Development of Techniques for Quantum-Enhanced
Laser-Interferometric Gravitational-Wave Detectors
byKeisuke Goda
B.S., University of California at Berkeley (2001)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 2007c Keisuke Goda, MMVII. All rights reserved.
!
The author hereby grants to MIT permission to reproduce and distribute publicly
paper and electronic copies of this thesis document in whole or in part.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
August 10, 2007

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nergis Mavalvala
Associate Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thomas J. Greytak
Associate Department Head for Education

2

Development of Techniques for Quantum-Enhanced
Laser-Interferometric Gravitational-Wave Detectors
byKeisuke Goda
Submitted to the Department of Physics
on August 10, 2007, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
A detailed theoretical and experimental study of techniques necessary for quantum-enhanced laserinterferometric gravitational wave (GW) detectors was carried out. The basic theory of GWs andlaser-interferometric GW detectors, quantum noise in GW detectors, the theory of squeezed states
including generation, degradation, detection, and control of squeezed states using sub-threshold
optical parametric oscillators (OPOs) and homodyne detectors, experimental characterization ofthese techniques (using periodically poled KTiOPO4 in an OPO at 1064 nm for the first time), key
requirements for quantum-enhanced GW detectors, and the propagation of a squeezed state in acomplex interferometer and its interaction with the interferometer field were studied. Finally, theexperimental demonstration of quantum-enhancement in a prototype GW detector was performed.
By injecting a squeezed vacuum field of 9.3 dB (inferred) or 7.4 ± 0.1 dB (measured) at frequencies
above 3 kHz and a cutoff frequency for squeezing at 700 Hz into the antisymmetric port of the
prototype GW detector in a signal-recycled Michelson interferometershot noise√ configuration, the √floor of the detector was reduced broadband from 7.0 × 10−17 m/ Hz to 5.0 × 10−17 m/ Hz while
the strength of a simulated GW signal was retained, resulting in a 40% increase in signal-to-noise
ratio or detector sensitivity, which is equivalent to a factor of 1.43 = 2.7 increase in GW detection
rate for isotropically distributed GW sources that are confined to the frequency band in whichsqueezing was effective. This is the first implementation of quantum-enhancement in a prototype
GW detector with suspended optics and readout and control schemes similar to those used in LIGO
and Advanced LIGO. It is, therefore, a critical step toward implementation of quantum-enhancement
in long baseline GW detectors.
Thesis Supervisor: Nergis Mavalvala
Title: Associate Professor of Physics

Acknowledgments
This thesis has been supported by a number of people. First of all, I would like to thank mysupervisor Nergis Mavalvala for her incredible support. Without a doubt, the 40m squeezing project
would not have been made possible without her constant support. Her leadership, intellectual talentand curiosity, and unlimited efforts about education for her students including myself have greatly
influenced and inspired me. I admire these aspects of hers and have learned many things from them.
It is not an overstatement that she is one of the greatest mentors at MIT. I am sure her otherstudents agree on this.
I have had the biggest fortune to work with many talented and inspiring people. First, I mustmention the name of the greatest Russian, Eugeniy E. Mikhailov at the College of William & Mary.
His superb technical skills and expertise about building experimental apparatus impressed me greatly
during his three years at MIT. The 40m squeezing experiment could not have been realized without
the squeezing-related techniques we acquired at MIT. His friendship has been invaluable. I have to
say that I very much enjoyed dining at good restaurants in and near Pasadena almost every night
while he visited Caltech.
Osamu Miyakawa at Caltech has been an incredible supporter as well. Although they weretough, I enjoyed midnight squeezing experiments with him. He is very good at explaining thingsand making difficult concepts easy to understand, which helped me a lot with understanding the
40m interferometer. Among few Japanese people in LIGO, he is a unique scientist with talent inmany aspects and it was the most fun to work with him. Without a doubt, he is a Japanese rolemodel.
I would like to thank my research collaborators. First, I am grateful to Shailendhar Saraf atRochester Institute of Technology for helping me with building important circuits for the squeezing
experiment such as the quantum noise-locking servo with higher stability. I also thank my Australiancollaborators, Kirk McKenzie, Warwick Bowen (currently at University of Otago), Ping Koy Lam,
David McClelland, and Malcolm Gray at Australian National University. Squeezing started in the
LIGO Quantum Measurement Group thanks to these people who told me how to squeeze during myfour-week visit at the university. I would also like to thank the Caltech 40m team, Bob Taylor andSteve Vass, for clean room work and 40m maintenance, and in particular, Alan Weinstein for hispatient support for the 40m squeezing project. His good nature has maintained the 40m lab healthy
and humorous.
I also thank my colleagues in the Quantum Measurement Group at MIT, Thomas Corbitt,
Christopher Wipf, David Ottaway, and Edith Innerhofer. I am grateful to Chris for helping me with
low-noise photodetector development for the 40m squeezing project, Dave for supervising me on the
wavefront sensor experiment and squeezing work at MIT, and Thomas for stimulating discussion
about squeezing. I remember my old days when Thomas, Chris, and I had to go through the painfulGeneral Exams at MIT.
I would also like to send my gratitude to other LIGO staff members at MIT, Keith Bayer forhis support about computer software and hardware and Ken Mason, for purchasing machined parts.
The squeezing project was based on their help. One of the greatest assistants is LIGO’s secretary,
Marie Woods. Her secretary work is more than excellent in that she does things on time without amistake. This was extremely important to do long distance work between MIT and Caltech for the
past two years. I am thankful to David Shoemaker and Scott A. Hughes for their support as thesis
committee members. Their constructive criticism assisted me with writing my thesis better.
I would also like to thank my collaborators on the research outside of the 40m squeezing project.
Kentaro Somiya at Albert Einstein Institute and Yanbei Chen at Caltech are impressive people andalso fun to work with on displacement-noise-free interferometers. I am also grateful for Kentaro’s
help on understanding squeezing in the two-photon picture. I also thank my personal researchcollaborators in the MIT Spectroscopy Lab, Gabriel Popescu (currently at University of Illinois at
Urbana-Champaign), Ramachandra R. Dasari, and Michael Feld, and Takahiro Ikeda (currently at
Hamamatsu Photonics). I would like to send my special thanks to Takahiro for showing me theinteresting labs and factory of Hamamatsu Photonics and introducing me to the president of thecompany, Teruo Hiruma. I personally admire Hamamatsu’s products since their quality is extremely
important for basic science research. I am also thankful for another personal research collaborator,
Eriko Watanabe, in Kodate Lab at Japan Women’s University. Her strong motivation has always
stimulated me. Seiji Kawamura at National Astronomical Observatory of Japan (NAOJ) kindly
gave me many seminar opportunities at NAOJ.
Outside work hours, there have been numerous people who have helped me. I would like tothank (in alphabetical order) Akira Okutomi (Nikon), Alexander Patrikalakis (Computer Science,
MIT), Arisa Watanabe, Ayako Komuro, Gautam Kene (Law, Columbia University), Haruka Tanji(Physics, Harvard University), Hiromi Sato, Kanako Ueno (Institute of Industrial Science, Universityof Tokyo), Kanna Watanabe (Massachusetts General Hospital), Kenji Taira (Olympus Corporation),
Kenya Suzuki (NTT Photonics Laboratory), Magnus Hsu (Quantum Optics, Australian National
University), Masaru Tsuchiya (Applied Physics, Harvard University), Misayo Matsumoto (Coach),Nami Yokofuke, Naoko Kurahashi (Physics, Stanford University), Noriko Shimodaira (XL Soft Corporation), Noriko Yogo (Sun International Group), Paula Popescu (Physics, Harvard University),
Rhea M. Dah-Nomoto (Medical, University of Southern California), Saori Kato, Saori Kitagawa, Tatsuya Takahashi (Silicon Library Inc.), Tomoko Tada (Neuroscience, MIT), Yasunori Chiaya (Gehry
Partners), Yasushi Takamatsu (Mitsui Chemicals), Yui Ikeshima (L.A. Walker), Yuka Oshimi (L.A.
Shokutsu Club), and my parents Emiko Goda and Masao Goda. I would like to cherish my memory
with my MIT colleagues who have spent a few to several years at MIT as graduate students with
me: Fuwan Gan (Research Lab. of Electronics, MIT), Bo Bai (Goldman Sacks), Ryan Rygg (Plasma

Fusion Center, MIT), Yoshiaki Kuwata (Aeronautics and Astronautics, MIT), Shinya Kurebayashi(Goldman Sacks), and Fumiaki Toyama (Advanced Micro Devices). I am sure they will be successful anywhere they are. In addition, I have enjoyed 5 years of soccer life at MIT with dozens of
teammates.
Finally, I thank SPIE, Optical Society of America (OSA), and New Focus Inc. for awards andscholarships. Their financial support helped me with attending conferences and meeting people
for discussion. SPIE and OSA also helped me establish SPIE and OSA student chapters at MIT
to encourage optics research on campus. I acknowledge National Science Foundation’s long-term
financial support for the squeezing research. This work has been supported by the National Science
Foundation Grant Nos. PHY-0107417 and PHY-0300345.
- Keisuke Goda August 8, 2007

8

Contents
Introduction

21

1 Background

25

1.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

1.2

Gravitational Waves in General Relativity . . . . . . . . . . . . . . . . . . . . . . . .

25

1.2.1

The Nature of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . .

25

1.2.2

Gravitational Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

1.2.3

Astronomical Sources of Gravitational Waves . . . . . . . . . . . . . . . . . .

29

Laser-Interferometric Gravitational-Wave Detectors . . . . . . . . . . . . . . . . . . .

33

1.3.1

The History of Gravitational-Wave Detectors . . . . . . . . . . . . . . . . . .

33

1.3.2

The Principle of Laser-Interferometric Gravitational-Wave Detectors . . . . .

34

Quantum Noise in Gravitational-Wave Detectors . . . . . . . . . . . . . . . . . . . .

36

1.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

1.4.2

The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . .

37

1.4.3

Quantum Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

1.4.4

Quantum Noise in LIGO and Advanced LIGO . . . . . . . . . . . . . . . . .

43

1.4.5

Quantum Limit by Quantum Fluctuations of a Vacuum Field . . . . . . . . .

47

Quantum-Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

1.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

1.5.2

Ideal Squeezed State Production and Quantum Enhancement . . . . . . . . .

48

1.5.3

Previous Experimental Efforts . . . . . . . . . . . . . . . . . . . . . . . . . .

52

The Goal of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

1.3

1.4

1.5

1.6

2 Theory of Squeezed States

55

2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

2.2

States of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

2.2.1

The Ball-on-Stick Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

2.2.2

Equivalence of Squeezed Light and Squeezed Vacuum . . . . . . . . . . . . .

59

9

2.3

Generation of Squeezed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

2.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

2.3.2

Quantization of Quadrature Field Amplitudes in Two-Photon Formalism . .

60

2.3.3

Generation of Squeezed States in Optical Parametric Oscillation . . . . . . .

63

2.3.4

The Ball-on-Stick Picture Revisited . . . . . . . . . . . . . . . . . . . . . . .

71

Second-Order Nonlinear Optical Processes for Squeezed State Production . . . . . .

74

2.4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

2.4.2

Atomic Polarization of a Dielectric Medium . . . . . . . . . . . . . . . . . . .

74

2.4.3

Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

2.4.4

Phase Matching Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

2.4.5

Quasi-Phase-Matching with Periodically Poled Materials . . . . . . . . . . . .

78

2.4.6

Second-Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

2.4.7

Classical Optical Parametric Oscillation . . . . . . . . . . . . . . . . . . . . .

86

2.4.8

Designing Stable Resonators with Nonlinear Media . . . . . . . . . . . . . . .

86

2.5

Degradation of Squeezed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

2.6

Detection of Squeezed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

2.6.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

2.6.2

Balanced Homodyne Detection . . . . . . . . . . . . . . . . . . . . . . . . . .

92

2.6.3

Unbalanced Homodyne Detection . . . . . . . . . . . . . . . . . . . . . . . . .

95

2.4

2.7

Control of Squeezed States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

2.7.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

2.7.2

Quantum Noise Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

2.7.3

Coherent Control of Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3 Experimental Generation, Detection, and Control of Squeezed States

107

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.2

Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.3

3.2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.2.2

Laser

3.2.3

Second-Harmonic Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.2.4

Optical Parametric Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2.5

Subcarrier Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.2.6

Homodyne Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.2.7

Quantum Noise Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.3.2

Spectrum of Scanned Squeezed Shot Noise . . . . . . . . . . . . . . . . . . . . 116
10

3.3.3Spectrum of Locked Squeezed Shot Noise . . . . . . . . . . . . . . . . . . . . 118

3.3.4

Verification of Quantum Correlations . . . . . . . . . . . . . . . . . . . . . . . 119

4 Requirements for Quantum Enhanced Gravitational Wave Detectors

121

4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.2

Squeezing in the Gravitational Wave Band . . . . . . . . . . . . . . . . . . . . . . . . 121

4.3

4.4

4.5

4.2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.2.2

Seed Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.2.3

Pump Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.2.4

Photothermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.2.5

Scattered Photon Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

High Level of Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.3.2

Low Optical Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.3.3

Recent Progress in Crystal Development . . . . . . . . . . . . . . . . . . . . . 128

4.3.4

Cavity Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Long-Term Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.4.2

Control of OPO Cavity Resonance . . . . . . . . . . . . . . . . . . . . . . . . 133

4.4.3

Control of OPO Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.4.4

Control of Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Frequency-Dependent Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.5.2

Cavities with Narrow Linewidths . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.5.3

Filters by Use of Electromagnetically Induced Transparency . . . . . . . . . . 139

4.5.4

Filters for Squeeze Amplitude Attenuation . . . . . . . . . . . . . . . . . . . . 145

5 Theory of Quantum Enhanced Gravitational Wave Detectors

149

5.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.2

Theory of a Quantum-Enhanced Signal-Recycled Michelson Interferometer . . . . . . 150

5.3

5.2.1

Equivalent Optical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.2.2

Quadrature Field Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.2.3

Quadrature Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Theory of Quantum-Enhancement in Advanced Interferometer Configurations . . . . 157

6 Demonstration of Quantum Enhancement in a Gravitational Wave Detector
6.1

159

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
11

6.2

Caltech 40m LIGO Interferometer Prototype . . . . . . . . . . . . . . . . . . . . . . 160

6.3

Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.4

6.3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3.2

Pre-Stabilized Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.3.3

Mode Cleaner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.3.4

Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.3.5

Squeezed Vacuum Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.3.6

Length Sensing Photodetector. . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.3.7

