + ωb ] + ejφb â†1 [−ω − ωb ] + Γm ξˆm(3.98)= jgr

where, as in the one-drive case, χc [ω] ≡κ

2 + j(ω − ω c )−1and χm [ω] ≡

Γ

m

2

+ j(ω − Ωm )

−1

are

the cavity and mechanical susceptibilities.
Unlike in the one-drive case, the many frequencies in these four equations mean that they
are an infinite set of coupled equations and can not be written in terms of only four variables, and67
thus cannot be solved directly for â1 [ω] and b̂1 [ω]. However, I can eliminate all of the â1 ’s andwrite them as equations for just the b̂1 ’s. This makes sense because I will first only be interested
in the state of the mechanics and will not care about the cavity fields until I calculate the output
spectrum. These equations for only the b̂1 ’s are the following:b̂1 [ω]χm [ω]= c0 [ω]b̂1 [ω] + b̂†1 [ω]− c∗1 [−ω]b̂1 [ω − ∆W ] + b̂†1 [ω − ∆W ]

(3.99)

+c1 [ω] b̂1 [ω + ∆W ] + b̂†1 [ω + ∆W ] + ĉnoise [ω],b̂†1 [−ω]χ∗m [ω]= c∗0 [ω] b̂†1 [−ω] + b̂1 [−ω] − c1 [−ω] b̂†1 [−ω + ∆W ] + b̂1 [−ω + ∆W ](3.100)+c∗1 [ω] b̂†1 [−ω − ∆W ] + b̂1 [−ω − ∆W ] + ĉ†noise [−ω],c0 [ω] = −gr2 (χc [ω + ωr ] − χ∗c [−ω + ωr ]) − gb2 (χc [ω + ωb ] − χ∗c [−ω + ωb ]) ,(3.101)c1 [ω] = −e−j(φb −φr ) gr gb (χc [ω + ωb ] − χ∗c [−ω + ωr ]) ,(3.102)ĉnoise [ω] = jgr e−jφr χc [ω + ωr ]X√

κi ξˆi [ω + ωr ]i+jgr ejφr χ∗c [−ω + ωr ]X√†

κi ξˆi [ω − ωr ](3.103)i+jgb e−jφb

χc [ω + ωb ]X√

κi ξˆi [ω + ωb ]i+jgb ejφb χ∗c [−ω + ωr ]X√†

κi ξˆi [ω − ωb ] +pΓm ξˆm [ω].

iHere, ∆W = ωb − ωr and I note that c0 [ω] = −c∗0 [−ω].
3.2.4

Assumptions used to simplify the 2-drive case

Equations 3.99 couple the field b̂1 [ω] to b̂1 fields and their daggers at all frequencies ω +s∆W ,for integer s. Therefore, in order to solve these equations for b̂[ω], I must approximate away terms
that are small to obtain a solvably small system of equations. I will eventually be interested in
calculating the mechanical spectrum near ω = Ωm . Thus, I am interested in solving for b̂[Ωm ] andb̂† [−Ωm ]. The fields b̂1 [ω] and b̂†1 [ω] are proportional to χm [ω] and χ∗m [−ω], respectively. Thus, ifχm [ω + s∆W ] is negligible for some integer s, then so is b̂1 [ω + s∆W ]. The only χm ’s that are
significant near ω = Ωm in the resolved sideband limit are χm [ω], χm [−ω + ∆W ], and their complex
conjugates. Thus, the only fields that contribute near ω = Ωm are b̂[ω], b̂† [−ω], b̂[−ω + ∆W ], and68b̂† [ω − ∆W ]. Note that if the system were not in the resolved sideband limit, it would not be
possible to truncate the equations like this and the system would not be easily solvable. The fieldb̂[ω] only couples to b̂† [ω − ∆W ] and b̂† [−ω] only couples to b̂[ω − ∆W ], so this reduces Equations
3.99 to two coupled equations to solve for b̂[ω] and two for b̂† [−ω]:b̂1 [ω]χm [ω]b̂†1 [ω − ∆W ]χ̃∗m [ω]b̂†1 [−ω]χ∗m [ω]b̂1 [∆W − ω]χ̃m [ω]= c0 [ω]b̂1 [ω] − c∗1 [−ω]b̂†1 [ω − ∆W ] + ĉnoise [ω],(3.104)= c̃∗0 [ω]b̂†1 [ω − ∆W ] − c̃1 [−ω]b̂1 [ω] + c̃ˆ†noise [−ω],(3.105)= c∗0 [ω]b̂†1 [−ω] − c1 [−ω]b̂1 [∆W − ω] + ĉ†noise [−ω],(3.106)= c̃0 [ω]b̂1 [∆W − ω] − c̃∗1 [−ω]b̂†1 [−ω] + c̃ˆnoise [ω].

(3.107)Here, I have defined tilded functions to make the expressions simpler: c̃0 [ω] ≡ c0 [−ω + ∆W ],c̃1 [ω] ≡ c1 [−ω + ∆W ], c̃ˆnoise [ω] ≡ ĉnoise [−ω + ∆W ], and χ̃m [ω] ≡ χm [−ω + ∆W ]. Solving theseequations results in the solutionsχm [ω] (1 − c̃∗0 [ω]χ̃∗m [ω]) ĉnoise [ω] − c∗1 [−ω]χ̃∗m [ω]c̃ˆ†noise [−ω]b̂1 [ω] =,(1 − c0 [ω]χm [ω]) (1 − c̃∗0 [ω]χ̃∗m [ω]) − c∗1 [−ω]c̃1 [−ω]χm [ω]χ̃∗m [ω]χ∗m [ω] (1 − c̃0 [ω]χ̃m [ω]) ĉ†noise [−ω] − c1 [−ω]χ̃m [ω]c̃ˆnoise [ω]b̂†1 [−ω] =.
(1 − c∗0 [ω]χ∗m [ω]) (1 − c̃0 [ω]χ̃m [ω]) − c1 [−ω]c̃∗1 [−ω]χ∗m [ω]χ̃m [ω](3.108)(3.109)Plugging in for the ĉnoise terms and re-writing in terms of coefficients, I get†b̂1 [ω] = p1 [ω]ξˆm[ω − ωb + ωr ] + p2 [ω]ξˆm [ω]rr
X κi
X κi †ˆξ [ω − ωb ] + p4 [ω]ξˆi [ω + ωr ],+p3 [ω]κ iκb̂†1 [−ω]

=

i
∗
†
ˆ
p1 [ω]ξm [−ω + ωb − ωr ] + p∗2 [ω]ξˆm[−ω]+p∗3 [ω]Xir(3.110)i(3.111)X
κi ˆξi [−ω + ωb ] + p∗4 [ω]
κ
i

r

κi ˆ†
ξ [−ω − ωr ].
κ i

69
3.2.5

Phonon spectrum and mechanical occupancy

The phonon spectrum for the two-drive scheme isEb̂† [−ω]b̂[ω](3.112)
DE
DE††= |p1 [ω]|2 ξˆm [−ω + ωb − ωr ]ξˆm[ω − ωb + ωr ] + |p2 [ω]|2 ξˆm[−ω]ξˆm [ω]E
X √κi κj D
2
+ |p3 [ω]|ξˆi [−ω + ωb ]ξˆj† [ω − ωb ](3.113)κi,jE
X √κi κj D †2+ |p4 [ω]|ξˆi [−ω − ωr ]ξˆj [ω + ωr ]
κ
i,j




2
2 th
2 th
(3.114)nth= |p1 [ω]|2 nth
c + 1 + |p4 [ω]| nc .
m + 1 + |p2 [ω]| nm + |p3 [ω]|Sbb [ω] =D

The coefficients pi [ω] can be messy to calculate in full. However, they are much simpler if I
make either of two assumptions, which seem to be valid for the cases I am interested in. The
first is that χm [−ω + ∆W ] = 0. This approximation is valid if the sidebands are separated bymany mechanical linewidths. This will be the case for both the sideband asymmetry and quantum
backaction measurements, where we want to be able to clearly distinguish the two sidebands.
However, this approximation will not be valid if the sidebands overlap. In that case, χm [−ω+∆W ] =χ∗m [ω] is clearly not zero. The alternative assumption is that c1 [ω] = 0. This assumption is valid if
the drives are symmetrical about the cavity frequency (regardless of whether the sidebands overlap
or not). The only case where I might like the sidebands to overlap would be a measurement of
quantum backaction evasion (see the discussion of this effect as a limiting case below), where thedrive powers are equal. This is likely best accomplished with the sidebands centered in the cavity,
so it seems reasonable that at least one of these two assumptions would be valid. If neither of them
are true (if the sidebands are overlapping and significantly detuned from the cavity frequency),
then the mechanical spectrum is very complicated and cannot be reduced to a Lorentzian.
Either of these assumptions leads to a substantial simplification of the mechanical field, letting
me easily identify the effective mechanical susceptibility:b̂[ω] =

χm [ω]ĉnoise [ω],
1 − c0 [ω]χm [ω]χm,eff [ω] =χm [ω].
1 − c0 [ω]χm [ω](3.115)

70
Either of these assumptions also leads to the following simplification of the spectrum:

2
2
2
th
2 th
Γm nthm + gb nc + 1 |χc [−ω + ωb ]| + gr nc |χc [ω − ωr ]|Sbb [ω] =|χm [ω]|2|1 − c0 [ω]χm [ω]|2
!
2

 x2
x
zp
zp
=
Γm nthSFb F [−ω] nth
SFr F [ω]nth|χm,eff [ω]|2m +c +1 +
c
h̄2
h̄2



 th 
0th0≈
Γm nth
+
Γ
n
n
+
1
+
Γ
n
+
1
nc |χm,eff [ω]|2opt,ropt,b m,b
m
c
m,r




th
0
th 0
Γm nth
+
n
+
n
+
2n
n
Γ
|χm,eff [ω]|2 .
≈
opt
m
c
m
c m

(3.116)(3.117)(3.118)(3.119)In the second step, I replaced the cavity susceptibilities with the force spectral density due to thered or blue drive alone. The total force spectrum is SF F [ω] = SFr F [ω] + SFb F [ω]. In the third
step, I approximated the force spectral densities by evaluating them at Ωm and writing them interms of Γopt,d and n0m,d for each drive alone. The total optomechanically induced linewidth isΓopt = Γopt,r + Γopt,b while the total mechanical occupancy in the absence of mechanical damping
and cavity photons is
Γopt n0m =
x2zp  r
b
S
[−Ω
]
=
S
[−Ω
]
+
S
[−Ω
]
= Γopt,r n0m,r + Γopt,b n0m,b . (3.120)
m
m
m
FF
FF
2 FF
2
h̄
h̄

x2zp

To get the final boxed expression for Sbb , I used the fact that n0m,r = 0 and n0m,b = −1. This resultsin the same expression as for the 1-drive case, now with the quantities Γopt and n0m including
contributions from both drives. The total mechanical occupancy is then also the same as in the
1-drive case:nth Γm +nfm = m
3.2.6


0
th 0
nth
c + nm + 2nc nm Γopt.
Γm + Γopt(3.121)

Important limits of the mechanical occupancy

Clearly, if one drive is turned off or substantially smaller than the other drive, this expression
limits to the one-drive case with only a red or blue detuned drive. The most notable thing that isdifferent than in the two-drive case happens if I apply red and blue-detuned drives that are equalin strength, gr = gb = g. Then the mechanical occupation becomesbathnfm = nthm + nm = nm +


4g 2  th
2nc + 1 .
κΓm

(3.122)

71
Once again, I find a situation in which the dynamical radiation pressure is absent and the quantum
backaction heating due to fluctuations of the input field is visible. Comparison to Equation 3.59, theon-resonance single drive case, shows that (in the resolved sideband limit), the quantum backaction
piece is stronger here by a factor of 4Ω2m /κ2 . This is because the quantum backaction is filtered by
the cavity and so should be much stronger when the sidebands are centered in the cavity rather
than off to the sides. Thus, in the resolved sideband limit, the mechanical oscillator will exhibit far
more quantum backaction motion when driven by two, equally strong, red and blue-detuned tones
than when driven by a single tone on resonance.

3.2.7

Output field spectrum

I will next calculated the output fields for the system. Similarly to how I wrote the frequencydomain Langevin equations in terms of only b̂1 ’s above, I can instead write them in terms of onlyâ1 ’s. Here I show the equation for â[ω] (just take it’s dagger to find the one for â† [−ω]):â1 [ω]χc [ω]= k0 [ω]â1 [ω] + k1,r â†1 [ω − 2ωr ] + k1,b â†1 [ω − 2ωb ] + k2 [ω]â†1 [ω − ωr − ωb ](3.123)+k3,b [ω]â1 [ω − ωb + ωr ] + k3,r [ω]â1 [ω − ωr + ωb ] + k̂noise [ω],k0 [ω] = −gb2 (χm [ω − ωb ] − χ∗m [−ω + ωb ]) − gr2 (χm [ω − ωr ] − χ∗m [−ω + ωr ]) ,(3.124)

k1,r [ω] = −e2jφr gr2 (χm [ω − ωr ] − χ∗m [−ω + ωr ]) ,(3.125)k2 [ω] = −ej(φb +φr ) gr gb (χm [ω − ωr ] − χ∗m [−ω + ωr ] + χm [ω − ωb ] − χ∗m [−ω + ωb ])(3.126),
k3,r [ω] = −ej(φr −φb ) gb gr (χm [ω − ωr ] − χ∗m [−ω + ωr ]) ,p †k̂noise [ω] = jejφr gr Γm ξˆm [ω − ωr ]χm [ω − ωr ] + ξˆm[ω − ωr ]χ∗m [−ω + ωr ]p †
+jejφb gb Γm ξˆm [ω − ωb ]χm [ω − ωb ] + ξˆm[ω − ωb ]χ∗m [−ω + ωb ]+X√

(3.127)(3.128)κi ξˆi [ω].

iAlso similarly to the b̂1 ’s, the field â1 [ω] is proportional to χc [ω]. I will only be interested infrequencies near the cavity frequency because that is where the sidebands of interest are centered.
The only χc ’s that are significant near ω = ωc are χc [ω], χc [−ω + ωr + ωb ], and their complex

72
conjugates. This leaves the following two equations and their daggers:â1 [ω]χc [ω]â†1 [ω − ωr − ωb ]χ̃∗c [ω]= k0 [ω]â1 [ω] + k2 [ω]â†1 [ω − ωr − ωb ] + k̂noise [ω],(3.129)ˆ†= k̃0∗ [ω]â†1 [ω − ωr − ωb ] + k̃2∗ [ω]â1 [ω] + k̃noise[−ω],(3.130)ˆ
where I have defined k̃0 [ω] ≡ k0 [−ω + ωr + ωb ], k̃2 [ω] ≡ k2 [−ω + ωr + ωb ], k̃noise [ω] ≡ k̂noise [−ω +ωr + ωb ], and χ̃c [ω] ≡ χc [−ω + ωr + ωb ] to make the expressions simpler. Solving these equations
gives


ˆ
1 − k̃0∗ [ω]χ̃∗c [ω] k̂noise [ω] + k2 [ω]χ̃∗c [ω]k̃noise [−ω]â1 [ω] =(1 − k0 [ω]χc [ω]) 1 − k̃0∗ [ω]χ̃∗c [ω] − k2 [ω]k̃2∗ [ω]χc [ω]χ̃∗c [ω]χc [ω]

(3.131)and its dagger. Substituting the solution for â1 into Equation 2.88 for âout,1 , writing out k̂noise , andwriting the expression in terms of coefficients gives (with N1 ↔ N2 for the reflection geometry)†âout,1 [ω] = N2 ξˆl [ω] + N1 ξˆr [ω] + h1 [ω]ξˆm[ω − ωb ] + h2 [ω]ξˆm [ω − ωr ]rr
X κi †X κiˆξ [ω − ωr − ωb ] + h4 [ω]ξˆi [ω],+h3 [ω]κ iκâ†out,1 [−ω]

=

i
i
∗ ˆ†
∗
†
∗ ˆ†
ˆ
[−ω + ωr ]N2 ξl [−ω] + N1 ξr [−ω] + h1 [ω]ξm [−ω + ωb ] + h∗2 [ω]ξˆm+h∗3 [ω]

X
i

r

X
κi ˆ
ξi [−ω + ωr + ωb ] + h∗4 [ω]κir(3.132)(3.133)κi ˆ†
ξ [−ω].
κ i

The output spectrum is thus (computed in a similar manner as for the one-drive case)




2 th
2
th
Saout aout [ω] = |h1 [ω]|2 nth
+
1
+
|h
[ω]|
n
+
|h
[ω]|
n
+
1
+ |h4 [ω]|2 nc
2
3
m
m
c
r
r 

κl
κl
2
∗
∗
+ |N2 | + N2 h4 [ω]+ N2 h4 [ω]nth(3.134)
l
κ
κ
r
r 

κl
κl
2
∗
∗
+ |N1 | + N1 h4 [ω]+ N1 h4 [ω]
nth
r .
κ
κ
Computing the coefficients here would just be messy and is straightforward to do if needed, so
I won’t write them here. Instead, I’ll specify to some interesting limiting cases to give someunderstanding of the important effects we’re interested in.

73
3.2.8

Limiting cases of the output spectrum

I assume from here on that the feed line has a purely real impedance: N2 = 1, N1 = 0.
I also set the mechanical susceptibilities for the upper blue and lower red sidebands are zero:χm [ω − ωb ] = χm [−ω + ωr ] = 0. I also always assume weak coupling.
I will first look at the case where the sidebands are separated in frequency by many mechanicallinewidths. In this case, there are three contributions to the output spectrum: the wide backgroundfeatures that are not accompanied by any χm ’s, the features associated with the upper red sideband
where χm [−ω+ωb ] = 0, and the features associated with the lower blue sideband where χm [ω−ωr ] =0. For the sideband features, I will assume the sidebands are well inside the cavity linewidth andthat I can evaluate χc [ω] at ω c . I also write the output in terms of the effective mechanicalsusceptibility. The spectrum is


th
th
n
−
n
|χc [ω]|2
Saout aout [ω] = nth
+
κκ
l
c
l
l

4g 2 κl Γtot  fth+ r 2|χm,eff [ω − ωr ]|2nm − 2nth
+
n
c
l
κ

4g 2 κl Γtot  fth|χm,eff [−ω + ωb ]|2 .
+ b 2
nm + 1 + 2nth
−
n
c
l
κ

(3.135)This solution displays all of the behavior I anticipated at the beginning of the section: the sideband
expressions are identical to those for the red or blue-detuned single drive, but now both sidebands
are measurable at the same time. If the red drive is larger than the blue drive, it is possible to cool
the mechanical oscillator to low occupancy while still observing the nfm , nfm +1 sideband asymmetry.
If the drives instead have equal amplitudes, the net dynamical backaction will be zero, allowing ameasurement of the quantum backaction by looking at either sideband.
The other case of interest is the one when the sidebands are exactly overlapped ωr = ω c − Ωmand ωb = ω c + Ωm and the number of drive photons from the two drives are equal gr = gb = g.
Now, χm [ω − ωr ] = χ∗m [−ω + ωb ] and the sidebands are completely inseparable. I once again assume
that the sidebands are well inside the cavity linewidth and that I can evaluate χc [ω] at ω c for the

74
sideband term. The output spectrum is
4g 2 κl Γm  th
2
th
th
+
κκ
Saout aout [ω] = nth
n
−
n
|χ
[ω]|
+
2n
+
1
|χm [ω − ωr ]|2 . (3.136)
c
l
c
m
l
l
κ2
Notice that the sideband height here is proportional to 2nthm + 1. In contrast, the spectrum withequal drives but not overlapping sidebands has a summed sideband height proportional to 2nfm +1, which includes a contribution from quantum backaction. Thus, the quantum backaction has
disappeared from the overlapping sideband measurement. The measurement scheme with two
drives and overlapping sidebands is thus termed ‘backaction evasion’. This effect arises because thenumber of photons in the cavity is modulated at 2Ωm , effectively measuring only one quadrature of
the mechanical motion. Therefore, the extra backaction motion will be added only to the quadratureorthogonal to the measured quadrature.

3.2.9

Dressed cavity response

As with the one-drive case, I have already done most of the work required to find the dressedcavity response for two drives. I can just use the solution for â1 (Equation 3.131), making the
replacements ξˆl [ω] → αp δ[ω − ωp ] and ξˆr,0,m = 0, and plug into Equation 2.88 to find the response:χc [ω] 1 − k̃0∗ [ω]χ̃∗c [ω]
Si1 [ωp ] = Ni − κl,(3.137)(1 − k0 [ω]χc [ω]) 1 − k̃0∗ [ω]χ̃∗c [ω] − k2 [ω]k̃2∗ [ω]χc [ω]χ̃∗c [ω]where again i = 1 for reflection and i = 2 for transmission geometry.
In the scenario where the sidebands are separated by many linewidths but still relatively
centered in the cavity, I can separate the contributions from the cavity and each sideband and use
the effective mechanical susceptibility to write an approximate expression for the cavity response:
Si1 [ωp ] = Ni − κl χc [ωp ] + κl χc [ωp ]2 gr2 χm,eff [ωp − ωr ] − gb2 χ∗m,eff [ωb − ωp ] .

(3.138)For the case where the sidebands overlap exactly and the two drives are centered about the cavity
frequency and of equal strength, I recover the bare cavity response
Si1 [ωp ] = Ni − κl χc [ωp ].

(3.139)75

3.3

Cavity optomechanics in the time-domain
To this point in this chapter, I have dealt with steady state solutions of Equations 2.85-2.88.

Now, I turn to the situation where the drive tones are turned on and off in time. If I apply astrong red-detuned drive tone, I will find that the linearized interaction Hamiltonian resembles abeam-splitter Hamiltonian. This means that photons and phonons can be exchanged. We havealready seen this in the steady state case of mechanical cooling: a red-detuned drive caused thephonons coupled to a relatively warm thermal bath to be exchanged with photons coupled to amuch cooler bath, thereby cooling the mechanical oscillator. In the time-domain, I can think about
turning this coupling on and off by turning the red-detuned drive (which I will call the transfer
field) on and off. I characterize the strength of the transfer by either the transfer optomechanicalcoupling gT [t] or the optomechanically induced mechanical damping rate ΓT [t] = 4gT [t]2 /κ.
One application that this control of the coupling allows is mechanical storage of information
initially contained in the input or cavity fields (this is investigated experimentally in Chapter 11).
The coupling is initially turned on to transfer the information from the cavity into the mechanicaloscillator. It is then turned off for some time, isolating the information from the cavity and storing
it in the mechanical oscillator. Lastly, the coupling is turned back on, transferring the informationback into the cavity or output fields. The choice of cavity vs. input/output (itinerant) fields
depends on the strength of the coupling tone. If the coupling between cavity and mechanics ismuch weaker than the cavity coupling, 2g < κ, then the state leaks into or out of the cavity fastenough that the state is never entirely contained in the cavity, and the transfer is essentially between
the mechanical oscillator and the itinerant fields. Conversely, if the coupling is very strong, 2g > κ,then the state of the mechanics is swapped with that of the cavity faster than information canenter or escape from the cavity. In our experiment, the mechanical loss rate is much slower thanthe cavity’s (Γm  κ). Therefore, we would like to skip the cavity entirely, and thus work in thelimit of weaker coupling.

76

Figure 3.2: Pulse storage timing diagram. The goal is to transfer an arbitrary preparation signal
at the cavity resonance (blue) into the mechanical oscillator and then later retrieve it. I apply astrong, red-detuned transfer drive (green) between times 0 and ta , which will be shaped in timeto optimally complete the transfer process. From time ta to tb , the signal will be stored in themechanical oscillator. At time ta + tb , the transfer drive will again be turned on to read out thestate of the mechanical oscillator. There will be some unwanted power that leaks out of the cavity
during the initial transfer process - this power normalized to the input preparation power is what
I call the leakage. The power that is retrieved from the cavity in the third process normalized tothe input preparation power is the efficiency of storage.

3.3.1

Simplification of the equations of motion

The problem solved in this section is how to shape the envelope of the transfer (strong, reddetuned) pulse in time in order to optimally store and retrieve an arbitrary known preparation(weak, on resonance) tone in the mechanics (see Figure 3.2). Very similar theoretical and experimental work has been done by many groups, both for storing and retrieving an itinerant state froma mechanical oscillator [82, 83, 84, 85, 86, 87], and by transferring a state from one electromagnetic
field to another, via a mechanical intermediary [88, 89, 90, 91].
I will optimize the transfer efficiency by comparing the energy stored and retrieved to the
input preparation field energy. The optimal envelope should not depend on whether the state isclassical or quantum, noisy or noiseless. These characteristics are likely to affect the fidelity of thestored and retrieved state, but not the total power conversion. I will thus ignore all noise terms
in the equations of motion. For simplicity, I will assume the feed line has no reactive components.
The Heisenberg-Langevin equations and output equation, without noise, in the reflection geometry

77
are:



√
− jωc a[t] + jg0 b[t] + b† [t] a[t] + κl αin [t],



2
1
Γm
,
− jΩm b[t] + jg0 a† [t]a[t] +ḃ[t] = −22√
aout [t] = ain [t] − κl a[t].
ȧ[t] = −κ

(3.140)(3.141)(3.142)

I have left off the hats because without noise, these equations are essentially classical. I will assume
a strong red-detuned drive and a weaker signal of interest at the cavity resonant frequency. The
cavity and output fields of interest will be near the cavity frequency, while the mechanical fields ofinterest will be near the mechanical frequency. Thus, the field assumptions are:αin [t] = αin,T [t]ejωT t + αin,p [t]ejωc t ,(3.143)a[t] = αT [t]ejφT ejωT t + a1 [t]ejωc t ,(3.144)b[t] = b0 [t] + b1 [t]ejΩm t ,(3.145)αout [t] = αout,T [t]ejωT t + αout,p [t]ejωc t ,(3.146)

where αin,T is the input amplitude for the large transfer drive and αin,p is the input amplitude
for the weaker preparation field. The cavity state amplitude due to the transfer drive is αT . The
αin ’s are complex, while φT accounts for the phase associated with the real αT . I will linearize theequations of motion around the large transfer drive (αin,T [t]  αin,p [t] and αT [t]  a1 [t]). The
steady state part of the mechanical state is b0 [t], while b1 [t] accounts for deviations from this.
The zeroth order values (with a1 [t], b1 [t], αin,p [t], and αout,p [t] set to zero in Equations 3.1403.142) are
jg0 (1 + 2αT [t]2 )b0 [t] =,Γm − 2jΩm√
αout,T [t] = αin,T [t] − κl αT [t]ejφT ,ejφT
αin,T [t] = √κlαT [t]+ α̇T [t] ,χc [ωT ](3.147)(3.148)where ω c = ωc + g0 (b0 + b∗0 ) and the linearized Langevin equations are√κȧ1 [t] = − a1 [t] + jejφT gT [t]b1 [t] + κl αin,p [t],2Γmḃ1 [t] = −b1 [t] + je−jφT gT [t]a1 [t].
2

(3.149)(3.150)78
Here, gT [t] = g0 αT [t] is the transfer coupling rate and I have made a rotating wave approximation
to get rid of fields at frequencies that are strongly suppressed by the cavity.
The next simplification I will make will be to adiabatically eliminate the cavity mode dynamics. Generally, this means removing the rapidly changing dynamics from the equations of motion,leaving equations to solve for the more slowly-varying quantities. A similar argument is presented
in reference [92, Appendix B], but for a different system of interest. The essential assumptions forthis approximation are:gT [t]  κ,

ġT [t] κgT [t],
2

Γm  κ,α̇in,p [t] καin,p [t].
2

(3.151)

I will re-write the fields in the equations of motion as a1 [t] = e−κt/2 A[t] and b1 [t] = e−Γm t/2 B[t] tomake them easier to integrate:√Ȧ[t] = eκt/2 jejφT gT [t]b1 [t] + κl αin,p [t]√= e(κ−Γm )t/2 jejφT gT [t]B[t] + eκt/2 κl αin,p [t],Ḃ[t] = e−(κ−Γm )t/2 je−jφT gT [t]A[t].
I can then integrate the A equation
Z t 

√
0
0
A[t] =e(κ−Γm )t /2 jejφT gT [t0 ]B[t0 ] + eκt /2 κl αin,p [t0 ] dt0 .

(3.152)(3.153)(3.154)(3.155)−∞

I can integrate both terms by parts, using the assumptions stated above to kill the α̇in,p and ġTpieces:d  κt/2eαin,p [t]=dtd  (κ−Γm )t/2
egT [t]B[t]=dtκ κt/2κeαin,p [t] + eκt/2 α̇in,p [t] ≈ eκt/2 αin,p [t],(3.156)22
κ − Γm (κ−Γm )t/2
egT [t]B[t] + e(κ−Γm )t/2 ġT [t]B[t] (3.157)2+e(κ−Γm )t/2 gT [t]Ḃ[t]

≈
Z t

0

eκt /2 αin,p [t0 ]dt0 ≈

−∞

Z t
−∞

0

e(κ−Γm )t /2 gT [t0 ]B[t0 ]dt0 ≈κ − Γm (κ−Γm )t/2
egT [t]B[t] + e(κ−Γm )t/2 gT [t]Ḃ[t], (3.158)
2
2 κt/2e
αin,p [t],(3.159)κ2e(κ−Γm )t/2 gT [t]B[t](3.160)κ − ΓmZ t20e(κ−Γm )t /2 gT [t0 ]Ḃ[t0 ]dt0 .
−κ − Γm −∞79
Substituting in the equation for Ḃ[t], I write A[t] as
A[t] =

Z t√
2 κl κt/2
2
2jejφT (κ−Γm )t/2
egT [t]B[t] +gT [t0 ]2 A[t0 ]dt0 .
eαin,p [t] +κκ − Γmκ − Γm −∞(3.161)

I could substitute this solution for A[t] into the last term and integrate by parts again to get thenext order terms in gT [t]/κ:A[t] =√
2 κl κt/2
2jejφT (κ−Γm )t/2eαin,p [t] +gT [t]B[t](3.162)eκκ − Γm√
4 κl gT [t]2gT [t] 4
gT [t]3(κ−Γm )t/2
κt/2
jφT
+
e
B[t] + O.
eαin,p [t] + 4jeκ κ(κ + Γm )(κ − Γm )3κHowever, the approximation that gT [t]  κ means that we can keep just the zeroth and first orderterms. Going back to the original variables a1 [t] and b1 [t] and making the approximation Γm  κ
gives

√
2  jφT
je gT [t]b1 [t] + κl αin,p [t] ,κ√κȧ1 [t] = − a1 [t] + jejφT gT [t]b1 [t] + κl αin,p [t] ≈ 0.
2
a1 [t] = e−κt/2 A[t] ≈(3.163)(3.164)

I find that the derivative of a1 [t] is negligible compared to κa1 [t]/2 in these approximations (seeFigure 3.3 to see how they compare for reasonable experimental parameters). Often, this result

Figure 3.3: Adiabatic approximation validity on different timescales. The different terms in theequation of motion for a1 [t] are plotted versus κt. The values of terms ȧ1 [t] (red) and κa1 [t]/2(blue) are plotted on the y-axis as a function of κt for experimentally reasonable parameters.
The derivative ȧ1 [t] is much smaller than κa1 [t]/2 for times greater than ≈ 2κ. The adiabatic
approximation involves setting ȧ1 [t] = 0, changing the value of κa1 [t]/2 to be the sum of the twoterms (shown in green).

80
is just stated as the approximation associated with adiabatic elimination (see reference [84]). The
physical interpretation of this is that the cavity field tracks the input and mechanical states anddoes not depend on its own history. This makes sense because κ, the rate that the cavity loses
information, is much larger than any other rate in the problem.
Substituting the solution for a1 into the Langevin equation for b1 leaves me with only one
equation of motion to solve:ḃ[t] +

Γm 2gT [t]2
+
2
κ



√
2je−jφT κl gT [t]αin,p [t]b[t] =.
κ(3.165)This is a first order differential equation for b1 of form ḃ1 [t] + P [t]b1 [t] = Q[t] which can be solved
via the method of integrating factors.

3.3.2

Integrating factor solutionsFirst order differential equations of the form ḃ1 [t] + P [t]b1 [t] = Q[t] can be solved using
integrating factors, as follows. I will assume there is some function µ[t] such that∂t (µ[t]b1 [t]) = µ[t]Q[t],(3.166)Z tµ[t]b1 [t] = µ[t0 ]b1 [t0 ] +µ[t0 ]Q[t0 ]dt0 ,(3.167)

t0

1
µ[t]

b1 [t] =Z tµ[t0 ]b1 [t0 ] +µ[t0 ]Q[t0 ]dt0.

(3.168)t0

I can then write a differential equation and solution for µ[t]:˙µ[t]=µ[t] =µ[t]Q[t] − µ[t]ḃ1 [t]µ[t]P [t]b1 [t]== µ[t]P [t],b1 [t]
b1 [t]
Z t
0exp
P [t ]dt .

(3.169)(3.170)t0

The boxed equations, along with Equations 3.163 and 3.142, allow me to find all of the fields given
some input transfer and preparation fields.
I will divide the problem of state storage in the mechanical oscillator into three sections intime: the transfer, the storage, and the retrieval, illustrated in Figure 3.2. I will treat these three
sections independently, using the final state of the system from one section as the initial condition of

81
the next. This is an approximation but is valid if the time for each section is long compared with the
dynamics of the fields. During the transfer section, for times 0 ≤ t ≤ ta , an arbitrary preparation
signal αin,p [t] is transferred into the mechanical oscillator by a time-dependent transfer field with
optomechanical coupling gT [t]. I assume the mechanical oscillator begins in state b1 [0] = 0, so thefields in the first time section areZ t 

Γm
2 2 0000µ[t] = exp+ gT [t ] dt ,
2
κ
0
Z t
√
κl 1
b1 [t] = 2je−jφTµ[t0 ]gT [t]αin,p [t0 ]dt0 ,κ µ[t] 0√κext 
21−2αout,p [t] =αin,p [t] − jejφT κl gT [t]b1 [t].
κκ

(3.171)(3.172)(3.173)For the storage section, ta ≤ t ≤ ta + tb , the storage and preparation fields are turned off, isolating
the mechanical oscillator from the cavity. The fields during this section of time are
Z t


Γm 0
dt = eΓm (t−ta )/2 ,ta 2(3.174)1
b1 [ta ] = b1 [ta ]e−Γm (t−ta )/2 ,µ[t](3.175)µ[t] = exp
b1 [t] =αout,p [t] = 0.

(3.176)

During the third section of time, ta + tb ≤ t ≤ ta + tb + tc , the signal is retrieved by turning thetransfer field back on, with constant transfer rate Γout . As the efficiency of the storage is only
dependent on the total retrieved power, I can make this readout transfer constant in time. If a
specific output pulse temporal profile were desired, I could instead shape the output transfer toachieve that profile. The fields during the third section of time areZ tµ[t] = exp

(Γm + Γout ) 0
dt = e(Γm +Γout )(t−ta −tb )/2 ,2
ta +tb1
b[ta + tb ] = e−(Γm +Γout )(t−ta −tb )/2 b[ta + tb ],µ[t]rΓout κlαout,p [t] = −jejφTb1 [t].
κb1 [t] =(3.177)(3.178)(3.179)Ideally, all of the preparation power would be transferred into the mechanical oscillator suchthat no power leaked out of the cavity during the first time section. That power would then be82
recovered in the third time section. I define the efficiency and leakage of the storage as
R ta +tb +tc
E=|αout,p [t]|ta +tbR ta
20 |αin,p [t]| dt

2

dt

R ta
and

0

|αout,p [t]|2 dt

0|αin,p [t]|2 dt

L = R ta.

(3.180)Note that L is just the measured leakage energy and there will also be energy lost via dissipation
through the internal port of the cavity. Thus, E and L do not in general add up to one. The
denominators of E and L are just a function of the preparation state, which is specified at the
beginning of the problem. Thus, optimizing the efficiency is equivalent to optimizing the numerator.
I can re-write the expression for the efficiency numerator in terms of the fields found above|αout,p [t]|2 ===Z ta +tb +tc|αout,p [t]|2 dt =ta +tb==Γout κl|b1 [t]|2κΓout κl −(Γout +Γm )(t−ta −tb )e|b1 [ta + tb ]|2κΓout κl −(Γout +Γm )(t−ta −tb ) −Γm tbee|b1 [ta ]|2 ,κZ ta +tb +tc
Γout κl −Γm tb
2e−(Γout +Γm )(t−ta −tb ) dte|b1 [ta ]|
κ
ta +t
 b
−(Γ
+Γ
)t
out
m
c
Γout 1 − eκl −Γm tbe|b1 [ta ]|2
Γout + Γmκ
Γout 1 − e−(Γout +Γm )tc
Γout + Γm
Z ta
2
4κ2l e−Γm tbµ[t0 ]
00
0
×gT [t ]αin,p [t ]dt .
κ30 µ[ta ](3.181)(3.182)(3.183)(3.184)(3.185)(3.186)

Here, I see that maximizing the efficiency is also equivalent to maximizing the state in the mechanRt
ical oscillator at time ta or optimizing the integral 0 a (µ[t0 ]/µ[ta ]) gT [t0 ]αin,p [t0 ]dt0 . This is thenan optimization problem of a functional, which can be solved by finding a stationary point of thefunctional with respect to gT [t].

83
3.3.3

Optimization of efficiency

I can re-write the function I want to optimize asµ[t0 ]
gT [t0 ]αin,p [t0 ]dt0 =µ[ta ]

 Z ta Γm
2 2 0000
exp −+ gT [t ] dt gT [t0 ]αin,p [t0 ]dt0 (3.187)
2
κ
0
t0
0
r Z taqκ0=eΓm t /2 αin,p [t0 ] h[t0 ]ḣ[t0 ]dt0(3.188)2 0
r Z ta hiκ=
F t0 , h[t0 ], ḣ[t0 ] dt0(3.189)2 0h
R t 2 00 00 i
Here, h[t] ≡ exp − κ2 t a gT[t ]dt is a function that allowed me to write the argument F of the inZ ta

Z ta

tegral as a function of only h[t], its derivative, and t. By calculus of variations, this integral will findqhi0a stationary point (optimum) of the integral only if F t0 , h[t0 ], ˙[h][t0 ] = eΓm t /2 αin,p [t0 ] h[t0 ]ḣ[t0 ]solves the Euler-Lagrange equation:
0 =

=

=


∂F
∂ ḣ[t0 ]
s
s
!
0]
0]
1 Γm t0 /2
1
h[t
ḣ[t
d
0
e
αin,p [t0 ]−eΓm t /2 αin,p [t0 ]2
h[t0 ] dt0 2ḣ[t0 ]

s
s
s
0]
0]
ḣ[t
h[t
h[t0 ]
Γ1 Γm t0 /2 
d
m
0
0
e
αin,p [t0 ]
−
α
[t
]
−
α̇
[t
]
− αin,p [t0 ] 0
in,p
in,p
0
0
0

2
h[t ]
2
dt
ḣ[t ]ḣ[t ]∂Fd−∂h[t0 ] dt0

a [t0 ]:
This gives me a first order differential equation and solution for gT
s
s
s
s
!
d
h[t0 ]
ḣ[t0 ] Γm h[t0 ] α̇in,p [t0 ] h[t0 ]
=
−
−
,
dt0
h[t0 ]
2ḣ[t0 ]
ḣ[t0 ] αin,p [t0 ] ḣ[t0 ]




d
1
2
Γm α̇in,p [t0 ]10=gT [t ] −+,dt0 gT [t0 ]κ2αin,p [t0 ] gT [t0 ]2Γm α̇in,p [t0 ]+gT [t0 ],ġT [t0 ] = − gT [t0 ]3 +κ
2αin,p [t0 ]reΓm t/2 αin,p [t]κr 
gT [t] =.
2 κ αin,p [0] 2
Rt
0
0
2
0
Γ
t
+ 2 0 e m αin,p [t ] dt
2
gT [0](3.190)(3.191)sh[t0 ]!ḣ[t0 ] .

(3.192)(3.193)(3.194)(3.195)Note here that the optimization procedure produced the optimal pulse shape given some initialcondition for the transfer field gT [0]. It did not produce an optimal pulse shape over all gT [0].
Therefore, for a given preparation pulse shape, I will still need to vary gT [0] to see which value
gives the best efficiency while keeping gT [t] within experimentally realizable values.

84
3.3.4

Pulse shaping examples

In this section I will use the above solutions to solve for the optimal transfer pulse shape andsystem fields for several trial preparation pulses: the rising exponential, the falling exponential,
the Gaussian, and the half wave sine squared pulse. These pulses and their optimal transfer fields
are shown in Figure 3.4 for an array of values of gT [0]. The efficiencies and maximum gT [t] forthese values are shown in Figure 3.5. All of these graphs are created using the following systemparameters: ta = 500 µs, κ = 2π × 370 kHz, Γp = 32 kHz (Γp is the preparation exponential
or Gaussian rate below). For the plots, I assume that the second transfer (readout) pulse is on
sufficiently long to read out the entire state (tc → ∞). I also set Γm = 0 for the plots in order tobetter compare the different pulse shapes. This removes the factor of mechanical decoherence andinstead makes the efficiency only dependent on the transfer process. Note that I keep Γm and tc
finite in the analytic expressions below.

3.3.4.1

Rising exponential

For the input field αin,p [t] = AeΓp (t−ta )/2 (Figure 3.4(a)), the efficiency is maximized for thetransfer coupling ratepκ(Γm + Γp )e(Γm +Γp )t/2
gT [t] = p.
2 e(Γm +Γp )t − 1 + δ −2

(3.196)pHere, δ ≡ 2gT [0]/ κ(Γm + Γp ) quantifies the initial transfer rate in dimensionless units. The
efficiency for the rising exponential is
E

=


2
2 e−Γm tb Γ
1 − e−(Γout +Γm )tc 1 − e−(Γp +Γm )ta δ 2
ηstout Γp(Γout + Γm )(Γp + Γm ) (1 − e−Γp ta ) 1 − e−(Γp +Γm )ta δ 2 + e−(Γp +Γm )ta(3.197)This efficiency is maximized for infinite δ, but in practice δ will be limited by experimental constraints. The efficiency is ultimately limited (even for ideal conditions {ta , tc } → ∞ and Γm = 0)
2 , where η = κ /κ is the state transfer efficiency of one transfer (there are two factors in theto ηststltotal efficiency, one for the initial transfer and one for the readout).

85

Figure 3.4: Optimal transfer pulse shaping results. The input preparation and optimal transfer
fields are shown as a function of time. The preparation field is shown in black for each pulse shapeand corresponds to the right axis. It is normalized to have an amplitude of one. The colored traces
are the optimal transfer pulse shapes for several different values of gT [0], shown in the legends.
(a) Rising exponential. (b) Decaying exponential. Note that for this graph only, the left y-axis isplotted logarithmically. Otherwise, it would be impossible to see the transfer pulses. (c) Gaussian.
(d) Sine squared half-wave pulse.

86

Figure 3.5: Transfer efficiency for different pulse shapes: Rising exponential (green), decaying
exponential (purple), Gaussian (red), and sine squared (blue). (a) Efficiency as a function of
2 on a logarithmic scale to highlightdifferent values of gT [0]. I have plotted the y-axis as 1 − E/ηstefficiencies near the maximum achievable value. The efficiency improves with larger gT [0] for all
four pulse shapes. However, this graph hides the fact that improvements in efficiency come at thecost of increasingly large transfer fields. (b) Maximum value of gT [t] as a function of different valuesof gT [0]. (c) Parametric plot of efficiency as a function of maximum value of gT [t]. This graph is themost useful as it displays efficiency achievable as a function of maximum gT [t] allowable. Keep inmind that the solutions I have derived are only valid in the weak coupling limit ΓT [t]/κ  1. This
plot shows that high efficiency is most easily achieved for the rising exponential and least easily
achieved for the decaying exponential.

3.3.4.2

Decaying exponential

For the input field αin,p [t] = Ae−Γp t/2 (Figure 3.4(b)), the efficiency is maximized for thetransfer coupling ratepκ(Γp − Γm )e−(Γp −Γm )t/2
gT [t] = p.
2 1 − e−(Γp −Γm )t + δ −2

(3.198)87pHere, δ ≡ 2gT [0]/ κ(Γp − Γm ) quantifies the initial transfer rate in dimensionless units. The
efficiency for the falling exponential isE2−(Γout +Γm )tc
2 e−Γm (ta +tb ) Γ
1 − e−(Γp −Γm )ta δ 2
ηstout Γp 1 − e .
(Γout + Γm )(Γp − Γm ) (1 − e−Γp ta ) 1 − e−(Γp −Γm )ta δ 2 + 1=(3.199)2.
Once again, this is maximized for infinite δ and is ultimately limited to ηst

3.3.4.3

Gaussian22

For the input field αin,p [t] = Ae−Γp (t−ta /2) /2 (Figure 3.4(c)), the efficiency is maximized forthe transfer coupling rate
p
2
2
2
2
κΓp e−Γp (t−ta /2) /2 eΓm (t−ta /2)/2 e−Γm /(8Γp )r h
gT [t] =ihi√Γm +Γ2p ta
Γm +Γ2p (ta −2t)
1/4
π
2 erf
−
erf
+ δ −22Γp2ΓpHere, δ ≡ gT [0] exph(3.200)√ p
2 i(Γm + Γ2p ta )/(2Γp ) /2 π 1/4 2/ κΓp quantifies the initial transfer rate indimensionless units and erf[x] is the error function. The efficiency for the Gaussian isE

3.3.4.4=−Γm ( t2a +tb − Γm2 )
4Γp
2Γ
ηst
out e

Γout + Γmhi2  h Γm +Γ2p ta iΓm −Γ2p ta
1 − e−(Γout +Γm )tc δ 2 erf
−
erf
2Γp
2Γp
i
h
i
 (3.201)hi  h
2t
2t
Γ
−Γ
Γ
+Γ
m
a
m
a
Γp ta
p
p
−
erf
+
1
2erf 2δ 2 erf2Γp2ΓpSine squared half-wave pulse

For the input field αin,p [t] = A sin2 [πt/ta ] (Figure 3.4(d)), the expressions for gT [t] is easy
to find by using Equation 3.195, but is more complicated (and less easy to display as a simple
expression) than for the previous examples. The integrals in Equation 3.187 can also not be done
analytically. However, everything can be solved numerically, given values for all of the systemparameters. This is what I have done to create the graphs in Figures 3.4(d) and 3.5.

Chapter 4

Theory III: Interferometric linear measurement

In Sections 3.1 and 3.2, I derived expressions for the field spectral density Saout aout [ω] immediately at the output of the optomechanical system. In this chapter, I will relate the voltage
spectral densities that we measure to Saout aout [ω].
In the following sections I use the fact that the cavity output fields in Section 3.1 took theformâout,1 [ω] =Xnoαi [ω]ξˆi [ω − ωi0 ] + βi [ω]ξˆi† [ω + ωi0 − 2ωd ] ,(4.1)

i={m,l,r,0}

where the sum is over the mechanical (m), left port (l), right port (r), and internal (0) cavity
0 = ω , and ω 0 = ω 0 = ω 0 = 0. The frequency dependent coefficients of each operator
operators, ωmdr0l

are contained in the αi ’s and βi ’s. The output fields in Section 3.2 took the same form, but with
0 = ω , and ω 0 = ω 0 = ω 0 = 0. As each ξˆ operator only enters the expression
ωd → (ωr + ωb )/2, ωm
r
i
r
0
l

for âout at a single frequency, there are only a few nonzero combinations of âout,1 and â†out,1 :DEâ†out,1 [ω1 ]âout,1 [ω2 ]= Saout aout [ω2 ]δω1 ,−ω2 ,DEâout,1 [ω1 ]â†out,1 [ω2 ]= (Saout aout [ω1 ] + 1)δω1 ,−ω2 ,(4.2)

hâout,1 [ω1 ]âout,1 [ω2 ]i = −e2jφd Scross [ωd − ω1 ]δω1 +ω2 ,2ωd ,DE∗â†out,1 [ω1 ]â†out,1 [ω2 ]= −e−2jφd Scross[ωd + ω2 ]δω1 +ω2 ,−2ωd ,

(4.4)

(4.3)

(4.5)

where for the single drive case, φd and ωd are the phase and frequency of the drive, and for the two
drive case, φd → (φr + φb )/2 and ωd → (ωr + ωb )/2. The spectral densities are related to the α’s

89
and β’s above by
Saout aout [ω] =

X

n
o
2 th
|αi [ω]|2 nth
+
|β
[ω]|
(n
+
1)
,
i
i
i

(4.6)

n
o
2 th
|αi [ω]|2 (nth,i + 1) + |βi [ω]| ni(4.7)

i={m,l,r,0}

Saout aout [ω] + 1 =X
i={m,l,r,0}Scross [ω] = −e−2jφd

X

n
o
th
αi [ωd + ω]βi [ωd − ω]nth
+
α
[ω
−
ω]β
[ω
+
ω](n
+
1)
.
i d
i d
i
i

(4.8)

i={m,l,r,0}

I have defined the terms Scross [ω] in a way that will highlight their interference with the
Saout aout [ω]’s. For the specific output field results found in Sections 3.1, the cross terms turn out tobe zero for a red or blue drive or any heterodyne measurement (ωLO 6= ωd ). The one case where
they do not vanish is when the detection is homodyne and there is a single, on-resonant drive.
Making only the approximations that N2 = 1, ωLO = ωd = ω c , and χm [−Ωm ] = 0 (true for ahigh-Q mechanical oscillator), the cross terms satisfy the identity∗
Scross [ω] + Scross[ω] = g 2 κl Γm (1 + 2nfm )|χc [ω c + ω]|2 |χm [ω]|2= Sagout aout [ω c − ω] + Sagout aout [ω c + ω].

(4.9)(4.10)

Here, I have used a superscript g to indicate the g-dependent part of the spectral density:
Sagout aout [ω] = Saout aout [ω] − Sag→0[ω].
out aout(4.11)

This division of notation is important, as the cross terms only interfere with the g-dependent part ofthe spectral density, which contains information about the mechanical sidebands and is proportional
to g 2 . The cross terms do not affect the g-independent part of the spectral density, which is thecontribution in the absence of optomechanical coupling (g → 0) and contains information only
about the cavity and input field thermal populations (it is zero in the absence of thermal photons).
The cross terms can also be nonzero in the case where there are two equal drives, one close
to optimal red-detuning ωr = ω c − Ωm −  and one close to optimal blue ωb = ω c + Ωm + ,with the sidebands separated by many mechanical linewidths but still within the cavity linewidth(Γm    κ). If the local oscillator frequency is centered between the two sidebands at ωLO = ω c ,90they are folded on top of each other and the measured frequency is ω ≈ . In this case, the crossterms take the form∗
Scross [ω] + Scross[ω] = g 2 κl Γm (1 + 2nfm )|χc [ω c + ω]|2 |χm [Ωm − ω + ]|2= Sagout aout [ω c − ω] + Sagout aout [ω c + ω]

(4.12)(4.13)

Note that these cross term relations are specific to the optomechanical model used in Sections 3.1
and 3.2.

4.1

Linear, interferometric measurement
I could imagine detecting the signal at the output of the cavity by just putting a microwave

voltage detector at the output of the cavity and directly measuring the voltage spectral density
at high frequency. However, it is technically difficult to digitize and store data taken at gigahertz
frequencies, so we typically use an interferometric scheme to mix the signal down to lower frequency(usually in the kilohertz or megahertz range). In a homodyne detection, a single input signal issplit into two parts traveling along different arms of an interferometer, the signal arm and the
local oscillator arm. The signal arm contains the system of interest, which imparts a phase shift
on the signal. Interfering this signal with the local oscillator (LO) allows the relative phase to bemeasured. Homodyne (ωLO = ωd ) and heterodyne (ωLO 6= ωd ) detection can both be done in either
the optical or microwave regime (Figure 4.1). In the optical case, the interference is done using
a beam splitter and the detector is a photodetector. In the microwave case, a frequency mixer isused to mix the signal with the local oscillator to produce a low frequency voltage signal which
can be digitized. Appendix D.1 details the relevant field transformations for beam splitters andphotodetectors in the optical case and explains how these elements can be combined to form asingle-quadrature or two-quadrature detector. As the physics behind the microwave and opticalcases is identical, I use this optical realization to model all interferometric detection. This chapter
relies heavily on the results in appendix D.1 and I thus recommend reading it first.
In the case of an on-resonant drive for an optomechanical system, homodyne detection folds

91

Figure 4.1: Interferometric detection. (a) Homodyne detection in the optical regime. A single
laser tone is split into two pieces, one of which interacts with the system of interest in the signal
arm of the interferometer. The other arm acts as a local oscillator. Using a photodetector tomeasure the interference of the two tones then allows a measurement of the phase shift due tothe system of interest in the signal arm. (b) Homodyne detection in the microwave regime. The
final beam splitter and photodetector are replaced by a frequency mixer and a voltage detector.
(c,d) Heterodyne detection in the optical and microwave regimes. These are the same as homodyne
detection except that the local oscillator is provided by a different source at a different frequency.
As explained in Appendix D.1, ηLO must be large (ηLO  1 − ηLO ) in order to avoid adding extra
noise to the measurement. In the homodyne case, ηsplit is not particularly consequential so long as
the local oscillator power is much larger than the signal power (which could also be accomplishedby attenuation or drive tone cancellation in the signal arm.)

the two sidebands on top of each other. Mathematically, this is evidenced by the cross terms
introduced above being nonzero, allowing interference that can result in a better signal to noise
than for heterodyne. In the case of the two driving tones considered in Section 3.2, the equivalentof homodyne detection is to place the LO at the cavity resonance, once again folding the sidebands
on top of each other. However, in many cases heterodyne detection is preferable - for instancewhen the signal frequency of interest from homodyne detection is too large, or when (as is the casefor a far-detuned drive) homodyne detection does not provide an improvement in signal to noise.
Heterodyne detection would also be necessary for a sideband asymmetry measurement because theinformation of interest is in the differences between the two sidebands. Therefore, from here on,
I only consider homodyne detection in the case where the single drive is near cavity resonance or
where the two drives are equal in magnitude. For all other cases, the heterodyne result is the onethat is relevant.

92

4.2

Single quadrature measurement
ˆ incident on a linear detector:
In Appendix D.1, I define two quadratures1 of the field d[ω]
ˆ + ωLO ] + dˆ† [ω − ωLO ],X̂[ω] = X̂ † [ω] = d[ω(4.14)

ˆ + ωLO ] − dˆ† [ω − ωLO ]).
Ŷ [ω] = Ŷ † [ω] = −j(d[ω(4.15)

A linear detector modeled as just a beam splitter followed by a photodetector measures an intensityIˆ only dependent on a single quadrature of the incident field (Equation D.9):
ˆ = Iˆ† [ω] = ALO
I[ω]



p
ηLO (1 − ηLO ) − sin[φLO ]X̂[ω] + cos[φLO ]Ŷ [ω] ,(4.16)

where ALO and φLO are the amplitude and phase of the local oscillator, and ηLO is the detection
efficiency. In this chapter, I calculate the measured intensity spectrum for a linear detection of theoutput field from an optomechanical system. I introduce a beam splitter with transmission η placed
between the system of interest and the detector to account for any loss2 . The field incident on the√√linear detector is then dˆin = ηâout + j 1 − η ξˆvac . Using the results in Appendix D.1 (specificallyEquation D.15) and writing in terms of Scross [ω], I findphoton
[ω] =
SII

1
1
+ (Saout aout [ωLO − ω] + Saout aout [ωLO + ω])
2η 2

1
∗
+ e2j(φLO −φd ) Scross[ω] + e−2j(φLO −φd ) Scross [ω] δωLO ,ωd ,2(4.17)

photonwhere SII[ω] is the intensity spectral density written in photon units. For a heterodyne mea-

surement, the cross terms are zero and the previous expression reduces tophoton
SII
[ω] =

=

1

1
1
+ (Saout aout [ωLO − ω] + Saout aout [ωLO + ω])
2η 2

1
1 g→0
+
Saout aout [ωLO − ω] + Sag→0
[ω
+
ω]
LO
a
out out2η 2
1+ Sagout aout [ωLO − ω] + Sagout aout [ωLO + ω] .
2

(4.18)(4.19)

Note that the quadratures I use are always defined as a sum of a field at one frequency and a daggered field at
a different frequency, where those two frequencies are symmetric about the LO frequency. Specifically, they are not
the real and imaginary part of the microwave or optical field at a single frequency.
2
The beam splitter transmission (or quantum efficiency) η can be related to an effective number of added noise
photons nadd via η = (1+2nadd )−1 such that the constant factor in front of the right-hand side of the single-quadrature
expressions for SII [ω] becomes 1/(2η) = 1/2 + nadd .

93For a homodyne measurement with an on-resonant single drive or two equal red and blue-detuned
drives, the cross terms can maximally interfere with the spectral terms (for the choice φLO = φd )3 :photon
SII
[ω] =


1
1 g→0
+
Saout aout [ωLO − ω] + Sag→0[ωLO + ω]out aout
2η 2(4.20)

+ Sagout aout [ωLO − ω] + Sagout aout [ωLO + ω] .
The homodyne and heterodyne cases both have the same half-photon contribution from added
noise and the same contribution from the Sag→0
out aout [ω] terms (which contain only information about
thermal photons). The difference in the two expressions is the factor of two enhancement of the
Sagout aout [ω] terms in the homodyne case. As these terms contain the information of interest about
the mechanical sidebands, this coefficient results in an obvious factor of two improvement in thesignal to noise of the measurement. However, there is another factor of two improvement when using
homodyne detection, as both Sagout aout [ωLO − ω] and Sagout aout [ωLO + ω] contain sideband information(as opposed to only one of these terms in the heterodyne case). This results in an overall factor
of four improvement in signal to noise when using homodyne over heterodyne single-quadraturedetection.

4.3

Two-quadrature measurement
In Appendix D.1, I model a linear detector measuring both quadratures of the incident field

as a 50/50 beam splitter followed by two single-quadrature linear detectors, one of which has an
LO that is ninety degrees out of phase. The two measured intensities then correspond to two
orthogonal quadratures of the incident field (Equations D.17 and D.19, ignoring added noise forthe moment):√
2IˆI [ω]p
ALO ηLO (1 − ηLO )√2IˆQ [ω]p
ALO ηLO (1 − ηLO )3= − sin[φLO ]X̂[ω] + cos[φLO ]Ŷ [ω],(4.21)= sin[φLO ]Ŷ [ω] + cos[φLO ]X̂[ω].

(4.22)In this dissertation, I only consider the choices φLO = φd and φLO = φd +π/2, which maximally constructively anddestructively interfere with the spectral terms. However, other choices for φLO can lead to complicated interferences.
ba −1/2
In particular, for cos[φd − φLO ] = ±(3 + 4nth
, the Scross terms will interfere to create maximal squeezingm + 4nm )of the vacuum noise. This is the effect observed in [57, 56].

94
I once again introduce a beam splitter with transmission η placed between the system of interestand the detector to account for any loss4 . Using Equation D.21 and writing in terms of Scross [ω],the intensity spectral densities for the I and Q quadratures and the quadrature combination Iˆθ = √ejθ IˆI + e−jθ IˆQ / 2 are[ω] =SIphotonI(Q) II(Q)[ω] =SIphotonθ Iθ

1 1(4.23)+ (Saout aout [ωLO − ω] + Saout aout [ωLO + ω])
η 2

1
∗
± e2j(φLO −φd ) Scross[ω] + e−2j(φLO −φd ) Scross [ω] δωLO ,ωd ,2
1 1 − sin[2θ]
1 + sin[2θ]+
Saout aout [ωLO − ω] +Saout aout [ωLO + ω]. (4.24)
η
2
2

The plus in the first equation corresponds to the in-phase intensity and the minus to the quadratureintensity. The equation for SIθ Iθ is only valid for heterodyne measurement.
For heterodyne measurement, the information of interest is likely only contained in either
Saout aout [ωLO + ω] or Saout aout [ωLO − ω] and thus the best signal to noise occurs when measuring the
combination intensity Iθ . A choice of θ = π/4 optimizes the measurement for a signal with original
frequency greater than ωLO , while a choice of θ = −π/4 optimizes the measurement for a signal
with frequency less than ωLO . This optimization leads to a result identical (up to an overall factor)
to the heterodyne single-quadrature measurement. This can be understood, as the two-quadrature
measurement results in twice as much noise, but also allows us to detect twice as much signal
information.
For homodyne measurement, the signal in the in-phase channel is optimized (while thequadrature measurement is furthest from optimized) at φLO = φd :SIphoton
[ω] =
I II


1 1 g→0+
Saout aout [ωLO − ω] + Sag→0
[ω
+
ω]
LO
a
out out
η 2(4.25)

+ Sagout aout [ωLO − ω] + Sagout aout [ωLO + ω] ,SIphoton
[ω] =
Q IQ


1 1 g→0+
Saout aout [ωLO − ω] + Sag→0[ωLO + ω] .
out aout
η 2(4.26)In the two quadrature expressions, I cannot just make the substitution η = (1 + 2nadd )−1 to write them in termsof added noise. As with two amplifiers in series, the noise from two beam splitters in series does not simply add. The
effective added noise n0add,2 from a second beam splitter with transmission η2 added in series with a beam splitter oftransmission η1 = (1 + 2nadd,1 )−1 is n0add,2 = nadd,2 /η1 . In the case of the two-quadrature measurement, η1 = 1/2
from the 50/50 beam splitter inside the linear detector, and η2 = η = (1 + 2nadd )−1 = (1 + 2n0add η1 )−1 = (1 + n0add )−1 .
The constant factor in front of the two-quadrature expressions then becomes 1/η = 1 + n0add .
4

95
The in-phase result is the same as the single-quadrature homodyne measurement, except with twice
the added noise. Thus, using a two-quadrature detector to make a homodyne measurement always
hurts the signal to noise by a factor of two.

4.4

Microwave detection and amplification
In the microwave regime, a frequency mixer and voltage detector replace the beam splittersand photodetectors in the discussion above (and in Appendix D.1). The voltage measured takes theplace of the intensity, as both are a linear combination of fields and their daggers. The only difference is a normalization constant, but this can be absorbed into the proportionality between SII [ω]photon[ω], which is not usually known (calibration of voltage to photon units or mechanicaland SII

displacement units is discussed in Section 6.2).
The most notable difference between microwave and optical measurements is that microwave
frequency mixers and detectors typically add many quanta of noise (the quantum efficiency η isvery small). To decrease the effective loss, we amplify the signal prior to the linear detector (seeFigure 4.2). The added noise at the input of the linear detector is then set by any loss (modeled asbeam splitter with transmission η 0 ) prior to the amplifier, which can in practice be much smaller

Figure 4.2: Model of amplification and added noise. Poor quantum efficiency η between the experiment and the detector can be overcome by adding an amplifier prior to that loss. The added noiseis then set by the quantum efficiency η 0 before the amplifier.

96
than the detector noise:1√dˆin = âout + j
Gηη 0

s

s
≈ âout + j

1
1 − η 0 ˆ0ξvac + j √
0
η
G

r

1−ηˆ
ξvac + amplifier noiseηη 0

1 − η 0 ˆ0
ξ + amplifier noise.
η 0 vac(4.27)(4.28)

0
For large gain, the ξˆvac term can be made much smaller than the combination of the ξˆvac
term andadded amplifier noise. Then the η dependence disappears and the added noise is set by only theloss prior to the amplifier. Note that the amplification will similarly make the half quanta of added
noise due to a two quadrature linear detector measurement negligible.

4.4.1

Phase sensitive amplification

A phase-sensitive amplifier (such as a Josephson parametric amplifier, JPA) amplifies only
one quadrature of the microwave field (as detailed in Appendix D.3.3):


J
Xin
[ω] = e−jπ/4 e−jφp ĉin,1 [ω] + jejφp ĉ†in,1 [ω − 2ωJ ] ,YinJ [ω] = −ejπ/4 e−jφp ĉin,1 [ω] − jejφp ĉ†in,1 [ω − 2ωJ ] ,JJXout[ω] = 2L[ω]Xin
[ω],

J
Yout
[ω] =

YinJ [ω],2L∗ [ω](4.29)(4.30)(4.31)

where ĉin(out),1 is the input or output field from the JPA and L[ω] is the direct amplitude gain.
When ωLO = ωJ and φLO = φJ − π/4, the detected in-phase intensity (or voltage) is proportional
to the amplified quadrature X J [ω + ωJ ]. Using a two-quadrature linear detector adds no additional
information over a single-quadrature detector, as the quadrature intensity IQ contains only deamplified information about Y J (which will be hidden by the large detector noise). Thus, the choice
to use a phase-sensitive amplifier sets the measurement as single quadrature, regardless of whether
a one or two quadrature linear detection is used, and the results in Section 4.2 are the relevant ones
for an interferometer containing a phase-sensitive amplifier.

97
4.4.2

Phase-insensitive amplification

A phase-insensitive amplifier (discussed in Appendix D.2) amplifies both quadratures of the
microwave field equally, requiring an extra half quantum of added noise to preserve commutation relations. This is very similar to the situation of the two-quadrature linear detector. Thus, regardlessof whether a single or two-quadrature linear detection is used after the amplifier, the two-quadrature
results in Section 4.3 are the relevant ones for an interferometer containing a phase-insensitive amplifier. Although, in measuring only the in-phase quadrature, a single-quadrature detector may
throw some of the available information away.

4.5

Quantum limits on detection
In this section I will evaluate the above single-quadrature intensity measurement expressions

for the several important cases of optomechanical drives detailed in Sections 3.1 and 3.2 and convert
them into displacement units. This can be accomplished, as we know that the displacement spectral
density of actual mechanical motion should befSxx[ω] = 2x2zp Γtot nfm |χm,eff [ω]|2 ,(4.32)

such that the number of mechanical quanta is
nfm =

1 1
2x2zp 2π

Z ∞
−∞

f
Sxx
[ω]dω.

(4.33)The measured intensity has a complicated proportionality to the displacement spectral density(which, in addition to mechanical and cavity parameters depends on the local oscillator amplitude,
amplifier gains, quantum efficiencies, etc.) However, the area (and thus height) of the mechanicalsideband in voltage or intensity units is proportional to the actual occupancy of the mechanicaloscillator (and the proportionality coefficient can be calibrated, see Section 6.2). An apparent (orimprecision) number of mechanical quanta can be found by comparing the height of the measured
background to the height of the sideband peak.
In this section, I will find the absolute limits on measurement imprecision and backaction forththeach drive case examined. I assume that there is no excess thermal noise (nthl = nr = nc = 0). This

98
makes all of the Sag→0
out aout [ω] terms zero. The results for an optimal two-quadrature measurement
can be found simply from the single-quadrature results, as they are always the same for heterodyne
measurement and a factor of two worse in imprecision for a homodyne measurement.

4.5.1

Single red-detuned drive

One very important drive case is that of the red-detuned drive, as this results in the coolingand damping of the mechanical oscillator. I will specify to the resolved-sideband limit, as damping
and cooling are both enhanced in this limit and mechanical ground state cooling is possible. The
output spectrum in the weak-coupling regime (Equation 3.76), is thenSaout aout [ω] = Sagout aout [ω] =4g 2 κl Γtot f
nm |χm,eff [ω − ωr ]|2 .
κ2(4.34)For a red-detuned drive in the resolved-sideband limit, there is no advantage to making a homodyne
measurement over a heterodyne (since the lower sideband is strongly filtered by the cavity response).
A heterodyne measurement will mix the upper sideband centered at ωr + Ωm down to frequency
ω = |ωLO − (ωr + Ωm + δ)|, where δ is the detuning from the sideband center frequency. The optimalheterodyne intensity spectral density at this frequency is (from Equation 4.18)
SII [ω] ∝1
2g 2 κl Γtot f+
nm |χm,eff [Ωm + δ]|2 .
2ηκ2(4.35)

Re-writing this measured intensity spectrum as an inferred mechanical displacement spectrum
gives
tot
imp
f
Sxx
[ω] = Sxx
[ω] + Sxx
[ω] =x2zp κ2+ 2x2zp Γtot nfm |χm,eff [Ωm + δ]|2 .
2g 2 κl η(4.36)

Evaluating this spectral density on resonance and writing in terms of number of mechanical quanta
gives
imp
f
ntot
m = nm + nm ,

where

nimp
m =

(1 + Cr )κ
.
4Cr κl η(4.37)

I have written the imprecision in terms of the red drive cooperativity Cr = 4g 2 /(κΓm ), which is
proportional to drive strength. The imprecision decreases with drive power until it saturates at99
κ/(4κl η), which is at minimum 1/4 for the reflection geometry or 1/2 for the transmission geometry.
These results are shown in Figure 4.3(a).

4.5.2

Single blue-detuned drive

The case of a single blue-detuned drive is very similar to the single red-detuned drive exceptthat the radiation pressure damping is opposite in sign, leading to amplification of mechanicalmotion. Once again specializing to the resolved-sideband limit where radiation pressure effects are
maximal, and assuming the weak coupling limit and absence of excess photon noise, the output
spectrum (Equation 3.77) is
Saout aout [ω] = Sagout aout [ω] =4g 2 κl Γtot f(nm + 1)|χm,eff [−ω + ωb ]|2 .
κ2(4.38)

Once again, there is no advantage to making a homodyne measurement over a heterodyne (since thistime the upper sideband is strongly filtered by the cavity response) and a heterodyne measurement
will mix the lower sideband centered at ωb − Ωm down to frequency ω = |ωLO − (ωb − Ωm − δ)|.
The inferred displacement spectral density and mechanical quanta are
x2zp κ2+ 2x2zp Γtot (nfm + 1)|χm,eff [Ωm + δ]|2 , (4.39)
2g 2 κl η(1 − Cb )κ
.
(4.40)nimpm =4Cb κl η

tot
imp
f
zp
Sxx
[ω] = Sxx
[ω] + Sxx
[ω] + 2Sxx
[ω] =
f
zp
ntot
= nimpm
m + nm + 2nm ,zp
and the zero point number of quanta is nzp
The zero point spectral density is Sxx
m = 1/2. The blue

drive cooperativity is Cb = 4g 2 /(κΓm ). This time, the number of imprecision quanta decreases tozero as the final mechanical occupation blows up at Cb = 1. These results for the blue drive areshown alongside those for the red drive in Figure 4.3(a).

4.5.3Single on-resonance drive, homodyne detection

The case of a single, on-resonance drive is somewhat different from the red and blue detuned
cases because there is no dynamical radiation pressure. Homodyne detection is often most preferable
because the two sidebands can be folded on top of each other, increasing the signal to noise due to100

Figure 4.3: Mechanical occupancy contributions for various drive schemes. For all graphs, η = 1,nthm = 50, κl = κ, and the best single-quadrature detection result is shown. Imprecisions (dashed),backactions (dash-dotted), and final or thermal and zero point (solid) occupancies are shown foreach case. (a) Single red or single blue drive in the far resolved-sideband regime. The phonon
occupancies are plotted as a function of cooperativity Cd = 4gd2 /(κΓm ) for d = r and d = b. (b)Single, on-resonant drive. Phonon occupancies are measured with a homodyne detection schemeand plotted as a function of cooperativity. The results are shown in the unresolved sideband regimeΩm  κ (purple) and the resolved-sideband regime Ωm = 5κ (green). (c) Two drive scheme withunequal drives. The occupancies are plotted as a function of red drive cooperativity with theblue drive cooperativity fixed at Cb = 0.5 (vertical black dotted line). (d) Two drive scheme withequal drive strengths. Phonon occupancies are plotted as a function of equal cooperativity. The
imprecision is that found in a homodyne, non-overlapping sidebands measurement. Comparison to(b) shows why the two drive scheme is preferable in the resolved-sideband regime.

101
their constructive interference. In the absence of excess photon noise, the output spectrum for an
on-resonant drive (Equation 3.79) is[ω] =Saout aout [ω] = Samech
out aout
4g 2 κl Γm  f
2
f
2
n
|χ
[ω
−
ω
]|
+
(n
+
1)|χ
[−ω
+
ω
]|
.
m
c
m
c
m
m
κ2 + 4Ω2m(4.41)The optimal homodyne single-quadrature intensity (Equation 4.20) and the inferred displacement
spectral density and occupancy arephotonSII[ω]=

8g 2 κl Γm1+ 2
2η κ + 4Ω2m



1
nfm +
2



|χm [Ωm + δ]|2 ,(4.42)

x2zp κ(1 + ν)1
+ 2x2zp Γm (nfm + )|χm [Ωm + δ]|2(4.43),
2CΓm κl η
2
1C(1 + ν)κ
th
zp
ba
+ nth
+
.
(4.44)= nimpm +m + nm + nm + nm =16Cκl η2 (1 + ν)

tot
imp
f
zp
Sxx
[ω] = Sxx
[ω] + Sxx
[ω] + Sxx
[ω] =

ntot
m

In the last line, the imprecision and backaction are written in terms of cooperativity C = 4g 2 /(κΓm )and a resolved sideband factor ν = 4Ω2m /κ2 . The product of the imprecision and backactionbanumbers of quanta is nimp
m nm = κ/(16κl η) ≥ 1/16. This minimum imprecision-backaction productis only met for a perfectly overcoupled cavity in the reflection geometry, measured with singlequadrature, quantum-efficient, homodyne detection. The minimum total added noise occurs whenSQLba
= 1/4 and is called ‘the standard quantum limit’. The minimum total number ofnimp
m = nm = nm
imp
zp
ba
quanta at the standard quantum limit is ntot
m = nm + nm + nm = 1. The cooperativity required toreach the standard quantum limit is C SQL = (1 + ν)/4. Figure 4.3(b) shows the different occupation
contributions resulting from an on-resonant cavity drive.
The standard quantum limit can be restated as a limit on the product of the imprecisiondisplacement spectral density and the backaction force spectral density (both of which are frequency
independent). These quantities, their product, and their values at the standard quantum limit are
imp
Sxx
=

SFbaF

=

imp ba
Sxx
SF F

=

SQL
Sxx
=

x2zp κ(1 + ν),
2CΓm κl η
ba [ω]
h̄2 Sxx2h̄2 Γm C=,
x4zp |χm,eff [ω]|2x2zp (1 + ν)h̄2 κ≥ h̄2 ,
κl η
2x2zp
,
Γtot

(4.45)(4.46)(4.47)SFSQLF =h̄2 Γtot.
2x2zp(4.48)102
SQL
Defining Sxx
in terms of Γtot rather than Γm is inconsequential in the on-resonant drive case(because Γtot = Γm ). However, this definition has the nice general property that
Sxx [Ωm ]
SQL
4Sxx

=

Γ2tot fn |χm,eff [Ωm ]|2 = nfm .
4 m(4.49)

SQL
Thus, plotting an output spectrum in units of Sxx [ω]/(4Sxx) allows the mechanical occupancy to

be directly read-off as the value on resonance.

4.5.4

Single on-resonance drive, heterodyne detection

While the minimum imprecision-backaction product for an on-resonant drive is achieved only
for a homodyne measurement, there are a few technical reasons it might be preferable to do aheterodyne measurement. One is that the mechanical oscillator’s resonant frequency may be toolarge to digitize easily. Another is that the amplifier may not have enough bandwidth to amplify
both sidebands (for instance if κJ < 2Ωm ). A third is that it is necessary to measure the two
sidebands independently in order to see sideband asymmetry. In these cases, it might be preferable
to do a heterodyne measurement even if the imprecision is slightly higher. Here, I assume that I
am interested in measuring the upper sideband. The lower sideband would give nearly identicalresults, with the replacement nfm → nfm + 1. The optimal heterodyne intensity spectral density,
inferred displacement spectral density, and number of phonons arephoton
SII
[ω] =
tot
Sxx
[ω] =

ntot
=
m

4.5.5

1
2g 2 κl Γm f
+ 2
n |χm [ω − ω c ]|2 ,2η κ + 4Ω2m m
2x2zp κ(1 + ν)
+ 2x2zp Γm nfm |χm [ω − ω c ]|2 ,Cκl Γm η
κ(1 + ν)
C
+ nth
.
m +
4Cκl η
(1 + ν)(4.50)(4.51)(4.52)

Generalized single-drive Heisenberg measurement relation

I can write the results of the previous few pages in a more generalized way. The general
imprecision for a single-quadrature homodyne measurement (ωLO = ωd , identical to a heterodyne

103
measurement for off-resonant drive frequency) is
nimp
m =

Γtot
.
2
4g ηκl |χc [ωd + Ωm ] − χ∗c [ωd − Ωm ]|2(4.53)The generalized backaction fluctuations are
nba
m =


g 2 κ(1 + 2nthc )|χc [ωd − Ωm ]|2 + |χc [ωd + Ωm ]|2 .
2Γtot(4.54)The product of these two quantities realizes a Heisenberg uncertainty relation between measurement
and perturbation of the motion:
ba
nimp
=
m nm

=

2
2
κ(1 + 2nthc ) |χc [ωd + Ωm ]| + |χc [ωd − Ωm ]|8κl η|χc [ωd + Ωm ] − χ∗c [ωd − Ωm ]|2κ(1 + 2nth
4∆2
1
4∆2
c )
1+ 2
≥
1
+
.
16κl η
κ + 4Ω2m
16
κ2 + 4Ω2m(4.55)(4.56)

This uncertainty relation can be re-expressed in terms of displacement and force spectral densities:
!



2
8x
4∆2
2h̄2 Γtot ba
zp
imp
2 imp ba2
imp ban
nm = 16h̄ nm nm ≥ h̄ 1 + 2.
(4.57)
Sxx SF F =Γtot mx2zp
κ + 4Ω2m
These relations recover the results obtained above for an on-resonant drive:
nimp
m =

(1 + ν)κ
,16Cκl ηnba
m =C(1 + 2nth
c )
,
1+ν

ba
nimp
m nm =

κ(1 + 2nth
1
c )
≥ .
16κl η
16(4.58)For an optimally red or blue-detuned drive they give:κ(1 + 4ν(1 ± C))
κ(1 ± C)≈,
16Cκl η(1 + ν)
4Cκl η
κ(1 + 2nth
1c )(1 + 2ν)
ba
nimp
≥ ,
m nm =
16κl η(1 + ν)
8
nimp
m =

nba
m =

C(1 + 2nth
C(1 + 2nthc )(1 + 2ν)c )≈(, 4.59)
1 + 4ν(1 ± C)
2(1 ± C)
(4.60)

where the pluses are for the red-detuned case and the minuses are for the blue-detuned case.