Readout and Control Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . 168

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.4.2

Broadband Squeezing-Enhancement in the Interferometer . . . . . . . . . . . 170

6.4.3

Increase in Signal-to-Noise Ratio by Squeezing-Enhancement . . . . . . . . . 172

7 The Future

173

7.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.2

Possible Future Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3

7.2.1

Bow-Tie Optical Parametric Oscillators . . . . . . . . . . . . . . . . . . . . . 173

7.2.2

Doubly-Resonant Optical Parametric Oscillators . . . . . . . . . . . . . . . . 174

Possible Future Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.3.1

Generation of Squeezed States in Optical Waveguides . . . . . . . . . . . . . 174

7.3.2

Generation of Squeezed States in Microcavities . . . . . . . . . . . . . . . . . 175

7.3.3

Metamaterial-Enhanced Optical Parametric Oscillation . . . . . . . . . . . . 175

7.3.4

Quantum-Enhanced Laser Interferometers in Resonant Sideband Extraction . 176

7.3.5

Quantum-Enhanced Laser Interferometers with an Output Mode Cleaner . . 176

7.3.6

Coherent Control of Squeezing with an Output Mode Cleaner . . . . . . . . . 177

Conclusion

179

Appendices181

A Tables of Constants

181

A.1 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.2 Astrophysical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
B Tables of Acronyms

183

B.1 LIGO Related Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
B.2 Squeezing Related Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
12

C Tables of Crystal Properties

185

C.1 Potassium Titanyl Phosphate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
C.2 Lithium Niobate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
D Expression of Noise in Decibels

187

E Publications

189

F Other Work191

F.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
F.2 Noninvasive Measurements of Cavity Parameters by Use of Squeezed Vacuum . . . . 191
F.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
F.2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
F.2.3 Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
F.3 Frequency-Resolving Spatiotemporal Wavefront Detection . . . . . . . . . . . . . . . 197
F.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
F.3.2 Description of Frequency-Resolving Wavefront Sensing . . . . . . . . . . . . . 198
F.3.3 Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
F.4 Displacement-Noise-Free Interferometers and Utility of Time-Delay Devices . . . . . 202
F.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
F.4.2 Detector Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
F.4.3 Detector Response to Gravitational Waves . . . . . . . . . . . . . . . . . . . . 206
F.4.4 Practicability of 3-D DFI Detectors. . . . . . . . . . . . . . . . . . . . . . . 207

F.4.5 Utility of Time-Delay Devices in DFI Detectors . . . . . . . . . . . . . . . . . 208
F.5 Quantitative Phase Imaging by Use of Stabilized Optical Interferometry . . . . . . . 208
F.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
F.5.2 Stabilized Hilbert Phase Microscopy . . . . . . . . . . . . . . . . . . . . . . . 209
F.5.3 Measurements of Cell Membrane Tension . . . . . . . . . . . . . . . . . . . . 211

13

14

List of Figures
1-1 A Michelson interferometer as a GW detector. . . . . . . . . . . . . . . . . . . . . . .

35

1-2 Shot noise and quantum radiation pressure noise in a Michelson interferometer. . . .

42

1-3 Interferometer configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

1-4 The strain sensitivity of the LIGO interferometers during the S5 science run. . . . .

45

1-5 The design strain sensitivity of Advanced LIGO. . . . . . . . . . . . . . . . . . . . .

46

1-6 A schematic of a Michelson interferometer with squeezing. . . . . . . . . . . . . . . .

48

1-7 The effect of squeezing on a Michelson interferometer. . . . . . . . . . . . . . . . . .

49

1-8 The effect of squeezing on Advanced LIGO. . . . . . . . . . . . . . . . . . . . . . . .

50

1-9 The quadrature variances of a1 and a2 as functions of φ. . . . . . . . . . . . . . . . .

51

2-1 Ball-on-stick pictures (first). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2-2 Ball-on-stick pictures (second). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

2-3 The signal and idler fields relative to the frequency of a carrier field. . . . . . . . . .

61

2-4 A model of an optical cavity with a second-order nonlinear medium. . . . . . . . . .

64

2-5 The production of a squeezed state in the ball-on-stick picture. . . . . . . . . . . . .

72

2-6 Wigner functions of a coherent state and squeezed states. . . . . . . . . . . . . . . .

73

2-7 The effect of the phase mismatch on the SHG conversion efficiency. . . . . . . . . . .

76

2-8 A homogeneous single crystal and a periodically poled crystal.

. . . . . . . . . . . .

78

2-9 The evolution of a second-harmonic field in BPM and QPM crystals. . . . . . . . . .

80

2-10 The SHG conversion efficiency of PPKTP against thermal expansion.

. . . . . . . .

83

2-11 A model of a SHG cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

2-12 The SHG conversion efficiency for various input coupler reflectivities. . . . . . . . . .

85

2-13 The parametric gain obtained through an OPO cavity. . . . . . . . . . . . . . . . . .

87

2-14 An optical cavity composed of two external mirrors and a crystal in between them. .

872-15 g1eff g2eff and ω0 as a function of d1 when d1 = d2 . . . . . . . . . . . . . . . . . . . . .

90

2-16 A beamsplitter model of squeezing degradation. . . . . . . . . . . . . . . . . . . . . .

902-17 A schematic of a balanced homodyne detector. . . . . . . . . . . . . . . . . . . . . .

92

2-18 A schematic of an unbalanced homodyne detector. . . . . . . . . . . . . . . . . . . .

9715

2-19 A schematic of a balanced homodyne detector with a noise-locking servo. . . . . . . 100
2-20 The squeezed quadrature variance compared to the noise-locking error signal. . . . . 101
2-21 The stability of the two locking points as a function of squeeze factor r. . . . . . . . 103
3-1 A schematic of the squeezer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3-2 A detailed schematic of the SHG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3-3 The temperature dependence of the SHG conversion efficiency. . . . . . . . . . . . . 110
3-4 The mode structure of the SHG cavity and corresponding second-harmonic modes. . 111
3-5 A detailed schematic of the OPO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3-6 The temperature dependence of the nonlinear interaction strength in the OPO. . . . 113
3-7 The dependence of the OPO parametric gain on the pump power.

. . . . . . . . . . 114

3-8 The stability of the noise locking technique. . . . . . . . . . . . . . . . . . . . . . . . 116
3-9 The fixed-frequency spectra of shot noise and squeezed shot noise. . . . . . . . . . . 117
3-10 The broadband spectra of shot noise and squeezed shot noise. . . . . . . . . . . . . . 118
3-11 The degradation of squeezing as a function of the subcarrier detuning. . . . . . . . . 119
4-1 The history of observed squeezing levels. . . . . . . . . . . . . . . . . . . . . . . . . . 125
4-2 The degradation of squeezing for various transmissivities or reflectivities. . . . . . . . 127
4-3 Comparisons in the generated level of squeezing between MgO:LiNbO3 and PPKTP. 129
4-4 The theoretically achievable levels of squeezing as a function of the pump power. . . 131
4-5 The optical configuration and readout scheme of Advanced LIGO. . . . . . . . . . . 136
4-6 Frequency-independent squeezing in a conventional GW detector. . . . . . . . . . . . 138
4-7 An EIT system and a proposed configuration for a squeeze EIT filter. . . . . . . . . 140
4-8 The noise spectral density of a GW detector with the effect of an EIT filter. . . . . . 143
4-9 A schematic of a squeeze amplitude filter cavity. . . . . . . . . . . . . . . . . . . . . 146
4-10 Filtered and unfiltered squeezing/anti-squeezing spectra. . . . . . . . . . . . . . . . . 148
5-1 A model of a Michelson interferometer with a two-photon correlator inserted. . . . . 150
5-2 An equivalent optical model of a Michelson interferometer. . . . . . . . . . . . . . . . 151
5-3 An equivalent optical model of a quantum-enhanced SRMI. . . . . . . . . . . . . . . 152
5-4 Quadrature rotation angles α+ and β+ for various SRC detunings. . . . . . . . . . . 155
5-5 The simulated squeezed quadrature variance of an interferometer output field. . . . . 156
6-1 A schematic of the quantum-enhanced prototype gravitational-wave detector. . . . . 161
6-2 The noise floor of the SRMI detector for different optical powers. . . . . . . . . . . . 164
6-3 A detailed schematic of the injection optics. . . . . . . . . . . . . . . . . . . . . . . . 165
6-4 A detailed schematic of the readout and control schemes. . . . . . . . . . . . . . . . 168
6-5 The noise floor of the SRMI detector with and without squeezing.
16. . . . . . . . . . 171

6-6 An increase in SNR by squeezing with a simulated GW signal. . . . . . . . . . . . . 172
7-1 A schematic of a bow-tie OPO cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . 174
F-1 Measured squeezed and anti-squeezed shot noise spectra and fits. . . . . . . . . . . . 196
F-2 A schematic of the wavefront sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
F-3 The frequency spectrum that was incident upon the wavefront sensor. . . . . . . . . 200
F-4 The measured amplitude and phase of the test sideband. . . . . . . . . . . . . . . . . 201
F-5 The DFI octahedron configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
F-6 The orthonormal system used to describe a generic plane GW. . . . . . . . . . . . . 205
F-7 The response of the DFI detector to GWs. . . . . . . . . . . . . . . . . . . . . . . . . 207
F-8 A schematic of the stabilized Hilbert phase microscope. . . . . . . . . . . . . . . . . 210

17

18

List of Tables
2.1

Nonlinear crystals often used in optical parametric oscillation. . . . . . . . . . . . . .

74

2.2

Sellmeier equation coefficients for flux-grown KTP. . . . . . . . . . . . . . . . . . . .

82

4.1

Comparison of the optical properties of different nonlinear media. . . . . . . . . . . . 126

4.2

Advantages and disadvantages of the quantum noise locking technique. . . . . . . . . 135

4.3

Advantages and disadvantages of the coherent control technique. . . . . . . . . . . . 137

A.1 Values of physical constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.2 Values of astrophysical constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
B.1 LIGO related acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
B.2 Squeezing related acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
C.1 Properties of Potassium Titanyl Phosphate (KTiOPO4 ). . . . . . . . . . . . . . . . . 185
C.2 Properties of Lithium Niobate (LiNbO3 ). . . . . . . . . . . . . . . . . . . . . . . . . 186

19

20

Introduction
Laser-interferometric gravitational wave (GW) detectors [1, 2] are designed to measure distancechanges on the order of 10−18 m caused by GWs from astronomical sources, such as coalescences ofneutron stars and black holes, supernova explosions, and the Big Bang, providing further verificationof Einstein’s General Theory of Relativity and opening an entirely new window onto the universe [3].
However, the sensitivity of laser-interferometric GW detectors is ultimately limited by quantum noisethat comes from the quantum statistics of photons due to the Heisenberg uncertainty principle. The
sensitivities of the currently operational GW detectors such as Laser Interferometer GravitationalWave Observatory (LIGO) [1], VIRGO [4], GEO600 [5], and TAMA300 [6] are already limited byquantum noise at high frequencies in the GW detection band (10 Hz - 10 kHz). Next generation GW
detectors such as Advanced LIGO [7], which are planned to be operational in the next few years,
are expected to be limited by quantum noise at almost all frequencies in the GW detection band.
This quantum limit can be circumvented by the injection of nonclassical, or squeezed, states oflight [8, 9] into the antisymmetric port of the interferometer [10]. Following the 1981 proposal of
Caves to improve the sensitivity of quantum-noise-limited laser interferometers by squeezed state
injection [10], a handful of experimental efforts have realized the proof-of-principle on the table-top
scale at MHz frequencies. The pioneering experiment was performed by Xiao et al. [11] using a MachZehnder interferometer a few years after the first observation of squeezed states by Slusher et al. in1985 [12]. Later, squeezing-enhancement in Michelson interferometer configurations similar to theoptical configuration used in the current and future large-scale GW detectors were demonstrated by
McKenzie et al. [13] and Vahlbruch et al. [14]. However, these demonstrations were not yet practical
for the implementation of squeezing-enhancement in large-scale GW detectors such as LIGO andAdvanced LIGO.
In this thesis, theoretical analysis and experimental demonstration of techniques necessary forquantum-enhanced laser-interferometric GW detectors are presented. The basic theory of GWs,
laser-interferometric GW detectors, and quantum noise in GW detectors, the theory of squeezed
states including generation, degradation, detection, and control of squeezed states in sub-threshold
optical parametric oscillators (OPOs) and homodyne detectors, experimental characterization ofthese techniques, requirements for quantum-enhanced GW detectors, and the propagation of a21

squeezed state in a complex interferometer and its interaction with the interferometer field arestudied. Finally, the first experimental demonstration of quantum-enhancement in a prototype GW
detector with suspended optics and control and readout schemes similar to those used in the currently operational LIGO detectors and envisioned for Advanced LIGO [7] is presented [15]. By
injecting a squeezed vacuum of 9.3 dB (inferred) or 7.4 ± 0.1 dB (measured) at frequencies above 3
kHz and a cutoff frequency for squeezing at 700 Hz [16] into the antisymmetric port of the prototype
GW detector in a signal-recycled Michelson interferometer (SRMI) configuration, the shot noise√√
floor of the detector is reduced broadband from 7.0 × 10−17 m/ Hz to 5.0 × 10−17 m/ Hz while thestrength of a simulated GW signal is retained, resulting in a 40% increase in signal-to-noise ratio
or detector sensitivity, which is equivalent to a factor of 1.43 = 2.7 increase in GW detection rate
for isotropically distributed GW sources that are confined to the frequency band in which squeezing
is effective [15]. This is a critical step toward implementation of squeezing-enhancement in long
baseline GW detectors.
Squeezed states are typically generated using nonlinear crystals such as lithium niobate (LiNbO3 )and periodically poled potassium titanyl phosphate (PPKTP) in sub-threshold OPOs. OPOs produce squeezed states of light by correlating the upper and lower quantum sidebands centered aroundthe frequency of a carrier field in the presence of an energetic pump field. Since all the GW detectors
presently use high power Nd:YAG lasers sources at 1064 nm, generating squeezed states at 1064 nm
is essential. Observation of squeezing in the GW detection band has been reported by McKenzie etal. [17] and Vahlbruch et al. [18].
This thesis is organized as follows: Chapter 1 describes the physics of GWs, possible sourcesof GWs, the basic theory of laser-interferometric GW detectors, and quantum noise in the GW
detectors, and briefly addresses the proposed scheme of quantum-enhancement in the GW detectors.
The goal of this thesis is also addressed. Chapter 2 describes the theory of squeezed states including
the generation, degradation, detection, and control of squeezed states using sub-threshold OPOs andhomodyne detectors in the two-photon formalism [19, 20]. Chapter 3 describes the experimental
characterization of the techniques discussed in Chapter 2 and presents experimental results. Chapter
4 discusses key requirements for quantum-enhanced GW detectors such as the frequency band,
level, long-term stability, and frequency-dependence of squeezing. Chapter 5 describes the theoryof quantum-enhanced GW detectors, especially in a quantum-enhanced SRMI configuration, using
a two-photon mathematical framework of quadrature field propagation. Chapter 6 describes theexperimental demonstration of quantum-enhancement in a prototype GW detector in the SRMI
configuration using the techniques that have been developed throughout the period of my Ph.D.
work. Finally, Chapter 7 discusses possible future improvements and investigations toward the actual
implementation of the quantum-enhancement in long baseline laser-interferometric GW detectors
such as Advanced LIGO. The appendices show tables of constants, acronyms, and crystal properties,
22

a list of publications, and other work which may or may not be related to squeezing.
This work has been done at both Massachusetts Institute of Technology (MIT) and California Institute of Technology (Caltech). The research and development of techniques necessary forquantum-enhanced GW detectors have been conducted at the MIT LIGO Lab while the experimental demonstration of quantum-enhancement in the prototype GW detector has been done at theCaltech LIGO 40m Lab. This work has been supported by the National Science Foundation Grant
Nos. PHY-0107417 and PHY-0300345.