4.5.6

Double drive scheme, sidebands not overlapping, heterodyne detection

In Section 3.2, I showed that in the far-resolved sideband regime both the sideband asymmetry
and the quantum backaction could be enhanced by using two microwave drive tones, one optimally
red-detuned and one optimally blue-detuned. For either of these measurements, it is preferable for

104
the sidebands to be spaced far apart compared to the mechanical linewidth. The output spectrum,
ignoring excess photon noise, (Equation 3.136) is
Saout aout [ω] =4gb2 κl Γtot f
4gr2 κl Γtot f
2
n
|χ
[ω
−
ω
]|
+
(nm + 1)|χm,eff [−ω + ωb ]|2 .
r
m,eff
m
κ2
κ2

(4.61)

Assuming the local oscillator is not at the average frequency of the two drives (ωLO 6= (ωr + ωb )/2),
the sidebands do not fold on top of each other and the results in the single-quadrature scheme forthe red upper sideband arephoton
SII
[ω] =
tot
Sxx
[ω] =

ntot
=
m

1
2g 2 κl Γtot f+ r 2
nm |χm,eff [Ωm + δ]|2 ,2ηκ
2x2zp κ+ 2x2zp nfm Γtot |χm,eff [Ωm + δ]|2 ,Cr κl Γm η
κ(1 + Cr − Cb )+ nfm .
4Cr κl η(4.62)(4.63)(4.64)The imprecision for the lower blue sideband is the same with Cr → Cb in the denominator. These imprecisions (and the accompanying final occupancies) are shown in Figure 4.3(c) for a fixed blue drive
power as red drive power is increased. The backaction contribution to motion and the backactionimprecision product for the red sideband arenbam =Cr + Cb,
2(1 + Cr − Cb )banimp
m,r nm =κ(Cr + Cb ).
8Cr κl η(4.65)

If the two drives are of equal strength Cr = Cb = C, the backaction takes the form nba
m = C (Equationba
3.122). The imprecision-backaction product is then nimp
m nm = κ/(4κl η) ≥ 1/4. This product may

not seem impressive compared with the product achieved with an on-resonant drive. However, thecooperativity required to reach this point is C = 1/2, making it considerably easier to achieve inthe resolved sideband limit than that required with an on-resonant drive, C = (1 + ν)/4.

4.5.7

Double drive scheme, sidebands not overlapping, homodyne detection

If the interferometric measurement of the two-drive scheme is done with the local oscillator atωLO = (ωr + ωb )/2 and the two drives are of equal strength gr = gb = g, the signal to noise can be
improved in exactly the same way it was for the single on-resonance drive case when a homodyne

105
measurement was used. The measured frequency will be ω = |(ωb − ωr )/2 − (Ωm + δ)| and theresults arephoton
SII
[ω] =
tot
Sxx
[ω] =

ntot
=
m

1
8g 2 κl Γm f
1+
(nm + )|χm [Ωm + δ]|2 ,
2
2η
κ
2
2
xzp κ
1
+ 2x2zp Γm (nfm + )|χm [Ωm + δ]|2 ,2CΓm κl η
2
κ
1
+ nth
+ C.
m +
16Cκl η2

(4.66)(4.67)(4.68)The imprecision-backaction product is then the same as for a single, on-resonant drive measuredbain the homodyne configuration: nimp
m nm = κ/(16κl η) ≥ 1/16. However, as with the heterodyne

two-drive detection above, this point is achieved at far less power (C SQL = 1/4) than with theon-resonance drive (C SQL = (1 + ν)/4) in the resolved sideband regime (compare Figure 4.3(b) and
(d)).

4.5.8

Double drive scheme, sidebands overlapping

As I showed in Section 3.2, overlapping the two sidebands from two equal drives results inbackaction evasion. This effect results from the fact that the mechanical state measured this way isstationary and the backaction force acts on the quadrature orthogonal to the one that is measured.
In this case, the output spectrum is (Equation 3.136)
Saout aout [ω] =
4g 2 κl Γm  th2n+1
|χm [ω − ωr ]|2m2
κ

(4.69)The measured heterodyne intensity, inferred displacement spectral density, and mechanical occupation arephoton
SII
[ω]

=

tot
Sxx
[ω] =

ntot
=
m



1
4g 2 κl Γm
1
th
+
nm +
|χm [Ωm + δ]|2 ,
2η
κ2
2


x2zp κ
12th+ 2xzp Γm nm +|χm [Ωm + δ]|2 ,CΓm κl η
2
κ
1
+ nth
m + .
8Cκl η
2(4.70)(4.71)(4.72)The imprecision number of phonons for the overlapping sidebands is only half of that with nonoverlapping sidebands. This makes sense from a quantum information standpoint because only one

106
quadrature of the mechanical field is measured (and thus an extra half quantum of added noisefrom measuring both quadratures is not required).

4.5.9

Summary of results

The imprecisions for situations in this section with large dynamical backaction (the single redand blue-detuned drives and two drives of unequal amplitude) are listed in Table 4.1(a). Imprecisionand backaction occupations for situations with zero net dynamical backaction (the on-resonancedrive and two drives of equal amplitude) are listed in Table 4.1(b).

(a)

drive

nimp
m

nba
m

ba
nimp
m nm

red

κ(1 + C)/(4κl Cη)C/(2(1 + C))

κ/(8κl η)blue

κ(1 − C)/(4κl Cη)C/(2(1 − C))κ/(8κl η)

double (NO)

κ(1 + Cr − Cb )/(4κl Cd η)(Cr + Cb )/(2(1 + Cr − Cb ))κ(Cr + Cb )/(8κl Cd η)

drive

meas

nimp
m

nba
m

ba
nimp
m nm

on-resheteroκ(1 + ν)/(4κl Cη)C/(1 + ν)

κ/(4κl η)

on-reshomoκ(1 + ν)/(16κl Cη)C/(1 + ν)κ/(16κl η)

double (NO)heteroκ/(4κl Cη)Cκ/(4κl η)

double (NO)homo

κ/(16κl Cη)Cκ/(16κl η)

double (O)heteroκ/(8κl Cη)

—

—

(b)

Table 4.1: Summary of imprecisions and backactions for different cavity drive cases. Quantities are
written in terms of the cooperativity C = 4g 2 /κΓm , the resolved sideband factor ν = 4Ω2m /κ2 , andthe interferometer quantum efficiency η. The column ‘drive’ details the nature of the optomechanical drive. ‘NO’ indicates not-overlapping sidebands while ‘O’ indicates overlapping sidebands. (a)Results for cases where the net dynamical backaction is large, relevant for heterodyne detection in
the resolved sideband limit. The two-drive result is thus for unequal amplitude drives and refers
to the imprecision of the sideband associated with drive d. The results here apply to both single
and two-quadrature detection as well as both homodyne and heterodyne detection. (b) Results
for cases where the net dynamical backaction is negligible. The column ‘meas’ indicates ‘homo’for homodyne measurement and ‘hetero’ for heterodyne. Thus, the two-drive cases are for equal
amplitude drives (gr = gb = g). The results shown are all for single-quadrature detection. Twoquadrature detection results are the same except that the imprecision is a factor of two larger for
homodyne cases.

Chapter 5

Device design and fabrication

In this chapter, I will discuss the different types of devices we have fabricated and measured.
I will first explain the types of measurements of interest and the device parameters needed to
optimize those measurements (Section 5.1). I will then discuss in detail the first major class ofdevices we made, namely ones with a wire or beam as the mechanical element (Section 5.2). As
I will show, these devices do not achieve the best optomechanical coupling to date; however, theyrequire less fabrication capabilities than the devices that have higher coupling and do achieve very
high force sensitivity. I will first discuss the optimal parameters for this type of device and thenexplain their fabrication and achieved parameters. Lastly, I will briefly discuss devices with asuspended drum-like membrane as the mechanical oscillator, as fabricated by our colleagues at
NIST (Section 5.3).

5.1

Quantities to optimize by fabrication
There are several quantities that can be designed on the device. For the cavity design, we

control the cavity capacitance C, inductance L, coupling capacitance Cc , and device geometry(transmission or reflection), or put in other terms, the cavity resonant frequency ωc , equivalentpparallel RLC impedance1 Zc = L/C, and coupling rates κl and κr (these quantities were relatedp
The impedance Zc = L/C is the cavity impedance of the parallel RLC circuit equivalent to the actual circuit
p ofinterest. This should not be confused with the characteristic impedance of a transmission line, which is Z0 = L/C
where L and C are the inductance and capacitance per unit length of the transmission line. For a shorted λ/4transmission line resonant circuit as in [10] and [45], Zc = 4Z0 /π, while for an open λ/2 resonant circuit as in [12],Zc = 2Z0 /π.
1108in Chapter 2).
For the mechanical design, we control the geometry and tension of the mechanical element.
For a wire device, the geometry includes the length l, width w, thickness t, and spacing s between thewire and the ground plane. For a suspended membrane device, it includes the size and shape of themembrane and the height above the substrate. These mechanical parameters control the mechanicalfrequency Ωm , mass m, and loss rate Γm . Both the cavity and mechanical parameters control
the optomechanical coupling G = dωc /dx. Ideally, the best device would have both the highest
optomechanical coupling and lowest mass possible. However, in practice, these two quantities trade
off depending on fabrication parameters. Thus, it is important to see how measured quantities,
such a force sensitivity or optomechanical control, depend on this compromise.

5.1.1

Optomechanical control

Many of the experiments of interest, such as dynamical radiation pressure cooling and amplification of mechanical motion (Sections 3.1-3.2), quantum backaction (Section 3.2), and preparation
and readout of the mechanical oscillator in time (Section 3.3) are optimized by having a large
single-photon optomechanical coupling g0 between the mechanical oscillator and the cavity. The
coupling can be written in terms of cavity and mechanical parametersdωcdωc dCtot−ωc dCtot−ωc2 Zc dCtot
=
=
=
,
dx
dCtot dx
2Ctot dx2
dxrh̄−ωc2 Zc dCtot= Gxzp =
,
2
dx
2mΩm

G =

(5.1)

g0

(5.2)

where Ctot is the entire capacitance of the circuit (Ctot = C + Cc ). From this expression, it isclear that it is preferable to have a large cavity frequency and large zero point motion (or small
mechanical mass and resonance frequency). It is easy to see why the quantities G and xzp trade off- in order to achieve a large change in capacitance per motion, the mechanical oscillator should havea large surface area. But in order to have a small mass, the mechanical oscillator should have small
dimensions. Within the microwave regime, it is not possible to dramatically increase the cavity
frequency above ∼10 GHz without making the energy comparable to the superconducting gap of

109
aluminum. For a given cavity frequency then, it is preferable to have large cavity impedance. This
makes sense because in a higher impedance cavity, the mechanical motion will modulate a larger−1
fraction of the total capacitance and thus have a higher participation ratio Ctot
(dCtot /dx).

I have so far only said that it is good to have a large g0 (and by association a large maximum√g = g0 nd ). But large compared to what? There are three important thresholds of coupling thatallow interesting physics. The easiest coupling benchmark to reach is that of large cooperativity
C = 4g 2 /(κΓm ) > 1. For a red-detuned drive in the resolved sideband limit, a cooperativity equal
to one allows doubling the mechanical linewidth and decreasing the mechanical occupancy by afactor of two. For a blue-detuned drive, it allows a decrease of the mechanical linewidth to zero andarbitrary amplification of mechanical motion. Cooperativity greater than or equal to one allows
access to the standard quantum limit, both for a single drive in the unresolved sideband regime orequal red and blue-detuned drives in the resolved sideband regime. Cooperativity equal to one isalso the coupling required to realize ideal conversion between itinerant photon and phonon energies
for a red-detuned drive applied to a reflection geometry system in the resolved sideband limit.
The second interesting benchmark for the coupling is the quantum-enabled regime: C˜ =4g 2 /(κΓm nthm ) > 1. This quantity is a ratio between the rate of coupling to the rate of mechanicalquantum decoherence (the rate that a single thermal phonon enters the mechanical oscillator).
Having C˜ > 1 is thus required to transfer, store, and retrieve microwave states with energies on theorder of a single photon. For a red-detuned drive in the resolved sideband regime, the quantum
enabled regime allows cooling to a mechanical occupancy less than one. The onset of the quantum
enabled regime is also the point where the quantum backaction motion of the mechanical oscillator
becomes larger than its thermal motion, for either a single drive in the unresolved sideband regime
or two equal drives in the resolved sideband regime.
The third coupling benchmark of interest is strong coupling: S = 4g 2 /κ2 > 1. This regime
allows the total mechanical linewidth to become equal to the cavity linewidth, causing the two
resonant structures to undergo normal mode splitting. The onset of strong coupling, S = 1, is also
the point where the equations of motion are critically damped. Below this coupling, the equations

110
of motion are overdamped and application of a drive tone swaps itinerant photons to mechanicalphonons and vice versa, but there is no oscillation of energy. For stronger couplings, the equationsof motion are underdamped, and application of a drive tone causes energy to oscillate back andforth between the cavity and mechanical oscillator faster than it can be lost from the cavity. For
example, strong coupling would be crucial for an experiment where the same cavity was coupled toboth a mechanical oscillator and a qubit and the qubit was used to prepare arbitrary states of themechanical oscillator. There are also many interesting things that can be done if an optomechanicalsystem is in the single photon strong coupling regime: 4g02 /κ2 > 1 (see reference [93]). However,this regime is currently out of the reach of experiments.
The three different coupling regimes, written in terms of device parameters are
4g 2
2h̄G2 nd=,κΓm
κmΩm Γm

C=

C˜ =

4g 2
2h̄2 G2 nd=,
kB T κmΓm
κΓm nthm

S=

4g 2
2h̄G2 nd=
. (5.3)κ2
κ2 mΩmHere, kB is Boltzmann’s constant and T is the temperature of the mechanical oscillator. These
expressions all benefit from large G, small κ, and small m. However, it is useful to see how theydiffer in their dependence on Γm and Ωm .

5.1.2

Force sensitivity

An important experiment with a different goal than optomechanical control is that of force
sensitivity. In this case, the idea is to detect some external force acting on the mechanical oscillator.
The sensitivity to this force will be limited by mechanical motion due to sources other than thisforce: namely thermal, backaction and imprecision motion. Assuming that the power used tomeasure is high enough to decrease the imprecision below the thermal noise but not yet highenough to create backaction motion comparable to the thermal motion, the force sensitivity will belimited by the thermal force spectrum: SFthF = 4kB T meff Γm , where meff is the effective mass of themechanical oscillator, discussed in Appendix E. The best force sensitivity is achieved by having thesmallest mass and mechanical loss and does not depend at all on the coupling strength (as long as
it is high enough to reach the regime where thermal noise is much larger than imprecision noise).

111
In experiments, we observe that this simple picture is not quite true. In practice, the mechanical oscillator heats up as a function of incident microwave power, Pin . Using the data fromreferences [10] and [76], it is possible to make an empirical law for this dependence (Figure 5.1).
0.35 for reference [10]Fitting the two sets of data to power laws (shown in red), I find Tm = 29Pin
0.14 , where T here is in millikelvin and P is in picowatts. Compromising between
and Tm = 71Pin
m
in
1/4

the two with a power law of Tm ∝ Pin(shown as a black line), I find a reasonable approximation

to the data. In this case, the extra noise due to this thermal backaction force on the mechanics will
hurt the force sensitivity as a function of power. It is therefore optimal to work at a power where
the imprecision and thermal backaction are in some sense balanced. The power dependence of thetwo contributions at the mechanical resonance goes as
1/4

SFthF = 4αmeff Γm Pin

and

SFimp
F =β

m2eff Ω2m Γ2m.
G2 Pin(5.4)

Here, I have used α to replace kB and the proportionality constant of the power to temperature

Figure 5.1: Power dependence of mechanical temperature in wire devices. (a) Mechanical temperature as a function of incident microwave power. The data is from references [10] (purple) and [76](green). The purple and green lines are power law fits to the data, while the black line is a compromise between the two power laws. As the microwave power increases, the mechanical oscillator
is parasitically heated. (b) Force sensitivity as a function of incident power when parasitic heating
is taken into account. The power dependent thermal force spectral density is shown as a blackline. The imprecision (dashed line) and total (solid line) force spectral densities are shown for two
different coupling strengths, G = 2π × 1 kHz/nm (red) and G = 2π × 30 kHz/nm (blue). The bestforce sensitivity and power required are both lower for larger coupling. This graph was made with
device parameters similar to those of reference [76].

112
relation and β to store the cavity parameters and geometry as well as the added noise of themeasurement. The power that minimizes the total force noise on resonance is as follows.

βm2eff Ω2m Γ2md  th−3/4
SF + SFimp= αmeff γm Pin
−
= 0,
2
dPin
G2 Pin
4/5
βmeff Ω2m Γm
optimal
Pin
=
,
αG2
 3


3 2/5imp
4/5 1/5 meff Ωm Γmth= 5α βSF + SF
.
G
min

(5.5)
(5.6)
(5.7)

In this case, I find that the force sensitivity once again depends almost linearly with small massand mechanical loss. But I now also find that the force sensitivity is improved by small mechanicalfrequency and large optomechanical coupling.

5.2

Wire devices
When we began studying optomechanics, our fabrication capabilities were limited. Thus,

we wanted to make a device that only required using planar fabrication techniques such as simple
lithography with no structures suspended above the chip. We opted to suspend the mechanicalelement by etching away the silicon substrate rather than creating extra sacrificial layers. To
form an optomechanical device in a planar geometry, the mechanical oscillator must modulate acapacitance between two pieces of metal in the same plane. A long wire is then the best choice for
maximizing surface length while also maximizing motion in the plane. We also chose to make thecavity and mechanical oscillator out of aluminum because it is superconducting at low temperaturesand unharmed by the silicon etching. In this section, I will talk about the best design parameters
for such a wire device, how it is fabricated, and the parameters we were able to achieve.

5.2.1

Wire device design

In this section, I will try to find relationships to predict the mechanical resonance frequency,
mechanical linewidth, and optomechanical coupling based on design parameters. That will allow me
to write the quantities of interest above in terms of design parameters to see how the experimental

113
quantities of interest can be optimized by design choices.

5.2.1.1

Mechanical resonance frequency

For a wire under high tension, the frequency of the fundamental mode can be found from the
speed v of propagation along the wire:v
1Ωm = 2π ×= 2π ×
2l
2l

r

τ
1
= 2π ×m/l
2l

s

EAl wt= 2π ×ρAl wt

1
2

s

EAl 
ρAl

!

1
l

(5.8)

where τ is the tension, EAl and ρAl are the Young’s modulus and density of aluminum, and  isthe strain experienced by the aluminum. In the absence of tension (see [94, pgs. 233-236] with theunderstanding that Iy is incorrect and should be Ix ), the resonance frequency issss!
EIx  c0 2
EAl (tw3 /12)  c0 2
EAl w
Ωm =
,
=
= 2π × 1.03
ρA l
ρAl (wt)l
ρAl l2(5.9)

where Ix is the polar moment of inertia about the direction of motion [94, pg. 204], A is the crosssectional area of the wire, and c0 = 4.73004 specifies the mode shape of the lowest mechanical modeof a doubly-clamped beam. Note that while the expression for high-tension does not depend on tor w, this expression does and I would find a slightly different expression for the frequency of thepout of plane mode: Ωm = 2π × (1.03 E/ρ)t/l2 . More generally [95], the resonance frequency forany suspended wire or beam issΩm = 2π ×

1.03

EAl
ρAl

!

w
l2

r
1+

l2
,
3.4w2

(5.10)

which reduces to the above expressions for very high or low tension.
The tension turns out to be a very important, controllable quantity. Without annealing, wefind that our aluminum films (and thus our wires) at room temperature are under compressive
stress. When such a wire is cooled down, the aluminum shrinks more than the silicon, making
the compressive stress smaller. The low-temperature resonant frequencies that we see for wires
fabricated without annealing are consistent with little to no tension. We fabricate high-tension
wires by annealing at a high temperature (specifically 340◦ C = 613 K). This makes the atoms

114
in the wire more mobile and they more quickly relax to their lowest energy configuration. There
will now be some high temperature where the aluminum and silicon are the same length with notension, and thus the aluminum will be under tension for lower temperatures. Because we cool thewires somewhat slowly, it is unlikely that this equilibrium temperature is 613 K, but it is above
300 K, as the annealed wires seem to be tensioned at room temperature. The equation governingthermal expansion is dl/dT = α[T ]l, where α[T ] is the coefficient of thermal expansion. At hightemperatures (T > 100 K), α is relatively constant, so I can integrate this equation to find the
expected amount of expansion/contraction:
Z T2
Z l2
dl
α
dT =
, l2 = l1 eα(T2 −T1 ) .
l
T1
l1(5.11)

The coefficients of thermal expansion for aluminum and silicon near room temperature are αAl =23 × 106 K−1 and αSi = 3 × 106 K−1 . At low temperatures, I can look up the measured relative
linear expansion coefficient ∆l/l referenced to room temperature in a book, such as in reference[96]. This gives nearly temperature independent values near zero Kelvin: ∆llAl [0 K] − lAl [300 K]== −0.004,
l Al
lAl [300 K]
 
lSi [0 K] − lSi [300 K]∆l== −0.0005.
l Si
lSi [300 K](5.12)(5.13)

If the aluminum and silicon are the same natural length l∗ at temperature T ∗ , then the aluminum
will want to shrink more than the silicon as it is cooled down. Because it is attached to the silicon,
it cannot do this and will end up being the length of the silicon. Thus, the aluminum will experiencea strain . The strains near room temperature (300 Kelvin) and near zero Kelvin are:∗

[300 K] =∆llSi [300 K] − lAl [300 K]l∗ eαSi (300−T )== ∗ α (300−T ∗ ) − 1llAl [300 K]
l e Al∗= e(αSi −αAl )(300−T ) − 1,
[0 K] ==∆llSi [0 K] − lAl [0 K](1 − 0.0005)lSi [300 K]==−1llAl [0 K](1 − 0.004)lAl [300 K](1 − 0.0005) (αSi −αAl )(300−T ∗ )e− 1.
(1 − 0.004)(5.14)(5.15)(5.16)(5.17)The strain of the wire is related to the stress σ by Hooke’s law σ = E, where E is the Young’s
modulus. The tension τ is this stress times the cross sectional area of the wire: τ = σwt = EAl wt.

115
Assuming a range for equilibrium temperature 300 K< T ∗ < 600 K, I find that the strain (which,unlike the tension, does not depend on the wire dimensions) is 0 < [300 K] < 0.006 and 0.004 < [15mK] < 0.010.

5.2.1.2

Mechanical loss rate

The mechanical damping is not particularly easy to model, but can be estimated empiricallyfrom some general experimental trends we have observed (see Figure 5.2). This figure shows that themechanical quality factor Qm ≡ Ωm /Γm increases at least linearly with Ωm as tension is changed.
It also indicates that for the lowest frequencies (in the absence of tension), Qm ∼ 3000, independentof length. So, I will make the assumptions that Qm [τ = 0] = 3000 and Γm [τ ] = Γm [τ = 0]. From
this I find thatΩm [τ = 0]Γm [τ ] == 2π ×
3000

1.03
3000

s

EAl
ρAl

!

w
w
≈ 11 2 .
2ll(5.18)

Figure 5.2: Measured mechanical quality factors of tensioned aluminum wire devices. The mechanical quality factor is shown as a function of mechanical resonance frequency for eight different
40 µm wires (blue squares) and eight 50 µm wires (red circles). The only difference between theeight devices in each data set is tension, which increases the mechanical frequency. The quality
factor increases at least linearly with frequency (the black line corresponds to the average Γm forall devices). The data is from reference [97].

116This expression, while empirical, gives some idea of the dependence of the mechanical linewidth
on mechanical oscillator parameters. At the end of this section, this will enable me to write thequantities of interest from the previous section in terms of device parameters. A more sophisticatedanalysis of damping for wire oscillators, which came out after my analysis of wire devices presented
here, can be found in reference [98].

5.2.1.3

Optomechanical coupling

The optomechanical coupling G for a mechanical oscillator coupled to a microwave circuit is(as found above)G≡dωc−ωc2 Zc dCwire
=
dx
2
ds

(5.19)

where Cwire and s are the capacitance and spacing between the wire and the ground plane. Achange in position x of the wire is equivalent to a change of the spacing s. There are a few different
ways to model the wire capacitance in order to find dCwire /ds. Here, I model the wire and groundplane as two parallel wires or a coplanar stripline (CPS) (Figure 5.3). I also find more specificnumerical results by simulating the full geometry.

Figure 5.3: Models for wire capacitance. (a) Geometry of the device, involving a finite thickness
wire of dimensions l ×w ×t spaced by distance s from a relatively infinite ground plane. (b) Parallelwire geometry. (c) CPS geometry.

117
Parallel wire model
I can model a wire near an infinite ground plane as just two parallel wires (Figure 5.3(b)). Thisshould give a lower bound on the capacitance for the geometry of interest. The formula for thecapacitance of two parallel wires is:Cwire = π0 l ln 

s+w
+
w

s

s+w
w

2

−1
− 1

,
#!−1
p
s + w + s(s + 2w)= π0 l ln
w
"
#!−2
p
s + w + s(s + 2w)
ωc2 Zc π0 l
G= p
ln
.
w
2 s(s + 2w)(5.20)"(5.21)CPS model
The capacitance between two uneven coplanar striplines (Figure 5.3(c)) is [99, pgs. 401-402]:
K 0 [k]Cwire = 20 l
,
K[k]

0

r

k =

w
d d→∞
−→
w+sd+srw.
w+s(5.22)

Here, K[k] is the complete elliptic integral2 of the first kind and K 0 [k] = K[k 0 ], where k 0 and k arepcomplements: k 2 + k 02 = 1; k = s/(w + s). It is possible to include the effect of finite thicknessof the striplines [99, pg. 415] by using an effective spacing and width:s → s = s − ∆,w → w = w + ∆,

∆≡




5t
4πw
1 + ln.
4πt(5.24)

Reference [99] only endorses this method for t/w ≤ 0.1, whereas we typically have t/w ≈ 1, so weshould be wary of results achieved this way, but they still may give some idea of how G scales witht. Obviously, this solution will be completely unusable for ∆ > s. The coupling G for the CPS
model isd
G = −0 lωc2 Zc
ds
2

"r
K

#
"r
#!
w
s
/K
.
w+s
w+s

(5.25)The definitions of the elliptical functions used in engineering texts are different from those in Mathematica due
to a difference in the definition of the incomplete elliptic integral of the first kind (Elliptic F) [100, 101, 102, 103]:
Z π/2
dθ
p
K[k]engineering == EllipticK[k2 ]mathematica .
(5.23)
01 − k2 sin2 θ118
Microwave Office simulation of CPS model
I can model the actual wire and ground plane CPS geometry with a finite thickness in microwave
office. The details of these simulations can be found in Appendix F. Simulating for many different
device parameters led to the following empirical formula for the coupling:G[s, l, t, w] =5.2.1.4ωc2 Zc 1.31 × 10−11 0.9
s + 16t l .
1.8
2s(5.26)

Wire device optimal parameters

There were several device optimization goals I set out at the beginning of the chapter. I
would like to know how to design a device that either minimizes the force noise or maximizes thecoupling to reach one of the three interesting coupling regimes. However, it is not always possible
to optimize all design quantities at once. For instance, tension increases the resonance frequency,
but also makes it much easier to make a successfully suspended device. Specifically, the longest
suspended untensioned device we have ever made was 50 µm long, whereas the longest tensioned
device was 150 µm long. Therefore, in some cases it may be preferable to add tension and increase
the resonance frequency if it means that a longer length can be achieved.
In the absence of parasitic heating of the mechanical oscillator due to the microwave drive,the force spectral density is SFthF ∝ meff Γm ∝ w2 t/l. This quantity is minimized by having awire of the longest length and smallest width and thickness. Because this quantity is independentof mechanical resonance frequency, a device made to optimize force sensitivity would be a good
example of one where we would increase tension to achieve a longer wire. When parasitic heating
is taken into account, the optimal force sensitivity is 7 3 2
5/2 m3eff Ωm Γ3m
w t s
Ωm
tot
∝
.
SF F,min∝4G

s + 16tl(5.27)

This quantity always benefits from long length and small width, thickness, and spacing. However,
it is important to look at the compromise between length and resonance frequency on the right sideof the expression. To understand the trade off, an untensioned wire with typical width w = 150
nm and longest length l = 50 µm would have the same force sensitivity as a tensioned one with119
a typical strain  ∼ 0.006 and length l = 85 µm. Thus, because adding tension allows wires oflength longer than l = 85 µm, it is desirable. A similar trade off can be made between l and s.
For example, it would be preferable to make the wire longer by a factor of 1.5 even at the cost ofincreasing the spacing by a factor of 2.5.
The first coupling value of merit was the cooperativity
4g 2
G2
C=
∝
∝
κΓm
mΩm Γm(s + 16t)2
w2 ts4l3.
Ωm(5.28)For this quantity, an untensioned wire of length l = 50 µm would be equivalent to a tensioned
one of length l = 95 µm. Thus, the possibility of making tensioned wires longer than l = 95 µm
makes adding tension desirable. The spacing and length trade off for a tensioned wire goes as
l4 /s2 or l4 /s4 , depending on the relationship between t and s. Thus, this case is more sensitive tospacing than the force sensitivity was. Here, unlike for force sensitivity where small thickness wasalways optimal, it is preferable to increase the thickness if it is larger than the spacing (to increase
coupling) and preferable to decrease the thickness if it is smaller than the spacing (to decreasemass).
The second coupling of merit was the quantum-enabled regime
C˜ =

4g 2
G2
(s + 16t)2 l3
∝
∝
.
mΓm
w2 ts4κΓm nthm(5.29)

This case does not depend on resonant frequency at all, so adding tension to increase the length isalways better. Again, it is preferable to increase the thickness if it is larger than the spacing andpreferable to decrease the thickness if it is smaller than the spacing.
Lastly, the strong coupling limit is
4g 2
G2
S= 2 ∝∝
κmΩm

(s + 16t)2
wts4



l
.
Ωm

(5.30)

To make a wire equivalent to the 50 µm long untensioned wire in this case would require a tensioned
one of length l = 190 µm, which has not been achieved. Thus, this is the only example where an
untensioned wire might ultimately be preferable over a tensioned one. Once again, it is preferable

120
to increase the thickness if it is larger than the spacing and preferable to decrease the thickness if
it is smaller than the spacing.

5.2.2

Wire device fabrication

In this section, I will describe the wire device fabrication process in an illustrative way. The
detailed step-by-step fabrication recipe can be found in Appendix G. The basic idea is to create a
microwave circuit and mechanical wire, all fabricated out of a single layer of aluminum on an silicon
substrate. A hole is then dry etched in the silicon under the wire in order to suspend (or release)
it. During the etch process, the rest of the chip is protected by a layer of silicon oxide. Tension isadded to the wire by annealing it at high temperature before the etch step.
Many of the steps in the following more detailed description involve photolithography orelectron beam (e-beam) lithography, so I will first explain this process. Lithography (literally‘stone writing’) is the process used to pattern a design onto the device. There are two main ways
to do this. The first is to add material in a pattern on top of the device. The other is to remove
material in a pattern from the device. Either way involves first making a kind of stencil out of apolymer resist. If material is to be added, it is then evaporated onto the device. When the resist
is then removed, it takes with it (or ‘lifts off’) the material not intended to end up on the device.
If instead material is to be removed from the device, the device is etched. The resist protects theareas where etching should not occur.
More specifically, the general lithographic process begins with baking the chip to remove any
moisture, spinning a layer of resist on the chip, and then baking it again to set the resist. For
photolithography, the chip is then aligned to the appropriately patterned photomask, making surethe chip is in contact with the photomask. The masked chip is exposed to ultraviolet light. For
e-beam lithography, the electron beam in the scanning electron microscope (SEM) is used to write
a pattern in the resist. Either of these processes makes exposed regions of the resist soluble in adeveloper. Thus, soaking the chip in developer causes the pattern to be left in the resist. The next
step is to either evaporate metal or to etch the chip. If the former, metal is then evaporated in121an e-beam evaporator, after a short O2 ash to promote the metal’s adhesion to the surface. Whenthe chip is then soaked in acetone, the resist is removed, leaving metal only in the places that were
exposed. If the resist has a pattern for etching instead of evaporation, the chip is then either put inthe etchant (for a wet etch such as hydrofluoric acid) or put in the reactive ion etcher (RIE) for adry etch (such as SF6 ). This etches the places that were exposed and leaves the unexposed regions
unharmed.
I begin the wire device fabrication with a 1 cm square silicon chip with 150 nm thermal siliconoxide. There are several steps of lithography, so I begin by photolithographically patterning andevaporating titanium/gold alignment marks (Figure 5.4(a,b)). The titanium is critical to keep thegold from peeling off of the chip, while the gold creates marks that are easily visible in both thephoto-aligner and the SEM.
I then photolithographically pattern small holes where the mechanical oscillator will eventually end up. As explained above, this starts by covering the chip with photoresist (Figure 5.5(a))
and exposing and developing the photoresist to leave the parts that were exposed bare of photoresist (Figure 5.5(b)). Next, I use buffered hydrofluoric acid to etch the silicon oxide, exposing thesilicon in the holes (Figure 5.5(c)). I then remove the photoresist, leaving the bare chip with holes(Figures 5.5(d) and 5.4(c,d)).
Next, I do a double-layer resist process to create the microwave circuit and mechanical oscillator in a single aluminum layer. The reason for this double layer process is that the large featuresof the microwave circuit are most quickly made using photolithography, while the tiny lines andspaces required for the mechanical wire and spacing to the ground plane are only achievable withe-beam lithography. It is important to do the evaporation all as a single step to avoid oxides
forming between the circuit and mechanical element which could prevent good electrical contact.
The double-layer process involves spinning a layer of photoresist on top of a layer of poly(methyl
methacrylate) (PMMA) (Figure 5.5(e)). I first photolithographically pattern the microwave resonator into the photoresist (Figure 5.5(f)). I then oxygen ash the device in the RIE. This eats away
at both layers of resist, transferring the resonator pattern into the PMMA (Figure 5.5(g)). I then122

Figure 5.4: Illustrative top view of the fabrication process. The materials are silicon oxide (green),gold (yellow), silicon (black), and aluminum (light grey). (a,b) Gold alignment marks allow goodalignment between holes, microwave resonators, and e-beam written wires. (c,d) Holes are patternedand etched in to the silicon oxide where the wire will end up. (e,f,g) The microwave circuit andmechanical element are patterned in a single layer of aluminum. The wire, ground plane adjacentto the wire, and wire supports are written together to ensure good alignment and spacing between
these elements.

123

Figure 5.5: Illustrative side view of the fabrication process. (a) A silicon/silicon oxide chip is covered
with photoresist. (b) Holes are exposed and developed in the photoresist where the wires will end
up. (c) Holes are etched into the silicon oxide. (d) The photoresist is removed, leaving just thechip with holes in the silicon oxide. (e) The chip is covered with a double-layer resist: photoresist
on top of PMMA. (f) The microwave circuit is exposed and developed into the photoresist. (g)The microwave circuit pattern is transferred to the PMMA by ashing. (h) The wire, supports, andground plane are e-beam written and developed into the PMMA. (i) Aluminum is evaporated on
the chip. (j) The PMMA is removed, leaving the circuit and wire as a single aluminum layer. (k)The silicon is etched out from under the wire, releasing it. The rest of the chip is protected fromthis etch by the silicon oxide.

124
write the wire, supports, and part of the ground plane nearest to the wire using e-beam lithography
and develop the resist (Figure 5.5(h)). Writing the last bit of the ground plane in the e-beam step
allows me to have fine control over the separation between the mechanical oscillator and groundplane. I then evaporate aluminum on the whole chip (Figure 5.5(i)). One key discovery was thatthe brief oxygen ash of the chip before evaporating the aluminum (which is generally good practice
before evaporating as it makes the metal stick better) resulted in problems with later suspending
the wires because they stuck to the substrate too much. The devices were far more successful when
the oxygen ash was omitted. I finally lift off the remaining resist and unneeded metal by dissolving
the PMMA in acetone, leaving the chip with the microwave resonator and mechanical oscillator ina single layer of aluminum (Figures 5.5(j) and 5.4(e,f,g)).
The next step is to anneal the chip by baking it in a 340 degree Celsius oven for half anhour. This causes the atoms in the chip to rearrange into a lower energy configuration at hightemperature. When the device is then cooled back down, the aluminum will want to shrink more
(a)

(b)

1 μm

Figure 5.6: SEM images of the wire suspension process. The images show one end of a wire, taken
at 30◦ rotation. Images that show the silicon and wire with good resolution tend to wash out thesubstrate, so I have false-colored the aluminum in red. The grey is the silicon oxide, while the roughblack is the etched silicon. A line appears in the silicon where the wire has shaded the etchingprocess. (a) Image of a wire which has only been partially released. The part of the wire touching
the line is still stuck. (b) Image of the same wire after more etching where it is fully released. The
wire is now parallel to the line on the bottom of the hole.

125
than the silicon, as its coefficient of thermal expansion is larger. Thus, the annealing process will
put the aluminum under higher tension.
The very last step was to suspend the wires. I did this by doing an SF6 etch in the RIE toetch away the silicon below the wires, releasing them (Figure 5.5(k)). It is possible to tell if a wireis released by looking at it in the SEM at an angle. There is usually a line where the silicon in
the hole has been slightly less etched due to the shadow of the wire (see Figure 5.6). If the wire is
(a)

(b)

55 mm

6 mm

(c)

(d)

25
mm
15

6 mm

mm

Figure 5.7: Images of device sample holders, which provide an interface between microwave cabling
and the on-chip device. (a,b) Original round sample holder with up to four ports. The chip, now
cut to a 6 mm square or half-square, is glued to a recessed area in the holder and the ground andcenter conductors of the holder are wire bonded to the chip. We also wire bonded unconnected
grounds on the chip because we found it suppressed other unwanted modes. The vertical piece of
metal on the circle board was implemented to break up modes of the sample box near the microwave
frequencies of interest. The board interfaces with microwave cables via SMP bullet style connectors.
(c,d) Improved sample holder for our experiment. It only has two microwave ports but has thebenefit of being smaller, requiring less fridge space and eliminating box modes near the microwave
frequencies of interest.

126
(a)

(b)

Wire

Silicon Line

Silicon LineGroundPlane

Wire

GroundPlane

Figure 5.8: SEM images of wire actuation, taken at a 40◦ side view. (a) Undriven SEM image of
the wire. It is suspended next to a ground plane and is parallel to the shadow line on the bottomof the silicon hole. (b) SEM image of the wire motion induced by a driving voltage of −4Vdc and
2Vpp at the mechanical resonance frequency of 882 kHz. When cooled down, this device had aresonance frequency of 1.5 MHz [45].

parallel to this line and not stuck to the ground plane, then it is released. The release process canbe further verified by driving the wire at room temperature in the SEM. We did this after the chip
was wire bonded to the sample holder (Figure 5.7). We could then attach an AC voltage source tothe center conductor of the chip and sweep the frequency until we found the mechanical resonance.
At the mechanical resonance frequency, the voltage would drive the oscillator strongly, an effectthat was visible in the SEM image (see Figure 5.8).
There were two visual concerns that we repeatedly noticed with my devices. One was thatthe aluminum film usually displayed many pinholes after annealing (Figure 5.9). This is potentiallybecause the annealing was performed in air, rather than a reducing environment. The other visualconcern that we noticed was that the double resist process created flagged edges on the aluminum
resonator (Figure 5.9). This is because the resist pattern did not maintain perfectly sharp edges
during the transferring of the resonator pattern to the PMMA. However, we did not abandon the
double resist process because otherwise the microwave resonator and mechanical oscillator would
have had to be evaporated separately, allowing for a layer of aluminum oxide to form between themand break electrical contact. Neither of these visual concerns seemed to prevent us from producing

127
(a)

(b)

5 μm

Figure 5.9: Images of visual fabrication concerns. (a) Optical image of a device after annealing.
The aluminum film displays many pinholes. (b) SEM zoom-in of some of the aluminum traces(darker grey). The edges of the aluminum film display flags where they was torn during liftoff.

Figure 5.10: Images of a device released using the focused ion beam (FIB). (a) False-color SEM
image of the wire. I have colored the silicon oxide green and the aluminum light grey. Remnantsof the aluminum supports are visible on the opposite side of the hole from the ground plane. (b)False-color SEM image looking down the wire at a 52◦ tilt. The remnants of a support post are
visible on the left, while imperfections from the cutting are left in the wire and ground plane where
the support used to attach. In the silicon below the wire appears a line due to the shadow of the
support during the RIE etch. (c) The e-beam attached to the FIB was superior to my typically-used
e-beam, so it offered an opportunity to asses the true dimensions of the wire and spacing. The wire
has a width of about 175 nm (determined by the single-pass line width of the e-beam writer). The
spacing, here designed to be 600nm, is actually smaller by about 100nm.