23

24

Chapter 1

Background
1.1

Overview

In this chapter, the background of quantum-enhancement in laser-interferometric gravitational wave
(GW) detectors is described. In Sec. 1.2, the physics of GWs in General Relativity, a basic theoryof gravitational radiation, and possible sources of GWs are described. In Sec. 1.3, the history of
GW detectors and the principle of laser-interferometric GW detectors are described. In Sec. 1.4, theorigin of quantum noise in GW detectors is described by deriving the quantum noise semi-classically.
The most rigorous derivation of quantum noise in GW detectors will be given in Chap. 5. In Sec.
1.4, it is also shown how quantum noise limits the sensitivities of the LIGO and Advanced LIGO
detectors. In Sec. 1.5, the proposed idea of quantum-enhancement by injection of squeezed states
into the antisymmetric port of GW detectors is introduced and previous experimental efforts toward
quantum-enhancement are described. Finally, the goal of the thesis is addressed in Sec. 1.6.

1.2

Gravitational Waves in General Relativity

1.2.1

The Nature of Gravitational Waves

The existence of GWs was predicted by Albert Einstein in 1916 in his famous General Theory ofRelativity [21, 22] which is based on Einstein’s earlier theory of Special Relativity and the equivalenceprinciple, and utilizes the mathematics of Riemannian geometry [23]. In contrast with Newton’s lawof gravitation, it postulates that gravitation is not due to a force in the conventional sense, but isan aspect of the geometry of space and time – or spacetime as he called it. GWs are ripples in the
fabric of spacetime caused by the motion of matter and travel at the speed of light as opposed to
Newton’s law of gravitation in which gravitation is instantaneous action.
In General Theory of Relativity, the infinitesimal spacetime interval ds is given in terms of the25infinitesimal coordinate displacement dxµ in the Cartesian coordinate system, xµ = (ct, x), with thetime coordinate t and the spatial coordinates x, by
ds2 = gµν dxµ dxν ,(1.1)

where the metric gµν is given by the Einstein field equations
18πG
Rµν − Rgµν = 4 Tµν .
2
c

(1.2)

Here Rµν is the Ricci curvature tensor, which can be regarded as the result of applying a secondorder nonlinear differential operator to the metric gµν , R = g µν Rµν is the scalar curvature, and Tµν
is the stress-energy tensor. The constants G and c are the gravitational constant and the speed oflight in vacuum, respectively. Eq. (1.2) can thus be regarded as a set of nonlinear partial differential
equations for the metric with Tµν as a source term.
If the curvature of spacetime is weak, the metric gµν can be approximated by a small perturbation
to flat spacetime, or Minkowski space,gµν % ηµν + hµν ,(1.3)

where ηµν is the Minkowski metric given in Cartesian coordinates by


−1



 0
ηµν = 

 0

0

0

0

1

0

0

1

0

0

0





0 


0 

1

(1.4)

and hµν is the metric perturbation away from Minkowski space. In this weak-field limit, the nonlinear Einstein field equations can be approximated as linear equations. In the transverse-traceless(TT) gauge, coordinates are marked out by the world lines of freely-falling test masses, and theperturbation metric hµν only has spatial components [24, 25]
hTT (t, x) = h+ (t − eZ · x/c) [eX ⊗ eX − eY ⊗ eY ] + h× (t − eZ · x/c) [eX ⊗ eY + eY ⊗ eX ], (1.5)
where (eX , eY , eZ ) is a spatial orthonormal set with the wave propagation direction eZ , h+ and h×are the two polarizations of the perturbation metric, and hTT (t, x) satisfies the weak-field limit of
the Einstein field equations
'

∇2 −

1 ∂2
c2 ∂t2(hTT (t, x) = 0.
26

(1.6)

Thus, the wave propagates in the direction eZ · x at the speed c.

1.2.2

Gravitational Radiation

Gravitational Quadrupole Radiation
Gravitational radiation is analogous to electromagnetic radiation in that GWs are produced byaccelerating masses just as electromagnetic waves are produced by accelerating charges, with one
difference that GWs are not radiated by the dipole moment unlike electromagnetic waves due toconservation laws. Energy conservation prohibits the monopole radiation of GWs since the energyof the source is constant or
Ė
d
=
c2
dt

)

dV ρ(r) = 0,

(1.7)

where ρ(r) is the mass density, r is the location of the source with respect to any origin we canchoose, and the integral runs over the volume of the source. Likewise, linear momentum conservationprohibits the gravitational analog of the electric dipole radiation since the gravitational analog ofthe electric dipole moment is constant or
ṗ =

d
dt

)

dV ρ(r)r = 0.

(1.8)

Furthermore, angular momentum conservation prohibits the gravitational analog of the magnetic
dipole radiation since the gravitational analog of the magnetic dipole moment is constant or
ṁ =

d
dt

)

dV ρ(r)r × v(r) = 0.

(1.9)

As relevant conservation laws run out, the lowest-order moment whose time derivative does notvanish is the quadrupole moment. The reduced quadrupole moment is defined by [24, 26]
Iµν ≡

)

dV

'

(
1
xµ xν − δµν r2 ρ(r).
3(1.10)

It is called “reduced” because it is the term used to describe the moment when the trace has been
removed. The gravitational luminosity LGW is given by [24]
LGW ≡

dE
1G
=
dt
5 c5

* 3
+
d Iµν d3 Iµν.
dt3 dt3(1.11)

Therefore, objects with quadrupole moments whose third-order time derivatives do not vanish radiate
GWs.
In general, GWs are radiated by objects whose motion involves acceleration, provided that the27

motion is not spherically symmetric (like a spinning, expanding, or contracting sphere) or cylindrically symmetric (like a spinning disk). A simple example is the spinning dumbbell. If it spins
along the axis between the two masses, it does not radiate. If it spins along an axis perpendicular to the line between the masses, it radiates. Some examples of objects that radiate GWs are
two objects orbiting each other in a quasi-Keplerian planar orbit (like a planet orbiting the Sun),a spinning non-axisymmetric planetoid (like a star with a large bump or dimple on its equator),
and a spherically asymmetric supernova. On the other hand, some examples of objects that do notradiate GWs are an isolated non-spinning solid object moving at a constant speed (due to the linear
momentum conservation), a spinning disk (due to the angular momentum conservation) although it
shows gravitomagnetic effects, and a spherically pulsating spherical star (due to Birkhoff’s theorem).

The Spinning Dumbbell Model
Suppose that a spinning dumbbell consists of two spherical masses of M1 and M2 attached to theends of a rod with a length of a between the centers of the masses. Assume that the mass of the rod
is negligible and the dumbbell spins at angular frequency Ω about the center of mass in the plane
perpendicular to the rod. The strongest allowed component of the metric perturbation caused by
the spinning dumbbell is given by [26]hµν (t) =2G ¨Iµν (t − R/c) ,Rc4(1.12)

where R is the distance from the source to the observer with the assumption of R ) a, and G is thegravitational constant. If we further assume that the orbital plane of the dumbbell is normal to thez direction, the components of the quadrupole moment are given by
Ixx
Iyy

(
R
1
= µa cos Ω t −
−
,
c
3
,
'
(
R
1
= µa2 sin2 Ω t −
−
,
c
3
2

,

2

'

(1.13)(1.14)

and
'
(
'
(
R
R
Ixy = Iyx = µa2 sin Ω t −cos Ω t −,cc(1.15)

where µ is the reduced mass given by µ ≡ M1 M2 /M and M is the total mass given by M ≡ M1 +M2 .
Plugging the second derivatives of Eqs. (1.13), (1.14), and (1.15) into Eq. (1.12) gives
hxx
hxy

'
(
R
= −hyy = −hGW cos 2Ω t −
,
c
'
(
R
= hyx = −hGW sin 2Ω t −,c
28(1.16)(1.17)

where hGW is the amplitude of the oscillation given by
hGW =4Gµa2 Ω2.
Rc4

(1.18)

If it is possible to have an experimental setup that allows extreme parameters such as M1 = M2 =1000 kg, a = 1 m, Ω/(2π) = 1 kHz, and R = 3000 km (the distance between the LIGO Hanford
Site and the LIGO Livingston Site), then the GW-induced strain at the detector is hGW ∼ 10−37 .
Suppose that a bar (or an interferometer with arms) of L = 100 km is used to detect the GW-induced
strain, and then, the induced length change of the bar (or the interferometer) is ∆L = hGW L ∼ 10−32
m. This is extremely small, and below the resolution of any presently known measurement system.
Therefore, it is not technically feasible to generate and detect GWs in a laboratory. For this reason,in order to detect GWs, we need violent astronomical events in the universe such as coalescences
and collisions of stars and supernova explosions. Astronomical sources of GWs will be described in
the next section.

1.2.3

Astronomical Sources of Gravitational Waves

Overview
A variety of objects and processes in the universe radiate detectable GWs on Earth. They include(1) inspirals of binary star systems such as neutron star (NS) and black hole (BH) binaries, (2)pulsars with spin precession, an excited NS oscillation mode, and small distortions of the NS shapeaway from axisymmetry, (3) bursts from sources such as stellar collapses, the mergers of compact
binary star systems, the generators of gamma ray bursts, and other energetic phenomena, and(4) stochastic background from the early universe (cosmological) and the random superposition ofmany weak signals from binary star systems (astrophysical). In this section, these GW sources are
discussed.

Inspirals of Binary Star Systems
Radio observations of pulsars confirm the existence of binary NS systems in our Galaxy [27, 28].
General Relativity predicts the decay of a binary orbit due to the emission of gravitational radiation.
The decay rate inferred from observations of PSR1913+16 agrees with the prediction within 0.3%
[29, 30, 31]. Other than PSR1913+16, there are a number of systems known to exhibit orbital decay
in accord with the quadrupole formula [29]. The orbital decay is easily modeled for compact binary
systems containing NSs or stellar mass BHs. The binary orbit is expected to evolve through the
LIGO detection band by the emission of GWs alone, making it possible to accurately compute theevolution without reference to complicated micro-physics.
29

When a compact binary system first forms, the orbit may be widely separated and highly eccentric. Gravitational radiation causes the orbit to shrink and circularize so that the binary components
eventually spiral together along a sequence of nearly circular orbits [28]. For binary NSs or stellar
mass BHs, the gravitational radiation eventually enters the frequency band of Earth-based GW
detectors. At this point, the orbit decays rapidly and the gravitational waveform chirps upward in
frequency and amplitude, sweeping through the LIGO detection band [32, 33].

For low-mass binary systems, the waveforms are well approximated by a post-Newtonian expansion in the LIGO detection band. If the dumbbell in Sec. 1.2.2 is a binary star system with masses
M1 and M2 with no rod, but the two stars are influenced by the gravitational force between eachother, then the gravitational interaction between the two stars gives a simple expression for the spin.
(orbiting) angular frequency Ω = GM/a3 . The metric perturbation is given in terms of more

physically intuitive quantities by

hGW =rS1 rS2,
aR(1.19)

where rS1 and rS2 are the Schwarzschild radii of the stars defined by rS1 ≡ 2GM1 /c2 and rS2 ≡2GM2 /c2 . A rough estimate for the amplitude of the metric perturbation in Eq. (1.19) is−21

hGW ∼ 10

'

10 km
a

('

10 Mpc
R

('

M1
M#

('

M2
M#

(

.

(1.20)For a NS/NS binary, the typical NS mass is near the Chandrasekhar mass, M1 % M2 % 1.4M# . Fora larger strain amplitude, a and R should be as small as possible.

For a binary star system with eccentricity e, the gravitational luminosity averaged over an orbital
period is given from Eq. (1.11) by
32 G4 µ2 M 3
f (e),
5 c5 a5(1.21)(
7
73 2 37 4
1 + e + e (1 − e2 )− 2
24
96(1.22)

LGW =where
f (e) ='

is the eccentricity correction function. As the binary system loses energy by gravitational radiation,
the stars spiral in toward each other. For circular orbits (e = 0), the total energy of the systemgiven by E = −GµM/(2a) decreases asdE
GµM da32 G4 µ2 M 3
.
=
=
−L
=
−
GW
dt
2a2 dt5 c5 a530(1.23)

Consequently, the separation between the stars evolves asa3 ȧ = −64 G3 µM 2,5c5(1.24)

which indicates that the rate of decrease in the separation ȧ increases as a decreases. Likewise, using.
the orbital angular frequency Ω = GM/a3 , the orbital period P = 2π/Ω evolves asṖ
96 G3 µM 2=−,P
5 c5 a4(1.25)

which indicates that P decreases as the separation a decreases.