128
devices with good microwave properties, so we did not fix them.
There is one other fabrication method that I used in an effort to make longer wires. As I
showed in the previous section, devices with longer wires and closer spacing to the ground plane
exhibit increased optomechanical coupling and are thus very desirable for our experiments. However, my experience was that the longer, closer wires tended to get stuck to the ground plane when
they were released (even if they were under tension). This was a particularly frustrating experience
because I was suspending them in the RIE, an environment with large and varying electric fields
that I could neither see nor control. The solution to this problem was to pattern the mechanicaloscillator with several orthogonal supports attaching it to the sides of the silicon oxide hole. Suspending the wire and supports in the RIE was then very easy and nearly always successful. The
last step was to use a focused ion beam (FIB) to cut through the supports (see Figure 5.10). In
this case, I could watch in real-time with an electron beam as the FIB cut the supports and control
the cutting process. It was therefore a far more successful last step in the fabrication process. I
found that the best strategy was to make a center support that went all the way across the holeand several others that only went half-way across. I would first cut all the short supports and lastly
cut the long support in the middle. This way, the last cut generally pulled the wire away from theground plane, making it much less likely to get stuck. The best device I was able to make this way
was 150 µm long and only 350 nm from the ground plane. In the next section, I will compare thisdevice to the ones we used for publications and a few others and will refer to it as the FIB device.

5.2.3

Parameters achieved with wire devices

In this section, I will describe the designs and parameters for our lab’s three published wire
devices, as well as those for my best wire device, discussed in the FIB section above (unpublished).
I will also compare two other notable wire devices from other research groups. For ease of reference,
I am going to name the wire devices A-F. Device G is not a wire device and will be introduced in
the next section. The measured parameters for each device are listed in Table 5.1.
Device A [10] was the first published wire device from our lab. As such, it had modest

129
A [10]

B [45]

C [76]

DE [12]

F [14]

G [48]

length l (µm)

50

100

150

150

30

8.5

width w (nm)

100

120

170

175

170

320

thickness t (nm)

130

130

160

130

140

150

spacing s (nm)

1000

1000

1000

350

75

10

mass m (pg)

1.76

4.2

11

9.2

2.1

1.1

48

ωc /2π (GHz)

4.9

5.22

7.49

6.4

7.5

6.98

7.54

Zc (Ω)

89

89

270

200

81

1266

569

κ/2π (kHz)

490

230

2880

490

600

6200

200

Ωm /2π (MHz)

0.237

1.525

1.04

1.039

6.3

32.5

10.56

Γm /2π (Hz)

103

5.1

1.8

1.2

6.3

500

32

G/2π (kHz/nm)

1.16

6.4

32

148

84

1,800

49,000

Table 5.1: Measured device parameters. Almost all parameters are either directly or indirectly
extracted from the publications (aside from device D).

Figure 5.11: Microwave circuit designs for different wire devices. Power is coupled on and off of thechip through a transmission line with either one or two ports. (a) Transmission line λ/4 resonant
circuit in the transmission geometry [10, 45]. (b) High impedance design in the transmission
geometry [76]. (c) High impedance design in the reflection geometry used for the best FIB-released
wire device.

130parameters. However, it is interesting as the only un-tensioned mechanical oscillator compared inthis section (because it was not annealed). The circuit design for device A was a λ/4 transmission
line resonant circuit in the transmission geometry (see Figure 5.11(a)).
Device B [45] was improved over device A by doubling the wire length and going to hightension where the mechanical linewidth was much smaller. This device was used to investigate
radiation pressure cooling of the mechanical oscillator, presented in Chapter 8.1.
Device C [76] was further improved by again extending the wire length. Rather than using atransmission line microwave resonator, we implemented a higher impedance lumped-element circuitdesign (Figure 5.11(b)). This concentrated the capacitance near the wire, resulting in an increased
participation ratio. This device was used to demonstrate a quantum efficient interferometric measurement of microwave fields, explained in Chapter 7.
Device D was created using the FIB to release the mechanical wire (as described above). This
allowed me to decrease the spacing between the wire and ground plane substantially. This device
also utilized a high impedance microwave circuit, but this time in the reflection geometry (Figure5.11(c)). This device was not studied as extensively as all of the published devices, so the numbers
quoted are from preliminary measurements.
Devices E [12] and F [14] are from the Schwab group at Cornell/Caltech and the Sillanpäägroup at Aalto University, respectively. I include them because these two devices offer a different
approach to optimizing the quantities of interest. While I have attempted to increase couplingby making longer and longer wires, they have kept using shorter wires and instead focused on
decreasing the spacing between the wire and ground plane. Their devices thus differ in many ofthe parameters and resulting quantities and it is interesting to see how all the different devices
compare on the figures of merit I laid out at the beginning of this chapter.
I compare the calculated/simulated parameters to those measured in Figure 5.12. Figure5.12(a) shows that the calculated mechanical resonance frequencies agree quite well with thosemeasured. Device A is the floppiest, as expected because it is at low tension. Figure 5.12(b)shows the calculated and measured mechanical linewidths. These do not agree nearly as well.

131

Figure 5.12: Comparison of device parameters for wire devices A-F and suspended membrane device
G. (a) Mechanical resonance frequency measured (black circles) and calculated using Equations
5.10 and 5.31 (red triangles). The strain was set to zero for device A and to 0.006 for B-G. (b)Mechanical linewidth measured (black circles) and calculated using Equation 5.18 (red triangles).
(c) Single-photon coupling rate measured (black circles) and calculated (see legend) using the threemodels and microwave office simulations presented in Sections 5.2.1.3 and 5.3 (and the zero point
motion). (d) Force sensitivity measured (black circles) and calculated (red triangles) with a common
mechanical oscillator temperature of 50 mK. (e) Number of drive photons required to reach eachof the three important limits on optomechanical coupling.

132
However, that should be expected as the empirical expression I used for the expected values wasfound empirically using only short wires. The best mechanical linewidths come from devices BE. Figure 5.12(c) compares all of the different models for wire to ground plane capacitance withthe coupling actually measured. The parallel wire model seems to underestimate the coupling forthe long wires while overestimating it for the shorter ones. The CPS model without thickness
always underestimates the coupling. The CPS model with thickness does a good job predicting
the coupling, but only in the regime for which it is applicable (it is invalid for the short wires).
Overall, the models seem to agree well with the measurements for all wires except the very shortone of device F. Device F exhibits by far the best single-photon coupling for a wire device. Figure5.12(d) shows the calculated force sensitivity, simply using SF F = 4kB Tm meff Γm and the measured
parameters. I have assumed that the mechanical oscillators are all in thermal equilibrium at 50
mK. Devices B-E should have very good force sensitivity, in large part because of their superiormechanical linewidths. Figure 5.12(e) displays the three coupling limits of merit. I plot the numberof photons that would be required to reach each of these limits. For device A, these are very large
numbers, and none of the three would be achievable. However, for devices B-F, substantial coolingwould be possible (4g 2 /κΓm > 1 at the highest powers).

5.3

Suspended membrane devices
In the previous section, I presented a class of planar devices that were conceived of with easeof fabrication in mind. As I showed, multiple groups have been able to push the parameters ofthese devices to quite impressive dimensions at the limits of fabrication, be it in terms of long wire
length or of tiny spacing to the ground plane. However, the optomechanical coupling of these wire
devices is ultimately limited by the fact that the wire makes up such a tiny fraction of the total
circuit capacitance. In order to make the next strides forward in optomechanical coupling, it was
thus necessary to move away from the planar geometry.
The first device to do this was made by our collaborators in Ray Simmonds’ group at NIST.
Using the vacuum gap capacitor technology that they have pioneered [104], they were able to133

Figure 5.13: Images of a suspended membrane device, used in [15, 48, 66]. The sapphire substrateis shown in blue and the aluminum is grey. (a) The circular membrane mechanical oscillator (nearthe bottom of the image) is coupled to a large spiral inductor. Power is coupled into and out of thedevice through an inductively coupled feed line (at top, with ports labeled 1 and 2. (b) Zoom-in of
the suspended membrane. Highlighted with yellow arrows are the vacuum gaps for the capacitor(left arrow), and a crossover of the inductor (right arrow).

create a device where the mechanical motion of a suspended membrane modulates nearly the entire
capacitance of the microwave circuit. Their technology builds up the device in multiple layers,
using a sacrificial layer between two layers of aluminum. When the sacrificial layer is etched away,
they are left with two layers of metal separated by a very tiny ‘vacuum gap’. This process can beused to suspend a membrane of metal, which can act both as a mechanical oscillator as well as the
top plate of a parallel plate capacitor. The process can also be used to make bridges such that two
traces of metal can cross each other, enabling the fabrication of a spiral inductor that has less stray
capacitance than the meander inductors used in my wire devices. Their device is shown in Figure
5.13.
I will not go into the design and parameters of the suspended membranes in nearly thedepth that I did for the wire devices; however, the mechanical resonance frequency and parallel
plate capacitance are fairly quick to calculate, so I do it here. For a circular aluminum suspended
membrane under high tension, the frequency of the fundamental mode can (as for the high tension

134wire) be found from the speed v of propagation:
s
s
r
α01
S
α01
σt
α01 EAl Ωm = 2π × vλ01 = 2π ×= 2π ×= 2π ×,R
m/(πR2 )
R
ρAl t
R
ρAl

(5.31)

where S is the surface tension, σ and  are the stress and strain, and EAl and ρAl are the Young’s
modulus and density of aluminum. The optomechanical coupling can be modeled using the formulafor the simple parallel plate capacitor: Cmembrane = 0 πR2 /s. The coupling is thenG=

ωc2 Zc 0 πR2.
2
s2(5.32)

Device G in Table 5.1 and Figure 5.12 is the suspended membrane device in references [15]and [48]. Figure 5.12 shows that the equations above provide good predictions of the mechanicalresonance frequency and optomechanical coupling for this device. This device has a larger mechanical resonance frequency and linewidth than many of the wire devices, but maintains a similar
mechanical quality factor. It does not make the best force sensor, as the mass of the membrane issubstantially larger than that of the wires. However, the impressiveness of this device is its large
optomechanical coupling of G = 2π × 50 MHz/nm. This enables it to reach all three of the coupling
regimes with far fewer photons than any of the wire devices, and thus allows exploration of physics
that was not within reach of the wire devices. This device was used to demonstrate strong coupling[15], cool the mechanical oscillator to the ground state ([48], Chapter 8.2), attempt to see sideband
asymmetry and quantum backaction (Chapters 9 and 10), and demonstrate transfer of coherent
itinerant microwave fields with the mechanical oscillator ([66], Chapter 11).

Chapter 6

Experimental calibrations

Several calibrations are similar between all of the experiments we perform. Here, I will
discuss how we go from the measured voltage spectral density in Chapter 4 to units related to thephoton[ω] on the right-hand side of the intensity
optomechanical system (either the photons units SII

expressions in Chapter 4, or mechanical displacement Sxx [ω]). The calibrations discussed eliminate
photon[ω] and determine the coupling g0 .
the unknown proportionalities between SII [ω] and SII

6.1

JPA gain calibration
The frequency dependence of the HEMT amplifier can generally be ignored, as it is close to flatover the frequency ranges present in our experiments. However, the JPA generally has a bandwidthof a few megahertz, meaning that it can be important to remove this frequency dependence. The
frequency dependence of the JPA can be determined independent of the optomechanical cavity byinjecting a probe tone at the input of the JPA (after the cavity) and then looking at the response at
the output of the experiment. This response reveals the resonant frequency and bandwidth of the
JPA (but not necessarily the absolute magnitude of JPA gain, as it also depends on the attenuationof the probe input line). In this way, the gain at any frequency can be referenced to the gain at
any other frequency. The absolute value of the JPA gain can then be removed by referencing theamplitude of this probe tone measured with the JPA on to the same measured amplitude with the
JPA off. This allows spectral data taken with the JPA to be put into effective ‘JPA off’ units.
Figure 6.1 shows these transformations. The raw voltage spectral densities measured with the JPA

136

Figure 6.1: Measured voltage spectral densities. The raw data measured with the JPA on andoff are shown in green and orange, respectively. The Lorentzian feature at ∼ 1.041 MHz is themechanical resonance, while the delta function at ∼ 1.047 MHz is a calibration tone. The green
data is scaled such that the calibration peaks will be equal in height, removing the JPA gain andresulting in the black data.

(green) and without the JPA (orange) are shown. Also included is the data with the JPA on, now
transformed to remove the frequency dependence of the JPA gain and scaled into ‘JPA off’ units,
making the two calibration tones of equal magnitude. The benefit of having the spectral data in‘JPA off’ units is that it can be directly related to measurements of the output classical drive takenwith the JPA off. These measurements cannot be taken with the JPA on, as the large drive tone
will saturate the JPA if it is not canceled first.

6.2

Inferring mechanical motion from measured voltage spectral density
This section explains how we transform measured voltage spectral density data (in ‘JPA

photonoff’ units, if relevant) into data with optomechanical units (either the photon units SII[ω] onthe right hand side of the equations in Chapter 4 or displacement spectral density units Sxx [ω]).
photonAs the measured voltage spectral density is proportional to SII[ω], I relate the theory to themeasurement with a proportionality constant λ:photonSV V [ω] = λSII
[ω].

(6.1)

137
In general, λ depends on quantum efficiencies and amplifier gains (and could thus change day to
day or even measurement to measurement).
The total drive power Pd can be measured by integrating the voltage spectral density about
the drive frequency (measured with the same digitizing setup as the spectral data but with the
JPA off if it is present):
Z0 Pd =≈ZZ
1
λphotonSVd Vd [ω]dω =
SII,d
[ω]dω
(6.2)
2π
2π
Z

λPout
λ
1
Saout,d aout,d [ω] + Saout,d aout,d [2ωLO − ω] dω =(1 + δωLO ,ωd ) , (6.3)
2π
2
2h̄ωd

where Pout is the theoretical expression for the cavity output power given in Equation 3.7. The
first two integrals are across the measured drive frequency |ωLO − ωd |, while the third integral isacross the true drive frequency ωd . In the second to last step, I assumed the drive power is large
compared to any added noise power. Alternatively, λ can instead be related to the derivative of
the measured drive voltage as a function of drive frequency:
dVd 2 λdαout 2 λdαout 2 λ(1 + δωLO ,ωd )|Ni |2 nd.
= (1 + δωLO ,ωd )= (1 + δωLO ,ωd ) −
=
dωd
2
dωd
2
dω c
2κl

(6.4)

The inferred spectrum in Chapter 4 photon units is thenphoton[ω] =SIISV V [ω]λSV V [ω]Pout (1 + δωLO ,ωd )2Z0 Pd h̄ωd|Ni |2 nd (1 + δωLO ,ωd )SV V [ω]=.
2κl|dVd /dωd |2
=

(6.5)
(6.6)

The inferred displacement spectral density can be found by relating the photon units to displacement spectral density units as done in Section 4.5. Note that this step depends on the details of thedrive and interferometer and thus must be calculated independently for each case. As an example,
if the voltage spectrum were measured using an on-resonant drive and a homodyne interferometercontaining a JPA and then converted to ‘JPA off’ units, and the drive power was measured with
the JPA off and ωLO 6= ωd , the displacement spectral density would be (from Equation 4.42)
Sxx [ω]κ2 + 4Ω2m photon
=
SII
[ω] =
x2zp
4g 2 κl

SVI VI [ω]Z0 PI,dκ2 + 4Ω2m|Ni κ − 2κl |2 .
2
2
32g0 κl

(6.7)

138
Note that SVI VI [ω]/(Z0 Pd ) and all of the other quantities in this expression are independently
measured optomechanical parameters. In particular, converting measured data into displacement
units does not depend on knowing the number of photons in the cavity.
As I said above, λ can change over time due to such things as amplifier gain changes. Rather
than repeating the drive measurements above for each data point, we sometimes apply a calibrated
force to the mechanical oscillator which lets us calibrate the relative change in λ between any
two measurements. We do this by applying both a DC and AC voltage to the feed line suchthat the force on the mechanical oscillator is F [ω] = VDC VAC [ω](dCfm /dx), where Cfm here is thecapacitance between the feed line and the mechanical oscillator. Let Λ be another proportionality
constant which directly relates the measured voltage spectral density to the displacement spectral
density (Λ ∝ λ but the relationship depends on the interferometer type). The voltage spectral
density in the presence of mechanical drive force F [ω] is thenSVI VI [ω] = ΛSxx [ω] =Λx4zp SF F [ω]h̄2|χm,eff [ω]|2 .

(6.8)

If the applied force is white (independent of frequency), then Λ can be related to the area A andlinewidth Γtot of the measured Lorentzian spectrum:
SVI VI [ω] =AΓtot
AΓtot h̄2
|χm,eff [ω]|2 , Λ =.
2π
2πx4zp SF F

(6.9)

The displacement spectral density for any new measurement written in terms of a fully calibrated
reference measurement is thenSxx [ω] =Aref Γtot,ref
SVI VI [ω]=SVI VI [ω],ΛAΓtot Λref(6.10)

where the critical assumption is that the force and zero point displacement do not change frommeasurement to measurement.

6.3

Optomechanical coupling calibration
The single-photon optomechanical coupling g0 is found by using a temperature sweep tocalibrate the mechanical output noise into quanta. The basic idea is that the area under the

139mechanical peak in the output spectra with an on-resonance or very low power drive is proportional
to g02 nthm and we expect that the mechanical oscillator is in equilibrium with the base temperatureof the dilution refrigerator at high temperature (Tcryo ∼ 100 − 300 mK). We can determine thenumber of thermal mechanical phonons from the Sxx [ω] data:
g02 nthm =th [Ω ]
g02 Sxx
m
SQL
4Sxx

.

(6.11)

Plotting this quantity as a function of fridge temperature Tcryo and using the equipartition theorem
allows us to identify g0 from the slope of the graph:g02 nthm =6.4

g02 kB
Tcryo .
h̄Ωm

(6.12)

Amplifier added noise and cable loss
We calibrate the added noise of amplifiers and loss in our interferometer by switching between

two calibrated noise sources (50 Ohm terminations), one at TH = 4 K and one at the fridge basetemperature Tcryo (see Figure 6.2(a)). The source is switched to the base temperature termination
during experiments to minimize the thermal input noise incident on the optomechanical system.
However, switching to the 4 K source allows us to calibrate the added noises of the interferometer.

If the switch is switched to the cold load at Tcryo , then the number of microwave noise quantaat the input of the amplifier (see Figure 6.2(b)) will benin [Tcryo ] =−11  h̄ω/(kB Tcryo )
+ e
−1
,
2

(6.13)

regardless of any cold attenuation between the cold load and the amplifier. By using two different base temperatures of the fridge, Tcryo,1 and Tcryo,2 , a Y-factor measurement can be made to
determine the gain and added noise of the amplifier:Y=nadd =

nout [Tcryo,1 ]nin [Tcryo,1 ] + nadd=,
nout [Tcryo,2 ]nin [Tcryo,2 ] + nadd
nin [Tcryo,1 ] − Y nin [Tcryo,2 ].
Y −1(6.14)(6.15)

140

Figure 6.2: Calibration of interferometer added noise. (a) Calibrated noise setup. The amplifier
input can be switched between one of two calibrated noise sources, one at TH and one at Tcryo .
Loss between the switch and amplifier is modeled as an attenuator with power attenuation A.
Superconducting cables (green) are used to losslessly transition between temperatures. The numberof microwave noise quanta at the input of the amplifier is nin , while the number of measured noise
quanta is nout . (b) Theoretical noise quanta for calibration measurements using ω = 2π × 7 GHz,
nadd = 0.3, A=1/4, TH = 4 K. With the switch switched to Tcryo (blue), the input and output noise
are independent of A and a measurement of the output noise reveals the amplifier gain (from theslope at high temperatures) and added noise (from the zero-temperature value). With the switch
switched to TH (red), the output noise is dependent on A and using the known values of G, nadd ,and TH allows a determination of A.

Here, Y and the nout ’s are measured values, while the nin ’s are calculated from the known temperatures via Equation 6.13.
To determine any cold attenuation between the calibrated noise source and the amplifier, theswitch is switched to the hot load. This fixes the input noise temperature of the attenuation at TH ,while the attenuation itself is at temperature Tcryo . The number of microwave noise quanta at theoutput of loss with power attenuation A (at the input of the amplifier) isnin [Tcryo ] =

1
A
1−A
+
+
.
2 eh̄ω/(kB TH ) − 1 eh̄ω/(kB Tcryo ) − 1(6.16)

Varying the base temperature of the fridge then varies only the temperature of the attenuation,
allowing us to identify the loss A between the switch and the amplifier.

Chapter 7

Experiment I: Quantum efficient microwave measurement

In this chapter, I detail our realization of a quantum efficient microwave interferometer,
which enabled us to make a measurement of our mechanical oscillator with an imprecision below
the imprecision at the standard quantum limit. This result is critical to the exploration of thequantum behavior of mechanical oscillators, as the quantum motion must be resolvable above
the measurement imprecision. Additionally, the improvement in interferometric measurement also
resulted in an exquisite force sensitivity of the optomechanical system, potentially leading to futureapplications in ultrasensitive detection.

7.1

Review of quantum limits on measurement
In Section 4.5.3, I found expressions for the imprecision and backaction numbers of mechanicalquanta in the presence of a single, on-resonance, cavity drive, using a homodyne interferometercontaining a phase-sensitive amplifier:
ba
th
zp
ntot
= nimpm
m + nm + nm + nm ,


κ
1+ν
nimp
=
(1 + 2nadd ),
m
16κl
C
C
nba
=
.
m
1+ν

(7.1)
(7.2)
(7.3)

zp
In these equations, nthm is the number of thermal quanta, nm = 1/2 is the half quanta of zero-point

motion, C = 4g 2 /(κΓm ) is the cooperativity and ν = 4Ω2m /κ2 is the resolved sideband factor. In

142terms of displacement spectral densities, these are
tot
imp
ba
th
zp
Sxx
[ω] = Sxx
[ω] + Sxx
[ω] + Sxx
[ω] + Sxx
[ω],

(7.4)

imp
SQL impSxx[ω] = 4Sxx
nm ,

(7.5)

ba(th,zp)
SQL ba(th,zp)
Sxx[ω] = 4Sxx
nm

Γ2m
|χm [ω]|2 .
4

(7.6)

Quantum limits are imposed on the product of the imprecision and backaction:
ba
nimp
=
m nm
imp ba
Sxx
SF F

κ
1
(1 + 2nadd ) ≥ ,
16κl
16

ba
2
= 16h̄2 nimp
m nm ≥ h̄ .

(7.7)
(7.8)

The minimum total number of phonons as a function of cooperativity occurs at C SQL = (1 + ν)/4,
where the imprecision and backaction phonon numbers are equal (nimp= nbamm = 1/4). This isSQL= 2x2zp /Γtot to be the= 1/4 or Sxx
called the standard quantum limit and thus I define nSQL
m

imprecision (or backaction) at this point.

7.2

Interferometric measurement setup
Our detailed homodyne interferometric scheme is shown in Figure 7.1. A single tone issplit into three parts. The first is used to drive the optomechanical system, the second is used
to pump the JPA, and the third provides the local oscillator to the mixer. The input lines areheavily attenuated to prevent thermal noise from room temperature from reaching the device. For
this experiment, the device we use is in the transmission geometry. This necessarily hurts theimprecision at any given power by a factor of two, as κl = κext /2 ≤ κ/2. The added noise of theamplifiers and loss of microwave lines and components can be calibrated by switching between two
different noise sources at the input of the experiment, as explained in Section 6.4. One source isthermalized to base temperature Tcryo (this is what we use while doing the actual experiment),
while the other is at TH = 4 K. Varying the base temperature of the cryostat with the switch
switched to the hot termination allows us to identify 6 dB of total loss between the switch and
JPA, 2 dB of which is estimated to be between the cavity and the JPA.

143

Figure 7.1: Interferometer diagram. A single microwave tone is divided in three parts to drive thecavity, pump the JPA, and provide the local oscillator to the mixer. A directional coupler is usedat base temperature to combine the highly attenuated cavity drive with calibrated noise from oneof two terminations determined by a switch. The drive signal then interacts with a transmission
geometry optomechanical system. The output signal from the cavity reflects off of the JPA before
being sent through an output chain of amplifiers to the mixer. Superconducting coaxial cables(green) are used between temperature stages to minimize loss. Cryogenic microwave circulators areused to prevent amplifier noise from reaching the optomechanical system.

Similar measurements (also explained in Section 6.4) allow us to identify that the added noise
HEMT = 24.5. When the JPAof the JPA is nJPA
add = 0.3 and the added noise of the HEMT alone is nadd

is on, the HEMT noise is effectively divided by ∼20 dB of JPA gain, contributing 0.25 quanta ofnoise. The effective number of added noise photons from the loss and amplifiers can then be found:


nHEMT
1
1
0.2 1
JPA
add
+ nadd = 10
+ nadd + JPA= + 1.16
(7.9)
2
2
G
2
Despite the heavy attenuation on the input lines, we estimate the thermal input noise to be nth
l =0.17. This noise will add to the imprecision of the measurement. However, it will contribute
negligibly to the thermal motion of the mechanical oscillator, as the occupancy in this experimentwill always be large compared to 0.17 (no cooling is employed). The total added noise of the144interferometer utilizing the JPA is thus estimated to be nadd + nth
l = 1.3. Without the JPA, loss
between the JPA and HEMT is important and the total added noise we estimate is nadd = 55. Thisadded noise value corresponds to a quantum efficiency of η = 0.9%. This means that without the
JPA, 99% of the signal is thrown away before the measurement - clearly a poor situation. With
the JPA, the interferometer’s quantum efficiency improves to η = 27%, an efficiency comparable tothose achieved in optical systems.

7.3

Device design and parameters
The device used for this experiment is the one labeled ‘Device C’ in Chapter 5. We realize

this optomechanical system by embedding an aluminum mechanical wire in a superconducting
microwave cavity. The wire has dimensions 150 µm ×170 nm ×160 nm, giving an estimated massof m = (11 ± 2) pg. We can look at the output spectra to determine the mechanical resonancefrequency Ωm = 2π × 1.04 MHz and mechanical linewidth Γm ≈ 2π × 1.8 Hz (depending on
power). The circuit is made from the same layer of aluminum as the wire and is designed tohave a higher impedance than a λ/4 transmission line cavity (see Figure 5.11(b)). The bare cavity
response is measured to extract the cavity resonance frequency ωc = 2π×7.49 GHz, cavity linewidth

Figure 7.2: Optomechanical coupling calibration. The integrated voltage spectral density SV V [ω]is proportional to g02 nfm ∼ g02 nthm through measurable proportionality constants. This quantity isplotted here as a function of cryostat temperature Tcryo . The slope of the linear (zero-offset) fit
reveals the single-photon optomechanical coupling g0 = 2π × 0.86 ± 0.3 Hz.

145
κ = 2π × 2.88 MHz, and external coupling κext = 2π × 2.5 MHz. The cavity linewidth is designed
to be comparable to the mechanical frequency (ν = 4Ω2m /κ2 = 0.52) to aid in achieving the best
possible measurement imprecision.
To calibrate the optomechanical coupling, we do a temperature sweep as described in Section
6.3. This is shown in Figure 7.2, where g02 nfm ≈ g02 nth
m is plotted verses cryostat base temperature. At
high temperatures, the mechanical oscillator tracks the cryostat temperature. At low temperatures,the mechanical occupancy saturates. We fit all but the lowest three temperatures to a line throughzero to deduce the single photon coupling g0 = 2π ×0.83±0.3 Hz. Plugging in the zero point motion
results in G = g0 /xzp = 2π × (32 ± 3) kHz/nm. The fractional uncertainty in G is larger than in
g0 because it includes the relatively large uncertainty in the mass. The value of G is independently
confirmed to be G = 2π × (30 ± 6) kHz/nm by measurements of the radiation pressure damping
using a detuned cavity drive.

7.4

Imprecision below the standard quantum limit
The use of a more quantum-efficient interferometer to read out an optomechanical system

clearly leads to an improvement in the measurement of mechanical motion. However, the quantitative figure of merit for efficient mechanical detection is the imprecision of the measurement in unitsof phonons. The imprecision number of phonons will be inversely proportional to the quantum efficiency of the interferometer and should decrease linearly with the number of drive photons. Thereis no theoretical limit on how small the imprecision can be. In this experiment, we investigate thisimprecision as a function of drive power and show that the imprecision can be pushed below aquarter quantum (the value at the standard quantum limit).
For a series of drive powers, we measure the output voltage spectrum and transform it (asdescribed in Section 6.2) into an inferred displacement spectral density. Figure 7.3(a) shows thehighest power measurement made using an on-resonance drive and our homodyne interferometer
including the JPA. Figure 7.3(b) shows the same measurement for a blue-detuned drive. Most
notably, the imprecision (frequency independent background) in spectral density units is the same in

146

Figure 7.3: Inferred mechanical spectra in displacement and phonon units, (a,b) Sxx [ω] and (c,d)SQL), respectively. (a,c) Mechanical spectra in the presence of an on-resonant drive. The
Sxx [ω]/(4Sxx
imprecision is below the imprecision at the standard quantum limit (dark blue dotted line). (b,d)
Mechanical spectra in the presence of a blue-detuned drive. The absolute imprecision is the same
as for the on-resonance drive, but the number of imprecision phonons can be arbitrarily small at
the cost of amplified mechanical motion.

SQL
both cases. However, it is somewhat more convenient to plot the spectra in units of Sxx [ω]/(4Sxx),

as in Figures 7.3(c,d). These units have the nice property that the number of phonons can be read
off simply by looking at the data on resonance. Thus, the height of the frequency-independent
tot
background is nimp
m and the total height of the Lorentzian peak is nm . For the on-resonant drive, themeasurement imprecision is below the imprecision at the standard quantum limit nSQL
= 1/4 (dashm
dotted line). Quantitatively, the number of thermal quanta is nth = 2600 and the measurement
SQL
imprecision is nimp
m = 0.2 ± 0.0075 < nm . The imprecision in displacement spectral density units√
imp 1/2
is (Sxx) = (4.8±0.4) fm/ Hz. This fractional uncertainty is larger than the uncertainty in nimpmbecause it includes the uncertainty in the mass of the mechanical oscillator. For the blue-detuned
drive, the imprecision number of phonons can be arbitrarily small (as expected, see Figure 4.3(a)),

147
but at the cost of amplifying the mechanical motion. In Figure 7.3(d), the mechanical linewidth isreduced to Γtot = 2π × 0.67 Hz, and the imprecision is nimp
m = 0.017. However, the real mechanicalmotion is increased to nfm = 5 × 104 .
While only the highest power point is required to demonstrate that the imprecision is belowthat at the standard quantum limit, the entire power dependence of the imprecision can be used
to confirm the excellent quantum efficiency of our interferometer. The number of quanta for the
on-resonant drive as a function of incident drive power is shown in Figure 7.4(a). The number ofthermal quanta (red circles) stays approximately constant as a function of power, while the numberof imprecision quanta (purple squares) decreases with power. The very highest power points aretaken from the data in Figure 7.3(c) and we once again see that the imprecision number of quanta
is below the imprecision at the standard quantum limit, nSQL
= 1/4. The expected imprecisions
mgiven a perfect quantum-efficient interferometer and an interferometer with nadd = 1.3 are shown
as solid and dashed lines, respectively. Both lines include the measured values of κ and Γm at each

Figure 7.4: Power-dependence of the imprecision. (a) Imprecision (purple squares) and total (redcircles) mechanical phonon numbers as a function of incident drive power. At the highest power(inferred from the data in figure 7.3(c)), the imprecision is below the imprecision at the standard
quantum limit (dark blue dotted line). Expected imprecisions assuming a shot noise limited interferometer and an interferometer with nadd = 1.3 are shown as solid and dashed lines, respectively.
(b) Cavity frequency noise limits the achievable imprecision. The JPA improves apparent cavity
frequency noise (black) over using only a HEMT amplifier (orange) by reducing the added noiseof the interferometer. However, when the cavity is driven, excess cavity frequency fluctuations
dominate the apparent cavity fluctuations in the JPA case (green).

148
power, which is why the lines are not perfectly straight. Most notably, the line for nadd = 1.3 agrees
well with our data, confirming our added noise calibration and the fact that our interferometer isnearly shot-noise limited.
The theory introduced so far in this thesis would predict that the imprecision could havebeen improved even further by increasing the microwave power. However, the imprecision in thisexperiment was limited by a source of technical noise. Ultimately, any information we know about
the mechanical position comes from our knowledge of the cavity frequency fluctuations (as the
optomechanical coupling is dependent on G = dωc /dx). The imprecision discussed in the theory
comes from the apparent cavity frequency fluctuations contributed by added microwave photons.
The measured cavity frequency fluctuations are shown in Figure 7.4(b) for the highest power usedin Figure 7.3. As expected, the frequency fluctuations measured using only a HEMT amplifier(orange) include a narrow Lorentzian peak at the mechanical frequency (corresponding to actual
mechanical motion) and are otherwise dominated by apparent fluctuations from the interferometer
added noise. Using the JPA but leaving the cavity drive off (black) improves these apparent
fluctuations, reducing the imprecision of the measurement. However, when the cavity drive isturned back on (green), excess cavity frequency fluctuations dominate the imprecision. Thesefluctuations do not decrease with larger drive power, and thus limit the imprecision from being
improved beyond what is presented above. This limitation could be decreased in future experiments
by choosing materials, fabrication techniques, or cavity geometry to decrease the cavity frequency
fluctuations or by increasing the optomechanical coupling g0 to make those frequency fluctuations
less important.

7.5

Force sensitivity
Once the imprecision is decreased below the thermal noise, a measurement of the optome-

chanical system is extremely sensitive to mechanical displacement and thus also any forces that
may act on the mechanical oscillator. The sensitivity to an external force applied to the center of149
the mechanical wire is limited by both the thermal and imprecision noise and is given by
!
!
imp
imp
2
2
2h̄
Γ
4n
h̄
Γ
4n
m
m
m
m
SFtotF [ω] ≈
+ nth
= 2
+ nth
m
m .
2x2zp
Γ2m |χm [ω]|2xzp
Γ2m |χm [ω]|2(7.10)The bold 2 comes from the use of the effective mass meff = m/2 at the center of the high-tension
wire (see Appendix E):
x2zp,eff =h̄2meff Ωm=2h̄= 2x2zp .
2mΩm(7.11)

Assuming the imprecision is smaller than the thermal occupation, the sensitivity to forces at themechanical resonance frequency is
SFtotF [Ωm ] = h̄2 Γh̄2 Γm  imp
m
th
n
+
n
≈ 2 nth= 2kB Tm mΓm = 4kB Tm meff Γm ,
m
m
x2zp
xzp m

(7.12)

where Tm is the thermal temperature of the mechanical oscillator. The bandwidth of the sensor(the region in frequency where the thermal contribution is larger than the imprecision one) isqimpΓm nthm /nm − 1.

Figure 7.5: Force sensitivity. (a) Displacement spectral density measured with the JPA on (green)and off (orange) at low incident power Pinc = 1 pW. Using the JPA significantly improves thedisplacement imprecision. (b) The same data as in (a), now plotted as a force spectral density
versus logarithmic frequency away from the mechanical resonance frequency. As the thermal motion
is much larger than the imprecision even with the JPA off, the force sensitivity on resonance is
not improved by turning the JPA on. However, the bandwidth of the force sensor (the region of
frequency over which the sensor is not limited by the imprecision) is improved by a factor of fivewhen the JPA is employed. This is indicated by a horizontal red dashed line at twice the force
sensitivity on resonance and vertical red dashed lines at the frequencies where this is equal to theththmeasured force sensitivity SFtotF [ω] ≈ SFimp
F [ω] + SF F = 2SF F .

150
Theoretically, the force sensitivity should be independent of power once the imprecision issmall compared to the thermal motion. However, we observe that both Tm and Γm increase withpower. Therefore, our best force sensitivity is achieved at low power Pinc = 1 pW, one of the lowest
powers used in Figure 7.4. The displacement and force spectral densities at this power are shownin Figure 7.5. The force sensitivity on resonance is SF F [Ωm ] = (0.26 ± 0.04) aN2 /Hz, both withand without the JPA. However, the bandwidth of the sensor improves fivefold from 2π × 1.85 Hz
to 2π × 10.4 Hz when the JPA is used (Figure 7.5(b), red dashed lines).
In this experiment, we were able to demonstrate the achievement of a nearly shot-noise limited interferometer in the microwave regime, an advance critical not only to our experiments in
optomechanics, but also essential for any experiment where the information of interest is containedin microwave fields. This improvement in efficiency allowed an on-resonant drive measurement ofa mechanical oscillator with an imprecision below the value at the standard quantum limit. Thisultrasensitive measurement of mechanical motion will be essential to future detection of quantum
states of the mechanical oscillator which could lead to squeezing of the mechanical mode, entanglement of mechanical motion and other quantum systems, and even investigations into fundamental
quantum mechanical theory.

Chapter 8

Experiment II: Radiation pressure cooling to the mechanical ground state

Radiation pressure cooling of a mechanical oscillator into its ground state was a long-standing
goal of the entire optomechanics community, as it is the first step toward realizing experiments
exploring quantum mechanics in macroscopic mechanical oscillators. In this chapter, I will detailour experiments on cooling the mechanical oscillator using the radiation pressure of microwave
photons. In Section 8.1, I detail our initial, proof of principle, cooling measurements from reference[45]. In Section 8.2, I explain how the advent of a device capable of reaching the strong-coupling
regime as well as the implementation of the nearly shot-noise limited interferometry describedin Chapter 7 allowed one of the first measurements of a macroscopic mechanical oscillator in its
quantum ground state [48].

8.1

Initial realization of radiation pressure coolingIn our initial experiments exploring radiation pressure cooling in 2008 [45], we observed theradiation pressure induced changes to the mechanical susceptibility and demonstrated cooling of
the mechanical oscillator from an occupancy of 700 down to 140 phonons. This initial attempt
was done with an interferometer containing only a HEMT amplifier and a comparatively weakly
coupled device to the one that follows.

152
8.1.1

Review of radiation pressure cooling theory in the weak-coupling limit

In Chapter 3.1, I found theoretical expressions for the radiation pressure induced damping
Γopt , mechanical resonance frequency shift Ωopt , final number of mechanical phonons nfm , and thecavity output operator spectrum Saout aout [ω], in the presence of an optimally red-detuned driveωr ≈ ω c − Ωm , in the resolved sideband limit (Ωm  κ). In the case where the added noise andth
final mechanical occupancy are relatively large compared to any photon occupancy, nthl and nc

can be ignored. Then these quantities in the weak coupling regime are (Equations 3.32, 3.57, and3.76)
Γtot = Γm + Γopt
Ωtot = Ωm + Ωopt
nfm




κ
κ
= Γm + 4g−,
κ2 + 4(∆ + Ωm )2 κ2 + 4(∆ − Ωm )2
∆ + Ωm∆ − Ωm= Ωm + 4g 2
+ 2
,
2
2
κ + 4(∆ + Ωm )
κ + 4(∆ − Ωm )2

th
0
th 0
nth
nth
m Γm + nc + nm + 2nc nm Γopt
m Γm
=
≈
,
Γm + Γopt
Γm + Γopt
2

Saout aout [ω] ≈ g 2 κl Γtot nfm |χc [ω]|2 |χm,eff [ω − ωr ]|2 .