Pulsars
Current models of stellar evolution suggest that our Galaxy contains of order 109 NSs and that oforder 105 are active pulsars [34, 35]. Periodic GWs in the LIGO detection band are produced by(1) NS spin precession [36, 37], and (2) small distortions of the NS shape away from axisymmetry[38, 39, 40, 41, 42]. At present, the second mechanism seems to be the most promising source ofdetectable GWs [43, 44].
A NS precesses or wobbles when its angular momentum is not aligned with any principal axis of its
inertia tensor. A wobbling NS emits GWs at the inertial frame precession frequency, which is nearly
the rotation frequency. While large-amplitude wobble could plausibly produce GW amplitudes
hGW ∼ 10−27 over short timescales, the problem with this mechanism is that dissipation shoulddamp NS wobble quickly. In addition, most millisecond pulsars have highly stable pulse shapes,
and typically appear not to be wobbling substantially. The single-pulse characteristics of PSR
J1939+2134 have been observed to be extremely stable.
In the case of GWs from small non-axisymmetries in the NS shape, assuming that the GW
emission mechanism is due to deviations of the NS shape from perfect axial symmetry and the GWs
are emitted at a frequency which is exactly twice the NS rotation frequency fr , the waveforms forthe two polarizations h+,× are given by
h+ = hGW

1 + cos2 ζ
cos Φ(t),2h× = hGW cos ζ sin Φ(t),(1.26)

where ζ is the angle between the NS spin axis and the direction of the GW propagation, Φ(t) is thephase evolution of the GW signal in the Solar System Barycenter (SSB) frame, and hGW is the GW
amplitude given by [44, 45]hGW =16π 2 G Izz fr2
+,
c4
R
31

(1.27)

where R is the distance from the observer to the NS, Izz is its principal moment of inertia about
the rotation axis, and + ≡ (Ixx − Iyy )/Izz is the equatorial ellipticity.
One possible source of ellipticity is tiny hills in the NS crust, which are supported by crustal
shear stresses. For low-mass X-ray binaries (LMXBs), lateral temperature variations in the crust oforder 0.5% or lateral composition variations of 0.5% in the charge-to-mass ratio could build up NS
ellipticities of order 10−8 −10−7 . Another possible source of NS ellipticity is strong internal magnetic
fields. Toroidal magnetic fields of order 1012 − 1013 G would lead to sufficient GW emission to halt
the spin-up of LMXBs and account for the observed spin-down of millisecond pulsars [46].
A simple model of a pulsar is 1.4M# NS of radius 10 km with the principal moment of inertia
about the rotation axis Izz = 1045 g cm2 and the magnetic field B ∼ 1012 G. Then, a rough estimate
for the GW amplitude is given by
hGW ∼ 10−31

'

fr
1 kHz

(2 '

10 kpcR(.

(1.28)

If pulsars are born rapidly rotating, then there should be several of the most recent pulsars withsuch an amplitude present in our Galaxy at any time.

BurstsGW bursts are expected to be produced from catastrophic phenomena in the universe such as stellar
collapses, the mergers of compact binary star systems as they form a single body, the generators of
gamma ray bursts, and other energetic phenomena [47, 48, 33, 49, 50, 51]. Perturbed or accreting
BHs, NS oscillation modes and instabilities, as well as cosmic string cusps and kinks are also potential
burst sources. The expected rate, strength, and waveform morphology for such events are poorly
known.

Stochastic BackgroundA stochastic background of gravitational radiation is analogous to the cosmic microwave backgroundradiation although its spectrum is unlikely to be thermal [52]. Sources of a stochastic background
could be cosmological or astrophysical in origin. Examples of the former are zero-point fluctuationsof the spacetime metric amplified during inflation, and first-order phase transitions and decaying
cosmic string networks in the early universe [52, 53, 54, 55]. An example of an astrophysical source isthe random superposition of an extremely large number of unresolved and independent GW emission
events [52, 53, 56].
The spectrum of a stochastic background is usually described by the dimensionless quantityΩ̃GW (f ) which is the GW energy density per unit logarithmic frequency ρGW , divided by the critical32

energy density ρc to close the universe,Ω̃GW (f ) ≡f dρGW,ρc df(1.29)

where the critical density is ρc ≡ 3c2 H02 /(8πG) and H0 is the present day Hubble expansion rate.
The measurable (one-sided) power spectrum of the GW strain S̃GW (f ) is given by [57]S̃GW (f ) =3H02 −3f Ω̃GW (f ).
10π 2(1.30)For a stochastic GW background with Ω̃GW (f ) ≡ Ω̃0 (constant) as is predicted in the LIGO detection
band by inflationary models in the infinitely slow-roll limit or by cosmic string models [58, 54], thepower in GWs falls off as 1/f 3 [57, 56]. A rough estimate for the GW strain amplitude in this caseis given byh̃GW (f ) =/−22

S̃GW (f ) = 5.6 × 10

h100

/

Ω̃0

'

100 Hzf(3/2

Hz−1/2 ,(1.31)

where h100 ≡ H0 /H100 and H100 ≡ 100 km/s/Mpc. Here we have assumed that the stochastic
background is isotropic, unpolarized, stationary, and Gaussian.

1.3

Laser-Interferometric Gravitational-Wave Detectors

1.3.1

The History of Gravitational-Wave Detectors

As stated already, GWs are ripples in the fabric of spacetime, and therefore, a passing GW changes
the separation of adjacent test masses. This tidal effect is the basis of all present GW detectors. The
great challenge of this type of detection, though, is the extraordinarily small effect that GWs would
produce on a detector. The amplitude of any wave falls off as the inverse of the distance from thesource. Thus, even GWs from merging binary BHs die out to a very small amplitude by the timethey reach the Earth. For GW sources occurring with a reasonable observation rate, the predicted
magnitude of the GW amplitude in the vicinity of the Earth is extremely small, on the order ofhGW ∼ 10−21 or lower. In fact, current theoretical models on the event rate and strength of such

events suggest that an amplitude sensitivity of 10−22 over a time scale of 1 ms is required to detecta few GW events per year (such as coalescing NS binaries). In other words, if the Fourier spectrumof a possible GW signal is considered, the required spectral density of the amplitude sensitivity for√√the 1 ms time scale is 10−22 / 1000 Hz % 3 × 10−24 / Hz over the frequency range of the signal. To
detect such a small effect, several detection techniques have been proposed: resonant bar detectors[59, 60] and interferometers such as Michelson interferometers [3, 1], Sagnac interferometers [61],atom interferometers [62, 63], and displacement-noise-free interferometers [25, 64].
33

A resonant bar detector or the so-called Weber bar [59] is composed of a large cylindrical bar withtransducers attached to detect any vibrations. This type of instrument was the first type of GW
detector. The idea is that a passing GW rings up the bar at its resonant frequency, which basically
amplifies the wave naturally. Alternatively, a nearby supernova might be strong enough to be seen
without resonant amplification. Modern forms of the bar detector are still operated, cryogenicallycooled, with superconducting quantum interference devices to detect the motion. Unfortunately,
one downside of bar detectors is that their sensitivity is limited to very narrow bandwidths.
The rough idea of a laser-interferometric GW detector was found by Pirani in 1956 [65]. Later,
the first explicit suggestion of such a detector was made by Gertsenshtein and Pustovoit in 1962 [66].
In the mid-1960s, Weber [59], unaware of the work done by Gertsenshtein and Pustovoit, reinvented
the idea, but left it in his laboratory notebook unpublished and unpursued. In 1970, Weiss at
MIT, unaware of the work done by Gertsenshtein and Pustovoit or Weber, reinvented the idea andcarried out a detailed design and feasibility study [67], which is currently used in the operational
GW detectors such as Laser Interferometer Gravitational-Wave Observatory (LIGO) [1], VIRGO[4], GEO600 [5], and TAMA300 [6]. The biggest advantage of interferometric detectors over bar
detectors is their broadband sensitivity that allows for detection of GWs at all frequencies in the
GW detection band.

1.3.2

The Principle of Laser-Interferometric Gravitational-Wave Detectors

The basic principle of a Michelson interferometer as a GW detector is given by the postulate thatthe infinitesimal spacetime interval ds for light is zero, or
ds2 = 0.

(1.32)

This is the key that makes it possible for us to detect GWs with a Michelson interferometer [1, 3, 26].
Consider a Michelson interferometer that consists of a laser at frequency ω, a beamsplitter, two
test mass mirrors with equal arm lengths of L, and a photodetector as shown in Fig. 1-1. Assume
that the beamsplitter is perfectly balanced and lossless and the mirrors are perfectly reflecting andlossless. Suppose, without the loss of generality, that the Michelson interferometer is set up suchthat its two arms are aligned along the x and y axes and then, only the h+ polarization of the GW is
considered. Moreover, assume that the curvature of spacetime where the Michelson interferometer
is located is weak so that the weak-field limit in Eq. (1.3) can be used.
Consider that the laser travels in the arm along the x axis. The infinitesimal spacetime interval
34

Test Mass y

L
L

LASER
Beamsplitter

Test Mass x

Photodetector

Figure 1-1: A Michelson interferometer that consists of a laser at frequency ω, a beamsplitter, two
test mass mirrors with an equal arm length of L, and a photodetector.

ds that the laser travels is given by
ds2 = 0= gµν dxµ dxν % (ηµν + hµν )dxµ dxν
= −c2 dt2 + [1 + hxx (t − eZ · x/c)] dx2 ,(1.33)

which indicates that the passing GW modulates the infinitesimal spacetime interval in the x arm bya fractional amount of hxx . The light travel time from the beamsplitter to the x test mass mirror
is, hence, given by
) τx1
0

1
dt =
c

) L.
) ,
1 L
1
1 + hxx (t − eZ · x/c)dx %1 + hxx (t − eZ · x/c) dx,
c 0
20(1.34)

where we have assumed that hxx + 1. Likewise, the light travel time from the x test mass mirror
to the beamsplitter (for the round-trip) is given by
) τx2
0

1
dt = −
c

) 0.
) ,
1 0
1
1 + hxx (t − eZ · x/c)dx % −1 + hxx (t − eZ · x/c) dx.
c L
2
L(1.35)

Therefore, the total round-trip time is found to beτxtot = τx1 + τx2 %

2L 1
+
c
c

) L
0

hxx (t − eZ · x/c) dx.

(1.36)

Similarly, the total round-trip time between the beamsplitter and the y test mass mirror for the35

laser that travels along the y arm is given byτytot %

2L 1
+
c
c

) L
0

hyy (t − eZ · x/c) dy.

(1.37)The photodetector at the interferometer output measures change in the interference of the twolaser beams induced by the passing GW. If the input electric field to the beamsplitter is E0 , assumingthat any common constant phase or reflection-induced constant phase shift is absorbed into theamplitudes of the electric fields, the interferometer output field at the beamsplitter is given byE=

1
1 0 iωτxtoti
E0 e− eiωτytot % E0 φGW ,22(1.38)whereω
φGW (t) = ω(τxtot − τytot ) %
c

) L
0

[hxx (t − eZ · x/c) dx − hyy (t − eZ · x/c) dy] .

(1.39)

is the GW-induced phase shift. If we assume that the GW propagates along the z axis so thateZ = ez , then hxx = −hyy ≡ hGW eiΩt and eZ · x = 0. Under this assumption, the phase shift canbe simplified to the expression [26]φGW (t) = hGW (t)4πL sin(ΩL/c) iΩL/c
e
λ
ΩL/c

(1.40)

where Ω/(2π) is the frequency of the GW and λ is the laser wavelength. Eq. (1.40) indicates thatin order to measure an extremely small value of hGW (t) on the order of 10−22 , large L and short λare required.

1.4

Quantum Noise in Gravitational-Wave Detectors

1.4.1

Introduction

Although there are several noise sources that limit the sensitivity of laser-interferometric GW detectors, such as seismic [68, 69, 70, 71, 72] and thermal noise [73, 74, 75, 76, 77, 78, 79, 80, 81],
the detector sensitivity is ultimately limited by quantum noise that comes from the quantum natureof photons due to the Heisenberg uncertainty principle. The limiting quantum noise sources of the
GW detectors are (1) photon shot noise that arises from uncertainty due to quantum mechanicalfluctuations in the number of photons at the interferometer output and (2) quantum radiation pressure noise that arises from mirror displacements induced by quantum radiation pressure fluctuations[82, 83]. Shot noise and radiation pressure noise1 dominate in different frequency bands. Radiation
1 Quantum radiation pressure noise is often shortened to radiation pressure noise in the GW community. It should
not be confused with classical radiation pressure noise.

36pressure noise exerts a force on the interferometer mirrors, which respond to the force only at low
frequencies due to the mechanical susceptibility. Shot noise dominates at higher frequencies where
the mirror response to radiation pressure noise becomes smaller. Both shot noise and radiationpressure noise are caused by quantum fluctuations of a vacuum electromagnetic field that enters theunused or antisymmetric port of the interferometers [84]. Shot and radiation pressure noises are
manifestations of the two quadratures of the vacuum.
The sensitivities of the currently operational laser-interferometric GW detectors such as LIGO
[1], VIRGO [4], GEO600 [5], and TAMA300 [6] are already limited by such quantum noise at highfrequencies in the GW detection band (10 Hz - 10 kHz). Next generation GW detectors such as
Advanced LIGO [7], which are planned to be operational in the next few years, are also expected tobe limited by quantum noise at almost all frequencies in the GW detection band [82].
In the following sections, the origin of the quantum noise in GW detectors is described. The
Heisenberg uncertainty principle is reviewed in the quantum optical picture, and then, photon shotnoise and quantum radiation pressure noise are derived based on it. The quantum noise in LIGO
and Advanced LIGO is also shown to see how it limits the detector sensitivities. Finally, the mostrigorous derivation of the quantum noise is briefly introduced and will be fully described in Chap.
5.

1.4.2

The Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle introduced by Heisenberg in 1927 states that it is fundamentally impossible to simultaneously obtain the precise knowledge of two non-commuting observables[85]. For observables A and B, if the commutation relation,[A, B] = AB − BA,(1.41)

is a non-zero constant, the product of the uncertainties of simultaneous measurements on the observables is bounded by∆A∆B ≥

1| -[A, B]. |,2(1.42)

where-O. = -Ψ| O |Ψ.

(1.43)

is the operator mean of observable O in the system state Ψ and
∆O ≡

23

(O − -O.)

2

4

=

37

/

-O2 . − -O.

2(1.44)

is the operator standard deviation of observable O in the system state Ψ.

The most common example of the Heisenberg uncertainty principle is the position and momentumof a free particle. The better position is known, the less well the momentum is known, and vice versa.
This principle also applies to a particle trapped in a harmonic potential. The particle oscillates in thepotential simple-harmonically. This situation is analogous to the oscillation of an electromagnetic
field at frequency ω. In the Heisenberg picture, an electromagnetic field characterized by its fieldannihilation operator a and its field creation operator a† satisfies the commutation relation [86, 87,88]
56a, a† = 1.

(1.45)

Following the commutation relation, the two observable quadrature fields (the so-called amplitude
and phase quadratures) defined by [88, 89]a1a2≡ a + a†Amplitude Quadrature,01≡ −i a − a†Phase Quadrature,(1.46)(1.47)

form an arbitrary quadrature operator,aθ ≡ ae−iθ + a† eiθ = a1 cos θ + a2 sin θ.

(1.48)

From Eq. (1.45), the commutation relation of the observables is given by[a1 , a2 ] = 2i.

(1.49)

Therefore, for the observables a1 and a2 , Eq. (1.42) leads to the uncertainty relation,∆a1 ∆a2 ≥ 1,

(1.50)

from which, we can see that it is impossible to simultaneously measure the amplitude and phasequadratures of an optical field with 100% accuracy. The ground state of an electromagnetic field is avacuum state or the state with no average photons in it, and is represented by |0. in the Schrödinger
picture. Even if there is no average number of photons in the vacuum state, it has fluctuations inboth amplitude and phase quadratures and their variances are equal since they are indistinguishable.
381.4.3

Quantum Noise Sources

Photon Shot Noise
Photon shot noise is the direct consequence of the quantum nature of photons. Measuring thepower of an electromagnetic field is equivalent to determining the number of photons arriving at a
photodetector during a measurement time interval. A coherent state |α. is defined as the eigenstateof the annihilation operator [90, 89],a |α. = α |α. .

(1.51)

The coherent state can be obtained by applying the displacement operator [89]
2

1

†

∗

D (α) ≡ e− 2 |α| eαa −α a(1.52)

D(α) |0. = |α. ,(1.53)

to a vacuum state |0., such thatwhich yields the coherent state in the basis of Fock or number states |n.,1|α. = e− 2 |α|∞7αn√ |n. .
n!
n=02

(1.54)The photon number probability distribution is therefore found to ben2

P (n) = |-n|α.| =-n. e−%n&,n!

(1.55)

8 9
where -n. = a† a = |α|2 is the average photon number. Eq. (1.55) is also called the Poisson
distribution or often referred to as counting statistics. When -n. ) 1, the Poisson distribution canbe approximated by a Gaussian distribution with the standard deviation given by
∆n =

/

2

-n2 . − -n. = |α| =

.

-n..

(1.56)In a Michelson interferometer as shown in Fig. 1-1, the average photon number per unit time at
the output of the interferometer is given by
*

dn
dt

+

=

λ
Pout ,
2πh̄c(1.57)and therefore, the standard deviation of the average photon number detected for a time interval of39τ is given by

∆n =

.

-n. =

:*

dn
dt

+

2

τ=

λPout τ.
2πh̄c(1.58)

Since we are using the power of the interferometer output as a monitor of the differential positionof the two test masses, we can obtain the displacement fluctuations by setting the optical powerfluctuations as equivalent to position difference fluctuations, given by the fractional photon number
fluctuation divided by the fractional output power change per unit position difference. Assumingthat we are interested in the linear response of the detector at the mid-fringe of the interferometer(Pout |mid−fringe = Pin /2) under the condition ΩL/c + 1 so that the detector response in Eq. (1.40)
is flat in the band, the standard deviation in the displacement L due to fluctuations in the average
photon number detected for a time interval of τ is given by∆Lshot =∆n/ -n.
1

Pout

=
dPout
dL

2

h̄cλ
.
4πPin τ(1.59)In the frequency domain defined byO(t) =

12π) ∞Õ(Ω)eiΩt dΩ

for an operator O(t),(1.60)−∞

where Ω is the sideband frequency relative to the carrier frequency, the strain noise caused by thephoton shot noise in terms of the equivalent GW noise is thus found to be∆L̃shot (Ω)
1h̃shot (Ω) ==L
L
where we have used a factor of

2

h̄cλ
,
2πPin

(1.61)√
2 from our preference for one-sided spectra. It is important to notethat Eq. (1.61) is frequency-independent and thus, flat in the frequency domain. Increasing the
input laser power Pin can improve the strain associated with the shot noise and thus enhance thedetector sensitivity at all frequencies. However, this is not the case due to the presence of quantum√
radiation pressure noise that scales as Pin , which will be discussed later in this section.

Shot noise is currently the limiting noise source at frequencies above 100 Hz in the LIGO detectors[91] and is also expected to be one of the dominant limiting noise sources in Advanced LIGO [7]. In
displacement-noise-free interferometers [25], shot noise is the only limiting fundamental noise source
and therefore, reducing the shot noise directly improves the detector sensitivity at all frequencies in
the GW detection band.
40Quantum Radiation Pressure NoiseThe optical measurement of the position of an interferometer test mass is analogous to the measurement of a particle with light, the so called Heisenberg microscope, in which the registration of
the arrival of a photon that bounces off the particle is a measure of the recoil of the particle causedby the change in the momentum of the photon upon reflection. In a Michelson interferometer asshown in Fig. 1-1, we register an arrival rate of photons that we interpret as a measurement of thedifference in phase between optical fields returning from the two arms.
The force exerted by an optical field of power Pm perfectly reflecting normally from a losslessmirror test mass in each interferometer arm is given by
Frad =

2Pm
Pin
=
,
c
c

(1.62)

where we have used Pm = Pin /2 since the beamsplitter splits the input power equally. The fluctuationin this force is caused by shot noise fluctuation in Pin given by∆Pin = h̄ω∆n = h̄ω

.

-n. =

2

2πch̄Pin τ
λ(1.63)as discussed early in this section, so that in the frequency domain, it is given by∆P̃in (Ω)∆F̃rad (Ω) ==c2

2πh̄Pin,cλ(1.64)

which is frequency-independent. Assuming that the beamsplitter is much more massive than thetwo mirror test masses, the mirror position fluctuation caused by the radiation pressure fluctuation
is given by1
1∆L̃rad (Ω) =∆F̃rad (Ω) =
2
mΩ
mΩ2

2

2πh̄Pin.
cλ(1.65)The power fluctuations in the two arms are anti-correlated, meaning that when one additional photon
goes into one arm, one less photon goes into the other arm, doubling the effect on the output of theinterferometer. Therefore, the strain noise caused by the radiation pressure noise in terms of theequivalent GW noise is found to be2∆L̃rad (Ω)
1h̃rad (Ω) =
=
L
mΩ2 L

2

8πh̄Pin,cλ(1.66)

which scales as Ω−2 due to the frequency response of the suspended mirrors, and is hence significant
at low frequencies. It is important to note that unlike the shot noise, the radiation pressure noise
increases with increasing Pin . Shot noise and radiation pressure noise are both shown in Fig. 1-2.
41

−19

10

−20

10

−21

Strain (Hz−1/2)

10Total Quantum Noise−22

10

Shot Noise
−23

10

−24

10

Radiation Pressure Noise
−25

10

−26

10

−3

10

−2

−1

10

Frequency (Hz)

10

0

10

Figure 1-2: Shot noise and quantum radiation pressure noise in a Michelson interferometer. The
test mass, input laser power, laser wavelength, and arm length used for the figure are m = 10 kg,
Pin = 5 W, λ = 1064 nm, and L = 500 km. At high frequencies, the shot noise dominates whereas
at low frequencies, the radiation pressure noise dominates due to its f −2 dependence.

Standard Quantum Limit
The total quantum noise of a Michelson interferometer, or the so-called optical readout noise, is thequadrature sum of the shot noise and radiation pressure noise,h̃tot (Ω) =/h̃2shot (Ω) + h̃2rad (Ω).

(1.67)

At low frequencies, the radiation pressure noise dominates whereas at high frequencies, the shotnoise dominates. In addition, the shot noise decreases as the input power Pin increases whereas theradiation pressure increases with larger Pin . If the total quantum noise is minimized with respect
to the input power Pin , the minimum noise or the so-called standard quantum limit (SQL) is foundto be [83, 92, 93, 94],h̃SQL (Ω) =

2

4h̄
,
mΩ2 L2(1.68)

at the optimum power PSQL = cλmΩ2 /(4π). Since this expression does not depend on Pin or λ, or
any other feature of the readout scheme, it hence places the ultimate quantum limit on precision
measurement.
42

1.4.4

Quantum Noise in LIGO and Advanced LIGOAdvanced Interferometer Configurations
Present day and next generation GW detectors utilize more complex interferometer configurations
than the simple Michelson interferometer. For example, as shown in Fig. 1-3, a power-recycled
Fabry-Perot Michelson interferometer (PRFPMI) configuration is used in LIGO [91] and a dualrecycled (power-recycled and signal-recycled) Fabry-Perot Michelson interferometer (DRFPMI) configuration is planned for Advanced LIGO [7].

(a)

(b)
Arm
CavityArm
CavityArm
CavityPRM

Arm
Cavity

PRM

LASER

LASER
BS

BS
SRM

Photodetector

Photodetector

Figure 1-3: Interferometer configurations: (a) a power-recycled Fabry-Perot Michelson (PRFPMI)
interferometer and (b) a dual-recycled (power-recycled, signal-recycled) Fabry-Perot Michelson(DRFPMI) interferometer. PRM: power-recycling mirror, SRM: signal-recycling mirror, BS: beamsplitter. The PRFPMI interferometer configuration is used in LIGO and the DRFPMI configurationis planned for Advanced LIGO.
Fabry-Perot cavities in the arms of the Michelson interferometer increase the light storage timeof the arms. The strain sensitivity of the Michelson interferometer at frequencies limited by shotnoise can also be enhanced by increasing the laser power incident on the beamsplitter. Since theinterferometer is operated on a dark fringe so that the laser power which is not lost to absorption,
scattering, or leakage out of the dark port goes back to the laser source, a partially transmitting
mirror2 can be inserted in between the laser source and the beamsplitter to recycle the laser backinto the interferometer. This mirror, which is conventionally called a power-recycling mirror (PRM),
and the interferometer form a power-recycling cavity (PRC). Likewise, a mirror can also be inserted
between the beamsplitter and photodetector to recycle the GW-induced phase shift back into theinterferometer (with optimization and limitations due to the losses in the interferometer). This
mirror, which is conventionally called a signal-recycling mirror (SRM), and the interferometer form
2 The mirror transmission is equal to the interferometer losses.

43

a signal-recycling cavity (SRC). Details of the advanced interferometer configurations can be foundin Refs. [82, 95, 96, 7, 1].

Conventional Gravitational Wave Detectors such as LIGO
For a conventional GW detector with arm lengths L and mirror masses m in the PRFPMI configuration, the quantum noise in the GW strain signal is given by [82]h̃SQLh̃(Ω) = √
2

2

1
,
K

(1.69)

8h̄
mΩ2 L2(1.70)K+

whereh̃SQL (Ω) =2is the SQL3 , and K is the effective coupling constant that relates motion of the mirrors to the output

signal, given by4

K=

2γ 4

Pin

Ω2 (γ 2 + Ω2 ) P.

(1.71)

SQL

Here γ is the linewidth of the arm cavities (typically γ ∼ 2π × 100 Hz), and PSQL is the input laser
power required to reach the SQL in the PRFPMI detector, given by PSQL = mL2 γ 4 /(4ω). Assumingthat Pin = PSQL , if Ω + γ, then K ) 1/K, and thus,γ1h̃(Ω) % h̃SQL = 2ΩΩ28h̄γ 2,mL2(1.72)

which indicates that the radiation pressure noise dominates in the band. If Ω ) γ, then 1/K ) K,
and thus,:Ω22h̄h̃(Ω) % h̃SQL 2 = Ω,2γ
mL2 γ 4(1.73)

which indicates that the shot noise dominates in the band. The frequency dependence of the shotnoise and radiation pressure noise in the PRFPMI detector is shown in Fig. 1-4.
Fig. 1-4 also shows the sensitivities of all the currently operational LIGO interferometers [97].
They are limited by shot noise at frequencies above ∼200 Hz. Radiation pressure noise is completely
masked by seismic noise [68], thermal noise [79, 80, 81], and control noise [98] at low frequencies.
3 This is a factor of√

2 different from the SQL in Eq. (1.68) because in this interferometer with Fabry-Perot cavity
arms, the effect is doubled by the cavities formed by two mirrors in each arm.
4 Since the power-recycling gain of a conventional GW detector is typically constant in the LIGO detection band
(10 Hz - 10 kHz), it is absorbed into the input laser power to the beamsplitter Pin , such that Pin is not the laser
power to the PRM, but to the beamsplitter.

44

−17

10

LHO 4km (2006.03.13)
LLO 4km (2006.06.04)
LHO 2km (2006.06.18)
Seismic Noise
Thermal Noise
Quantum Noise
LIGO Goal 4km (Total Noise)

−18

10

−19

Strain (Hz−1/2)

10

−20

10

−21

10

−22

10

−23

10

−24

10

1

2

10

10

3

Frequency (Hz)

10

4

10

Figure 1-4: The strain sensitivity of the LIGO interferometers during the S5 science run. LHO 4km:
LIGO Hanford 4 km Detector, LHO 2km: LIGO Hanford 2 km Detector, LLO: LIGO Livingson
Detector. The neutron star binary inspiral range for 1.4M# /1.4M# neutron stars is 14.5 Mpc for
LHO 4km, 15.1 Mpc for LLO 4km, and 7.4 Mpc for LHO 2km [97]. For all the detectors, shot noiseis the limiting noise source at frequencies above 200 Hz.
Since the radiation pressure noise is not significant, increasing the laser power can simply increase
the sensitivity at frequencies above ∼200 Hz although higher laser power can lead to problems withthermal loading (deformations of the interferometer mirrors due to absorbed laser power) [1, 99, 100].