(8.1)
(8.2)
(8.3)
(8.4)

In Chapter 4, I derived expressions to relate the intensity (or voltage) spectral density measured by an in-phase, linear, two-quadrature detector to the Saout aout [ω]’s (the relevant result
for an interferometer only containing a HEMT amplifier). The measured spectrum at frequency
ω = |ωLO − (ωr + Ωm + δ)|, where δ is the detuning from the mechanical sideband peak, is[ω] =SIphoton
I II
≈

8.1.2

1 1
+ (Saout aout [ωLO − ω] + Saout aout [ωLO + ω])
η 2
1 1 2
+ g κl Γtot nfm |χc [ωr + Ωm ]|2 |χm,eff [Ωm + δ]|2 .
η 2 r

(8.5)
(8.6)

Device parameters and interferometric setup

The optomechanical device for this experiment is described in Chapter 5 as ‘Device B’. The
mechanical oscillator is an aluminum wire with dimensions 100 µm ×130 nm ×120 nm, resulting in a
mass1 of 4.2 pg. The bare mechanical frequency and linewidth (determined from the output spectra)are Ωm = 2π × 1.525 MHz and Γm ∼ 2π × 5 Hz. The microwave circuit is a λ/4 transmission line
resonator (see Figure 5.11(a)) in the transmission geometry (κl = κext /2. The bare cavity response

153

Figure 8.1: Optomechanical coupling calibration. The integrated voltage spectral density SV V [ω]is proportional to g02 nfm ∼ g02 nthm through measurable proportionality constants. This quantity isplotted here as a function of cryostat temperature Tcryo . The slope of the linear (zero-offset) fit
reveals the single-photon optomechanical coupling g0 = 2π × 0.21 Hz.

indicates a cavity resonance frequency and linewidth ω c = 2π×5.22 GHz and κ = 2π×230 kHz. The
cavity linewidth is designed to be much smaller than the mechanical frequency, putting this device
soundly in the resolved sideband limit (quantified by the resolved-sideband factor ν = 4Ω2m /κ2 =176) and theoretically enabling cooling to the ground state (via the requirements outlined in Section
3.1.6).
To calibrate the optomechanical coupling, we do a temperature sweep as described in Section
6.3. This is shown in Figure 8.1, where g02 nfm ≈ g02 nth
m is plotted verses cryostat base temperature.
This measurement is done with a red-detuned drive, but at low enough drive power (nd ≈ 9 ×
105 , C ≈ 0.14) where radiation pressure effects should be negligible. At high temperatures, themechanical oscillator is in equilibrium with the cryostat temperature. At low temperatures, themechanical oscillator temperature saturates to ∼ 50 mK, or nthm ∼ 700. We fit all but the lowest
two temperatures to a line through zero to deduce the single photon coupling g0 = 2π × 0.21 Hz.
Dividing by the zero point motion results in G = g0 /xzp = 2π × 5.7 kHz/nm.
1This is different from the number published in [45], as the mass used consistently throughout the publishedanalysis was erroneously 6.2 pg rather than the true 4.2 pg. However, neither the single-photon coupling g0 , nor theradiation pressure induced damping, nor the final mechanical occupancy depend on mass. Therefore, consistently
using the incorrect mass changes only the value for G and the units Sxx . The published achievements in damping
and occupancy are unaffected.

154This experiment was done with a heterodyne interferometer containing a HEMT amplifier
only (Figure 8.2). The most notable component that was not contained in the interferometer inthe previous chapter is a low frequency mechanical drive line that allows us to drive the mechanicaloscillator near its resonance frequency. This low-frequency drive is combined with the microwave
drive through a bias T, allowing it to modulate the voltage along the feed line, exerting a force on
the mechanical oscillator. Sweeping this mechanical drive frequency then allows us to measure theresponse of the mechanical oscillator to this force.

Figure 8.2: Interferometric measurement setup. The heterodyne interferometer is formed by interrogating the system of interest (here an electromechanical system) with a microwave cavity drive
and then mixing the resulting microwave signal with a local oscillator at the output of the experiment. A bias T combines the cavity drive with a low-frequency (∼ 1 MHz) mechanical drive. Onceagain, cryogenic circulators are used to block amplifier and thermal noise at 4K from reaching theoptomechanical system.

155
8.1.3

Observed change in mechanical susceptibility

The presence of radiation pressure changes the mechanical susceptibility by changing theresonant frequency and linewidth of the mechanical oscillator (Equation 3.32). We investigate
this by observing the change in the driven mechanical response as a function of microwave drive
frequency while adjusting the incident power to keep the number of drive photons in the cavity
constant at nd = 8 × 106 (C ≈ 1.2), essentially measuring the effective mechanical susceptibility. At
this power, the radiation pressure damping is slightly larger than the intrinsic mechanical damping.
The mechanical resonance frequency and linewidth are plotted as a function of drive frequency in

Figure 8.3: Optomechanically induced change in mechanical susceptibility. The optomechanicallyinduced mechanical resonance frequency (a) and the total mechanical damping (b) are plotted
as a function of drive frequency with the number of drive photons held constant. The intrinsic
damping and frequency are shown as horizontal dashed lines. Vertical dashed lines indicate thecavity resonance frequency, as well as the optimally red and blue detuned frequencies ω c ± Ωm .
The two quantities are fit concurrently (solid blue line) to Equations 3.32 with nd as the only freeparameter.

156
Figure 8.3 and fit with the theoretical equations found above. The only adjustable parameter inthese fits is nd and the value recovered agrees with estimates for the incident power within the 3 dB
uncertainty of the input drive line attenuation. There are data points missing near ωd = ω c + Ωm ,as expected, because this is where the mechanical oscillator is amplified to self-oscillation.

8.1.4

Cooling to 140 mechanical quanta

The radiation pressure damping measured in the previous section suggests that the mechanical oscillator was also being cooled by the radiation pressure. Theoretically, the final mechanicaloccupancy should be inversely proportional to the mechanical damping: nfm = nthm Γm /Γtot . However, to measure this thermal occupancy, we need to look at the output spectral density without amechanical driving force. Here, I present the results of this thermal occupancy measurement as afunction of red-detuned drive power.
One detail of this measurement is that the cavity becomes nonlinear at the highest microwave
powers due to the non-linear electrodynamics of the cavity’s aluminum superconducting film. As
in the case of the optomechanical nonlinearity described in Section 3.1.2, this nonlinearity shifts
the cavity resonance down in frequency and eventually causes the cavity to become bistable. Thisnonlinearity (prior to the bistability) does not prevent cooling measurements, but must be taken
into account when finding the optimal red-detuned frequency. In order to determine this frequency,
we make similar mechanical response measurements to the one in Figure 8.3, but now at constant
incident power (rather than constant photon number) and only over the range of frequencies near
optimal red-detuning (see Figure 8.4(a-b)). This allows us to determine the frequency of maximal
damping (and thus maximal cooling) for each incident power, plotted in Figure 8.4(c).
Once the optimal drive frequency for cooling is established, we measure both the voltage
spectrum without a mechanical drive and the driven mechanical response for each of a series of
red drive powers. The spectrum without the mechanical drive is converted to displacement units
by using the driven mechanical response to reference to the well-calibrated temperature sweep(which was calibrated by measuring the derivative of the output drive voltage as a function of drive

157

Figure 8.4: Calibration of optimal detuning for radiation pressure damping and cooling. Optomechanically induced mechanical resonance frequency (a) and total effective mechanical linewidth (b)as a function of drive frequency for the highest incident powers. The intrinsic (low power) optimal
red-detuned frequency is shown as a vertical dashed line. The incident powers plotted are 1 dBsteps from Pinc = 16 nW (right, green) to Pinc = 100 nW (left). As the power increases, thepoint of maximal damping (and therefore the cavity resonance frequency) shifts lower in frequency.
The cavity linewidth (as inferred from the width of the damping curves) decreases as the powerincreases. The frequency point of maximal damping is chosen as the operating point for the coolingdata. (c) Extracted cavity resonance frequency from the data in (a) and (b), now plotted as thenumber of drive photons in the cavity when optimally detuned.

frequency, see Section 6.2). The displacement spectral density is plotted in Figure 8.5 for several
different drive powers. As the power is increased, the mechanical linewidth increases while the final
occupancy decreases. The imprecision of the measurement also decreases with power for all powers,
indicating that it is not limited by cavity frequency fluctuations as in references [10] and [76].
In Figure 8.6, the mechanical linewidth and final occupancy are plotted as a function of
number of drive photons. The mechanical linewidth increases by a factor of 30 and at high power
becomes linear in number of photons, as expected for radiation-pressure dominated damping. The

158

Figure 8.5: Mechanical spectra for several microwave drive powers. The displacement spectral
density of the mechanical sideband is shown for drive photon numbers between 9×105 and 1.4×108(0.1 < C < 21). The highest four powers are drawn in the same four colors as in Figure 8.4. As
the power is increased, the frequency-independent imprecision decreases. At the same time, themechanical oscillator is damped (the linewidth increases) and cooled (the total area under the
Lorentzian peak decreases - note the logarithmic scale).

Figure 8.6: Summary of cooling data. (a) Total mechanical linewidth and (b) final mechanicaloccupancy as a function of number of drive photons. The mechanical oscillator is cooled from an
initial occupancy of 700 quanta to a final occupancy of 140 quanta.

159
mechanical occupancy is cooled from 700 phonons to 140 ± 7 phonons, a factor of five. The
discrepancy between this factor of 5 and the factor of 30 increase in linewidth can be understoodbecause the large drive powers used may heat the mechanical oscillator thermal bath. This effectis compounded by the fact that we observe the intrinsic mechanical linewidth Γm to increase withtemperature, thus coupling the oscillator more strongly to its warmer bath. The maximum numberof inter-cavity drive photons, and thus the minimum final occupancy, is limited by the non-linearityof the cavity.
This experiment showed that radiation pressure dominated damping and cooling in the resolved sideband regime are feasible in a microwave electromechanical system containing a macroscopic mechanical oscillator. The cooling was limited by the number of allowed drive photons;
thus, an improvement in optomechanical coupling could allow cooling to the ground state. Anothernecessary improvement for observing ground state cooling would be to integrate the system witha nearly shot noise limited measurement capable of discerning the final sub-phonon mechanicaloccupancy. The second half of this chapter presents the exciting results that came from these
improvements.

8.2

Cooling to the quantum ground state
In the previous section, I described our initial experiment with radiation pressure damping

and cooling. The experiment detailed in this chapter improves upon those results significantly
by using a much more strongly coupled optomechanical system, thereby reducing the number ofdrive photons in the cavity required to cool the mechanical oscillator. For this experiment, we
also implemented the nearly shot-noise limited interferometer presented in Chapter 7, enabling
ultrasensitive detection of mechanical motion even as the mechanical oscillator was cooled to its
ground state.

160
8.2.1

Review of radiation pressure cooling theory in the strong-coupling limit

The optomechanical device used in this experiment can reach the strong-coupling regime [15](2g > κ), so it will be generally best to use the expressions for the final number of mechanicalphonons nfm and the cavity output operator spectrum Saout aout [ω] valid in the strong-coupling limit(Equations 3.63 and 3.75):
nfm =

th
2
th
(1 + Γopt /κ)Γm nth(4g 2 + κ2 )Γm nth
m + Γopt nc
m + 4g κnc,=
κ(4g 2 + κΓm )
Γtot

2 th
Saout aout [ω] = |1 − N1 |2 nth
l + |N1 | nr

(8.7)
(8.8)



κl |χc [ω]|2
th
2
th
th
g 2 Γm (nthm − nl )|χm [ω − ωr ]| + κ(nc − nl )|1 + g 2 χc [ω]χm [ω − ωr ]|2thκl (nthl − nr )Re[N1 χ∗c [ω](1 + g 2 χc [ω]χm [ω − ωr ])]+|1 + g 2 χc [ω]χm [ω − ωr ]|22thκl |χc [ω]|2 g 2 Γm nth
m |χm [ω − ωr ]| + κnc
≈
.
(8.9)
|1 + g 2 χc [ω]χm [ω − ωr ]|2+th
In the last step, I have set nth
l = nr = 0, as is found to be the case for this experiment (this will

be discussed below).
In Chapter 4, I found the expression for the intensity (or voltage) spectral density measured byan in-phase, linear, single-quadrature detector (the relevant result for an interferometer containing
a JPA). In units of photon number (as in the right hand side of expressions in Chapter 4), thisspectrum at frequency ω = |ωLO − (ωr + Ωm + δ)| isphoton
SII
[ω] =

≈

1
1
+ (Saout aout [ωLO − ω] + Saout aout [ωLO + ω])
2η 2

2
th
κl |χc [ωr + Ωm + δ]|2 g 2 Γm nth1
m |χm [Ωm + δ]| + κnc+.
2η
2|1 + g 2 χc [ωr + Ωm + δ]χm [Ωm + δ]|2

(8.10)(8.11)The measured spectral density can be related to photon units by comparing with the measured
red-detuned drive power (as explained in Section 6.2):
SV V [ω]2h̄ωr photon
2κl
=
SII
[ω] ≈
S photon [ω].
2
2 II
Z0 Pr
Pout,r
|N2 | nthr Ωm
The measured red drive power is Pr = (2πZ0 )−1R(8.12)SV V [ω]dω, where the integral is across themeasured drive frequency. For low drive powers, when the coupling is weak and the cavity photon

161
occupancy is still negligible compared to the number of thermal phonons, the output spectrum canbe transformed into mechanical displacement units and fit with a simple Lorentzian:
κ2 x2zp photon|N2 |2 κ2 Ω2m SV V [ω]Sxx [ω] ≈ 2 SII
[ω] =
g κl
Z0 Pr
2G2 κ2l


SQL
imp
2
f
4nm + Γtot nm |χm,eff [Ωm + δ]|2 .
= Sxx(8.13)(8.14)

The cavity parameters at each power can easily be extracted from the response of the cavity.
At high power, the number of drive photons in the cavity (required to determine the cavity photon
noise occupation nthc ) can also be extracted from the dressed response. At low power, the dressed
response does not reveal the number of drive photons, but this number can be inferred by comparing
the measured output power to a power where the number is known. Once the cavity parameters
and number of drive photons is known, the expressions in this section allow us to extract the cavity
and mechanical occupancies from measurements of the voltage spectral density.

8.2.2

Device parameters and interferometric setup

In this experiment, we use a heterodyne interferometer including a Josephson parametricamplifier (see Figure 8.7). One microwave generator supplies the optomechanical drive and cancellation tones, both of which are heavily attenuated before reaching Tcryo ∼ 15 mK. The drive
tone interacts with the optomechanical system, picking up sidebands with information about themechanical oscillator. It is then destructively interfered with the cancellation tone, so as not to
overwhelm the JPA. This cancellation is maintained to the part-per-million level by using a feedback
loop to control a voltage variable attenuator and phase shifter in the cancellation arm. Anothermicrowave generator provides the JPA pump tone and the local oscillator to the interferometer.
The JPA pump is 1.3 MHz higher in frequency than the cavity frequency so that the measurement
is heterodyne, and it is also heavily attenuated before reaching the weakly coupled port of the JPA.
The drive-canceled signal coming from the optomechanical system is amplified as it reflects off thestrongly coupled port of the JPA and is subsequently amplified by HEMT and room temperatureamplifiers before being mixed down and detected. A switch connects one of two calibrated noise162

Figure 8.7: Interferometric measurement setup. Two microwave tones are each divided in two parts
to form a heterodyne interferometer. The first tone is used to drive the transmission geometryelectromechanical system and then cancel that drive tone prior to amplification. A filter cavity(purple) is used at the output of this generator to attenuate generator phase and amplitude noise.
The second is used to pump the JPA and provide the local oscillator to the mixer. A switch isused to switch between two terminations, one at Tcryo and one attached to a variable temperature
stage with temperature Tvar . A superconducting coaxial cable (green) is used between temperature
stages to minimize loss. Directional couplers (20 dB) are used at base temperature to minimize
power being dissipated on the cold stage. Cryogenic circulators are used to define the input andoutput ports of the JPA and to isolate the experiment from amplifier and thermal noise at 4K.

sources (50 Ω terminations) to the input drive line. Varying the temperatures of these terminations allows Y-factor measurements of the amplifier added noises and microwave component losses(see Section 6.4). This allows us to determine that the total added noise of the amplifiers in thisexperiment is namps
add ∼ 0.5. Combining this with the estimated 2.5 dB loss between the cavity and
JPA gives a total added noise and interferometer quantum efficiency of nadd = 1.3 and η = 0.38.
As in the experiment in Chapter 7, the introduction of a JPA into our interferometer vastly
improves the quantum efficiency (and thus the signal to noise) of the measurement. The difference
between a measurement with the JPA and one with only a HEMT is perhaps most apparent when
the two inferred displacement spectral densities are plotted on a linear scale (Figure 8.8(a)). Using163
the JPA improves the imprecision of the measurement by more than a factor of 30, reducing the
integration time by a factor of 103 (as, from the Dicke-Radiometer formula, the uncertainty in the√
power P goes as σP = P/ Bτ where B is the measurement bandwidth and τ is the integrationtime).
At high powers in the measurements that follow, we see a nonzero cavity noise occupancy(nthc ∼ 0.3), which is much too large to be accounted for by the cryostat temperature (which would
give nthc = 0.04). In an effort to determine whether this effect was caused by excess generator noise,
we inserted a home-made filter cavity directly after the drive generator. We tune the filter cavity
to maximize transmission at the drive frequency and suppress it by 40 dB at the optomechanicalcavity frequency (see Figure 8.8(b)). The phase and amplitude noise of the generators we use
are specified to be less than -150 dBc at 10 MHz detuning from the drive frequency. The filter
cavity decreases this specification to sufficiently below the the microwave drive tone shot-noise level.
Therefore, we are confident that the observed cavity noise is not a result of microwave generator
noise. Additionally, we do not observe a difference in the cavity noise occupancy between when thecavity filter is present or removed, confirming that even without the filter cavity, generator noise
is not the cause of the cavity occupancy. Moreover, the wide cavity part of the spectral density(ignoring the mechanical sidebands) should take the form
2 th
2 th
Saout aout [ω] = κl |χc [ω]|2 κ0 nth
0 + |N1 − κl χc [ω]| nr + |N2 − κl χc [ω]| nl(8.15)thth
th
where κ0 nth
0 = κnc − κl nl − κr nr and κ0 = κ − κl − κr . The cavity occupancy we see is always apeak, not a dip, and always Lorentzian in form (despite the cavity response taking a Fano form).
Thus, the cavity noise occupancy we see is best modeled as a warm thermal bath coupled to thethcavity, and we set nthr = nl = 0.

The optomechanical device for this experiment is described in Chapter 5 as ‘Device G’. Unlike
the previous experiments using wires as the mechanical oscillator, this device uses the drum-headvibrational mode of a suspended aluminum membrane. The circular membrane has diameter 15 µm
and thickness 100 nm, resulting in a mass of 48 pg. The bare mechanical parameters are found164

Figure 8.8: Interferometer and device calibration. (a) The interferometer quantum efficiency (andthus the measurement imprecision in number of mechanical phonons) improves by a factor of 30
when using the JPA (green) over only using the HEMT (orange). Here, the spectral density of amechanical sideband is shown in both cases using the same drive power and integration time. (b)Filter cavity response. A resonant cavity is introduced following the microwave drive generator
and tuned to be transparent at the drive frequency and attenuating at the cavity frequency (blueshaded region). The generator noise at the cavity frequency is suppressed by 40 dB, removing it asa potential cause of excess optomechanical cavity noise. (c) Optomechanical coupling calibration.
The integrated voltage spectral density SV V [ω] is proportional to g02 nfm ∼ g02 nthm through measurable
proportionality constants and is plotted here as a function of cryostat temperature Tcryo . The slopeof the linear (zero-offset) fit reveals the single-photon optomechanical coupling g0 = 2π × 201 Hz.

by measuring the output spectra of the device at low power and are Ωm = 2π × 10.56 MHz andΓm = 2π × 32 Hz. The factor of 10 increase of resonance frequency over the wire devices isbeneficial, as the number of initial thermal phonons in the mechanical oscillator is then a factor of
10 less. The membrane is suspended 50 nm above another aluminum layer, forming a large parallel
plate capacitor. This capacitor, in parallel with an inductor, leads to a bare microwave resonance165
frequency, linewidth, and external coupling rate ωc = 2π × 7.54 GHz, κ = 2π × 200 kHz, andκext = 2π × 133 kHz. The resolved-sideband factor ν = 4Ω2m /κ2 = 1.1 × 104 puts this device far
into the resolved-sideband regime, where cooling to the ground state is possible. Power is coupledin and out of the microwave resonator via a transmission geometry feed line inductively coupled tothe spiral inductor.
To calibrate the optomechanical coupling, we do a temperature sweep as described in Section
6.3. The voltage spectral density is measured at several different temperatures. This measurement
is done with a very weak red-detuned drive (nd ∼ 3, C ∼ 0.08), where radiation pressure effects are
negligible. The voltage spectral density is converted into displacement units by also measuring thedrive power at each point (as described in Section 6.2 and reviewed in the theory section above).
The resulting occupancies are shown in Figure 8.8(c), where g02 nfm ≈ g02 nth
m is plotted verses cryostat
base temperature. At all temperatures including Tcryo = 15 mK, the mechanical oscillator is inequilibrium with the cryostat temperature, indicating that the initial mechanical occupancy at basetemperature is nthm ∼ 30. We fit the data to a line through zero to deduce the single photon couplingg0 = 2π × 201 Hz. Dividing by the zero point motion results in G = g0 /xzp = 2π × 49 ± 2 MHz/nm.

8.2.3

Damping and cooling the mechanical oscillator to the quantum ground state

To investigate the effects of radiation pressure on the optomechanical system, we infer thecavity and mechanical states as a function of drive power while regulating the cryostat temperature
at Tcryo = 20 mK. At each power, we measure the dressed response of the cavity to determine thecavity resonance frequency ω c , linewidth κ, Fano parameter N2 , and (for high powers only) the
optomechanical coupling g. We then measure the output spectrum near the mechanical sideband
and convert it into units directly related to the optomechanical system (SV V [ω]/(Z0 Pd ) or Sxx [ω]).
This allows extraction of the total mechanical resonance frequency Ωtot and linewidth Γtot , as wellfth
as the mechanical and cavity occupancies nimp
m , nm , and nc .

Figure 8.9 shows the inferred spectral densities from our measurements. For low drive powers
with 18 < nd < 4.5 × 103 (0.5 < C < 113), the spectra can be plotted in units of displacement

166
spectral density and fit to a simple Lorentzian (Figure 8.9(a)). Noting the logarithmic scale, we seethat the imprecision of the measurement decreases with power, the mechanical linewidth increases,
and the total area under the curve (shaded area) decreases. The final occupancy can easily be
SQL
determined as nm = Sxx [Ωm ]/(4Sxx). For larger drive powers with 4.5 × 103 < nd < 1.8 × 105(113 < C < 4500), the cavity occupancy becomes important and the mechanical occupancy can no
SQL
longer be simply determined from Sxx [Ωm ]/(4Sxx). Instead, a correct inference of occupancies

requires measuring the spectrum over a larger frequency range (of order κ rather than Γtot ) and

Figure 8.9: Inferred mechanical spectra in the presence of radiation pressure cooling and damping.
(a) Inferred displacement spectral densities and fits for low powers. As the number of drive photons
is increased, the imprecision (frequency-independent background) decreases, the mechanical damping (the Lorentzian linewidth) increases, and the mechanical temperature (shaded area under thepeak) decreases. (b) Inferred spectra proportional to photon units and fits for high powers. Here,the data is shown over a larger frequency range where the nonzero cavity occupation is visible asa Lorentzian with the cavity linewidth and frequency. The lowest power data is the same as thehighest power data in (a), and the frequency span of (a) is indicated for comparison. As the power
is increased, the mechanical and cavity peaks interfere, eventually normal-mode splitting by g/π.

167
using the full expression for SV V [ω]/(Z0 Pr ) to fit the data and extract nfm and nth
c . As the powerthis increased past C = 4g 2 /(κΓm ) > nthm /nc , the mechanical sideband destructively interferes withthe cavity occupancy (this is called squashing) and eventually, for powers above S = 4g 2 /κ2 > 1/2,
the two harmonic oscillators undergo normal mode splitting.
The optomechanical parameters extracted from the spectra in Figure 8.9 are shown as functions of drive photon number in Figure 8.10. Figure 8.10(a) shows the imprecision in phonon units.
The imprecision initially decreases as 1/nd but then saturates to nimp
= 1.9 as the mechanicalmlinewidth becomes radiation-pressure dominated. Comparing to the expected value for a transmission geometry device, nimp
= κ/(2κext η), gives a value of the quantum efficiency η = 0.39,mwhich is consistent with the expectations from amplifier added noise and cable loss. The imprecision achieved with the JPA is critical to measuring mechanical motion with occupancies near one,3
as without the JPA it would be nimp
m = 70 (a factor of ∼ 10 in integration time, as mentioned
imp
previously). In displacement units, the minimum imprecision is Sxx= 5.5 × 10−4 fm2 /Hz and is

not limited by cavity frequency fluctuations as in [10] and [76]. Figure 8.10(b) shows the various
coupling rates as a function of drive photon number. The optomechanical coupling g increases as√nd (as expected), while the total cavity linewidth κ is only weakly power-dependent. At low

powers, the total mechanical linewidth Γtot is dominated by the intrinsic mechanical damping Γm ,while at high powers, it is dominated by radiation pressure damping Γopt . The black line throughthis data is not a fit, but is rather calculated as Γtot = Γm + 4g02 nd /κ, using the measured value of κat each point. At the highest powers, Γtot becomes comparable to κ, and the cavity and mechanicalmodes hybridize into two normal modes.
Figure 8.10(c) shows the final mechanical and cavity occupancies as a function of drive
photon number. At low powers, the mechanical oscillator is in equilibrium with the cryostat
temperature Tcryo = 20 mK, resulting in nfm = nthm = 40. No cavity noise photons are visible at
low power within our 0.5 quanta measurement uncertainty. As the drive power is increased, themechanical occupation decreases, at first following the weak-coupling expression for final occupancy,
nfm = Γm nth
m /Γtot , which decreases inversely to number of photons once the damping is radiation-

168

Figure 8.10: Results of radiation pressure damping and cooling as a function of drive photon
number. (a) The imprecision number of photons decreases at low power but saturates at highpower to 1.9 quanta, confirming the quantum efficiency of the interferometer as 39%. (b) The√
optomechanical coupling goes as nd , while the cavity damping κ is relatively constant. The totalmechanical linewidth follows the expected theoretical dependence (black line), given the measured
values of κ, Γm , and g0 . (c) The radiation pressure damping in (b) is accompanied by coolingfrom nfm = 40 to 0.34 ± 0.05 phonons (shown in purple and fit with the black line), initializing themechanical oscillator in its ground state. The cooling is ultimately limited by cavity noise photonsnthc (shown in green).

pressure dominated. We observe the mechanical oscillator with an occupation of a single quantaat nd ∼ 4 × 103 photons (C ∼ 100). At higher powers, the onset of normal mode splitting andthe presence of cavity noise photons slows the cooling rate until the final occupancy reaches itsthththeoretical lower limit nfm = Γm nth
m /κ + nc ≈ nc . We observe cooling to a final mechanical169
occupancy of 0.34 ± 0.05 quanta, the first realization of ground-state radiation-pressure cooling ofa macroscopic mechanical oscillator.
This experiment also achieved sensitivity near the Heisenberg limit found in Section 4.5. As
derived there, the imprecision-backaction product for a red-detuned drive issqκ(1 + 2nc )1
imp banm nm =≥ √ ≈ 0.35.
8κl η
8(8.16)

Given the parameters in our experiment, we would expect this product to be approximately
q
ba
ba
nimp
m nm = 1.13. We can measure an upper bound on this quantity by realizing that nm =qq
imp baf + 1/2. Therefore,fnfm + 1/2 − (nth
n
n
≤
nimp
+
1/2)Γ
/Γ
≤
n
m
m (nm + 1/2). For our
m
tot
m
m
m
= 1.9. Substituting these numbers in rehighest power point, we measure nfm = 0.34 and nimpmq√ba
veals an upper limit on the imprecision-backaction product of nimp
m nm ≤ 1.26 = 3.57/ 8. Thismeasured product is the closest experimental approach of the Heisenberg limit to date.
In this experiment, we utilized an electromechanical system capable of reaching the strongcoupling regime (which is shown in much more detail in reference [15]) to achieve ground statecooling of a macroscopic mechanical oscillator. This achievement was a long-standing goal of the
field of optomechanics, as it is a critical first step toward using an optomechanical system forquantum information processing. This achievement opens the door for exciting future experiments,
such as coherently storing microwave information in the mechanical oscillator, entangling photons
and phonons, and coupling optical and microwave cavities via a mechanical oscillator.

Chapter 9

Experiment III: Two-drive sideband asymmetry

In Chapter 8, I described how we used the radiation pressure of microwave photons to damp
and cool a macroscopic, megahertz frequency, mechanical oscillator to a phonon occupancy below
a single quanta. This realization was an extremely important result, as it put the mechanicaloscillator into the regime where quantum effects should be important. However, the measurementof average occupancy less than one did not by itself demonstrate that the mechanical oscillator
behaved quantum mechanically. In fact, even if the system behaved completely classically down
to zero temperature, it would still be possible to measure an average occupancy less than one(or energy less than some number h̄Ωm ). Therefore, the next step is to observe something moresuggestive of the quantum nature of the mechanical oscillator.
In this chapter, I present our efforts to measure the asymmetry between the upper and lower
mechanical sidebands. As I found in Sections 3.1 and 3.2, the upper sideband should be proportional
to the number of mechanical phonons nfm , while the lower sideband should be proportional to that
number plus one (nfm + 1 = nfm + 2nzp
m ). This asymmetry can be understood as a consequence of theexistence of the quantum mechanical ground state or as the result of commutation relations of themechanical noise operators. It is still a semi-classical result; however, because the same effect could
be created without quantum mechanics if, for example, each of the three cavity baths contributed
exactly half a quantum of classical photon noise.
For this experiment, we measure the same suspended membrane device as in Chapter 8(Device ‘G’ in Chapter 5). This device is far in the resolved-sideband regime, so a measurement of

171
the sideband asymmetry is best accomplished by using two drive tones, one approximately optimally
red-detuned (at ωr ≈ ω c − Ωm ) and one approximately optimally blue-detuned (at ωb ≈ ω c + Ωm ).
The drives will be situated so that the upper red and lower blue sideband frequencies are separated
by many mechanical linewidths (|ωb − ωr − 2Ωm |  Γtot ), making the occupancies of the two
sidebands separately measurable. The two-drive scheme has a double benefit over a single drive inthis regime - the radiation pressure control of the mechanical oscillator is maximal, while the readoutof the mechanical sidebands is enhanced by their location near the cavity resonance frequency. The
blue drive will have power large enough to enable readout of the lower mechanical sideband, but
small enough compared to the red drive that the mechanical oscillator can still be damped andcooled to low occupancy.
In theory, this sideband asymmetry measurement should not be much more difficult than thecooling measurement in Chapter 8, as the spectral measurements should be calibrated in the same
way and the theoretical form of Saout aout [ω] for two drives is derived in Section 3.2. However, thisexperiment turns out to be much more difficult than the cooling one, as any calibration errors tend
to have a differential effect on the sideband occupancies, ruining the asymmetry. In particular,
determining the number of photons in the cavity from each drive has proven to be difficult to thepoint of prohibiting a rigorously defensible observation of the asymmetry. In this chapter, I will
discuss our efforts to do this experiment and the complications encountered (and relegate the fulldetails of the calibrations to Appendix H). I will conclude that the asymmetry is visible in our
measurements, provided that we are allowed to adjust the relative photon number calibration in areasonable, but not completely rigorous, way.

9.1

Review of two-drive measurement theory
As this is the same device and very similar measurement to that in Chapter 8.2, I will assumeththat there is no thermal photon measurement port occupancy: nthl = nr = 0. I also specify tosingle-quadrature heterodyne measurement and assume the weak coupling limit, as I will not end
up cooling into the strong coupling regime. However, I will not make the assumption that the

172sidebands are exactly at the cavity resonance frequency or that the cavity is completely Lorentzian(as I did in Section 4.5).
Ignoring the lower red and upper blue sidebands (looking near ω = ω c and thus settingχm [ωr − ω] = χm [ω − ωb ] = 0), the dressed response, the measured spectrum in units of photons,
the cavity output spectrum, and the final mechanical occupancies areκl χc (1 − k̃0∗ χ̃∗c ),(1 − k0 [ω]χc )(1 − k̃0∗ χ̃∗c ) − k2 k̃2∗ χc χ̃∗c
1
1
+ Saout aout [ω],2η 2
κl |χc |2

S21 [ω] = N2 −photonSII[|ωLO − ω|] =Saout aout [ω] =

(9.1)
(9.2)

(9.3)
2
(1 − k0 χc )(1 − k̃0∗ χ̃∗c ) − k2 k̃2∗ χc χ̃∗c
n
2
2
th
∗ 2
∗
2 2
2 ∗
∗ 2
× κnthc 1 − χ̃c (gb χm,r − gr χm,b ) + gr gb κ(nc + 1) |χ̃c | χm,r − χm,b2
2 ∗2
+gb2 Γm nth
m,b |χm,b | 1 − (gb − gr )χ̃c χm,r2
+gr2 Γm nth
m,r |χm,r |nfm,b =
nfm,r =22
1 − (gb2 − gr2 )χ̃∗c χ∗m,b

o

,

th
th
Γm nthm,b − nc Γopt,b + (nc + 1)Γopt,r

,
Γtot
th
th
Γm nthm,r − (nc + 1)Γopt,b + nc Γopt,r
.
Γtot

(9.4)
(9.5)

These expressions are written in terms of several short-hand symbols:
k0 = k0 [ω] = −gr2 χm,r + gb2 χ∗m,b ,k̃0∗ = k0∗ [−ω + ωr + ωb ] = gb2 χm,r − gr2 χ∗m,b ,k2 k̃2∗ = k2 [ω]k2∗ [−ω + ωr + ωb ] = −gr2 gb2 (χm,r − χ∗m,b )2 ,
χm,r = χm [ω − ωr ],thnthm,b = nm + 1,

χ∗m,b = χ∗m [ωb − ω],nfm,b = nfm + 1,

χc = χc [ω],thnthm,r = nm ,

(9.6)
(9.7)

χ̃∗c = χ∗c [−ω + ωr + ωb ],(9.8)

nfm,r = nfm .

(9.9)

photonThe measured spectral density SV V [ω] can be related to the spectrum in photon units SII[ω]through a parameter λ:photonSV V [ω] = λ[ω]SII[ω],Z
2
λ[ωd ]Pout,d1
ndN2
Z0 Pd =SVd Vd [ω]dω == λ[ωd ]− κl .
2π2h̄ωd2κl χc [ωd ](9.10)(9.11)

173
If λ is frequency independent (as was the assumption made in all previous experiments, including
cooling to the ground state), then this unit conversion can be simply made using only the measured
output drive power and cavity parameters. In most experiments this is a reasonable assumption, as
the HEMT amplifier gain is nearly frequency-independent over frequency ranges of order Γtot or κ.
However, in this experiment the two drives are separated by 20 MHz, a frequency difference overwhich the HEMT gain, and thus λ, can differ by several percent1 .

Understanding the frequency

dependence of λ is then critical for measuring sideband asymmetry, as ignoring the frequency
dependence can have a differential effect on the sidebands. The most blatant signature of making
this error is that the two inferred sideband occupancies will disagree at high occupancy (where
both the quantum and classical models predict that they are equal). Measuring the full frequency
dependence of λ is not an easy task, as a response measurement of the HEMT gain and output
cables will necessarily be convolved with any frequency dependence of the input cables. As an easy
approximation, I will therefore assume that the frequency dependence of λ[ω] is linear:λ[ω] =(ωr − ω)λ[ωb ] + (ω − ωb )λ[ωr ].
ωr − ωb(9.13)

Given this assumption and knowledge of the values of λ[ωr ] and λ[ωb ], any measured voltage
spectrum can then be put into photon units and fit to the theoretical expressions reviewed above.

9.2

Calibration of the two-drive experiment
In this section, I will discuss the calibration of the frequency-dependent parameter λ[ω] at

ωr or ωb . Knowing one of these values is the same as knowing the number of photons in the cavity
due the red or blue drive, as these quantities are related only by measurable cavity and drive
parameters. As λ[ω] contains information about the amplifier gains, which can change over time,1
This frequency dependence would also have been present for the cooling experiment of Section 8.2. Ignoring thefrequency dependence of λ leads to inferred quantities that are slightly different than the true values:

η
ηtrue

=

λ[ωc ],
λ[ωr ]g02
2
g0,true=λ[ωr ],λ[ωc ]nthcthnc,true=λ[ωr ],λ[ωc ]

thnfm ≈ nfm,true + 2(nthc − nc,true ).

(9.12)

However, these differences are not large enough to affect the main result of the cooling experiment - namely coolingto a mechanical occupancy of less than one quanta.

174
the assumption that it is constant from day to day is probably not a good one; however, it does
seem to be constant over measurement sweeps taken on timescales of hours. I therefore re-calibrate
λ[ω] for each separate sweep of temperature or power.

9.2.1

Calibration of red-drive voltage to photon unit conversion√
The number of red-drive photons in the cavity (or gr = g0 nr ) is fairly straightforward
to extract (in the same way as was done for the cooling experiment in Section 8.2) from the
dressed cavity response measured with a high-power, red-detuned drive and the blue drive off. Thiscalibration is discussed with some example data in Appendix H.1 and proves to be a straightforward
way to determine gr and λ[ωr ]. Assuming that ωr is held fixed and that λ[ωr ] has not changed since
the calibration, this value (as well as the bare cavity parameters) allows us to go back and forth
between the measured output power and the number of photons due to the red drive, regardless of
the red and blue drive powers.

9.2.2

Calibration of blue-drive voltage to photon unit conversion

The number of blue-drive photons in the cavity proves to be much more difficult to determine
than the number of red-drive photons. The different approaches we have attempted are detailed inAppendix H.2. We conclude there that the two-drive dressed response is not a reliable method of
calibration, as the resulting data does not always match the predicted form from my model, is notalways described by parameters consistent with those extracted from spectral data, and even in thebest cases reveals more scattered values for the number of blue-drive photons than other methods.
We also conclude that using the onset of mechanical parametric oscillation as an indication of equal
red and blue photon numbers is neither adequately repeatable nor accurate. The most consistentmethod found for determining the number of blue-drive photons in the cavity is to measure thespectral sidebands of the mechanical oscillator at high-occupancy (when the net radiation pressure
effects are small). In this case, we expect that the occupancies of the upper red and lower blue
sidebands are approximately equal (nfm,r = nfm,b = nfm ) and that the thermal cavity occupancy

175
affects them negligibly. Thus, even without calibrating the spectral data into optomechanical units,a measurement of the relative sideband amplitudes reveals gr2 /gb2 (and thus combined with the reddrive calibration reveals gb or λ[ωb ]). Unfortunately, the better measurements of asymmetry (twoof which are presented later in this chapter) were all calibrated using the parametric oscillation
method and must be adjusted to achieve reasonable results (this is discussed more in Appendix H
and when these results are presented).

9.3

Interferometric setup and device
The interferometric setup (Figure 9.1) for this experiment is nearly identical to that of theground state cooling experiment, except that there are now two input drives. The JPA pump is
located about 600 kHz above the cavity frequency. As the interferometric setup at low temperatures
has not changed from the ground state cooling experiment, the quantum efficiency is approximatelythη = 0.38, and we assume that nth
l = nr = 0, even without the filter cavity.

The optomechanical device for this experiment is also the same one as for the ground statecooling experiment (Device ‘G’ in Chapter 5). However, the device was thermally cycled between
experiments, so some of the optomechanical parameters have slightly different values. The baremechanical resonance frequency is Ωm = 2π × 10.472 MHz, the cavity resonance frequency is ωc =2π × 7.5887 GHz, and the power dependent damping of the cavity is 2π × 200 kHz < κ < 2π × 500
kHz.
To calibrate the optomechanical coupling, we again do a temperature sweep as describedin Section 6.3. This sweep is done with both a red and a blue-detuned drive, but at low enoughpowers where radiation pressure effects should be small (nr ≈ nb ≈ 17, Cr ≈ Cb ≈ 0.13). The
voltage spectral density of both the upper red and lower blue sidebands are measured four times at
each of several different temperatures. The calibrated values of λ[ωr ] and λ[ωb ] for the temperature
sweep (included in the calibration summary Figure H.6) are used to convert the measured voltage
spectral density into displacement units. The resulting occupancies are shown in Figure 9.2, where
g02 nfm ≈ g02 nth
m is plotted verses cryostat base temperature for each sideband. The frequencies of the176

Figure 9.1: Interferometric measurement setup. We realize a heterodyne interferometer that measures only a single microwave quadrature in the same way as for the ground state cooling experimentin the previous chapter. However, now there are two input drives and two cancellations, one red
detuned from the cavity resonance frequency and one blue detuned. A probe tone can be introduced
into either the input line (for dressed response measurements of the cavity) or the cancellation line
(for measurements of the JPA response and gain).

sidebands are shown relative to the cavity lineshape in the upper left of the figure. In the main
figure, we see that the number of phonons is linear with cryostat temperature down to about 25 mK.
Fitting the upper red sideband points (red triangles) to a line through zero and using equipartition
results in a single-photon optomechanical coupling g0 = 2π × 188.8 Hz (red dashed line). Fitting
only the lower blue sideband points (blue circles) results in g0 = 2π × 185.7 Hz (dashed blue line).
The fact that these values are very close to each other indicates a very good calibration of the λ’s.
From now on, I use the average coupling value g0 = 2π × 187.25 Hz (shown as a black solid line).

177

Figure 9.2: Optomechanical coupling calibration. The integrated voltage spectral density SV V [ω]can be converted to g02 nfm ≈ g02 nthm using measurable proportionality constants. This quantity isplotted here as a function of cryostat temperature Tcryo for the upper red sideband (red triangles)
and lower blue sideband (blue circles). The average slope of the linear (zero-offset) fits reveals the
single-photon optomechanical coupling g0 = 2π × 187.25 Hz.

9.4

Measurements of sideband asymmetry
In this section I present two representative measurements of sideband asymmetry, labeled asdata sets 11 and 14 to be consistent with the numbering in Appendix H. For each data set, the blue
drive generator strength is kept constant while the red drive generator power is increased in stepsof 1 dB, damping and cooling the mechanical oscillator to lower occupancy. The blue generator
drive power is 3 dBm for both data sets, resulting in similar blue photon numbers of order nb ∼ 500(C ∼ 5). The number of red photons ranges from nr ∼ 500 to nr ∼ 104 (5 < C < 200) within eachdata set. The insets of Figure 9.3 show the cavity lineshapes for the lowest (solid line) and highest(dashed line) red drive powers as well as the upper red and lower blue sideband frequencies. The
main difference between these two data sets is that data set 11 has sidebands that are detuned byabout κ/3 above the cavity, while data set 14 has sidebands nearly resonant with the cavity.
The conversion parameter λ[ω] is calibrated for each data set (as explained earlier in this178
chapter and in Appendix H) and used (along with measured bare cavity parameters and output drive
powers) to convert a raw voltage spectral density into photon units. The data is then fit according
to the theory reviewed at the beginning of this chapter to extract the inferred final and imprecision
occupancies for each sideband and the thermal cavity occupancy. These occupancies are plottedfor each data set as a function of the red drive photon number in Figures 9.3(a,b). As expected, thefinal number of mechanical phonons extracted from each sideband decreases with red power. The
cavity occupancy does not behave the same way that it did in the ground state cooling experiment.
In that experiment, the cavity occupancy was negligible at low power and increased with red drive
photons. Here, it seems to be surprisingly large when the two drives are similar in power anddecreases with number of red photons, remaining small compared to the final occupancy of the

Figure 9.3: Measured sideband asymmetry as a function of red-drive photon number for data set
11 (a,c) and data set 14 (b,d). The cavity response and sideband frequencies for each data set
are shown as insets. (a,b) Inferred final mechanical occupancies, imprecision, and thermal cavity
occupancy as a function of number of red-drive photon number. (c,d) Comparison of upper red
and lower blue final sideband occupancies. The lower blue occupancy is scales such that the redand blue occupancies are equal for the lowest power. The red line is the red data plus one.

179
mechanical oscillator even at the highest red powers. In Chapter 4, I found that the imprecision forthe lower red sideband should be nimp
m,r = κ(1 + Cr − Cb )/(2κext Cr η) and limit to κ/(2κext η) at highred power. The blue lower sideband imprecision should be nimp
m,r = κ(1 + Cr − Cb )/(2κext Cb η), which
increases as κnr /(2κext nb η) ∝ nr at high red power. The measured imprecisions agree well withthese expectations. The blue sideband imprecision at high red power is much larger than the final
mechanical occupancy of that sideband. Further loss of signal-to-noise prevents us from measuring
the blue sideband occupancy for larger red drive powers.
Figures 9.3(c,d) show the same results as Figures (a,b), but include only the final mechanicaloccupancies from each sideband, the true quantities of interest for this experiment. As discussed
in Appendix H, I sadly do not trust the relative calibration of λ[ωb ]/λ[ωr ] for either of these data
sets, as the values for both sets were extracted using the parametric oscillation calibration method.
I therefore make the best approximation to a valid high-occupancy calibration by scaling all of the
final blue occupancies by a common factor (one for each data set) such that the highest occupancy
points agree. This means of calibration is not particularly rigorous, as it uses the data set tocalibrate itself. However, it is a reasonable thing to do if we want to understand what asymmetry
we might see if we had an accurate calibration. The inferred final mechanical occupancies areplotted as points and a red line equal to the red data plus 1 has been added as a guide for the eye.
We observe that the sideband asymmetry is visible and agrees well with our expectations.
In conclusion, we have observed a preliminary asymmetry of one between the occupanciesof the upper red and lower blue sidebands using a two-drive scheme, but a rigorous quantitative
determination of this asymmetry is highly dependent on careful, independent calibration of they-axis, which we have not achieved. It could be possible, by combining the high-occupancy spectral
calibration method with an asymmetry sweep using one of the above input drive configurations,
to make a definitive claim of asymmetry. However, the fact that this device is highly power
dependent, is not described by my theoretical model in the presence of two drives, and requires
sensitive calibration of frequency dependent gains makes continuing to pursue this measurement
more an exercise in calibration rather than an exploration of new and interesting physics. It might

180
instead be a wiser choice look for asymmetry with a different device in the un-resolved sideband
regime.

Chapter 10

Experiment IV: Quantum backaction and backaction evasion

In this brief chapter, I discuss an interesting experiment that initially seemed possible with oursuspended membrane device, but exhibited a very surprising limitation in the form of parametricoscillation. I will show only our very preliminary attempts to observe quantum backaction andbackaction evasion and describe a likely explanation for the extremely low power limitation on
backaction evasion due to mechanical parametric oscillation.
As the measurement of a mechanical oscillator becomes more and more sensitive, quantum
mechanics dictates that this measurement necessarily imparts a backaction force onto the mechanical motion. This basic premise of quantum measurement is discussed more fully in the introductionof this dissertation, so I will just give a brief explanation here. The necessity that a strong measurement be accompanied by a backaction force comes about because the position and momentumof the oscillator are non-commuting observables. The Heisenberg uncertainty principle then requires that a measurement of the position with uncertainty ∆xm,meas be accompanied by a random
perturbation of the momentum ∆pm,perturb large enough to satisfy ∆xm,meas ∆pm,perturb ≥ h̄/2. As
the position and momentum are not constants of motion, this uncertainty in momentum evolves
into an uncertainty in position a quarter of an oscillation later, leading to an uncertainty relation
between measurement uncertainty and perturbation of the total mechanical motion. This can bebaexpressed as nimp
m nm ≥ 1/16.

The addition of random backaction motion nba
m due to a strong measurement was seen mathematically in the equations for the final mechanical occupancy in the absence of large radiation

182

Figure 10.1: Mechanical occupancy contributions near the point where backaction motion becomesdominant. (a) Thermal (green), backaction (purple), and final (black) mechanical occupancies in
the case where the sidebands are not overlapping. The case of ideal power-independent mechanicalbath occupancy is shown as solid lines, and the case where the bath temperature rises with power isshown with dashed lines. (b) Contributions to the mechanical occupancy for a backaction-evading(overlapping sidebands) scheme. No backaction motion is visible, so the thermal contribution ismeasured directly.

pressure effects (with a single, on-resonance cavity drive or with two, equal-amplitude, red andblue-detuned drives). In this chapter, I focus on the two drive case, as it is the optimal scheme
for observing quantum backaction with our far-resolved-sideband-regime device. Assuming thetwo drives have equal cooperativity Cr = Cb = C = 4g 2 /(κΓm ), the final mechanical occupancy isththba
nfm = nthm + nm = nm + C(2nc + 1).

The quantum backaction is real mechanical motion (as opposed to the apparent motion that
we call imprecision) and is thus filtered by the mechanical response, making it hard to distinguishfrom the thermal motion of the oscillator. Figure 10.1(a) shows the thermal (green), backaction(purple), and final (black) mechanical occupancies for the two-drive scheme as a function of cooperativity. In order to observe the quantum backaction, the measurement strength must be strong
enough for the backaction contribution to dominate the thermal contribution, causing an nfm ∝ Cdependence. Assuming nc  1, this requires cooperativity C > nthm . Our device can accommodate
drive photon numbers up to nd ≈ 105 (as in [48]), enabling C ≈ 2 × 103 . The thermal occupancy
is only nthm ≈ 40, so the quantum backaction seems like a reasonable effect to observe.

183
One complication to the above picture is that the mechanical oscillator thermal bath often
heats up at large drive powers. This classical backaction that we observe is sub-linear in cooperativity (for example, as shown as dashed lines in Figure 10.1(a)), so the nfm ∝ C dependence of thequantum backaction should still be observable at high enough cooperativity. However, its presence
may increase the cooperativity requirement for observing quantum backaction and may make thetransition between thermal and backaction dominated motion less stark.
A more definitive way to distinguish the classical from quantum backactions is to comparea measurement of quantum backaction to a backaction evasion measurement. In Section 3.2, I
considered the output spectrum when the two drives were of equal strength (gr = gb = g) andthe sidebands were overlapped. In this case, I found that the output spectrum was proportionalf
to nthm , not nm ! Thus, despite backaction still being present on the oscillator, no sign of thatbackaction is present in the measurement. This effect arises because the overlapping-sidebands
scheme measures only one quadrature of the mechanical motion, X̂1 = x̂ cos[Ωm t] − p̂ sin[Ωm t].
Because this quadrature and the orthogonal quadrature X̂2 = x̂ sin[Ωm t] + p̂ cos[Ωm t] do not commute ([X̂1 , X̂2 ] = [x̂, p̂] = jh̄), a measurement of X̂1 is accompanied by the necessary backaction
force on X̂2 . However, unlike x̂ and p̂, in the resolved sideband limit the quadratures X̂1 and X̂2
are constants of motion, meaning that they do not evolve into each other. Thus, a continuous
measurement of X̂1 can be made without contributing extra backaction motion to that quadrature.
The mechanical occupancy contributions measured with a backaction evasion scheme are shown in
Figure 10.1(b). No backaction motion is measurable, so the thermal motion (including any classical backaction heating) is easily determined. Thus, by comparing two measurements, one withthe sidebands separated and one with them overlapping, a clear signature of quantum backaction
should be visible.
Figure 10.2 shows some preliminary measurements we made in attempt to show that comparing results using a non-overlapping-sidebands scheme and a backaction evasion scheme would
be feasible. In each part of the figure, the number of drive photons is fixed and the drive tone detunings are changed, going from well resolved sidebands (purple), to nearly overlapping sidebands

184

Figure 10.2: Parametric instability in the backaction evasion scheme. Each figure shows resolved
sidebands (purple), slightly overlapped sidebands (black), and fully overlapped sidebands (red).
(a) For C ≈ 0.6, the sidebands remain stable when overlapped. (b) For C ≈ 2.3, overlapping the
sidebands causes parametric amplification of the mechanical motion.

(black), to completely overlapping sidebands (red). In Figure 10.2(a), the number of photons is
nr = nb ≈ 28 (C ≈ 0.6). The overlapped sidebands appear to be about twice the height of theresolved sidebands, as expected. Conversely, in Figure 10.2(a), where nr = nb ≈ 115 (C ≈ 2.3),
the sidebands seem well-behaved even as they are slightly overlapped, but the mechanical oscillator
becomes unstable as the sidebands are fully overlapped.
Parametric instabilities during backaction-evasion are not particularly surprising. A parametric instability can arise any time the mechanical spring constant (and thus resonance frequency)
is modulated at 2Ωm and the strength of the modulation of Ωm is larger than the mechanical damping. The total number of drive photons in the cavity (averaged over a cavity oscillation) due to185
two equal-strength drives separated by exactly 2Ωm is
nd [t] ==CV [t]2
C(V0 cos[(ω c + Ωm )t] + V0 cos[(ω c − Ωm )t])2
=
h̄ωc
h̄ωc
4CV02
cos[Ωm t]2 cos[ωc t]2 = nd (1 + cos[2Ωm t]) ,h̄ωc(10.1)(10.2)

where V [t] is the voltage across the circuit capacitance C, and nd without an argument is the usual
time averaged number of drive photons. Thus, if the mechanical spring constant is dependent on
number of photons in any way, it will be modulated at 2Ωm , and a parametric instability can arise.
In fact, we had expected to see such an instability due to the second-order optomechanicalcoupling λ = d2 ωc /dx2 (as observed in [105]). Including the second-order optomechanical coupling,
the displacement-dependent part of the system energy is
1
1
E[x] = ks x2 +
2
2



dC
1 d2 C 2C+x+x V [t]2 .
dx
2 dx2

(10.3)The effective mechanical spring constant (the coefficient of the x2 term) is thenks,eff = ks +1 d2 Ch̄ωc d2 C
2
V
[t]
=
k
+
nd (1 + cos[2Ωm t]),s
2 dx2
2C dx2(10.4)

which is modulated at 2Ωm with amplitude equal to the time-averaged static correction to ks . This
modulation will create parametric oscillation of the mechanical oscillator if the static resonance frequency correction (and therefore the modulation) is larger than Γm . The time-averaged mechanicalresonance frequency (to first order) isrΩm,eff =ks,effks,eff − ksh̄ωc nd d2 Cωc λ
κ 3≈ Ωm 1 += Ωm +=Ω+
Γ
C−.(10.5)m
m
m
2ks
4mΩm C dx2
ωc 4 4G2

For a suspended membrane device, λ/G2 = −1/ωc , the resonant frequency change is Γm Cκ/ωc , andthe cooperativity required for instability is C ≥ ωc /κ. For our device, this instability should occur
at C ≈ 4 × 104 , far above the C ≈ 2 where we observe parametric oscillation.
The cause of the parametric oscillation we do observe is likely an effect that was investigatedin detail in reference [16]. They observed an instability due to microwave power dissipated on thedevice heating the mechanical oscillator and changing its resonance frequency (and thus its spring

186
constant):Ωm,effks,effdΩm dTm dPdissnd [t],dTm dPdiss dnd
dΩm dTm dPdiss≈ ks + 2mΩm
nd (1 + cos[2Ωm t]).
dTm dPdiss dnd= Ωm +(10.6)(10.7)

Here, Tm is the mechanical bath temperature and Pdiss is the power dissipated at the device. We
have not investigated this model in any detail to determine if the onset of our observed instability
quantitatively matches this model; however, we do generally see heating of the mechanical oscillator
with drive power and changes in mechanical resonance frequency with temperature. Additionally,
the onset of instability observed in reference [16] occurred at C = 2.8, a similar onset cooperativity
to that of our observation. Therefore, this effect seems like a good candidate for explaining ourobserved parametric oscillation.

Chapter 11

Experiment V: State transfer between coherent itinerant microwave fields anda mechanical oscillator

The past several experimental chapters have detailed our efforts in pushing both measurement of mechanical motion and that mechanical motion itself into the quantum regime. These
achievements were longstanding goals of the field, as they are critical prerequisites toward using
optomechanical systems as tools in quantum information applications. Mechanical elements haveexciting potential for quantum information, as their lifetimes can be very long, enabling storageof information. Additionally, mechanical oscillators can couple to many different kinds of systems,
enabling transfer of quantum states between systems at vastly different frequencies. In particular,a mechanical oscillator could be used to couple a microwave and optical cavity, joining the quantum state preparation and manipulation strengths of superconducting circuit experiments with thequantum state storage and transmission strengths of the optical world.
The experiment described in this chapter [66] reveals our efforts to temporally control our
optomechanical system, with the goal of transferring a microwave state into the mechanical oscillator, storing it there, and then retrieving it at a later time, a sequence that I analyzed the efficiency
of in Chapter 3.3. We achieve coherent state transfer between itinerant microwave fields and themechanical oscillator, even for itinerant fields with amplitudes of a single quanta. This transfer
occurs on timescales far shorter than the quantum state lifetime of the mechanical oscillator, indicating the possibility of using this state transfer in the future to couple together different quantum
systems or perform more complicated protocols.

188
In this experiment, we control the microwave fields incident on the cavity by applying a weak
preparation field at the cavity resonance frequency. We also control the optomechanical couplingin time by turning on and off a much stronger, optimally red-detuned, transfer field. Most of
this experiment is done with a weak-coupling limit transfer field (2gT  κ). In this case, itinerant
microwave fields leak into and out of the cavity fast enough that the state is never fully contained in
the cavity and the transfer is essentially between itinerant microwaves and the mechanical oscillator.
At the end of the chapter, I will show one measurement with a strong-coupling limit transfer field(2gT  κ) that enables transfer of states directly between the cavity and mechanical oscillator;
however, having the state occupy the cavity for any length of time is undesirable due to the relatively
large coupling of the cavity to external and internal ports. Therefore, the best transfer is achievedin the weak-coupling regime (2gT  κ).
The sequence of transfer and preparation fields for this experiment is illustrated in Figure11.1. An initial preparation pulse is transfered into the mechanical oscillator by applying a constanttransfer field. The transfer field is turned off at time ta , isolating the mechanical oscillator and stateof interest from the cavity. The transfer field is turned back on after a delay time tb , transferring the

Figure 11.1: Sequence of fields for preparation, storage, and retrieval of an itinerant state. 0 < t <ta : The microwave field at ω c is prepared as a rising exponential to be optimally transferred into themechanical oscillator by a constant transfer field at ωr = ω c − Ωm . Any microwaves not capturedby the mechanical oscillator will reflect from the cavity and be measured at ω c . ta < t < ta + tb :Both input microwave fields are turned off, isolating the mechanical oscillator and state of interest
from the cavity and output. ta + tb < t < ta + tb + tc : The transfer field is turned back on, coupling
the mechanical state to the cavity and output environment, allowing read-out of the mechanicalstate.

189
mechanical state back out of the optomechanical system so that we can read it out. Measurementof the output of the system at the cavity frequency reveals the part of the preparation field not
captured by the mechanical oscillator (for 0 < t < ta ) and the final retrieved state (for ta + tb <t < ta + tb + tc ).
The device and interferometer we use for this experiment are essentially the same as for the
ground state cooling and the sideband asymmetry measurements. However, we shorted one end of
the microwave feedline, creating a boundary condition for one port of the cavity that results in thereflection geometry. Most of the device parameters remained the same, but the cavity couplings
for the device in this configuration are κl = κext = 2π × 275 kHz and 320 kHz < κ/(2π) < 360 kHz.

11.1

Review and details of state transfer theory
The most critical requirement for coherent state transfer at quantum levels is that the transfer

time be much faster than the rate of rethermalization from the mechanical bath. The transfer rate
Γl between itinerant microwaves and the mechanical oscillator can be read off from the equationof motion for the oscillator (found in Chapter 3.3, assuming weak coupling, ΓT  κ, where ΓT =2 /κ):
4gT

1ḃ1 [t] + (Γm + ΓT )b1 [t] = je−jφT
2

r

ΓT κl
αin,p [t]
κ

⇒

Γl =

ΓT κl
.
κ

(11.1)The total rate of transfer of itinerant photons to all channels (including the internal cavity bath)is ΓT . The quantum decoherence rate Γqu for the mechanical oscillator to rethermalize by one
phonon can be found by solving the mechanical equation of motion in the absence of coupling to
the cavity (but in the presence of the mechanical bath) and solving for the time required for the190
final occupancy to increase by a quantum:ḃ1 [t] = −pΓmb1 [t] + Γm ξm,1 [t],2

(11.2)Z t

p(11.3)
eΓm t1 /2 ξm,1 [t1 ]dt1 ,b1 [t] = e−Γm t/2 eΓm t0 /2 b1 [t0 ] + Γmt0
DE
nfm [t] =b†1 [t]b1 [t] = e−Γm (t−t0 ) nfm [t0 ] + (1 − e−Γm (t−t0 ) )nth
(11.4)
m,
 
−1
f
nth
m − nm [t0 ]f−1f
nm [t0 + Γqu ] = nm [t0 ] + 1 ⇒ Γqu = Γm log th≈ Γm nth
m . (11.5)
nm − nfm [t0 ] − 1f
In the last line, I have made the approximation that nthm  {nm , 1}. The comparison between thestate transfer rate and quantum decoherence rate is most easily quantified by the cooperativity
2 /(κΓ nth ). When C̃  1, cooling of the mechanical ground state isparameter C̃ = ΓT /Γqu = 4gT
m m

possible, as is preparation of the mechanical oscillator in a coherent state. However, in order tofaithfully transfer more complicated quantum states into the mechanical oscillator, it must also betrue that the rate of transfer is much larger than the rate for a single thermal phonon to enter themechanical oscillator (Γl /Γqu  1). For our experiment, Γqu = 2π × 1.75 kHz. The transfer rateused for most of the experiment is ΓT = 2π × 7 kHz and the maximum possible transfer rate islimited by the onset of strong coupling to ΓT = κ = 2π × 340 kHz. The ratio κl /κ = Γl /ΓT = 0.8.
Thus, both C̃ = ΓT /Γqu  1 and Γl /Γqu  1 are conditions easily met by our experiment. We
therefore assume in the rest of the analysis that the mechanical bath can be ignored during statetransfer (for 0 < t < ta and ta + tb < t < ta + tb + tc ). However, we include and observe the effectof the mechanical bath during the storage time (when the mechanical oscillator is isolated from the−Γm tb nf [t ].
cavity), such that nfm [ta + tb ] = (1 − e−Γm tb )nth
m +e
m a

In Chapter 3.3, I solved the equations of motion in the time domain for each of the fields
in each section of time (assuming weak coupling, ΓT  κ, and no sources of noise). I found thata rising exponential preparation field αin,p [t] = AeΓp (t−ta )/2 is nearly optimally transferred to themechanical oscillator by a constant transfer pulse. Here, I extend this analysis to include sourcesof noise: for 0 < t < ta and ta + tb < t < ta + tb + tc , I include the various sources of photon
noise but continue to treat the mechanical oscillator noiselessly (based on the assumption that themechanical rethermalization rate is smaller than the transfer rate). For ta < t < ta + tb , I include

191
the mechanical bath, as the mechanical oscillator is decoupled from the cavity. The solutions forthe mechanical and output fields in each section of time for the rising exponential preparation andconstant transfer fields are then:√0 ≤ t ≤ ta :

κΓT
,
2

αin,p [t] = AeΓp (t−ta )/2 ,rΓκl
2je−jφTΓp (t−ta )/2(1 − e−(Γ+Γp +Γm )t/2 ),b1 [t] = Ae
Γ + Γ p + Γmκ√
2 κl κi ˆ2κl ˆ
aout,p [t] = 1 −ξl,1 [t] −ξi,1 [t]κκgT [t] =+AeΓp (t−ta )/2
ta ≤ t ≤ ta + tb :1−

(11.6)

2κl (Γp + Γm + Γe−(Γ+Γp +Γm )t/2 )κ(Γ + Γp + Γm )(11.7)!
,

gT [t] = αin,p [t] = 0,b1 [t] = e−Γm (t−ta )/2 b1 [ta ],√
2 κl κi ˆ2κl ˆ
aout,p [t] = 1 −ξl,1 [t] −ξi,1 [t],κκ(11.8)−Γm tb f
nfm [ta + tb ] = (1 − e−Γm tb )nth
nm [ta ]m +e(11.10)(11.9)

4A2 ΓT κl−Γm tb= (1 − e−Γm tb )nth,m +e
(ΓT + Γp + Γm )2 κ√κΓT,
αin,p = 0,
ta + tb ≤ t ≤ ta + tb + tc :gT [t] =2
b1 [t] = e−(Γ+Γm )(t−ta −tb )/2 b1 [ta + tb ],√
2 κl κi ˆ2κl ˆ
aout,p [t] = 1 −ξl,1 [t] −ξi,1 [t]κκrΓκl −(Γ+Γm )(t−ta −tb )/2
jφT
−je
e
b1 [ta + tb ].
κ(11.11)(11.12)(11.13)−1
Assuming that {ΓT , Γp }  {Γm , t−1
a , tc }, the efficiency and leakage of the storage process takethe approximate forms
R ta +tb +tc

E

=

L =

|hαout,p [t]i|2 dt2Γp κlR ta2(ΓT + Γp )κ
0 |αin,p [t]| dt
R ta

2
2
2Γp κl0 |hαout,p [t]i| dt
.
≈ 1−
R ta
(ΓT + Γp )κ
|αin,p [t]|2 dt
ta +tb

ΓT
≈
Γp

2

e−Γm tb ,(11.14)(11.15)0

The efficiency is maximized for ΓT = Γp (note that this is not the point of minimum leakage forκl 6= κ). When this condition is realized, the efficiency E = (κl /κ)2 e−Γm tb is composed of the192mechanical state decay during the storage time and a squared state transfer efficiency ηst = κl /κ
(one factor for the capture and one for the release).
Calculating the efficiency of the transfer process reveals the total energy transfered; however,
we would also like to understand the noise properties of the output state (and thus both the coherent
and thermal (noise) contributions of the inferred mechanical state at time ta + tb ). Taking the
traditional Fourier transform of the time-dependent voltage does not make sense for this experimentgiven the decaying exponential nature of the output fields. However, we can make an analogoustransformation of the output voltage using optimally chosen basis functions:Z ∞V̆1 [ω] =e−ΓT t/2 cos[ωt]V [t + ta + tb ]dt,

(11.16)e−ΓT t/2 sin[ωt]V [t + ta + tb ]dt.

(11.17)0Z ∞V̆2 [ω] =0

In the limit where ΓT → 0, we recover the traditional voltages in the frequency domain:ZV̆1 [ω] =cos[ωt]V [t]dt = Re[V [ω]],(11.18)sin[ωt]V [t]dt = Im[V [ω]],(11.19)V̆1 [ω]2 + V̆2 [ω]2 = SV V [ω].

(11.20)ZV̆2 [ω] =Using the derived expression above for the recovered output field aout,p [t], the relation between√
output and measured fields V [t] = λV photon [t] (see Section 6.2), and the heterodyne singlequadrature interferometer transformations (see Chapter 4), the measured voltage can be written
in terms of the mechanical state quadratures at time ta + tb (ignoring all noise sources for now).
The V̆ ’s at the sideband frequency can then be related to these mechanical quadratures (similarly
SQL
to how we related the output voltage spectral density SV V [ωLO − ωsideband ] to nfm = Sxx [Ωm ]/Sxx

in the steady state experiments):s
1  −jφ2ΓT
X1 [ta + tb ] = − e b1 [ta + tb ] + ejφ b†1 [ta + tb ] =V̆1 [ωLO − ω c ],(11.21)2ληstsj  −jφ2ΓT
jφ †X2 [ta + tb ] =e b1 [ta + tb ] − e b1 [ta + tb ] =V̆2 [ωLO − ω c ],(11.22)
2
ληst
E
2ΓT D
nfm [ta + tb ] = X1 [ta + tb ]2 + X2 [ta + tb ]2 =V̆1 [ωLO − ω c ]2 + V̆2 [ωLO − ω c ]2 (11.23)
.
ληst

193
Here, the mechanical quadratures measured are chosen by φ = φLO − φT + (ωLO − ω c )(ta + tb ) andthe brackets indicate ensemble averages. Including sources of cavity and interferometer noise andcalculating the basis function transformation with frequency ω 6= ωLO − ω c results inE Γ2 nf [t + t ]
1
2ΓT D
a
b
+
.
V̆1 [ωLO − ω c − δ]2 + V̆2 [ωLO − ω c − δ]2 = T m
2
2
ληst
ηηst
ΓT + δ

(11.24)The inferred mechanical state is filtered by a Lorentzian profile with width 2ΓT and rides on an
imprecision background set by the combination of the state transfer efficiency and the interferometer
efficiency. This total efficiency can be found by calculating this quantity at |δ|  ΓT . The totalenergy in the inferred mechanical quadratures (leaving the quadrature argument [ta + tb ] implied
from here on) is thenX12 + X22 = nfm [ta + tb ] +

1
.
ηηst

(11.25)The coherent and noisy contributions to nfm can be distinguished by calculating the relevant statistical quantities:
2
4A2 ΓT ηst−Γm tb A ηst≈e,(ΓT + Γp )2
ΓT
1
1= (1 − e−Γm tb )nth.
= nnoisem +m [ta + tb ] +
ηηst
ηηst

−Γm tbhX1 i2 + hX2 i2 = ncohm [ta + tb ] = e(X1 − hX1 i)2 + (X2 − hX2 i)2(11.26)(11.27)We therefore know how to infer all of the properties of the mechanical state at time ta + tb fromthe voltage measured at the output of the experiment. The only unknown quantities in need ofcalibration are λ, which can be calibrated using a temperature sweep, and ηηst , which can be
determined using the detuned basis functions.

11.2

Coherent state transfer
We first demonstrate coherent state transfer of itinerant microwave states by applying thesequence of microwave fields shown in Figure 11.1 with fixed delay time tb = 25 µs. This time ischosen to be short enough that the total transfer and retrieval efficiency is degraded by less than
1%, but still long enough that any photons left in the cavity during storage should leak out long

194
before readout (in κ−1 = 500 ns). The exponential rising rate of the preparation field is fixed at
Γp = 2π × 6.4 kHz and we vary the strength of the transfer field ΓT . Figure 11.2(a) shows the(normalized) measured voltage V [t]/A as a function of time for four different transfer strengths.
This measurement does not require any calibration, as the voltages are normalized to the input
preparation amplitude A, found by measuring the output voltage of the same preparation tone
detuned away from the cavity. The time axis of all plots in this chapter has been shifted for easeof inspection such that ta = 0. When the transfer field is much weaker than the preparation riserate (ΓT < 2π × 100 Hz llΓp ), the incident fields are badly impedance matched to the mechanicaloscillator and they are simply reflected and measured in the first segment of time. For ΓT =2π × 3.9 kHz ≈ Γp , the systems are well matched and the output field in first time segment
is indistinguishable from that that would be measured if the system were replaced by a 50 Ω
termination. However, when the readout field is applied, the information is recovered, indicatingthat it was stored in the mechanical oscillator. When the transfer field is much larger than the

Figure 11.2: Coherent state transfer between itinerant photons and mechanical phonons. (a) Normalized output voltage measured as a function of time for four transfer strengths, each offset by 2
from the previous trace. Leakage is observed for t < ta when the preparation and transfer fields arepoorly matched. Efficient transfer of the preparation state to the mechanical oscillator and backis observed at ta + tb < t < ta + tb + tc when the preparation rise rate and transfer field strength
are similar. (b) Efficiency E (green) and leakage L (purple) as a function of transfer strength.
The maximum efficiency occurs when ΓT = Γp (grey dashed line). The expected values given
independently determined parameters (black lines) agree well with the data.

195
preparation rise rate (ΓT = 2π × 550 kHz > Γp ), the transfer field is strong enought to transfer theitinerant state to the mechanical oscillator and back to the ininerant field before being capturedby the mechanical oscillator. This once again results in energy being retrieved in the first sectionof time. Figure 11.2(b) shows the measured efficiency E and leakage L as a function of transfer
drive strength. The efficiency is maximum when ΓT = Γp , where it reaches 65%, implying a state√transfer efficiency of ηst = 0.65 = 0.81 ± 0.02. This value agrees well with the cavity decay rates
that we measure independently: ηst = κl /κ = 0.80 ± 0.01. The full ΓT dependence of E and L
agrees well with the theoretical expressions presented above (plotted as black lines with no freeparameters) and only deviates from this where the tranfer strength approaches the strong coupling
regime.

11.3

Calibration of inferred mechanical state
In the above theory section, I detailed the transformations required to infer the coherent,noise, and apparent contributions to the number of mechanical quanta from the measured output
voltage. The only quantities that required calibration were λ and ηηst . These two quantities canbe calibrated using a temperature sweep of the mechanical oscillator in thermal equilibrium andinferences of the mechanical quadratures made with |δ|  ΓT . Figure 11.3(a) shows the inferredmechanical phonon number as a function of δ, the offset frequency of the basis functions used to
optimally extract nfm . The red data shows this quantity when the mechanical state is in equilibriumwith its bath at 25 mK. Even if the y-axis were not calibrated (if λ were not known), the ratioof the height of the Lorentzian peak to the background would give a measure of nthm ηηst . Figuref2
2
11.3(b) shows the measured number of thermal phonons nth
m = nm [ta + tb ] = X1 + X2 − 1/(ηηst )as a function of cryostat temperature. Each point on this graph is derived from 2000 inferencesof the mechanical state (as shown in the inset). The delay time is set to tb = 75 ms  Γ−1m ,such that the mechanical state at ta + tb is in equilibrium with the mechanical bath. The inferredphonon number follows the cryostat temperature linearly between 35 and 80 mK and saturates to
25 mK (50 quanta) at the base temperature of 15 mK. The range of linear dependence allows us

196

Figure 11.3: Calibration of phonon units and apparent mechanical motion. (a) The apparent motion(imprecision) can be calibrated by calculating the inferred mechanical occupancy with the basis
functions detuned by |δ|  ΓT from the frequency of optimal inference. (b) The conversion factor
between measured voltage and inferred mechanical occupancy can be calibrated by measuring themechanical oscillator in equilibrium with its bath as a function of cryostat temperature. (inset)
Each point in the temperature sweep consists of 2000 inferences of the mechanical state.

to determine λ, thereby calibrating all future measurements into phonon units. Once λ is known,
we can return to Figure 11.3(a) and find that the linear background of the 25 mK thermal state
gives (ηηst )−1 = 3.1. This indicates an interferometer efficiency of η = 0.39, consistent with thevalues obtained in the previous experiments with the same interferometer. Also shown in Figure11.3(a) is the inferred thermal phonon contribution for a much lower entropy state (blue data).
= 0.9 and (ηηst )−1 = 3.1, confirming that the
The extracted contributions for this state are nnoisemimprecision of this measurement does not depend on the details of the measured state.

11.4

Phase-coherent state transfer at the single quanta level
We next investigate the noise properties of the inferred mechanical state to demonstratepreservation of the phase and amplitude of the prepared signal, even at single-photon itinerant
input state amplitudes. We also estimate the quantum state lifetime of the mechanical oscillator
by observing its coherent and thermal phonon contributions as a function of storage time.
In Figure 11.4(a), I again show the 2000 inferences of the mechanical quadratures for a 25
mK thermal state (now in phonon units) achieved by setting the delay time to tb = 75 ms. This

197
distribution has no coherent component and obeys a two-dimensional Gaussian distribution. If we
return to a delay time of tb = 25 µs and set the preparation rise rate and transfer strength toΓp = 2π × 6.4 kHz and ΓT = 2π × 7 kHz, respectively, we can further investigate states with acoherent contribution. We make 2000 inferences of the mechanical quadratures for each of four
preparation phases, separated by 90 degree increments (shown as four different colors in Figure

Figure 11.4: Phase-coherent state transfer. (a) Distribution of mechanical state inferences for amechanical oscillator at equilibrium at 25 mK. (b) Distribution of mechanical state inferences givena state prepared with one of four π/2-rotated preparation phases (indicatedp by different colors).
+ 1/(ηηst ) while
The radius of the large circle on the yellow data is the standard deviation nnoisemthe radius of the small circle is the minimum possible standard deviation given η = ηst = 1 andnnoise= 0. (c) Coherent and noise components of the mechanical occupancy as a function of storagemtime. The error bars are standard error of the mean for 500 measurements per point. The blacklines are the theoretical expectations, given Γm = 2π × 35 Hz as an independently determinedparameter. (d) Mechanical state inferences as in (b), but with preparation amplitudes at the single
photon level. The coherent component from 2000 inferences is shown as a colored dot surrounded
by a circle with radius the standard error of the mean. The crosses show the expected values given
measurements of the preparation fields for the known state transfer efficiency (black) and for ideal
state transfer efficiency ηst = 1 (grey).

198
11.4(b)). The phase of the preparation pulse is preserved through the storage and recovery of thestate. The inferred coherent and noise components of the final occupancy are ncoh
m = 28.7 ± 0.8 andnnoise= 0.9 ± 0.1. In the ground state cooling experiment (reference [48]), we found that the finalmmechanical occupancy was nfm = 0.5 for a cooling field with strength Γopt = 2π × 7 kHz. Therefore,
we would expect that the final occupancy after the 25 µs storage would be nfm = 0.8, in reasonableagreement with the observed value. Figure 11.4(b) demonstrates that we can prepare and read out
a mechanical state with less than one phonon of energy in a phase coherent way.
Figure 11.4(c) shows the evolution of the final mechanical state as a function of storage time.
We choose the amplitude of the preparation pulse such that the coherent contribution at time ta is
2
th
ncoh
m [ta ] ≈ A ηst /ΓT = 50 = nm . The evolution of both the coherent and noise components of theoccupancy agree with the theoretical expectations, as ncohm [t] exponentially decays in time at rateth
Γm and nnoisem [t] returns to nm at the same rate. This measurement strengthens our theoreticalth −1 =
expectations that the quantum state lifetime of the mechanical oscillator is Γ−1
qu = (Γm nm )90 µs.
In order to test the preservation of phase and amplitude for transfered microwave states withamplitudes of less than one quanta, we make the same measurements as in Figure 11.4(b), but
decrease the preparation field amplitude to the single (photon) quanta level. In Figure 11.4(d), weplot the average values (coherent component) of 2000 inferences of mechanical quadrature, again
for each of four phases (colored dots). We also plot the expected results from measurements ofthe preparation field and knowledge of the state transfer efficiency (black crosses). The deviationof the measured values is consistent with statistical error and is below 0.1 quanta. Therefore, thismeasurement demonstrates phase-coherent state transfer and storage at the single quanta level.