Next Generation Gravitational Wave Detectors such as Advanced LIGO
For a next generation GW detector in the DRFPMI configuration with a detuned SRC, the quantum
noise in the GW strain signal is given by [96]h̃(Ω) = h̃SQL:(C11 sin ζ + C21 cos ζ)2 + (C12 sin ζ + C22 cos ζ)2,
2Kt2SRM |D1 sin ζ + D2 cos ζ|2(1.74)

where ζ is the homodyne phase5 ,
C11
C12
D1

'
(
K
= C22 = (1 + rs2 ) cos 2φ + sin 2φ − 2rs cos[2(β + Φ)],2

= −t2s (sin 2φ + K sin2 φ),;<= − 1 + rs e2i(β+Φ) sin φ,

C21 = t2s (sin 2φ − K cos2 φ),;<
D2 = − −1 + rs e2i(β+Φ) cos φ,(1.75)(1.76)(1.77)

5 This definition of the homodyne phase is slightly different in meaning from that in the field of quantum optics.

45

and rs and ts are the amplitude reflectivity and transmissivity of the SRM. Here 2β = 2 tan−1 (Ω/γ)
is the net phase gained by the sideband frequency Ω while traveling one way in the arm cavity,
φ = ωLSRC /c is the phase gained by the carrier frequency ω while traveling one way in the SRC witha SRC length of LSRC , and Φ = cot−1 K is the additional phase gained by the sideband frequencyΩ while traveling one way in the SRC. The quantum noise in the interferometer is shown in Fig.
1-5 using Advanced LIGO parameters. In the figure, it is not straightforward to distinguish thecontributions of shot noise and radiation pressure noise to the strain sensitivity due to the coupling
between the shot noise and radiation pressure noise in the case of the detuned SRC. The two dips
at 60 Hz and 300 Hz are due to the radiation-pressure-induced optical spring effect [96, 101, 102]
and the resonance of the arm cavities [95].

−21

10

Quantum Noise
Seismic Noise
Suspension Thermal Noise
Silica Brownian Thermal Noise
Coating Brownian Noise
Coating Thermorefractive Noise
Gravity Gradients
Total Noise−22

Strain (Hz−1/2)

10

−23

10

−24

10

−25

10

1

10

2

10

3

Frequency (Hz)

10

4

10

Figure 1-5: The design strain sensitivity of Advanced LIGO. It is anticipated to be limited byquantum noise at almost all frequencies in the GW detection band. At low frequencies, radiationpressure noise dominates while at high frequencies, shot noise dominates. In the mid-band, coating
Brownian noise dominates. The dips at 60 Hz and 300 Hz are due to the radiation-pressure-induced
optical spring effect and the resonance of the arm cavities.

Fig. 1-5 also shows the design strain sensitivity of Advanced LIGO [7] as well as the main
contributors to its noise budget. The goal of Advanced LIGO is to improve the sensitivity by morethan a factor of 10 at all frequencies in the GW detection band. Advanced LIGO is anticipated tobe limited by quantum noise at almost all frequencies in the GW detection band. At frequencies
below ∼40 Hz, radiation pressure noise dominates, whereas at frequencies above ∼200 Hz, shot noise
dominates. In the mid-band, thermal noise dominates.
46

1.4.5

Quantum Limit by Quantum Fluctuations of a Vacuum Field

Although in Sec. 1.4.3, we have semi-classically derived shot noise and radiation pressure noise ina Michelson interferometer, the most proper explanation of the origin of the quantum noise in theinterferometer is given by the quantum-mechanical two-photon formalism [84, 10] (which will be
described in Chap. 5). The two-photon formalism shows that when the Michelson interferometeris operated on a dark fringe, its shot noise and radiation pressure noise are caused by quantum
fluctuations of a (uncorrelated) vacuum electromagnetic field that enters the unused or antisymmetricport of the interferometer [84, 10]. This is intuitively appealing since the Michelson interferometer ona dark fringe acts like a mirror if seen from the interferometer output port, and therefore, any vacuum
field that enters the interferometer from the output port returns to the same output port and beatswith the GW-induced interferometer output field, yielding quantum noise. In the quantum optical
picture, uncorrelated quantum sidebands that enter the antisymmetric port of the interferometer
return to the output port and beat with the GW-induced interferometer output field.

1.5

Quantum-Enhancement

1.5.1

Introduction

In 1981, Caves proposed that the quantum limit of laser interferometers for precision measurements
such as laser-interferometric GW detectors can be circumvented by the injection of nonclassical orsqueezed states of light [8, 9] into the antisymmetric port of the interferometers [10]. A squeezed
state is a state whose uncertainties in the two quadrature field amplitudes are unequal and will be
described in detail in Chap. 2. The proposed squeezing injection scheme and the effect of squeezing(with a proper frequency-dependent squeeze angle, which will be described in detail in Chap. 4) in a
Michelson interferometer are respectively shown in Figs. 1-6 and 1-7. Fig. 1-8 also shows the effectof squeezing on Advanced LIGO when a squeezed state with a proper frequency-dependent squeeze
angle is injected into the antisymmetric port.
More rigorously, a squeezed state is a state in which two photons at frequencies ω ± Ω centeredaround the carrier frequency ω are correlated, and can be generated by a two-photon correlator suchas an optical parametric oscillator (OPO). In the quantum optical picture, the injection of a squeezed
state into the antisymmetric port of the interferometer is equivalent to the replacement of a random
vacuum field with the squeezed state, or in other words, the replacement of the uncorrelated vacuum
field with the correlated (squeezed) field. Since the interferometer is essentially a two-photon device[82, 103, 96, 104, 102] and the OPO is a two-photon correlator that correlates upper and lower
quantum photons (sidebands) [105], the quantum-mechanical two-photon formalism gives the mostproper explanation of quantum noise reduction or quantum-enhancement in the interferometer by47

Michelson
Interferometer
LASER
Squeezed
State

OpticalCirculator

Photodetector

Figure 1-6: A schematic of a Michelson interferometer with squeezing injected into the antisymmetricport of the interferometer via the optical circulator. The squeezing-enhanced interferometer fieldwith a possible gravitational wave signal that beats against it is detected by the photodetector.

squeezing, which will be described in detail in Chap. 5.

1.5.2

Ideal Squeezed State Production and Quantum Enhancement

In this section, a squeezed state, or more rigorously a squeezed coherent state, is conceptually obtainedby use of operators. Suppose that we have a two-photon squeeze operator for two photons (upperand lower sideband photons) or the so-called signal and idler fields [as (ω + Ω) and ai (ω − Ω)] withrespect to the carrier frequency ω (without worrying about the physical origin of the operator forthe moment) [105]† †∗Ssi (ξ) ≡ eξ as ai −ξas ai ,(1.78)

where ξ = reiφ . Here r is the squeeze factor and φ is the squeeze angle. In addition, the displacement
operators for the signal and idler fields [Ds (αs ) and Di (αi )] are given in Eq. (1.52). A squeezed
coherent state is obtained by acting with the displacement operators [Ds (αs ) and Di (αi )] on the
two-photon vacuum state |0, 0. followed by the squeeze operator Ssi (ξ) [106, 105], such that
|αs , αi , ξ. = Ssi (ξ)Ds (αs )Di (αi ) |0, 0. .
48

(1.79)

−19

10

−20

Strain (Hz−1/2)

10

−21

10

−22

Quantum Noise−23

Squeezed Quantum Noise

10

10

−24

10

−3

−2

10

10

−1

Frequency (Hz)

10

0

10

Figure 1-7: The effect of squeezing on a Michelson interferometer when a squeezed vacuum fieldwith a proper frequency-dependent squeeze angle (which will be described in detail in Chap. 4) is
injected into the antisymmetric port. The test mass, input laser power, laser wavelength, and arm
length used for the figure are m = 10 kg, Pin = 5 W, λ = 1064 nm, and L = 500 km. The effect ofsqueezing is assumed to be quantum noise reduction by 20 dB or increase in strain sensitivity by afactor of 10.

Using the useful unitary transformation properties of the squeeze operator [89, 88]†Ssi(ξ)as Ssi (ξ)= as cosh r − a†i eiφ sinh r,(1.80)†
Ssi(ξ)ai Ssi (ξ)= ai cosh r − a†s eiφ sinh r,(1.81)

we find the quadrature variances of a1 and a2 [105],
V1 (r, φ)V2 (r, φ)2Ω2sinh 2r cos 2φ,
ω2
2
8
9
Ω2
2
≡ (∆a2 ) = |a2 − -a2 . |2 = cosh 2r + 1 − 2 sinh 2r cos 2φ.
ω
2

3

2

≡ (∆a1 ) = |a1 − -a1 .|4= cosh 2r −1−

(1.82)(1.83)

Fig. 1-9 shows the quadrature variances of a1 and a2 as a function of φ. The periodic dependenceof squeezing and anti-squeezing on φ is evident.

If Ω/ω + 1, which is often the case since Ω/(2π) ∼ 10 kHz at most and ω/(2π) = c/λ, where λ
49

−21

10

−22

Strain (Hz−1/2)

10

−23

10

Quantum Noise−24

10Squeezed Quantum Noise

−25

10

1

2

10

10

3

Frequency (Hz)

10

4

10

Figure 1-8: The effect of squeezing on Advanced LIGO when a squeezed vacuum field with a proper
frequency-dependent squeeze angle (which will be described in detail in Chap. 4) is injected into theantisymmetric port. The effect of squeezing is assumed to be quantum noise reduction by 20 dB ora factor of 10 increase in strain sensitivity.

is the laser wavelength which is typically 1064 nm, then Eqs. (1.82) and (1.83) reduce toV1 (r, 0)= V2 (r, π/2) % e−2r ,(1.84)

V2 (r, 0)= V1 (r, π/2) % e2r .

(1.85)

A practical method of the generation of squeezed states as well as the physical origin of the twophoton squeeze operator will be described in the next chapter. Moreover, in this limiting case, thephoton number distribution of a squeezed state is given by
P (n) = | -n|α, ξ. |2 ,(1.86)

where-n|α, ξ. =

2

,
'
(
1
einφ tanhn r
10 2
α + α∗ eiφ tanh r∗2 iφ√
exp − |α| + α e tanh r Hn,
2n n! cosh r
22eiφ tanh r(1.87)

where Hn are the Hermite polynomials. The photon number distribution of a squeezed state with asqueeze factor r > 0 can be narrower than that of a coherent state. The average photon number of50

Quadrature Variance Relative to Shot Noise (dB)

10
8

V1

6
4
2
0
−2

V2

−4
−6
−8
−10
0

0.2

0.4

0.6

0.8

1
φ (π)

1.2

1.4

1.6

1.8

2

Figure 1-9: The quadrature variances of a1 and a2 as functions of φ. r = 1 and Ω/ω = 1 × 10−5 areused for the figure. The periodic oscillation of squeezing and anti-squeezing is evident.

a squeezed state is given by01-n. = |α|2 cosh2 r + sinh2 r − α∗2 eiφ sinh r cosh r − α2 e−iφ sinh r cosh r + sinh2 r
,
'
(
'
(φ
φ
= |α|2 e−2r cos2 θ −+ e2r sin2 θ −+ sinh2 r,(1.88)
22
where α = |α|eiθ has been used. The equation indicates that even if the coherent amplitude is zero(|α| = 0), the squeezed state contains some photons and is, therefore, no longer the true vacuum.
The photon number variance is given by

∆n =

:

,
'
(
'
(φ
φ
|α|2 e−4r cos θ −+ e4r sin θ −+ 2 sinh2 r + 2 sinh4 r,2
2(1.89)

which can be approximated, when the photon number is large and the squeeze factor is moderate,as:∆n % |α|

'
(
'
(
φ
φ
e−4r cos θ −+ e4r sin θ −.
22(1.90)

If θ − φ/2 = 0 is chosen, then ∆n = |α|e−2r is squeezed.
If this squeezed state is injected into the antisymmetric port of a Michelson interferometer with asqueeze angle set to reduce the shot noise of the interferometer, the shot noise and radiation pressure
51

noise in Eqs. (1.61) and (1.66) become [10, 96, 82]1h̃shot (Ω) =L2h̄cλ −r
e ,
2πPin

1
h̃rad (Ω) =
mΩ2 L

2

8πh̄Pin re .
cλ(1.91)

Since the shot noise and radiation pressure noise are the manifestations of the orthogonal quadratures, if the squeeze factor r is constant, the shot noise and radiation pressure noise cannot be
reduced simultaneously. This issue will be discussed in Chap. 4.

1.5.3

Previous Experimental Efforts

Following the 1981 proposal of Caves [10], a handful of experimental efforts have realized the proofof-principle on the table-top scale at MHz frequencies. The pioneering experiment was performedby Xiao et al. [11] using a Mach-Zehnder interferometer a few years after the first observation ofsqueezed states by Slusher et al. in 1985 [12]. Later, squeezing-enhancement in table-top realizationsof Michelson interferometer configurations similar to those used in the current and future large-scale
GW detectors was demonstrated [13, 14]. McKenzie et al. [13] and Vahlbruch et al. [14] showedsqueezing-enhancement in power-recycled and dual-recycled Michelson interferometers on the tabletop scale at MHz frequencies, respectively.
However, these demonstrated techniques were not yet practical for the implementation of squeezingenhancement in large-scale GW detectors with suspended optics and complex readout and control
schemes such as LIGO and Advanced LIGO. This is the starting point of the thesis. The goal of thethesis will be stated in the next section based on the previous experimental efforts.

1.6

The Goal of this Work

The goal of this thesis work is a detailed theoretical and experimental study of techniques necessaryfor quantum-enhancement in laser-interferometric GW detectors. The work focuses on techniques
for the practical realization of quantum-enhancement in large-scale GW detectors with suspended
optics and readout and control schemes compatible with next generation GW detectors such as
Advanced LIGO.
Chap. 2 describes the theory of squeezed states including the generation, degradation, detection, and control of squeezed states using sub-threshold OPOs and homodyne detectors in the
two-photon formalism [19, 20]. Chap. 3 describes the experimental characterization of the techniques discussed in Chap. 2 and presents experimental results. Chap. 4 discusses key requirements for quantum-enhanced GW detectors such as the frequency band, level, long-term stability,and frequency-dependence of squeezing. Chap. 5 describes the theory of quantum-enhanced GW
detectors, especially in a quantum-enhanced signal-recycled Michelson interferometer (SRMI) con52

figuration, using a two-photon mathematical framework of quadrature field propagation. Chap. 6
describes the experimental demonstration of quantum-enhancement in a prototype GW detector in
the SRMI configuration using the techniques that have been developed throughout the period of
my Ph.D. work. Finally, Chap. 7 discusses possible future improvements and investigations toward
the actual implementation of the quantum-enhancement in long baseline laser-interferometric GW
detectors such as Advanced LIGO. The appendices show tables of constants, acronyms, and crystal
properties, a list of publications, and other work which may or may not be related to squeezing.