11.5

State transfer in the strong coupling regime
Lastly, we investigate the maximum achievable transfer rate by increasing the transfer fieldstrength into the strong coupling regime (ΓT > κ). In this regime, the state of interest is fully
contained in the cavity before and after transfer, making the transfer process between cavity pho-

199

Figure 11.5: State transfer between the cavity and mechanical oscillator. (a) Sequence of fields
for preparation, storage, exchange, storage, and retrieval of an itinerant microwave state. The
exchange pulse transfers the state between the mechanical oscillator and the cavity. (b) The
coherent component of the mechanical occupation alternately increases and disappears, indicating
whether the energy ended up in the mechanical oscillator or the cavity at the end of the exchange.
Several exchanges of energy are visible.

tons and the mechanical oscillator rather than between itinerant photons and the oscillator. To
investigate this effect, we do the same (weak coupling limit) preparation and readout operations as
before, but now introduce a strongly coupled, transfer (or exchange) pulse of duration tsc during
the storage segment (as illustrated in Figure 11.5(a)). We wait 100 µs before and after the exchange pulse to let the cavity fields decay away. Thus, the signal measured during the readout only
contains information about the portion of the state that ended up in the mechanical oscillator after
the exchange. The coherent contribution to the mechanical state can be inferred just as for theprevious measurements in this chapter and is shown in Figure 11.5(b). By varying the time tsc , wesee exchanges of energy between the mechanical oscillator and cavity. The solid line is a fit using
the equations of motion for the system and agrees well with the data for ΓT = 2.98κ. Therefore,
this measurement demonstrates that the transfer rate can exceed κ, but that the transfer is thenbetween the cavity and the mechanical oscillator rather than itinerant photons and the oscillator.

200
The maximum rate for itinerant photon to mechanical oscillator transfer is κ.
In this chapter, I have described our efforts to transfer a prepared itinerant microwave state
into the mechanical oscillator, store it there, and then retrieve it at a later time. We have demonstrated that such state transfers can take place on timescales far faster than the quantum decoherence rate of the mechanical oscillator. We have also shown that this process is phase coherent, evenfor input states at the level of a single photon. This achievement already provides an important
tool - namely storage - to the world of microwave quantum information processing. Furthermore,
it is a first step toward transferring quantum states between microwave and optical frequencies via
a mechanical intermediary.

Chapter 12Conclusions and future directions

This dissertation has focused on optomechanical systems, systems where the collective motional state of many atoms is coupled to an electromagnetic field. One of the two primary featuresof such systems is that mechanical motion changes the phase of the electromagnetic field, enabling
sensitive inferences of that mechanical motion to be made. The other feature is that the electromagnetic field exerts a radiation pressure force back on the mechanical oscillator, enabling controlof the mechanical state.
In addition to the general optomechanical theory presented to describe many of the interesting effects seen in optomechanical systems, this dissertation describes several of our experimental
achievements using an electromechanical system. At the outset of my PhD work in 2007, our group
had shown a proof of principle demonstration of electromechanical coupling between a nanomechanical wire oscillator and a microwave resonator [10]. Since then, we have improved on theultrasensitive detection realized by our electromechanical system by realizing a quantum efficient
microwave interferometer (Chapter 7, [76]). This interferometer is applicable not only for electromechanical experiments but also for any experiment that imprints information of interest onto
the phase of a microwave field. Quantum efficient measurement allowed us to detect the motionof a mechanical oscillator with imprecision below that at the standard quantum limit and infer aforce sensitivity of SF F = 0.26 aN2 /Hz.
We have also made many improvements toward control of the mechanical state using theradiation pressure force of the microwave field. We showed initial radiation pressure cooling to 140

202
phonons [45] using a wire mechanical oscillator. We then improved upon the microwave interferometer and device employed to cool a suspended membrane mechanical oscillator to 0.34 phonons[48], the first achievement of radiation pressure cooling of a macroscopic mechanical oscillator to
the ground state. We further demonstrated coherent state transfer between a mechanical oscillator
and itinerant microwave fields, even at single photon signal amplitudes [66].
All of these improvements (and similar results in many of the other experiments in the field)
have set the stage for investigations into optomechanical systems as tools for quantum information
and basic tests of quantum physics. The immediate next step for our electromechanical experimentwill be to create entangled states of photons and phonons, as discussed theoretically in [106].
Another parallel effort will be to devise an optomechanical device capable of simultaneously coupling
to both microwave and optical fields [67]. This would allow quantum information to pass between
these two different frequency regimes, making it possible to utilize the storage and long-distancetransmission capabilities at optical frequencies while also taking advantage of the exquisite state
preparation and manipulation available with superconducting qubits at microwave frequencies. We
are also interested in making steps toward the realization of a system capable of exploring theintersection of quantum mechanics and gravitational theory, as proposed in [70, 71, 72, 73].

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Appendix A

Symbol definitions√
j = − −1 = −iimaginary constantδi,jKronecker delta function

δ[t − t0 ]Dirac delta function

Sxy [ω]κ=power spectral density,

R∞

−jωt
−∞ e

x̂† [0]ŷ[t] dt

ωccavity resonance frequency

κicavity loss rate to port i = {l, r, 0}

P

i κi

total cavity loss rateNicomplex coefficient for port i = {1, 2}

â, âin , âoutannihilation operators for cavity and itinerant photons

th th thnthl , nr , n0 , nc

left, right, internal, and total thermal cavity photon numbersndnumber of cavity photons due to coherent drivemmechanical oscillator mass213
Γm , Γopt , Γtot

intrinsic, optomechanically induced, total mechanical linewidthsΩm , Ωopt , Ωtot

intrinsic, optomechanically induced, total mechanical frequenciesb̂phonon annihilation operatorx̂ = xzp (b̂ + b̂† )mechanical position operator

xzp =

p
h̄/(2mΩm )zero point mechanical motion

imp bafnthm , nm , nm , nm

thermal, final, imprecision, and backaction phonon numbers

ξˆinoise operator for bath i = {l, r, 0, m}

G = dωc /dx

cavity frequency coupling to mechanical displacement

g0 = Gxzp

single photon optomechanical coupling√g = g0 nd

total optomechanical couplingωdcavity drive frequency

ωpprobe frequency in response measurements

ΓTstate transfer rate in pulsed experiments

Γppreparation state rise rate in pulsed experimentsηinterferometer efficiency

χc [ω]

cavity susceptibility, [κ/2 + j(ω − ωc )]−1

χm [ω]

bare mechanical susceptibility, [Γm /2 + j(ω − Ωm )]−1

χm,eff [ω]effective mechanical susceptibility, [Γtot /2 + j(ω − Ωtot )]−1

Appendix B

Conventions

B.1

Fourier transform conventions
Most of the time, I use the following Fourier Transform conventions:Z ∞f [ω] =e−jωt f [t]dt

(B.1)

Z−∞
∞

1
ejωt f [ω]dω
2π −∞
Z ∞
†
f [ω] =e−jωt f † [t]dt.

f [t] =

(B.2)
(B.3)

−∞

√
Here, I use j = − −1 = −i. Many physicists use the definitions with −i substituted for j. It
does not matter so long as the definition is consistent throughout a calculation. However, theabove convention for imaginary numbers is already chosen for us when we use circuit notation for
impedances (say, of an inductor): Zinductor = jωL. This is because the impedance notation comes
directly from the Fourier transform of the time-dependent circuit equation relating the voltage V [t]to the current I[t]:dI[t]V [t] = L
Z ∞dt
Z ∞
dI[t]
−jωt
V [ω] =eV [t]dt = L
e−jωt
dt
dt
−∞
−∞


Z ∞
∞
= L e−jωt I[t] −∞ + jω
e−jωt I[t]dt = (jωL)I[ω]

(B.4)
(B.5)
(B.6)

−∞

I will use the above Fourier transform definitions for all cases except when calculating spectral
densities and operator expectation values. While it is still possible to use regular Fourier transforms

215
for spectral densities (the math will still work out), the expressions are less physically intuitive
because they involve carrying around Dirac delta functions and canceling them at the end. Instead,
I use the more physical windowed Fourier transforms:
Z T /2
1
f [ω] = lim √e−jωt f [t]dt
T →∞T −T /2√ Z T /2Tejωt f [ω]dωf [t] = lim
T →∞ 2π −T /2
Z T /2
1†
f [ω] = lim √e−jωt f † [t]dt
T →∞T −T /2

B.2

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

Operator expectation values and spectral densities
I have defined all of the noise operators to have the following expectation values:DEξˆi† [t2 ]ξˆj [t1 ]= nthi δi,j δ[t2 − t1 ]DEξˆi [t2 ]ξˆj† [t1 ]=nthi + 1 δi,j δ[t2 − t1 ](B.10)(B.11)

where the deltas with subscripts are Kronecker deltas and those with square brackets are Diracdeltas.

B.2.1

Regular Fourier transforms

As I said in the previous section, it is more natural here to use windowed Fourier transforms.
However, here I’ll present the way to use regular Fourier transforms so that the differences areclear. Note that this is not what I use throughout my dissertation. The expectation values in thefrequency domain using regular Fourier transforms are then:
Z ∞Z ∞
D
E
D
E
†
ˆ
ˆ
ξi [ω2 ]ξj [ω1 ]=e−jω2 t2 e−jω1 t1 ξˆi† [t2 ]ξˆj [t1 ] dt2 dt1
−∞
Z−∞
∞ Z ∞
=
e−jω2 t2 e−jω1 t1 nthi δij δ[t2 − t1 ]dt2 dt1
−∞ −∞
Z ∞
th
= ni δij
e−j(ω1 +ω2 )t2 dt2(B.12)(B.13)(B.14)−∞= 2πnthi δi,j δ[ω1 + ω2 ]DE
ξˆi [ω2 ]ξˆj† [ω1 ]= 2π(nthi + 1)δi,j δ[ω1 + ω2 ](B.15)(B.16)

216
Very similarly, the expectation values of the mechanical and output operators reveal expressions for their respective spectral densities:
Z ∞Z ∞
DE
DE†b̂ [ω2 ]b̂[ω1 ]=e−jω2 t2 e−jω1 t1 b̂† [t2 ]b̂[t1 ] dt2 dt1
−∞
Z−∞
D
E
∞ Z ∞
e−jω2 t2 e−jω1 (t2 +τ ) b̂† [t2 ]b̂[t2 + τ ] dt2 dτ
=
−∞
Z−∞
D
E
∞ Z ∞
e−jω2 t2 e−jω1 (t2 +τ ) b̂† [0]b̂[τ ] dt2 dτ
=
−∞
Z ∞
Z−∞
D
E
∞
−j(ω1 +ω2 )t2
e−jω1 τ b̂† [0]b̂[τ ] dτedt2=

(B.17)(B.18)(B.19)(B.20)−∞−∞= 2πδ[ω1 + ω2 ]Sbb [ω1 ]E
1 D †b̂ [−ω]b̂[ω]Sbb [ω] =2πδ[0]E
1 D †Saout aout [ω] =âout [−ω]âout [ω] .
2πδ[0](B.21)(B.22)(B.23)In the third step, I used the fact that the autocorrelation function Gxy [τ ] = x̂† [t2 ]ŷ[t2 + τ ] isinvariant to time-translation of t2 . I also used the definition of the spectral density:
Z ∞Sxx [ω] =

e

−jωt

Z ∞
Gxx [t]dt =−∞DEe−jωt x̂† [0]x̂[t] dt(B.24)−∞As stated previously, the expressions derived in this section will work (the operator expectation values will have Dirac deltas of 0 to cancel the ones in the denominators); however, using
windowed Fourier transforms leads to much cleaner and more physical expressions for the spectral
densities.

B.2.2

Windowed Fourier transforms

Windowed Fourier transforms can be used to calculate the expectation values and give nearly
identical results to regular Fourier transforms. The one benefit is that the pesky Dirac delta

217
functions found with regular Fourier transforms end up as Kronecker delta functions here:DEξˆi† [ω2 ]ξˆj [ω1 ]=1lim
T →∞ T=1
T →∞ TZ T /2 Z T /2
−T /2 −T /2
Z T /2 Z T /2

lim

−T /2

1

−T /2
Z T /2

T →∞ T

−T /2

= nth
i δi,j lim

DE
e−jω2 t2 e−jω1 t1 ξˆi† [t2 ]ξˆj [t1 ] dt2 dt1(B.25)e−jω2 t2 e−jω1 t1 nthi δi,j δ[t2 − t1 ]dt2 dt1(B.26)e−j(ω1 +ω2 )t2 dt2(B.27)= nthi δi,j δω2 ,−ω1

(B.28)

DE(B.29)
ξˆi [ω2 ]ξˆj† [ω1 ]= (nthi + 1)δi,j δω2 ,−ω1
Z
Z
D
E
E
1 T /2 T /2 −jω2 t2 −jω1 t1 D †b̂† [ω2 ]b̂[ω1 ]
= lim
e
e
b̂ [t2 ]b̂[t1 ] dt2 dt1(B.30)
T →∞ T −T /2 −T /2ZZE
1 T /2 T /2−t2 −jω2 t2 −jω1 (t2 +τ ) D †
= lim
e
e
b̂ [t2 ]b̂[t2 + τ ] dt2 dτ (B.31)
T →∞ T −T /2 −T /2−t2ZZE
1 T /2 ∞ −jω2 t2 −jω1 (t2 +τ ) D †eeb̂ [0]b̂[τ ] dt2 dτ
(B.32)= lim
T →∞ T −T /2 −∞
Z
Z ∞
D
E
1 T /2 −j(ω1 +ω2 )t2= lim(B.33)edt2
e−jω1 τ b̂† [0]b̂[τ ] dτT →∞ T −T /2−∞= δω2 ,−ω1 Sbb [ω1 ]DESbb [ω] =b̂† [−ω]b̂[ω]DE
Saout aout [ω] =â†out [−ω]âout [ω] .

(B.34)(B.35)(B.36)

Using windowed Fourier transforms instead of the regular versions will not change the final outcomeof a calculation in any way (assuming one or the other is used consistently). However, getting ridof the canceling Dirac deltas makes using windowed Fourier transforms a bit more physical.

B.3

Convolutions
The definition of a convolution isf [ω] ∗ g[ω] ≡

1
2π

Z ∞
−∞

f [ω 0 ]g[ω − ω 0 ]dω 0 .

(B.37)

218
Thus, the product of two functions in the time domain is the convolution of those two functions in
the frequency domain:
Z ∞

−jωt

f [t]g[t]e

Z ∞
dt =

−∞

=
=
=

 Z ∞
Z ∞1
1
0 jω 0 t
000 jω 00 t
00
f [ω ]e dωg[ω ]edω e−jωt dt(B.38)
2π −∞−∞ 2π −∞
Z ∞
Z ∞
Z ∞
1
0
00
0
00
00
0
ej(ω +ω −ω)t dt(B.39)f [ω ]g[ω ]dωdω
2(2π) −∞
−∞
−∞
Z ∞
Z ∞
1
0
f [ω 0 ]g[ω 00 ]δ[ω 0 + ω 00 − ω]dω 00(B.40)
dω
2π −∞
−∞
Z ∞
1
f [ω 0 ]g[ω − ω 0 ]dω 0(B.41)
2π −∞= f [ω] ∗ g[ω](B.42)

If one of those functions is a complex conjugate, we getZ ∞−∞f [t]g ∗ [t]e−jωt dt = f [ω] ∗ g ∗ [−ω].

(B.43)

Appendix C

Discussion of different Fano resonance models

Different strategies have been employed to understand response data that displays a Fano
resonance attributed to wire bonds. One, presented in [74], attempts to extract the resonant
frequency and linewidth of the cavity that would have been found had there not been reactive
components in the feed line. This is a sensible strategy if the goal is to characterize the bareresonator - for instance if you wanted to study many resonators as a function of some fabrication
parameter, such as dielectric thickness. The other strategy, the one that I develop in Section
2.1.3, attempts to find the resonant frequency and linewidth of the cavity that now characterize its
behavior. I want to be able to infer the state of a mechanical oscillator from the cavity readout,
so it is important that I know the cavity parameters that the mechanical oscillator ‘sees’. Thus, I
want the extracted parameters to correspond to the response of the voltage Ṽ across the capacitor
C, and I do not really care what the cavity parameters would have been had there not been reactive
components in the feed line.
If I re-write the cavity response calculated in reference [74] in my own variables (and set thereactances in different ports equal), I find:S21 =1
κext /2−.
1 + jB κ/2 + j (ω − ω0 + κext B/2)

(C.1)

This is identical to Equation 2.39, but with ω0 called the resonance frequency rather than ωc . If
I use both equations to fit the Fano resonance shown in Figure C(a), I get the exact same fit line
and parameters other than resonance frequency; however, Equation 2.39 results in a resonancefrequency of 5.22058 GHz, while Equation C.1 results in a resonance frequency of 5.22072 GHz.

(a)

220

(b)

0.14
0.12
0.10
2
S 21 0.08
0.06
0.04
0.02
5.220090Γ m (Hz)

80
70
60
50
5.21865.2204 5.2208 5.2212
Frequency (GHz)5.21905.2194
Drive Frequency (GHz)

Figure C.1: Fano resonance fits. (a) Response and fits from a resonant circuit in the transmission
geometry, displaying a Fano response. The data (red) is fit using Equation C.1 (black) and Equation
2.39 (teal, dashed). (b) Mechanical linewidth as a function of microwave drive frequency, alongwith a fit to Equation C.2.

To see which resonance frequency interacts with the mechanics, I show data of the mechanicallinewidth as a function of red-detuned microwave drive frequency, measured on the same device
at a similar power (Figure C(b)). What we see is a Lorentzian lineshape, demonstrating that themechanical oscillator ‘sees’ a Lorentzian voltage response, even if the cavity response is Fano. The
mechanical resonance frequency for this device is 1.5249 MHz, putting it in the resolved sideband
limit. Thus, the theoretical expression for the mechanical linewidth with an approximately reddetuned drive isΓtot = Γm +4g 2 κ
.
κ2 + 4(ωd − ωc + Ωm )2(C.2)

From this I can extract a cavity frequency of 5.22055 GHz, consistent with the value extracted fromfitting the cavity response with Equation 2.39.

Appendix D

Component responses for interferometric detection

D.1

Beam splitters and photodetectors as linear detectors
In Chapter 4, I discuss heterodyne and homodyne interferometric detection. I model thesemeasurement schemes in the optical domain, as beam splitters and photodetector are easily modeled
and understood. However, the physics behind interferometric detection is identical in the opticaland microwave regimes, so all the results obtained are directly relevant to our experiments. In
this section, I introduce the beam splitter and photodetector transformations and show how theseelements can be combined to make either a single or two-quadrature linear detector.
ˆ that are linear
A simple beam splitter (Figure D.1(a)) creates two output fields, ĉ and d,
combinations of the input fields â and b̂. For a beam splitter with transmission coefficient η, thefields are related by√p
1 − ηâ[t],

(D.1)

p
√
ˆ
d[t]
=
ηâ[t] + j 1 − η b̂[t].

(D.2)

ĉ[t] =η b̂[t] + j

A simple photodetector (Figure D.1(b)) measures the intensity Iˆ due to the incoming field

222

Figure D.1: Measurement models in the optical regime. (a) A beam splitter with transmission
ˆ (b) An opticalη transforms two input signals â and b̂ into two linear combinations ĉ and d.
ˆphotodetector detects the intensity I of an incident field â. (c) Single quadrature linear field
detector. A linear measurement of field dˆin can be made by combining it with a strong coherent
local oscillator dˆLO and detecting the intensity Iˆ of the output field dˆout . (d) Two quadrature linear
field detector. Both quadratures of the input field can be detected by first splitting the signal inhalf and then performing a linear measurement of each piece. The quadrature component is foundby using a local oscillator that is ninety degrees out of phase. Additional noise ξˆvac must be addedin this process through the vacuum port of the initial beam splitter.

ˆ = â† [t]â[t]. The average value of this intensity isâ: I[t]DE
DE Z ∞DE†ˆI[t]=â [t]â[t] =â† [t]â[t + τ ] δ[τ ]dτ
−∞

Z ∞D
E 1 Z ∞â† [0]â[τ ]=e−jωτ dω dτ
2π −∞−∞Z ∞ Z ∞
Z ∞
D
E 
1
1
†
−jωτ
=
â [0]â[τ ] dτ dω =eSaa [ω]dω,
2π −∞
2π −∞
−∞

(D.3)
(D.4)
(D.5)

where, as in Appendix B, I have used the fact that the autocorrelation function x̂† [t2 ]ŷ[t2 + τ ]is invariant to time-translation of t2 . The result of these equations is that the average value ofthe intensity is the integral of the number spectral density, which is just the total photon number
incident on the detector, as expected.
A linear measurement of a signal of interest dˆin [t] can be performed by combining that signal
on a beam splitter of transmission ηLO with a local oscillator dˆLO [t] = ALO ejφLO ejωLO t + ξˆLO [t](see Figure D.1(c)) and then detecting the output on a photodetector. The field incident on the223
photodetector will be√dˆout [t] = ηLO dˆin [t] + jp
1 − ηLO (ALO ejφLO ejωLO t + ξˆLO [t]).

(D.6)

Any noise ξˆLO [t] accompanying the strong coherent LO field will add to the signal of interest unlessηLO  1 − ηLO . Making this assumption, and the assumption that the local oscillator power isDEmuch larger than the signal power (A2LO (1 − ηLO )  dˆ†in [t]dˆin [t] ηLO ), the detected intensity isˆI[t]= dˆ†out [t]dˆout [t]≈ A2LO (1 − ηLO ) + jALO

(D.7)


p
ηLO (1 − ηLO ) ejφLO ejωLO t dˆ†in [t] − e−jφLO e−jωLO t dˆin [t] (, D.8)

ˆI[ω]= A2LO (1 − ηLO )2πδ[ω]

(D.9)


p
+jALO ηLO (1 − ηLO ) ejφLO dˆ†in [ω − ωLO ] − e−jφLO dˆin [ω + ωLO ] .


The average value of the intensity is

D

E
ˆ
I[t]
= A2LO (1 − ηLO ), which is just the number of co-herent photons. The AC part of the intensity can be re-written in terms of two orthogonal field
quadratures, X̂ and Ŷ :
ˆ + ωLO ] + dˆ† [ω − ωLO ],X̂[ω] = X̂ † [ω] = d[ω(D.10)

ˆ + ωLO ] − dˆ† [ω − ωLO ]),Ŷ [ω] = Ŷ † [ω] = −j(d[ωpˆ = Iˆ† [ω] = ALO ηLO (1 − ηLO ) − sin[φLO ]X̂[ω] + cos[φLO ]Ŷ [ω] .
I[ω](D.11)(D.12)

This expression illustrates that this detection scheme can only detect a single quadrature at a time.
Which quadrature it detects is determined by the local oscillator phase.
Returning to the intensity in terms of fields, the spectral density of the intensity (ignoring
the part at DC and referencing back to the input state to put the spectrum in photon units) is:DEˆ† [−ω]I[ω]ˆI
SII [ω]photonSII[ω] ==(D.13)
2A2LO ηLO (1 − ηLO )2A2LO ηLO (1 − ηLO )E D
E
1 nD ˆ
=din [−ω + ωLO ]dˆ†in [ω − ωLO ] + dˆ†in [−ω − ωLO ]dˆin [ω + ωLO ](D.14)
2D
E
D
Eo−e2jφLO dˆ†in [−ω − ωLO ]dˆ†in [ω − ωLO ] − e−2jφLO dˆin [−ω + ωLO ]dˆin [ω + ωLO ]=

1 1+ (Sdin din [ωLO − ω] + Sdin din [ωLO + ω])(D.15)
2 2
D
E
D
E
1
− e2jφLO dˆ†in [−ω − ωLO ]dˆ†in [ω − ωLO ] + e−2jφLO dˆin [−ω + ωLO ]dˆin [ω + ωLO ]2

224
There is half a quanta of overall added noise, two spectral terms now converted to low frequencies
and folded on top of each other, and two extra terms which for certain conditions may be nonzero
and cause interference.
As I showed above, the detection scheme in Figure D.1(c) only measures one quadrature of
the field of interest. In order to measure both quadratures, the signal can be split in half on abeam splitter and then each half can be sent to a single-quadrature linear detector, one of which
has a local oscillator ninety degrees out of phase from the other (Figure D.1(d)). This allows
detection of both the in-phase and quadrature components of the field; however, it comes at thecost of additional added noise that comes in through the vacuum port of the first beam splitter.
This extra noise is in fact required of any measurement of both quadratures of the field due to thenon-commutativity of the field quadratures. The intensities measured for each quadrature of thetwo-quadrature measurement are
ALOIˆI [ω] = j √2=IˆQ [ω] =

=



n
p
†
[ω − ωLO ](D.16)
ηLO (1 − ηLO ) ejφLO dˆ†in [ω − ωLO ] − j ξˆvaco−e−jφLO dˆin [ω + ωLO ] + j ξˆvac [ω + ωLO ]


ALO p
√
ηLO (1 − ηLO ) sin[φLO ](−X̂[ω] + Ŷξ [ω]) + cos[φLO ](Ŷ [ω] + X̂ξ [ω]) ,(D.17)
2
n


ALO p
†
√
ηLO (1 − ηLO ) ejφLO dˆ†in [ω − ωLO ] + j ξˆvac[ω − ωLO ](D.18)2o+e−jφLO dˆin [ω + ωLO ] − j ξˆvac [ω + ωLO ]


ALO p
√
ηLO (1 − ηLO ) sin[φLO ](Ŷ [ω] − X̂ξ [ω]) + cos[φLO ](X̂[ω] + Ŷξ [ω]) . (D.19)
2

Here, X̂ξ and Ŷξ are quadratures of the vacuum noise field. Ignoring the noise terms, the in-phase detector measures (− sin[φLO ]X̂[ω]+cos[φLO ]Ŷ [ω]), the same field quadrature as the single-quadraturedetector above. The quadrature detector measures the orthogonal combination, (sin[φLO ]Ŷ [ω] +cos[φLO ]X̂[ω]). The intensity spectral densities from two-quadrature measurement (returning to225field notation) areDphotonSI(Q)I(Q)
[ω] =

E
†
IˆI(Q)[−ω]IˆI(Q) [ω]A2LO ηLO (1 − ηLO )1= 1 + (Sdin din [ωLO − ω] + Sdin din [ωLO + ω])2DE1∓ e2jφLO dˆ†in [−ω − ωLO ]dˆ†in [ω − ωLO ]2DE
+e−2jφLO dˆin [−ω + ωLO ]dˆin [ω + ωLO ] .

(D.20)(D.21)The negative sign is for the in-phase intensity, while the positive sign is for the quadrature intensity.
The in-phase part is identical to what I found for the single-quadrature measurement, except that
there is now an extra half quantum of added noise. For some cases, the last two terms will benonzero and can interfere maximally and oppositely for the two quadratures. However, for cases
where these terms are zero, the information is split equally between the two quadratures. In thiscase, it is optimal to measure the spectral density of a combination of the in-phase and quadrature √intensities: Iˆθ = ejθ IˆI + e−jθ IˆQ / 2. The spectral density in this case isEDIˆθ† [−ω]Iˆθ [ω]SIphoton[ω] =(D.22)θ Iθ
A2LO ηLO (1 − ηLO )1 − sin[2θ]
1 + sin[2θ]= 1+
Sdin din [ωLO − ω] +Sdin din [ωLO + ω].
(D.23)
2
2
A choice of θ essentially filters away frequencies above or below the local oscillator.

D.2

Linear, phase-insensitive amplifiers
In this section, I look at the kind of linear amplifiers most frequently employed in microwave

experiments - those whose amplification is not sensitive to the phase of the signal being amplified.
The most state-of-the-art phase insensitive microwave amplifiers are High Electron Mobility Transistor (HEMT) amplifiers. I will show that a phase insensitive amplifier must necessarily add noise
to the signal it is amplifying. A very similar but more extensive discussion can be found in [107,Chapter 2.4].
A phase insensitive amplifier must amplify all quadratures of the microwave field by the sameamount. However, the output fields ĉout cannot simply be an amplified version of the input fields

226ĉin . If they were, the field commutation rules would be violated:hiĉout [ω1 ], ĉ†out [ω2 ] = δω1 ,−ω2 ,h√i√Gĉin [ω1 ], Gĉ†in [ω2 ] = Gδω1 ,−ω2 ,ĉout [ω] 6=(D.24)(D.25)√
Gĉin [ω].

(D.26)In order for the commutation relations to be preserved, noise due to the internal modes of theamplifier must be added to the amplified input field:√√†ĉout [ω] =Gĉin [ω] + j G − 1ξˆamp
[ω],
h
i
h√
i
√
√
√
†
ĉout [ω], ĉ†out [ω] =Gĉin [ω] + j G − 1ξˆamp[ω], Gĉ†in [ω] − j G − 1ξˆamp [ω]ihhi= G ĉin [ω], ĉ† [ω] + (G − 1) ξˆ† [ω], ξˆamp [ω]inamp= G − (G − 1) = 1.

(D.27)(D.28)(D.29)(D.30)The effective output signal (normalized by the gain G) isp†ĉout,eff [ω] = ĉin [ω] + j 1 − G−1 ξˆamp
[ω],
i
h
1
ĉout,eff [ω], ĉ†out,eff [ω] = .
G(D.31)(D.32)For sufficiently large gain, the commutation of the effective microwave fields will be zero, indicatingthat the fields can be thought of classically.

D.3

Josephson parametric amplifiers (linear, phase-sensitive amplifiers)
In this section, I solve the equations of motion for the Josephson Parametric Amplifier (JPA)

and arrive at the transformation between incident signal fields and their amplified outgoing counterparts. Much of this analysis follows [108] (but retains my definitions of Fourier transforms andstays in the lab frame).

D.3.1

Equation of motion and solution for the JPA

I model the JPA as a cavity resonator with a Kerr nonlinearity, coupled to a single input/output port with coupling rate γext (see figure D.2). The total linear cavity dissipation rate is

227

Figure D.2: Model of the Josephson Parametric Amplifier. A nonlinear resonant cavity withresonant frequency ω0 and Kerr constant K is coupled via rate γext to a single input/output port.
The total linear loss rate for the cavity is γ. The operators for internal, input, and output modes
are ĉ, ĉin , and ĉout , respectively.

γ. I am only interested in the response of the system, so I will ignore all noise sources. The system
Hamiltonian isĤsystem [t] = h̄ω0 ĉ† [t]ĉ[t] +h̄ † †
Kĉ [t]ĉ [t]ĉ[t]ĉ[t],2(D.33)

where ω0 is the bare cavity resonance frequency, K is the Kerr constant, and ĉ† (ĉ) is the creation(annihilation) operator for the cavity. Including the input field ĉin , I arrive at the HeisenbergLangevin equation of motion:i √γjhĉ[t], Ĥsystem [t] + γext ĉin [t] − ĉ[t]
h̄
2
γ

√
= −
− jω0 ĉ[t] + jKĉ† [t]ĉ[t]ĉ[t] + γext ĉin [t].
2

˙
ĉ[t]
=

(D.34)(D.35)

I assume that the fields incident on the JPA are a large coherent pump field at frequency ωpwith amplitude cin,0 and a relatively smaller signal field with amplitude ĉin,1 . I can thus write thesefields in the following form:ĉin [t] = cin,0 ejωp t + ĉin,1 [t],(D.36)ĉ[t] = c0 ejφp ejωp t + ĉ1 [t],(D.37)ĉ† [t] = c0 e−jφp e−jωp t + ĉ†1 [t].

(D.38)

Here, the coherent amplitude cin,0 is generally complex, but c0 is defined to be real with φp accounting for phase. The number of pump photons np in the cavity is c20 . The zeroth order (coherent)
solution to the equation of motion (ĉin,1 = ĉ1 = 0) is
cin,0 = c0 ejφp

γ + 2j(Kc20 − z),√
2 γext(D.39)

228
where z = ω0 − ωp + 2Kc20 . Linearizing the Heisenberg-Langevin equations of motion about the
zeroth order solution givesγ√ĉ˙1 [t] = −− j(ωp + z) ĉ1 [t] + je2j(ωp t+φp ) Kc20 ĉ†1 [t] + γext ĉin,1 [t],2√ĉ˙1 [t] + (W − jωp ) ĉ1 [t] + e2jωp t V ĉ†1 [t] = γext ĉin,1 [t].

(D.40)(D.41)

Here, W = γ/2 − jz and V = −je2jφp Kc20 . Equation D.41 is equivalent to the one before it, but
is arranged with the inter-cavity fields on the left of the equation and the input field on the right.
Making a transformation to take x = Equation D.41 to ẋ + (W ∗ − jωp ) x − e2jωp t V x† gives a newequationĉ¨1 [t] + (W + W ∗ − 2jωp )ĉ˙1 [t] + ((W − jωp )(W ∗ − jωp ) − |V |2 )ĉ1 [t]√= γext ĉ˙in,1 [t] + (W ∗ − jωp ) ĉin,1 [t] − e2jωp t V ĉ†in,1 [t] .

(D.42)The reason for re-writing the equation of motion this way is that it is now in terms of only thecavity annihilation operator - the creation operator has been eliminated. Fourier transforming into
the frequency domain now easily results in a solution for ĉ1 , and thus the output field, in terms of
the input fields:ĉ1 [ω] (W + j(ω − ωp ))(W ∗ + j(ω − ωp )) − |V |2(D.43)
√= γext (W ∗ + j(ω − ωp ))ĉin,1 [ω] − V ĉ†in,1 [ω − 2ωp ] ,ĉ1 [ω] =√γext(W ∗ + j(ω − ωp ))ĉin,1 [ω] − V ĉ†in,1 [ω − 2ωp ],(W + j(ω − ωp ))(W ∗ + j(ω − ωp )) − |V |2√ĉout,1 [ω] = ĉin,1 [ω] − γext ĉ1 [ω] = L[ω]ĉin,1 [ω] + M [ω]ĉ†in,1 [ω − 2ωp ],(D.44)(D.45)ĉ†out,1 [ω] = = L∗ [−ω]ĉ†in,1 [ω] + M ∗ [−ω]ĉin,1 [ω + 2ωp ].

(D.46)

Here, L[ω] is the direct (amplitude) gain and M [ω] is the intermodulation (amplitude) gain.

229
D.3.2

JPA direct and intermodulation gains

The direct gain of the JPA is
L[ω] ==(W − γext + j(ω − ωp ))(W ∗ + j(ω − ωp )) − |V |2(W + j(ω − ωp ))(W ∗ + j(ω − ωp )) − |V |2(γ/2 − γext + j(ω − ωp − z))(γ/2 + j(ω − ωp + z)) − K 2 n2p,(γ/2 + j(ω − ωp − z))(γ/2 + j(ω − ωp − z)) − K 2 n2p(D.47)(D.48)

where I have written the expression in terms of number of photons np = c20 . As a function of
frequency, the maximum gain should be achieved at ω = ωp . The optimal pump frequency can thenbe found by maximizing the direct gain |L[ω]|2 with respect to z. This occurs at z = 0, indicatingthat the pump frequency and photon number should be chosen such that ωp = ω0 + 2Knp to
optimize gain. From now on, I assume that z = 0. The direct gain on resonance (ω = ωp ) is then
4K 2 n2p − γ(γ − 2γext )L[ωp ] = L0 =4K 2 n2p − γ 2(D.49). Rewriting the frequency dependent direct gain in terms of L0 rather than the number of pump
photons gives
L[ω] =

L0 γγext − 2j(ω − ωp )(L0 − 1)(γ − γext + j(ω − ωp )).
γγext − 2j(ω − ωp )(L0 − 1)(γ + j(ω − ωp ))(D.50)If I make the substitutions ωp = ωJ , γext = L0 κJ and γ = L0 κ̃J and then expand to linear order in
1/L0 (large gain approximation), I findL[ω] ≈L0 κJ.
κJ − 2j(ω − ωJ )(D.51)

This is a Lorentzian response with linewidth κJ and resonant frequency ωJ . The amplifier bandwidthκJ = γext /L0 decreases with increasing on-resonant gain (or photon number). This expression makes
clear the fact that the gain-bandwidth product L0 κJ = γext is always constant (since γext is a fixed
function of cavity geometry).
The intermodulation gain of the JPA is√− γext V
M [ω] =
(W + j(ω − ωp ))(W ∗ + j(ω − ωp )) − |V |2pγγext (L0 − 1)(L0 − 1 + 2γext /γ)
2jφp
= je
.
γγext − 2j(ω − ωp )(L0 − 1)(γ + j(ω − ωp ))(D.52)(D.53)

230
If I make the substitutions ωp = ωJ and γ = γext = L0 κJ and then expand to linear order in 1/L0
(large gain approximation), I find
M [ω] ≈ je2jφp

D.3.3

L0 κJ
≈ je2jφp L[ω].
κJ − 2j(ω − ωp )(D.54)

JPA quadrature amplification

I have so far found how the incident and reflected creation and annihilation operators areaffected by the JPA. However, the transformations performed on orthogonal quadrature representations of the input and output fields is more interesting. Two orthogonal field quadratures can be
defined byJ
Xin(out)[ω] = e−jφ ĉin(out),1 [ω] + ejφ ĉ†in(out),1 [ω − 2ωJ ],(D.55)J
Yin(out)[ω] = −je−jφ ĉin(out),1 [ω] + jejφ ĉ†in(out),1 [ω − 2ωJ ].

(D.56)

Choosing the quadrature phase to be φ = φp + π/4, the JPA transformations to these quadratures
are
J
J
J
Xout
[ω] = (L[ω] − je−2jφp M [ω])Xin[ω] ≈ 2L[ω]Xin
[ω],
J
Yout
[ω] = (L[ω] + je−2jφp M [ω])YinJ [ω] ≈YinJ [ω].
2L∗ [ω](D.57)(D.58)

To get these results, I have used the equivalences L∗ [−ω + 2ωJ ] = L[ω] and M ∗ [−ω + 2ωJ ] =−e−4jφp M [ω]. To get the approximate expressions, I plugged in the full expressions for L and M ,expanded to lowest order in 1/L0 , and then re-wrote the result in terms of L. These expressions
make it clear that the Y J quadrature is deamplified by the same amount that the X J quadrature is
amplified, such that the product of their power gains is always unity. This means that no added noiseis required in order to preserve commutation relations between input and output fields. Because
the JPA amplifies the two quadratures by different amounts, it is a phase-sensitive amplifier.
If I had instead used the approximate expressions for the direct and intermodulation gains,
J but I would have found Y J = 0. In practice, the Y
I would have found the same result for Xout
outquadrature will be deamplified below the noise floor, so these are fine approximations to make for231
calculating measured quantities, even though they do not preserve the product of the quadrature
power gains.

Appendix E

Effective mass

When calculating the force sensitivity of an optomechanical system, we are often most interested in its sensitivity to a force at the mechanical oscillator’s point of maximum displacement (atthis point, a force will result in the largest change of motion and thus the most sensitive detectionof that force). For a wire’s fundamental mode, this means a force at its midpoint. For a circularmembrane’s fundamental mode, this means a force at the center. For other modes, the position of
optimal force sensitivity might be located at some other point.
In order to determine the sensitivity to a force near the mechanical resonance frequency at amechanical wire’s midpoint, z = l/2, I want to model that wire as an effective point mass meff ona spring, where the mass’ displacement ũ is the displacement at the center of the wire u[l/2] (seeFigure E.1). The total energy of the mass on a spring will then be meff Ω2m ũ2 , while the energy of

Figure E.1: Equivalent effective mechanical oscillator. (a) Illustration of a mechanical wire of lengthl, mass m, and displacement u[z], where z points along the wire. The wire is acted on by a force
F [z]. (b) Illustration of a point mass on a spring of mass meff , displacement ũ, and force F̃ . The
spring constant can be chosen such that both oscillators have the same resonant frequency Ωm .

233
the wire isRl2
2
0 (m/l)Ωm u[z] dz. Equating the energies of the two systems, I findRlm 0 u[z]2 dz
meff =.
lu[l/2]2(E.1)

For a wire under high tension, the displacement of the first eigenmode is u[z] = A sin[πz/l]. Fora wire under low tension, the displacement of the first mode is given in reference [94, pg. 235]:
u[z] = a(cos[βz] − cosh[βz]) + b(sin[βz] − sinh[βz]), where βl = 4.73 and a = −1.0178b for the firstmechanical mode. Integrating these distributions, I find the effective masses in the two cases:
meff = m/2

wire with tension

meff = 0.396m

wire without tension

(E.2)
(E.3)

The effective force that will be detected given a force anywhere on the wire can be found by
setting the work done in these two pictures equal. The work done on the mass on a spring by aRlforce F̃ is F̃ ũ. The work done on the wire by any force along its length is 0 (F [z]/l)u[z]dz. Settingthese equal would give a measured forceRlF̃ =0 F [z]u[z]dzlu[l/2].

(E.4)

Thus, if the wire were acted on by a force constant along the wire’s length, F [z] = F , the force
measured would be F̃ = 2F/π for a tensioned wire. However, we are more interested in the casewhen the force is just a point force at the center of the wire F [z]/l = F δ[z − l/2]. Then themeasured force is the actual force F̃ = F .
A similar analysis can be done for a circular tensioned membrane of radius R. Here, theradially-symmetric modes are Bessel functions of the first kind of order zero, u[r, θ] = J0 [αi r/R],
where αi is the ith root of the Bessel function. The energy of the lowest order mode is
Z 2π Z R2Ωmdθr(m/(πR2 ))u[r, θ]2 dr = mΩ2m J1 [α1 ]2 .
0

(E.5)

0

Equating this with the energy of the mass on a spring at the center of the membrane, meff Ω2m , andplugging in α1 = 2.4048, I findmeff = mJ1 [α1 ]2 = 0.27m

circular membrane with tension .

(E.6)

Appendix F

Microwave office simulations of wire capacitanceIn this appendix, I will describe how I used Microwave Office to simulate the capacitance ofa mechanical wire of dimensions l × w × t, spaced by distance s from a ground plane. I simulated
both wires similar to those in references [10, 45, 76] (Section F.1) as well as short wires similar tothe one in reference [14] (Section F.2).
For all simulations, I modeled the wire and ground plane as two coupled striplines using
Microwave Office’s built-in schematic element for 2 Edge Coupled Striplines [109]: “S2CLIN’. It is
possible for Microwave Office to model this system as a schematic because it is a problem describedby a set of integral equations which can be numerically solved [110]. S2CLIN takes separate widths
W 1 and W 2 for each conductor, the gap spacing S, the length L, and a substrate definition (seeFigure F.1). The substrate definition I used is a Two-Layer Stripline Substrate Definition [111]:

Figure F.1: Geometry for Microwave Office simulations. This is the layout of the S2CLIN schematic
built in to Microwave Office.

235“SSUBL”. It takes a relative dielectric constant r and thickness H for dielectrics above and below
the conductors, and a conductor thickness T . To model our geometry, I made W 2 and the H’smuch larger than the other dimensions and set r = 1 to have vacuum as the dielectrics. In theschematic, I grounded the infinite conductor and connected a 50 Ohm port to the other conductor(the wire). The capacitance between the conductors was found by measuring the admittance Y11
at some frequency and then dividing by that frequency: C = Im[Y11 ]/ω.

F.1

Microwave Office simulations for long wires
The fabrication dimensions for the best wire devices I made are(t0 , l0 , w0 ) = (130nm, 150µm, 150nm),s ≥ 350nm.

(F.1)

I therefore took sweeps of 50nm < s < 1µm as a function of either l, w, or t with the other two
variables fixed to these reference values. The dependencies on each of these variables individuallyis discussed in the following subsections and a full formula including all variables is found at theend of this section. As I fit to non-integer power laws in the following sections, it is important toknow that all variables in the equations are in SI units.

F.1.1

Width dependence for long wires

In Figure F.2(a), the simulated wire capacitance is plotted as a function of spacing from theground plane for several values of the width w with t = t0 and l = l0 fixed. The simulation for(t0 , l0 , w0 ) is in black in all plots. In this plot, the analytic result for the zero thickness coplanar
stripline model with t = t0 , l = l0 , and w = w0 (long black dashes) estimates less capacitancethan the simulation and agrees less and less well for small spacings. The parallel wire model (short
black dashes) also estimates less capacitance, but seems to have a similar spacing dependence.
The coplanar stripline model with thickness (dash-dotted black line) agrees well for large spacingsand becomes invalid for small spacings. Looking at the simulations for different widths, there is
not much width-dependence. These simulations can be fit to power laws Cwire = y0 + A[w]sp[w] .

236

Figure F.2: Width dependence for long wires. (a) Wire capacitance as a function of spacing. The
dots are simulations for t = 130 nm, l = 150 µm, and four values of width: w = 50 nm (blue),w = 100 nm (red), w = 150 nm (black), and w = 200 nm (green). The colored lines are power law
fits to the data with spacing larger than 50 nm. The black lines are the analytical models presented
in Section 5.2.1.3: the parallel wire model (long dashes), the zero thickness coplanar stripline model(short dashes), and the coplanar stripline model with thickness (dash-dotted line). (b) Derivativeof wire capacitance with spacing. The derivative (and hence the coupling) is independent of width.
The colors are the same as in (a).

Figure F.3: Length dependence for long wires. (a) Wire capacitance as a function of spacing. The
dots are simulations for t = t0 , w = w0 , and four values of length: l = 50 µm (blue), l = 100 µm(orange), l = 150 µm (black), and l = 200 µm (green). The colored lines are power law fits to thedata with spacing larger than 50 nm. (b) Power law coefficients A (red, left axis) and p (blue, rightaxis) as a function of length. The power p is essentially independent of length, while the coefficient
A is essentially linear in length.

237
The derivative of capacitance with spacing is then −dCwire /ds = A[w]p[w]sp[w]−1 (shown in figure
F.2(b)). This figure makes it clear that dCwire /ds is essentially independent of width.

F.1.2

Length dependence for long wires

In figure F.3(a), the simulated wire capacitance is plotted as a function of spacing from theground plane for several values of the width l with w = w0 and t = t0 . I once again fit this data
to power laws Cwire = y0 + A[l]sp[l] . I find that p[l] is essentially the same for the different lengths(figure F.3(b)). The coefficient A[l] is essentially linear in l, so the total capacitance, and therefore
also dCwire /ds, is linear in length.

F.1.3

Thickness dependence for long wires

In figure F.4(a), the simulated wire capacitance is plotted as a function of spacing from theground plane for several values of the thickness t with w = w0 and l = l0 . I once again fit this data
to power laws Cwire = y0 + A[t]sp[t] . However, Cwire is neither independent nor linear in t (unlike
for for w or l). In figure F.4(b), I plot the derivative of capacitance −dCwire /ds = A[t]p[t]sp[t]−1 asa function of thickness for several values of spacing. This dependence is essentially linear in t with
an offset: −dCwire /ds = α[s] + β[s]t. In figure F.4(c), I plot α[s] and β[s] as a function of spacing
and fit to a power law. This gives me an expression for dCwire /ds:
−dCwire1.97 × 10−15 0.9= 1.97 × 10−15 s−0.9 + 3.12 × 10−14 s−1.8 t =
s
+
16t
.
ds
s1.8
F.1.4

(F.2)

Final formula for long wires

The coupling for long wires appears to be independent of w and proportional to l, at least
for cross sections along (l0 , t0 , w) and (l, t0 , w0 ), respectively. Assuming these dependences are true
for the whole parameter space, the final formula for −dCwire /ds is−dCwire
ds

=
=


l
1.97 × 10−15 0.9
s + 16t−6
1.8
150 × 10s
−11
1.31 × 10
s0.9 + 16t l .
1.8
s

(F.3)
(F.4)

238

Figure F.4: Thickness dependence for long wires. (a) Wire capacitance as a function of spacing.
The dots are simulations for l = l0 , w = w0 , and seven values of thickness: t = 0 nm (pink), t = 30
nm (purple), t = 70 nm (blue), t = 100 nm (green), t = 130 nm (black), t = 160 nm (yellow),
and t = 200 nm (red). The colored lines are power law fits to the data with spacing larger than
50 nm. (b) Derivative of capacitance with respect to spacing, plotted as a function of thickness.
Several logarithmically spaced values of the spacing from s = 10 nm (red) to s = 1000 nm (blue)
are plotted. The lines are linear fits. (c) Plot of linear fit coefficients α[s] (red, left axis) and β[s](blue, right axis) as a function of spacing. The lines are power law fits with no offset.

F.2

Microwave Office simulations for short wires
The approximate fabrication dimensions for short wire devices are:(t0 , l0 , w0 ) = (130nm, 5µm, 150nm),s ∼ 18nm.

(F.5)

I therefore took sweeps of 4nm < s < 100nm as a function of each of l, w, and t, with the other two
variables fixed to the reference values. The dependencies on each of these variables individually is
discussed in the following subsections and a full formula including all variables is found at the endof this section.

239

Figure F.5: Width dependence for short wires. (a) Wire capacitance as a function of spacing. The
dots are simulations for t = t0 , l = l0 , and four values of width: w = 50 nm (purple), w = 100
nm (red), w = 150 nm (black), and w = 200 nm (green). The colored lines are power law fits tothe data. The black lines are the analytical models presented in Section 5.2.1.3: the parallel wiremodel (long dashes) and the zero thickness coplanar stripline model (short dashes). (b) Derivativeof wire capacitance with spacing. The derivative (and hence the coupling) is independent of width.
The colors are the same as in (a).

Figure F.6: Length dependence for short wires. (a) Wire capacitance as a function of spacing. The
dots are simulations for t = t0 , w = w0 , and four values of length: l = 1 µm (purple), l = 5 µm
(black), l = 10 µm (red), and l = 15 µm (green). The colored lines are power law fits to the data.
(b) Power law coefficients A (red, left axis) and p (blue, right axis) as a function of length. The
power p is essentially independent of length, while the coefficient A is essentially linear in length.

240
F.2.1

Width dependence for short wires

In figure F.5(a), the simulated wire capacitance is plotted as a function of spacing fromthe ground plane for several values of the width w. Once again, simulations for (t0 , l0 , w0 ) arein black. Both the parallel wire model for t = t0 , w = w0 , and l = l0 (short black dashes)
and the zero thickness coplanar stripline model (long black dashes) estimate a lower capacitancethan the simulation and agree less and less well for small spacings. The coplanar stripline modelwith thickness is not appropriate for these dimensions, so it is not plotted. The simulated data
can be fit to power laws Cwire = y0 + A[w]sp[w] . The derivative of capacitance with spacing isthen −dCwire /ds = A[w]p[w]sp[w]−1 (shown in figure F.5(b)). Just as for long wires, dCwire /ds isessentially independent of width.

F.2.2

Length dependence for short wires

In figure F.6(a), the simulated wire capacitance is plotted as a function of spacing from theground plane for several values of the width l with t = t0 and w = w0 . I once again fit this data to
power laws Cwire = y0 + A[l]sp[l] . As in the long wire case, I find that p[l] is essentially the same forthe different lengths (figure F.6(b)) and A[l] is essentially linear in l. Thus, the total capacitance,
and therefore also dCwire /ds, is linear in length.

F.2.3

Thickness dependence for short wires

In figure F.7(a), the simulated wire capacitance is plotted as a function of spacing fromthe ground plane for several values of the thickness t with l = l0 and w = w0 . I once again fit
this data to power laws Cwire = y0 + A[t]sp[t] . In figure F.7(b), I plot the derivative of capacitance−dCwire /ds = A[t]p[t]sp[t]−1 as a function of thickness for several values of spacing. This dependence
is once again essentially linear in t with an offset: −dCwire /ds = α[s] + β[s]t. In figure F.7(c), I
plot α[s] and β[s] as a function of spacing and fit to a power law. This gives me an expression for

241

Figure F.7: Thickness dependence for short wires. (a) Wire capacitance as a function of spacing.
The dots are simulations for l = l0 , w = w0 , and seven values of thickness: t = 0 nm (pink), t = 30
nm (purple), t = 70 nm (blue), t = 100 nm (green), t = 130 nm (black), t = 160 nm (yellow), andt = 200 nm (red). The colored lines are power law fits to the data. (b) Derivative of capacitancewith respect to spacing, plotted as a function of thickness. Several logarithmically spaced valuesof the spacing from s = 1 nm (red) to s = 100 nm (blue) are plotted. The lines are linear fits. (c)Plot of linear fit coefficients α[s] (red, left axis) and β[s] (blue, right axis) as a function of spacing.
The lines are power law fits with no offset.

dCwire /ds:
−dCwire1.28 × 10−20 0.6= 1.28 × 10−20 s−1.4 + 4.65 × 10−17 s−2 t =s + 3600t .
2
ds
s

F.2.4

(F.6)

Final formula for short wires

The coupling for short wires appears to be independent of w and proportional to l, at least
for cross sections along (l0 , t0 , w) and (l, t0 , w0 ), respectively. Assuming these dependences are true
for the whole parameter space, the final formula for −dCwire /ds is−dCwirel
1.28 × 10−20 0.6
2.56 × 10−15 0.6
=
s
+
3600t
=
s + 3600t l .
−6
2
2
ds
5 × 10
s
s

(F.7)

242

Figure F.8: Check of empirical simulation formula for long wires. The two wires shown are ofdimensions t = 30 nm, w = 150 nm, l = 50 µm (red) and t = 200 nm, w = 150 nm, l = 200 µm(blue). (a) Wire capacitance as a function of spacing, with power law fits. (b) Derivative of
capacitance with respect to spacing. The solid lines are the simulation. The long and short dashedlines are the predictions from the long and short wire formulas, respectively.

Figure F.9: Check of empirical simulation formulas for short wires. The two wires shown are ofdimensions t = 30 nm, w = 150 nm, l = 1 µm (red) and t = 200 nm, w = 150 nm, l = 15 µm (blue).
(a) Wire capacitance as a function of spacing, with power law fits. (b) Derivative of capacitancewith respect to spacing. The solid lines are the simulation. The long and short dashed lines are
the predictions from the long and short wire formulas, respectively.

F.3

Check of simulation formulas
I can check the formulas for the long and short wires and see how they differ by applying

them to simulated wires at the edges of short and long wire parameter space. The simulated wires
and the long and short formulas expectations are shown in figures F.8 and F.9. Surprisingly, boththe long and short wire formulas provide reasonably good agreement with the simulations for both

243
long and short wires. I will therefore only use the long wire formula as the empirical simulation
form of coupling in Chapter 5.

Appendix G

Wire device fabrication recipe

Chips used:• High resistivity wafer with 150 nm thermal oxide, cut into 1 cm squares
Chemicals/resists used:• ACE (acetone), IPA (isopropanol), DI (deionized water), N2 (nitrogen gas)
• S1813 photoresist (Shipley)• RD6 photoresist developer (Futurrex)• Buffer-HF Improved: Ammonium Fluoride, Hydrofluoric Acid, DI (Transene)• 950 PMMA A7 in anisole (MicroChem)• 1:3 MIBK/IPA (MicroChem)
General comments:• Vent RIE 3x after SF6 etch and clean using O2 ash (50 sccm, 500 W, 5 min)
• Always place chip on quartz wafer in RIE• Always agitate chip slightly while developing/etching• Always spin resist at 4000 rpm, max acceleration, 45 sec

245• S1813 resist (4000 rpm): ∼ 12000 to 14000 Å
• PMMA (4000 rpm): ∼ 4000 Å
• NPGS patterns: ∼ 15 nC/cm dose, ∼ 25 Å spacing for lines, ∼ 425 µC/cm2 , ∼ 150 Å forareas• Evaporating faster gives compressive stress (floppier mechanical elements)
Recipe:(1) Clean chip
• Blow of dust, soak in ACE w/ sonication, IPA, N2 dry(2) Alignment mark pattern• Bake 100◦ C, 1 min, cool 1 min• Spin S1813 (5 drops)• Edge bead removal with ACE, N2 dry• Bake 100◦ C, 90 sec• Align to mask and expose 6 sec• Develop 35 sec in 15 mL RD6, 10 sec DI, 10 sec fresh DI, N2 dry(3) Evaporate alignment marks• O2 ash 50 sccm, 250 W, 30 sec• Evaporate 5 nm Ti, density 4.5, ∼ 5 Å/s, ∼ 100 mA• Evaporate 40 nm Au, density 19.3, ∼ 5 Å/s, ∼ 85 mA• Liftoff in ACE > 1 hr, squirt with ACE, sonicate in ACE, IPA, N2 dry(4) SiO2 holes

246• Bake 100◦ C, 1 min, cool 1 min• Spin S1813 (5 drops)• Edge bead removal with ACE, N2 dry• Bake 100◦ C, 90 sec• Align to mask and expose 6 sec• Develop 35 sec in 15 mL RD6, 10 sec DI, 10 sec fresh DI, N2 dry• Hard Bake 115◦ C, 2 min, cool 1 min• HF 1 min 35 sec, 10 sec DI, 10 sec fresh DI, N2 dry• Sonicate in ACE, IPA, N2 dry(5) Double layer resist• Bake 180◦ C, 1 min, cool 1 min• Spin PMMA (2 drops)• Edge bead removal with ACE, N2 dry• Bake 180◦ C, 5 min, cool 1 min• Spin S1813 (5 drops)• Edge bead removal with ACE, N2 dry• Bake 50◦ C, 90 sec (higher temp can ruin PMMA)• Align to mask and expose 4 sec• Develop 30 sec in 15 mL RD6, 5 sec fresh RD6, 10 sec DI, 10 sec fresh DI, N2 dry• If you screw up:∗ Flood expose 4 sec∗ Develop 40 sec in RD6, 10 sec DI, 10 sec fresh DI, N2 dry∗ Bake PMMA 180◦ C, 30 sec, cool 45 sec

247∗ Spin S1813 (5 drops) and continue as above(6) Transfer resist pattern to PMMA
• O2 ash 50 sccm, 400 W, 240 sec plus more time until smooth surface is visible in allashed areas• Flood expose 4 sec• Develop 40 sec in 15 mL RD6, 10 sec DI, 10 sec fresh DI, N2 dry(7) E-beam write• Write at 35kV, 8mm, probe current 14, current 10 pA• Write all wires, focusing between each write• Develop 90 sec in 15 mL 1:3 MIBK/IPA, 20 sec IPA, 20 sec fresh IPA, N2 dry(8) Evaporate aluminum• DO NOT DO AN O2 ASH!!
• Evaporate Al, density 2.7, ∼ 5 Å/s, ∼ 35 mA• Liftoff in ACE, squirt with ACE, sonicate in ACE, IPA, N2 dry(9) Cut chip
• Spin S1813 (5 drops)• Mount chip, sharpen blade, cut using saw speed 2, remove chip
• DI, ACE, IPA, N2 dry(10) Anneal• Bake in oven at 340◦ C for 30 min• Let cool > 15 min(11) Suspend wires

248• SF6 etch 18 sccm, 450 W, start with ∼ 120 sec• Look in SEM at 28kV, 15mm, to see if mechanics is suspended. If not, repeat.
(12) Mount on sample mount• Attach chip to board with small dot of rubber cement and allow to dry > 5 min• Wire bondAppendix H

Calibration details for two-drive experiments

Measurements of sideband occupation in the two-drive scheme (such as those presented in
Chapter 9) rely sensitively on the values used for the number of photons in the cavity due to thered and blue drives (or equivalently, the values of the calibration parameter λ[ωr ] and λ[ωb ]). In
the following sections, I discuss the methods I use to calibrate these photon numbers, including
example data sets.

H.1

Calibration of red drive voltage to photon unit conversion
The number of red drive photons can be found by fitting the magnitude squared of the dressed

response of the cavity in the presence of a strong red-detuned drive:|S21,meas [ω]|2 = G N2 −2κl χc [ω]Pnoise.
+2
1 + gr χc [ω]χm [ω − ωr ]
Pin

(H.1)

Here, G is some gain coefficient and Pnoise /Pin is a separately determined background1 due to thenoise of the measurement. This background is usually negligible compared to the signals of interest
but could be important as features of the dressed response (especially in the two drive scheme) can
approach zero transmission. It is not a free parameter of the fits. The value of λ[ωr ] can be easily
1

If the voltage measured at the output of the experiment has average and fluctuating contributions, V = V + δV ,2then the measured output power is P = V +2V δV +(δV )2 . The average power, assuming that the voltage fluctuations
2
have mean zero, is hP i = V + (δV )2 = Pout + Pnoise and the system response is |S21 |2 = Pout /Pin + Pnoise /Pin .

2The variation of the power, averaged over Navg measurements, is σP2 = P 2 − hP i2 /Navg = 4V (δV )2 /Navg =2
2
4Pout Pnoise /Navg . The noise floor of a measurement of |S21 | is then Pnoise /Pin = Navg σ|S21 |2 /(4|S21 |2 ). This quantity
can be determined experimentally by looking at the value and standard deviation of response data off resonance,
where it is constant as a function of frequency.

250
calculated from the extracted coupling gr , the bare cavity response parameters, and the measured
output drive power:
λ[ωr ]
2κl Z0 Pr
.
=
2
2
g0
gr |N2 /χc [ωr ] − κl |2(H.2)

In addition to determining gr , a measurement of the dressed response magnitude reveals thebare cavity response parameters, ω c , κ, κl , and N2 = 1/(1 + jB), as well as the effective mechanicalparameters, Ωtot and Γtot . Figure H.1 shows measurements of the dressed response and the raw
voltage spectrum in the presence of a range of red-detuned drive strengths, along with fits to thetheoretical expressions. Figure H.1(a) shows the dressed response over a frequency range where thebare response of the cavity is visible. At the lowest drive power, no mechanical feature is present,
while at the highest drive power, the cavity and mechanical modes have begun to hybridize. Figures
H.1(b-d) show zoom-ins on the mechanical features from the same data. These figures show thatthe observed cavity and mechanical features are well described by the theoretical function for the
dressed response. Figures H.1(e-f) show the raw voltage spectral densities measured with the same
red-detuned drives. Even without a full calibration into optomechanical units, these spectra reveal
the total mechanical resonance frequency and linewidth.
Figure H.2 shows the parameters extracted from the data in Figure H.1. Figures H.2(a)and (b) show the cavity resonance frequency and linewidths as a function of generator power.
The parameters extracted using the full dressed response fit expression are shown as circles, while
those extracted using only a bare cavity fit are shown as crosses. These parameters agree well
for all except the highest powers, where the cavity and mechanical modes begin to hybridize andthe notion of bare cavity parameters breaks down. Figures H.2(c) and (d) show the extracted
mechanical parameters from both the dressed response (black solid circles) and the raw spectral
data (red open circles). The spectral data is more reliable at low powers, where the responsefeature is small, while the response data is more reliable at high powers, where the spectral data issmall and squashing must be taken into account. However, the parameters extracted from the two

251

Figure H.1: Dressed response and spectra with a red-detuned drive only. The y-axes of all plots
have been offset by 0.5 for each trace to enable clearer examination. All data for a given power isshown in the same color. (a) Dressed response measured for red-detuned generator drive powers
in 6 dB steps from 25 nW (magenta) to 100 mW (red). (b-d) Zoom in on the upper red sideband
features from (a). (e-f) Raw voltage spectral densities of the mechanical sideband for the same
drive frequencies and powers as in the color-corresponding response data.

252

Figure H.2: Extracted parameters from dressed response and spectral data in the presence of asingle red-detuned drive. (a) Cavity resonance frequency extracted from both bare (crosses) anddressed (circles) cavity fits. The upper red sideband frequency is shown as a dashed line, indicatingthat the drive stays approximately red-detuned (compared to the cavity linewidth) for all powers.
(b) Cavity linewidths extracted from both bare (crosses) and dressed (circles) cavity fits. Theinternal (and thus total) linewidth changes drastically with power. (c-d) Effective mechanicalresonance frequency and linewidth extracted form dressed response (black solid circles) and raw√
voltage spectral density (red, open circles) data. (e) Optomechanical coupling gr = g0 nr (blacksolid circles, left axis) and measured output drive power Z0 Pr (blue open circles, right axis). (e)Calculated voltage to photon unit conversion factor λ[ωr ]. The average over the mid-power sectionof data is shown as a black line.

253
types of data agree, especially over the mid-range of powers. Figure H.2(e) shows the extracted
values of gr at each power (black solid circles, left y-axis) as well as the measured output drive
power (red open circles, right y-axis). As expected, the output power follows an approximately√
linear dependence on generator power, while gr = g0 nr follows a square root dependence. Figure
H.2(f) shows the calculated values for the parameter λ[ωr ]. Ignoring the lowest three values (where
the measurement of gr is less consistent) and the highest two values (where the system begins tohybridize), the average value (black line) is λ[ωr ]/g02 = 3.0617 × 10−21 .

H.2

Calibration of blue drive voltage to photon unit conversion
Determining the number of photons in the cavity due to the blue-detuned drive (or equiva-

lently determining λ[ωb ]) has proved to be substantially more difficult than determining the number
from the red-detuned drive. This section discusses three potential methods I have investigated for
doing this calibration: using the dressed response to directly extract gb , using spectral data with
two nearly equal drives to make a relative measurement of gb /gr , and using the onset of mechanicalparametric oscillation to determine the point where gr = gb . I will find that the dressed response
is often inconsistent with either the expectations found in Section 3.2 or with spectral data takenwith the same input fields. These discrepancies likely indicate that our model of the system isincomplete. I avoid these difficulties by measuring spectral data at high occupancy (where any
asymmetry should be negligible) and assuming that nfm,r = nfm,b , allowing me to determine therelative value gb /gr . This seems to be the best method for determining the value of λ[ωb ]. I also
use the onset of mechanical parametric oscillation to try to determine the point where gr = gb . I
find that this method seems to result in a less reproducible calibration than using high occupancy
spectral data.

H.2.1

Direct calibration of blue photon number from the dressed cavity response

The number of photons in the cavity due to a blue-detuned drive can theoretically be extracted directly from a measurement of the two-drive dressed response (the dressed response cannot

254
be measured with only a blue drive at the powers of interest because the mechanical oscillator would
self-oscillate). However, I often find that two-drive dressed response measurements do not matchthe theoretical expectations from my model. One reason for the discrepancy could be dynamic
effects from the power dependence of the internal cavity loss, similar to the dynamic photo-thermorefractive frequency shift described in the supplemental information of reference [65]. This effectis a likely candidate because the internal loss decreases with increasing drive photon number, presumably due to saturation of two-level fluctuators. If this saturation and accompanying decrease inloss are instantaneous, the cavity resonance frequency will decrease ninety degrees in phase later.
This will change the number of photons in the cavity, potentially leading to a feedback effect on
the cavity resonance frequency.
Additionally, I sometimes see that the mechanical parameters extracted from response andspectral data do not agree. Two examples of this are shown in Figure H.3. These points are thelowest and highest temperature points of a temperature sweep with gb ∼ gr ∼ 4×104 (Cr ∼ Cb ∼ 15).
In both figures, the red upper sideband feature in the dressed response is shown in orange (lefty-axis) and the red upper spectral sideband is shown in purple (right y-axis). They have beenplotted so as to match the backgrounds and peaks of these features, enabling easy comparison of
the linewidths and resonance frequencies. In Figure H.3(a), the spectral feature is 40% wider thanFigure H.3: Examples of inconsistent response (orange) and spectral (purple) features. Sometimes
the mechanical resonance frequencies and linewidths extracted from response and spectral data do
not agree. Even without fitting to a model, this discrepancy is clear from a visual comparison of
the sideband features.

255
the response feature, while in Figure H.3(b), the response feature is 25% wider. The resonancefrequencies also seem slightly different. These discrepancies make it difficult to rely on the twodrive dressed response as a means for calibrating the number of drive photons in the cavity. I also
sometimes see discrepancies between the cavity resonance frequency and linewidth extracted from
response data (bare or dressed) and the frequency and linewidth of the thermal cavity photon peakin the spectral data.
Despite these difficulties, I have found some drive situations for which two-drive dressed
response measurements do seem to be described by my model. Figure H.4 shows a set of measurements for which this is the case. The red and blue drive powers were chosen to be approximately
the same, so as to enhance both sideband features, but with the red drive slightly larger thanthe blue drive, so as to avoid self-oscillation of the mechanical oscillator. The different traces arelabeled by point number because both drive powers change between points, as listed in Table H.1.
These generator powers correspond to approximate values of cooperativity 0.1 < C < 600.
Figure H.4(a) shows a wide view of the dressed response for the even numbered points, alongwith fits to the two drive dressed response model. The highest powers (points 0 and 2) display theonset of normal mode splitting while the lowest power (point 14) shows no mechanical sideband
features. Zoom-ins of the the intermediate power sideband features are shown in Figures H.4(b)and (c). The model agrees well with both the cavity and mechanical features.
Figure H.5 shows the parameters extracted from the two-drive dressed response in Figure H.4
as well as from raw voltage spectral data. Even without calibration of the y-axis, the raw voltage
spectral data at high occupancy allows us to extract the mechanical parameters Ωtot and Γtot , aswell as the ratio of red to blue photon numbers (described in more detail in the next section).
Figures H.5(a) and (b) show the mechanical parameters extracted from both the response (purple
circles) and spectrum (green triangles). These parameters agree well, unlike for the data presented
in Figure H.3. Figure H.5(c) shows the voltage to photon unit conversion parameter λ[ω] at thetwo drive frequencies, calculated using the measured output drive power and the photon numbers
extracted from the two-drive dressed response. The parameter extracted at the red drive frequency,

256

Point
Pgen,r (dBm)
Pgen,b (dBm)

0
17
19

1
14
17

2
11
14

3
8
11

4
5
8

5
2
5

6
-1
2

7
-4
-1

8
-7
-1

9
-7
-2

10
-10
-5

11
-13
-8

12
-16
-11

13
-19
-14

14
-22
-17

Table H.1: Red and blue drive generator powers used in the blue calibration example. The powers
were adjusted to keep the number of photons in the cavity due to the two drives approximatelybalanced.

Figure H.4: Dressed cavity response in the presence of two drives, one red-detuned and one bluedetuned, for the even numbered points in Table H.1 (colored consistently in all parts). Each dressed
response is offset by 0.25 for clarity of display, except those for points 2 and 4, which are offset by
0.5. (a) Wide view of the dressed response. (b,c) Narrow views of the data in (a) showing the red
upper and blue lower sideband features.

257

Figure H.5: Summary of parameters extracted from two-drive dressed response and spectral measurements. (a,b) Mechanical resonance frequency and linewidth extracted from the two-drive
dressed response and the raw voltage spectral density. (c) Extracted λ at the red and blue drive
frequencies from the dressed response. (d) Ratio of λ’s at the two drive frequencies extracted bothfrom the dressed response and spectral data. (e) Normalized quantities that should be proportional
to the generator power for both drives. The number of blue photons (or gb2 ) extracted from the
dressed response is less consistent than the blue photon number extracted from spectral data.

λ[ωr ], is far more consistent from point to point than the parameter at the blue drive frequency,
λ[ωb ]. In Figure H.5(d), the ratio of these parameters, λ[ωb ]/λ[ωr ] is shown. The purple circles
come directly from the data in Figure H.5(c), while the green triangles are calculated using gb /grfrom the spectral data. This ratio calculated with the spectral data seems far more consistent point

258
to point than that from the dressed response. The accuracy of the values for gb extracted from thetwo types of data can be seen another way, looking at Figure H.5(e). In this figure, the various
measures of power (generator power, measured power, and photon number) have been normalized
by arbitrary factors to permit comparison. The number of red photons clearly follows both the red
generator and measured red drive powers. The measured blue power also follows the blue generator
power, but the number of blue photons extracted from the dressed response (solid blue line) has
several noticeable deviations away from this dependence. The number of blue photons inferred fromthe number of red photons and the spectral photon ratio (dashed blue line) follows the generator
and measured powers much more faithfully.
I conclude that fitting the two-drive dressed response does not provide an adequate determination of the number of blue drive photons in the cavity, as this measurement does not always
agree with my model of the system, does not always agree with spectral measurements, and isless reproducible than calibrating with high-occupancy spectral measurements. The insufficiencyof my model to explain the data can likely be attributed to dynamic effects due to the highly
power-dependent nature of the internal cavity loss.

H.2.2

Relative photon number calibrations using high-occupancy spectra or mechanical parametric oscillation

Rather than measuring the number of cavity photons due to the blue-detuned drive directly,
we can calibrate it by making a relative measure of the number of red and blue photons and thenusing the red-drive calibration above to find the number of blue photons (or equivalently λ[ωb ]).
One way to do this is to measure the voltage spectral density of the upper red and lower blue
mechanical sidebands when the mechanical occupancy is large. In this case, both the asymmetry
and any thermal photon occupancy contributions to the sidebands should be negligible and we canmake the assumption that nfm,r = nfm,b = nfm . Assuming that the sidebands are separated by many259
mechanical linewidths, the spectral density takes the limiting form
2
2
2
2
f
Saout aout [ω] = κκl nthc |χc [ω]| + κl gr Γtot nm |χc [ω]| |χm,eff [ω − ωr ]|(H.3)+κl gb2 Γtot nfm |χc [ω]|2 |χm,eff [ωb − ω]|2 .
The ratio of the red to blue sideband amplitudes in this case is gr2 /gb2 . Thus, even without convertingthe (high mechanical occupancy) measured voltage spectrum into photon or displacement units,
the ratio of red to blue photon numbers can be found by fitting the spectrum with the form
SV V [ω] = A0 + |χc [ω]|2 Acav + Ar |χm,eff [ω − ωr ]|2 + Ab |χm,eff [ωb − ω]|2 ,(H.4)

where the A’s are some amplitude coefficients. The ratios of photon numbers and λ’s are then
gb
=
gr

r

Ab
Ar

and

λ[ωb ]
g 2 Pb |N2 /χc [ωr ] − κl |2= r2.
λ[ωr ]
gb Pr |N2 /χc [ωb ] − κl |2(H.5)

Another calibration method that I tried was to use the parametric oscillation of the mechanical oscillator to determine the point where the red and blue photon numbers were the same. At
high power, the mechanical linewidth is given byΓtot = Γm +4(gr2 − gb2 )4gr2 4gb2
−
≈
.
κ
κ
κ

(H.6)

Therefore, the linewidth reaches zero and the mechanical oscillator parametrically oscillates atapproximately the point where the two drive photon number are equal. I found this point by
fixing the blue drive power and decreasing the red drive power until parametric oscillation wasobserved. I then returned the red drive to the lowest power for which the mechanical oscillator did
not parametrically oscillate and measured the output drive powers and bare cavity response. Thiscalibration method is appealing over the high-occupancy spectral calibration, as it does not require
spectral measurements (each of which includes measurements of the spectrum with the JPA on,
the spectrum with the JPA off, and the JPA response) and can therefore be performed much more
quickly.
In order to determine the calibration parameters λ[ωr ] and λ[ωb ] (which we assume couldchange day-to-day) for a data set of interest, I make several (usually four) identical calibration

260
measurements before the data set and several (usually four) more after the data set. Averaging
over these extracted values, I then assume the calibration parameters are constant throughout thedata set. Figure H.6 shows the results obtained from the calibrations associated with the example
red calibration above, the example blue calibration above, the temperature sweep used in Chapter
9, and twelve sweeps of the red drive power in an attempt to observe asymmetry similar to thetheoretical plot shown in Figure 4.3(c). I call these sweeps asymmetry sweeps 3-14. The frequencies
used for each data set are shown in Figure H.6(a), where the upper red and lower blue sideband
frequencies are shown relative to the cavity response, which changes from the solid to dashed line
throughout the data set. Figure H.6(b) shows the calibration factor λ[ωr ] extracted from the eachmeasurement of the red-drive-only dressed response (colored markers). The average for each data
set is shown as a colored horizontal line. The calibrated values usually seem consistent acrossa single data set (with the exception of those from data set 13, where the first four points are
noticeably different than the last four, indicating that the gain probably changed during the dataset). The results presented in this plot confirm my assumptions that the calibration is relatively
constant throughout a data set, but can change substantially from day to day.
Figure H.6(c) shows the relative calibration parameters λ[ωb ]/λ[ωr ] extracted from each calibration measurement as the colored open markers (ignore the black dots for now). No data is shownfor the red calibration example above, as no data was taken with the blue drive on. The values
shown for the blue calibration example are those extracted from the high-occupancy spectral data.
The temperature sweep and asymmetry sweeps 2-8 were also calibrated using the high-occupancy
spectral method, while asymmetry sweeps 9-14 were calibrated using the parametric oscillation
method. Once again, the averages of the measured values associated with each data set are shownas horizontal lines. The first conclusion I make from this data is that the values extracted fromthe high-occupancy spectral calibration method exhibit far less scatter than those extracted fromthe parametric oscillation method. They are also more closely grouped about one (as we might
expect because λ should only be weakly frequency dependent). In an effort to check whether thesecalibrations were valid, I also analyzed any high-occupancy points (points with nfm > 20) from the261
asymmetry sweeps as if they were calibrations and plotted them as the black dots in the figure (remember that some of the calibration points were measured before the asymmetry sweep and somewere measured after, so these points were actually taken in between the designated calibrations).
Not surprisingly, the black dots for the data sets calibrated using the high-occupancy spectral
method are very consistent with the calibration (as there is essentially no difference between the

Figure H.6: Summary of parameters extracted from many different relative calibration measurements. For each data set, I plot (a) the cavity lineshape and the upper red and lower blue sideband
frequencies, (b) the red-drive calibration parameter λ[ωr ], and (c) the relative calibration parameter λ[ωb ]/λ[ωr ]. Values from each individual calibration are shown as colored symbols, with theaverages (which were the employed calibration values) for each data set indicated as lines. The
black dots in (c) are extracted from high-occupancy points in the actual asymmetry sweep.

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two types of data). However, there are substantial differences between the black dots and the calibrated values found using the parametric oscillation method. The last thing to notice about Figure
H.6(c) is that all of the relative calibration parameters extracted using the high-occupancy spectral
calibration method (either as intentional calibration points or as the black dots) are grouped aboutone, but can vary by about 10% (a frequency dependence that seems reasonable for the HEMT gain
over 20 MHz). I therefore conclude that using mechanical parametric oscillation to find the point ofequal red and blue photon numbers is both less repeatable and less reliable than the high-occupancy
spectral method.
Unfortunately (in retrospect), some of the best asymmetry sweeps are data sets 9-14. This
is not because of the calibration choice made (as the actual power sweep does not depend on thischoice), but is instead a consequence of the input drive choices made for each sweep. Asymmetry
sweeps 3 and 4 are different from all of the other sweeps because the sidebands of interest (seeFigure H.6(a)) are further apart in frequency. In theory, this should not affect the asymmetry.
However, in practice the asymmetry may be degraded, as it will be far more sensitive to errors in
the cavity resonance frequency, linewidth, and thermal occupancy, as well as the JPA gain frequency
dependence. Keep in mind that errors in the extracted cavity parameters might not be surprising,as I am using the cavity response function derived in the theory chapters to fit the bare cavity
response, which likely does not not fully describe the system. In particular, we sometimes observe
discrepancies between the bare cavity response and the thermal cavity spectral shape, making it
unclear which values are relevant for converting between different quantities. Asymmetry sweep
5 exhibits similar asymmetry to that seen in data sets 9-14, but required a lot of point-specificanalysis to remove non-ideal effects like JPA gain drift. The other asymmetry sweeps calibrated
with the high occupancy spectral method (sweeps 6-8) are so far detuned that the power required
to produce a significant asymmetry saturates the JPA (although the fact that no asymmetry isobserved for these sweeps confirms our expectations). In Chapter 9, I therefore present asymmetry
sweep 11, where the sidebands of interest are detuned by about κ/3 from the cavity resonance, andasymmetry sweep 14, where the sidebands are nearly on-resonance. To rectify the poor calibration

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of these data sets, I scale the extracted blue occupancies by a common factor (one for each set) suchthat the highest occupancy point has equal red and blue occupancies. This method of calibration
is clearly not very rigorous, as it uses data points in the data set to calibrate the data set. However,doing this does allow us to understand what the asymmetry would look like if we did have a valid
independent calibration.