53

54

Chapter 2Theory of Squeezed States
2.1

Overview

In 1927, Dirac proposed the wave-particle duality of light to account for interference phenomena andyet explain the excitation of a specific atom located along a wavefront absorbing one photon of energy.
In this quantum theory of radiation, each mode of a radiation field can be understood by a quantizedsimple harmonic oscillator [87, 86, 107]. An important consequence of the quantization of radiationis the fluctuations associated with the zero-point energy or the so-called vacuum fluctuations. Thesefluctuations have no classical analog and are responsible for many interesting phenomena in quantum
optics. The quantization of radiation fields is well described in Refs. [87, 86, 107, 89], and thus, notfully derived in this thesis.
In classical physics, an electromagnetic field consists of waves with a well-defined amplitude andphase. This is not the case when we treat the field quantum-mechanically. There are fluctuations
associated with both the amplitude and phase of the field. An electromagnetic field in a number
state |n. has a well-defined amplitude, but completely uncertain phase, while a field in a coherent
state has equal amount of uncertainties in the two quadratures that are conjugate to each other. The
uncertainties satisfy the Heisenberg uncertainty principle such that the product of the uncertainties inthe two quadrature field amplitudes is equal to or greater than half the magnitude of the expectation
value of the commutator of the variables as discussed in Sec. 1.4.2.
A squeezed state is a state in which fluctuations are reduced below the symmetric quantum
limit in one quadrature component. This is possible at the expense of increased fluctuations in the
canonically conjugate quadrature so that the Heisenberg uncertainty principle is not violated. A field
in a squeezed state has attractive applications in optical communication, photon detection, noisefree amplification [108, 109], correlated twin-beam generation [110, 111], quantum teleportation[112, 113], entanglement [114, 115], quantum imaging [116, 117, 118], quantum cryptography [119],55

spectroscopic measurement [120, 121], polarization measurement [122], and precision measurement[10, 13, 11, 122, 16, 15].
In this chapter, the concept of the quantization of radiation is used to explain states of lightsuch as coherent and squeezed states. In Sec. 2.2, the ball-on-stick picture is introduced to visualize various quantum states. Furthermore, the generation, degradation, detection, and control ofsqueezed states are theoretically introduced in Secs. 2.3, 2.5, 2.6, and 2.7, respectively, and thenexperimentally demonstrated in the next chapter. Second-order nonlinear optical processes for thegeneration of squeezed states are discussed in Sec. 2.4. In this thesis, we focus on the productionof squeezed states using sub-threshold optical parametric oscillators (OPOs) with second-order nonlinearity since the OPO-based squeezing has been the most commonly used technique (due to its
reliability and large squeezing level) for the generation of squeezed states. Other methods of squeezing include Kerr squeezing [123, 124], quantum radiation pressure driven ponderomotive squeezing[125, 126] and electromagnetically induced transparency (EIT) induced squeezing [127, 128], butthese techniques are not covered in this thesis.
It is important to note that throughout this chapter, we use the operator-linearization formalism
in which the linearization of operators is used to facilitate calculations of nonlinear equations of
motion. In the rest of the thesis, all equations are linear and the linearization of operators is not
used. Although the creation and annihilation operators are denoted δa† and δa in this chapter, theamplitude and phase variances (∆a1 and ∆a2 in the time domain or Ṽa1 and Ṽa2 in the frequency
domain1 ) obtained from δa† and δa are essentially equivalent to ∆a1 and ∆a2 or Ṽa1 and Ṽa2
obtained from a† and a used in the other chapters. In other words, δa† and δa in this chapter are
equivalent to a† − ā∗ and a − ā in the other chapters.

2.2

States of Light

2.2.1

The Ball-on-Stick Picture

Overview
Various states of light and vacuum can be visually understood by use of the so-called ball-on-stick
picture of an electromagnetic field. This is analogous to the phasor diagram in classical physics where
the orthogonal axes are the real and imaginary parts of an electromagnetic field. The similarity isthat the stick represents the classical steady state coherent amplitude of the field whereas thedifference is that the ball on the stick represents noise that can be classical or quantum mechanical.
The noise ball can be regarded as a representation of the contour height of a probability distribution
1 Ṽ
a1 and Ṽa2 will be defined later in this chapter.

56

u2

u2

u2

a2

a2
a1

a1

∆ a2

∆ a2

∆ a1

∆ a1 ?
∆ a1

ωt

ωt

u1

u1

∆ a2 ?

u1

Figure 2-1: Ball-on-stick pictures. Left: A coherent state with the minimum uncertainty. Center :A noisy state both classically and quantum-mechanically compared with a coherent state. Right: A
vacuum state that is a coherent state of vacuum. In the left and center pictures, a1 and a2 [definedin Eqs. (1.46) and (1.47)] are the orthogonal amplitude and phase quadrature axes that rotate at
the field frequency ω with respect to the universal amplitude and phase quadrature axes with zerofrequency, denoted u1 and u2 respectively. In the right picture, the question marks in the vacuum
quadrature variances reflect the fact that the amplitude and phase quadratures are undefined in thiscase.
function – the Wigner quasi-probability distribution function of an electromagnetic field given by [9]WG (a1 , a2 ) =,21(a1 − 2α1 )2(a2 − 2α2 )2
exp −
−
,
π ∆a1 ∆a2
2(∆a1 )22(∆a2 )2

(2.1)

where we have only considered Gaussian statistics, and α1 = (α + α∗ )/2 = |α| cos θ, and α2 =(α − α∗ )/(2i) = |α| sin θ. Wigner functions corresponding to a coherent state, a vacuum state,

amplitude and phase squeezed states are shown in Figs. 2-1 and 2-2. The Wigner function of any
Gaussian state can be completely defined by its standard deviation contour and a coherent amplitude
stick.
In this section, several common states such as a coherent state, a vacuum state, an amplitude
squeezed state of light, a phase squeezed state of light, and a squeezed state of vacuum are plottedin the ball-on-stick picture to intuitively understand the various states. In Sec. 2.3.4, we will revisit
this ball-on-stick picture to visualize how an OPO generates a squeezed state.

The Coherent StateIn quantum mechanics, a coherent state is a specific kind of quantum state of the quantum harmonic
oscillator whose dynamics most closely resemble the oscillating behavior of a classical harmonic
oscillator system. For instance, a coherent state describes the oscillating motion of a particle in aquadratic potential well. In quantum optics, the coherent state |α. is defined in Eq. (1.54) [90, 89].
As shown in Fig. 2-1, the uncertainty on the stick is spread in all directions equally. A field ina coherent state is a minimum-uncertainty state with equal uncertainties in the two quadrature
57

u2

u2

u2

a2

a2
a1

a1

∆ a2

∆ a2

∆ a1 ?

∆ a1

∆ a1
ωt

∆ a2 ?

ωt

u1

u1

u1

Figure 2-2: Ball-on-stick pictures. Left: An amplitude squeezed state compared with a coherent
state. Center : A phase squeezed state compared with a coherent state. Right: A squeezed state of
vacuum compared with a vacuum state. In the left and center pictures, a1 and a2 [defined in Eqs.
(1.46) and (1.47)] are the orthogonal amplitude and phase quadrature axes that rotate at the field
frequency ω with respect to the universal amplitude and phase quadrature axes with zero frequency,
denoted u1 and u2 respectively. In the right picture, the question marks in the vacuum quadrature
variances reflect the fact that the amplitude and phase quadratures are undefined in this case.
components so that ∆a1 = ∆a2 = 1.
The Noisy StateA state can be noisy both classically or quantum-mechanically. As shown in Fig. 2-1, the area of
the noise ball is always greater than unity and its shape can be both elliptic and circular so that∆a1 > 1 and ∆a2 > 1 simultaneously, but its semiminor axis can never be less than the radius of
the noise ball of the coherent state.
The Vacuum State
A vacuum state is a coherent state of light with no coherent steady state amplitude and the minimum
uncertainty. Since the vacuum has no coherent amplitude, its amplitude and phase quadrature axes
are undefined. This means it is meaningless to discuss amplitude and phase quadratures since they
are nonexistent unless it beats with a coherent field. To reflect this, we use a question mark in ournotation, so that ∆a1 ? and ∆a2 ? denote the undefined quadrature variances, which are shown in
Fig. 2-1.
The Amplitude Squeezed State of LightAn amplitude (coherent) squeezed state of light is obtained when a coherent state of light is squeezedin the amplitude quadrature. The semiminor axis of the squeezed noise ball is less than the radiusof the noise ball of the coherent state. However, the area of the noise ellipse must remain equal to or
greater than unity so that the Heisenberg uncertainty principle is not violated. The state is shown58

in Fig. 2-2. In this case, ∆a1 < 1 and ∆a2 > 1, but ∆a1 ∆a2 = 1 (in the case of perfect squeezing).
The Phase Squeezed State of LightA phase (coherent) squeezed state of light is obtained when a coherent state of light is squeezed in
the phase quadrature. Similar to the amplitude squeezed state of light, the area of the ellipse ballmust be equal to or greater than unity. The state is shown in Fig. 2-2. In this case, ∆a1 > 1 and∆a2 < 1, but ∆a1 ∆a2 = 1 (in the case of perfect squeezing).
The Squeezed State of Vacuum
A squeezed state of vacuum is obtained when a vacuum state is squeezed. Just like a vacuum state, asqueezed vacuum state has no coherent amplitude, and therefore, its quadrature axes are undefined.
The quadrature axes can be defined only when the state interacts with a coherent field. However,this interaction converts the squeezed vacuum state to a squeezed state of light, and therefore, it is
no longer a squeezed state of vacuum. The state is shown in Fig. 2-2. Here, similar to the amplitude
or phase squeezed state of light, ∆a1 ? < 1 and ∆a2 ? > 1, but ∆a1 ?∆a2 ? = 1 (in the case of perfectsqueezing).

2.2.2

Equivalence of Squeezed Light and Squeezed Vacuum

A squeezed state of light or a squeezed state of vacuum are often shortened to be squeezed light orsqueezed vacuum, respectively. In addition, in the previous section, a squeezed state of light anda squeezed state of vacuum are separately described. However, they are essentially identical since
they are vacuum fluctuations on top of the coherent stick that are squeezed, not light. The lightoften refers to the coherent amplitude or the stick.

2.3

Generation of Squeezed States

2.3.1

Introduction

In classical optics, the sideband model is widely used to visually understand amplitude and phasemodulation. Amplitude- or phase-modulated fields can be considered as the sum of correlated
coherent sidebands relative to the frequency of a carrier field. Likewise, in quantum optics, thecoherent state can be the sum of uncorrelated quantum sidebands which are a pair of creation
and annihilation operators in the ground state of a simple harmonic oscillator, h̄ω/2, at every
frequency and beat against the carrier field. For a coherent state, all quantum sidebands haverandom amplitude and phase and therefore, the result is broadband white noise or shot noise inboth the amplitude and phase quadratures. In other words, quantum noise is distributed equally59

(randomly) in both the amplitude and phase quadratures. A squeezed state can be considered as
the sum of correlated quantum sidebands relative to the carrier frequency, just as amplitude- andphase-modulated fields are the sum of correlated coherent sidebands relative to the carrier frequency.
At frequencies where squeezing occurs, the upper and lower quantum sidebands around the carrier
frequency become correlated or synchronized with each other. This quantum sideband correlation
picture can be understood most rigorously by the two-photon formalism [19, 20]. In the formalism,
a squeezed state is obtained by acting with displacement operators on a vacuum state followed by a
two-photon squeeze operator, as described in Sec. 1.5.2.
Among many methods of generating squeezed states in practice, optical parametric oscillation(OPO)2 has been one of the most common techniques that do the operation described above since
it was first observed by Wu et al. in 1986 [129]. An OPO is a two-photon correlator that correlates
upper and lower quantum sidebands centered around the frequency of a carrier field or half thefrequency of an energetic pump field. Squeezing is basically equivalent to the sum of correlated
quantum sidebands, and thus, must be treated using the two-photon formalism. The generation ofsqueezed states is described in many textbooks, often using degenerate OPO (DOPO) as a modelof squeezing-production. However, this sometimes leads to the misunderstanding of squeezing. In
the following sections, the production of squeezed states using nondegenerate OPO (NDOPO) is
introduced to properly understand the origin and physical meaning of squeezed states. The proper
understanding of squeezed states in the two-photon formalism is necessary to understand the injection of squeezed states into laser-interferometric gravitational wave (GW) detectors since they are
essentially two-photon devices and squeezed states are two-photon correlated states produced by
OPOs as two-photon correlators.

2.3.2

Quantization of Quadrature Field Amplitudes in Two-Photon Formalism

Consider a carrier field at ω and a pair of sidebands relative to the carrier at ω ± Ω. The upper andlower sidebands are the so-called signal and idler of the carrier field [130]. The total electric field isgiven by integrating the signal and idler fields over all positive and negative frequencies around zerofrequency,
E(t)

=

)

=
>
dΩ Ẽ(ω + Ω)eiΩt + Ẽ(ω − Ω)e−iΩt eiωt)=>
+ dΩ Ẽ † (ω + Ω)e−iΩt + Ẽ † (ω − Ω)eiΩt e−iωt .

(2.2)

2 Throughout the thesis, OPO stands for both optical parametric oscillation and optical parametric oscillator
although this abbreviation might be confusing.

60

Using the standard operator expressions, Ẽ(ω + Ω) and Ẽ(ω − Ω) can be written asẼ(ω + Ω) = E

2ω+Ωãs (ω + Ω),
2ωẼ(ω − Ω) = E2ω−Ωãi (ω − Ω),
2ω

(2.3)

where
E=

2

h̄ω
,
2π+0 cA(2.4)

and the subscripts (s and i) stand for the signal and idler. Here h̄ is Planck’s constant, +0 is thepermittivity of vacuum, c is the speed of light in vacuum, and A is the effective cross section of the.
field. It is important to note that the coefficients, (ω ± Ω) /2, are necessary for the correct energy

eigenvalues, h̄ (ω ± Ω). As Ω → 0, both equations reduce to the field equations in the degeneratecase. as and ai satisfy the commutation relations,=>"as , a†s"% 2πδ(Ω − Ω ),>="% 2πδ(Ω − Ω ).
ai , a†i"

(2.5)
(2.6)

Re(E)

a s†
ai †
-ω
-ω- Ω

Carrier

-ω+Ω
ω+Ω

ai

ω- Ω ω

CarrierIm(E)asFrequency

Figure 2-3: The signal and idler fields relative to the frequency of a carrier field. The signal andidler fields can be classical or quantum-mechanical.

Eq. (2.2) can be rewritten in terms of two orthogonal quadrature electric fields asE(t)= E1 (t) cos ωt + E2 (t) sin ωt
)
)
=
>
=
>
†
iΩt
−iΩt
=
dΩ Ẽ1 (Ω)e + Ẽ1 (Ω)ecos ωt + dΩ Ẽ2 (Ω)eiΩt + Ẽ2† (Ω)e−iΩt sin ωt, (2.7)
61whereẼ1 (Ω)
Ẽ2 (Ω)= Ẽ(ω + Ω) + Ẽ † (ω − Ω),=>= −i Ẽ(ω + Ω) − Ẽ † (ω − Ω) .

(2.8)
(2.9)

Defining the amplitude and phase quadrature fields byã1 (Ω)
ã2 (Ω)

2

2
ω+Ω
ω−Ω †
=
ãs (ω + Ω) +ãi (ω − Ω)Amplitude Quadrature, (2.10)
ω
ω
?2
@
2
ω+Ω
ω−Ω †
= −i
ãs (ω + Ω) −ãi (ω − Ω)Phase Quadrature, (2.11)
ωω

Eqs. (2.8) and (2.9) can be rewritten asẼ1 (Ω)=

Ẽ2 (Ω)=

E√ ã1 (Ω),
2
E
√ ã2 (Ω).
2

(2.12)(2.13)

Hence, the electric field in Eq. (2.7) is given in terms of a1 and a2 by
E(t)

)
)
E
E
iΩt
dΩ (ã1 cos ωt + ã2 sin ωt) e + √dΩ(ã†1 cos ωt + ã†2 sin ωt)e−iΩt
= √
2
2
,')
(
')
(
E
iΩt
iΩt
= √
ã1 e dΩ cos ωt +ã2 e dΩ sin ωt
2
(
')
(
,')
E
† −iΩt† −iΩt+√ã1 edΩ cos ωt +ã2 edΩ sin ωt ,(2.14)
2

which clearly indicates that a1 and a2 are the orthogonal quadratures. The second component isoften ignored since it is the complex conjugate of the first component.
As Ω → 0, Eqs. (2.10) and (2.11) reduce to the amplitude and phase quadrature operators in
the degenerate case,ã1ã2= ã + ã† ,01= −i ã − ã† .

(2.15)(2.16)

In most cases, Ω + ω. For example, for an optical field at λ = 1064 nm, ω/(2π) % 1015 Hz. The
frequency of interest Ω is typically on the order of MHz or lower, and therefore, the coefficients.
1 ± Ω/ω in Eqs. (2.10) and (2.11) are often reduced to unity3 .
3 However, this simplification often leads to the misunderstanding of the commutation relations in Eqs. (2.5) and(2.6), and then Eqs. (2.15) and (2.16) are incorrectly used as if the carrier was at zero frequency.

62

2.3.3Generation of Squeezed States in Optical Parametric Oscillation

Overview

We now turn to the discussion of how to generate squeezed states in practice or more specifically how
to do the two-photon squeeze operation described in Sec. 1.5.2. Although there are many methodsof generating squeezed states, we focus on OPO with second-order nonlinearity, which is one of themost common and reliable methods of squeezing. In this section, a model of optical cavities withsecond-order nonlinear media in the form of quantum Langevin equations of motion is introduced.

Equations of Motion for Optical Cavities with Second-Order Nonlinear MediaAs shown in Fig. 2-4, consider that three different fields as (ω + Ω), ai (ω − Ω), and b(2ω) entera system that involves a second-order nonlinear interaction, the most general Hamiltonian for thesystem is given by [131, 132, 133]
H = Hsys + Hu + Hint ,(2.17)

where Hsys is the Hamiltonian for the internal modes in the closed system (cavity) given by
Hsys = h̄(ω + Ω)a†s as + h̄(ω − Ω)a†i ai + h̄(2ω)b† b,(2.18)

Hu is the Hamiltonian for the open system that interacts with the universe given by
Hu = ih̄

) ∞=
−∞

;
< .
;
<>
.
0
1 .
2γsu ũ†s ãs − ã†s ũs + 2γiu ũ†i ãi − ã†i ũi + 2γbu ũ†b b̃ − b̃† ũb dΩ,(2.19)and Hint is the Hamiltonian for the nonlinear interaction given byHint =;<
1ih̄+ a†s a†i b − as ai b† .
2

(2.20)

Here as and ai are signal and idler fields with respect to the fundamental frequency ω, b is asecond-harmonic field at the second-harmonic frequency 2ω, + is the second-order nonlinear coupling
constant, and us , ui , and ub are the external modes coupled with the associated internal modes. The
origin of the second-order nonlinearity will be described in Sec. 2.4.
Using the quantum Langevin approach [131, 132], the Heisenberg equations of motion for the
intra-cavity fields [as (ω + Ω), ai (ω − Ω), b(2ω)] are respectively given in terms of all the input fields
63δvsl δvil δvbl
γsl γil γblasaibγsin γiin γbin

γsout γiout γboutε

A sin A iin B in

A sout A iout B out
δvsout δviout δvbout

Figure 2-4: A model of an optical cavity with a second-order nonlinear medium with three input
ininfields [Ains (ω + Ω), Ai (ω − Ω), B (2ω)], three intra-cavity fields [as (ω + Ω), ai (ω − Ω), b(2ω)], threeout
out
output fields [As (ω + Ω), Ai (ω − Ω), B out (2ω)], vacuum fields associated with the corresponding
output coupling losses [δvsout (ω + Ω), δviout (ω − Ω), δvbout (2ω)], and vacuum fields associated with thecorresponding intra-cavity losses [δvsl (ω + Ω), δvil (ω − Ω), δvbl (2ω)].

and cross-couplings with them by [131, 132, 133]
ȧs

=

ȧi

=

ḃ

=

/
.
.
1
i(ω+Ω)tout δv out +[as , Hsys + Hint ] − γstot as + 2γsin Aine+2γ2γsl δvsl , (2.21)
s
s
s
ih̄
/
/
/
1
i(ω−Ω)tout δv out +[ai , Hsys + Hint ] − γitot ai + 2γiin Aine+2γ2γil δvil , (2.22)
i
i
i
ih̄
/
/
/
1
[b, Hsys + Hint ] − γbtot b + 2γbin B in ei(2ω)t + 2γbout δvbout + 2γbl δvbl .
(2.23)
ih̄

Transforming into the rotating frame of each field withas→e−i(ω+Ω)t as ,−i(ω+Ω)t in
Ain
As ,
s →e

(2.24)ai→e−i(ω−Ω)t ai ,−i(ω−Ω)t in
Ain
Ai ,
i →e

(2.25)b→e−i(2ω)t b,

B in → e−i(2ω)t B in ,(2.26)

we simplify Eqs. (2.21), (2.22), and (2.23) to be [134]
ȧs
ȧi
ḃ

/
.
.
0
1
1
out δv out += − γstot − iωsd as + +a†i b + 2γsin Ain+
2γ
2γsl δvsl ,
s
s
s
2
/
/
/
0
1
1
out δv out += − γitot − iωid ai + +a†s b + 2γiin Ain+
2γ
2γil δvil ,
i
i
i
2
/
/
/
0
1
1
= − γbtot − iωbd b − +as ai + 2γbin B in + 2γbout δvbout + 2γbl δvbl ,2(2.27)(2.28)(2.29)

inin
where as , ai , and b are intra-cavity fields, Ain
are input fields, δvsout , δviout , and δvbout
s , Ai , and B

64

are output coupling vacuum fields, δvsl , δvil , and δvbl are fields associated with intra-cavity losses
characterized by the associated cavity decay constants γ’s respectively, and γstot , γitot , and γbtot are
the sum of all the decay constants in each mode so that γstot = γsin + γsout + γsl , γitot = γiin + γiout + γil ,and γbtot = γbin + γbout + γbl . Here ωsd , ωid , and ωbd are the cavity detunings given by ωsd = ω + Ω − ωsc ,ωid = ω − Ω − ωic , and ωbd = 2ω − ωbc , where ωsc , ωic , and ωbc are the cavity resonance frequencies
in the corresponding modes. We have assumed that the nonlinear coupling constant + which isusually temperature dependent is optimized and is therefore real4 . Throughout the rest of thechapter, we will assume that the cavity detunings and the frequency of interest Ω are much smaller
than the associated decay rates and therefore, we will ignore them, such that γstot − iωsd % γstot ,γitot − iωid % γitot , and γbtot − iωbd % γbtot . This general system of equations accounts for all secondorder nonlinear interactions including DOPO and NDOPO and second-harmonic generation (SHG).
Which interaction to choose depends on the phase-matching condition, which will be described inSec. 2.4.4.

Quadrature Field Amplitudes
The most common method of generating the analytic form of squeezed quadrature variances is toexpand the operators about their steady state values and then linearize the resulting expressions
to first-order in the fluctuation terms [136]. To linearize the equations of motion, we make thesubstitution for the annihilation and creation operators for as and a†s ,a†s = ā∗s + δa†s

as = ās + δas ,(2.30)

inin(similarly for ai , b, Ains , Ai , and B ). ās is the complex expectation value -as . (so that ās = -as .)
in†
in†
in
and δas is the operator for the fluctuations in as so that -δas . = 0. δAins , δAs , δAi , and δAisatisfy the commutation relations,
=
=

in†
Ain
s , As"in†
Ain
i , Ai">>"= 2πδ(Ω − Ω ),"= 2πδ(Ω − Ω ),(2.31)(2.32)

and all others vanish.
We then separate Eqs. (2.27), (2.28), and (2.29) into the time-dependent fluctuation component
and time-independent coherent amplitude component. The coherent components of the equations
4 The nonlinear coupling constant ! is a function of the phase mismatch parameter ∆k, so that ! =κ0 zei∆kz/2 sinc(∆kz/2), where κ0 is a constant and z is the nonlinear interaction length. The dependence of ∆kon the temperature of the nonlinear medium will be described in Sec. 2.4. Refs. [134, 135] describe the cases of cavity
detunings and complex nonlinear coupling constants.

65

are given by.
1= −γstot ās + +ā∗i b̄ + 2γsin Āin
s ,
2
/
1
= −γitot āi + +ā∗s b̄ + 2γiin Āin
i ,
2
/
1
= −γbtot b̄ − +ās āi + 2γbin B̄ in .
2

0
0
0

(2.33)(2.34)(2.35)

Assuming that the pump field is undepleted, as is often the case, (+ās āi + γbtot b̄), Eq. (2.35) reduces
to0 % −γbtot b̄ +/2γbin B̄ in .

(2.36)

Solving for the intra-cavity states yields

where b̄ =

/

ās
ā∗i






= 

γstot
− 12 +b̄∗− 12 +b̄
γitot

−1  .

2γsin Āin
s 
  .
,
in∗
in
2γi Āi(2.37)

2γbin B̄ in /γbtot . The fluctuation components of Eqs. (2.27), (2.28), and (2.29) aresimilarly given by
δ ȧs
δ ȧi
δ ḃ

/
.
.
1
1
out δv out += −γstot δas + +ā∗i δb + +b̄δa†i + 2γsin δAin+
2γ
2γsl δvsl ,
s
s
s
2
2
/
/
/
1
1
out δv out += −γitot δai + +ā∗s δb + +b̄δa†s + 2γiin δAin+
2γ
2γil δvil ,
i
i
i
2
2
/
/
/
1
1
= −γbtot δb − +ās δai − +āi δas + 2γbin δB in + 2γbout δvbout + 2γbl δvbl .
22(2.38)(2.39)(2.40)In terms of frequency components defined in Eq. (1.60), Eqs. (2.38), (2.39), and (2.40) can be
rewritten in a compact form,iΩx̃c = Mc x̃c + Min x̃in + Mout ṽout + Ml ṽl ,(2.41)

where the field vectors are defined by


δãs







 δã†s  δãi ,x̃c ≡  δã†i  δ b̃ 


δ b̃†



δ Ãin
s







 δ Ãin†

s




 δ Ãin
i,x̃in ≡ in†  δ Ãi  δ B̃ in 
δ B̃ in†δṽsout δṽsout†  δṽiout ,ṽout ≡  δṽiout†  δṽbout δṽbout†
66



δṽsl







 δṽsl†  δṽil ,ṽl ≡  δṽil†  δṽbl δṽbl†(2.42)

and the coupling matrices are defined by

Mc

≡

Min

≡

Mout

≡

Ml

≡

−γstot

0

0

1
2 +b̄

1
∗
2 +āi



0





1 ∗
1


0
−γstot
+
b̄
0
0
+ā
i
2
2




1
1
tot
∗

−γi
0
0
0 
2 +b̄
2 +ās

,
 1 ∗

1
 2 +b̄00−γitot
0
+ā
s
2


 1

1
tot
 − 2 +āi
0 
0− 2 +ās
0
−γb


0
− 12 +ā∗i
0− 21 +ā∗s0−γbtot
'.
(
/
/
/
/
.
in
in
in
in
in
in
diag
2γs , 2γs , 2γi , 2γi , 2γb , 2γb ,
'
(
/
/
/
/
.
.
out
out
out
out
in
out
diag
2γs , 2γs , 2γi , 2γi , 2γb , 2γb
,
(
'/
/
/
/
/
/
diag
2γsl , 2γsl , 2γil , 2γil , 2γbl , 2γbl .

(2.43)(2.44)(2.45)(2.46)

Thus, the cavity field vector is found to be−1x̃c = (iΩI − Mc )(Min x̃in + Mout ṽout + Ml ṽl ) ,(2.47)

where I ≡ diag(1, 1, 1, 1, 1, 1) is the identity matrix. The output field vector defined by
x̃out

≡

;

out†
out†
δ Ãout, δ Ãout, δ B̃ out , δ B̃ out†s , δ Ãsi , δ Ãi

<T,

(2.48)

where T is the transpose, is thus found to be [131, 132]x̃out

= Mout x̃c − ṽout−1= Mout (iΩI − Mc )=>−1
Min x̃in + Mout (iΩI − Mc ) Mout − I ṽout−1+Mout (iΩI − Mc )Ml ṽl ,(2.49)

which indicates that the output field x̃out is determined by the input field x̃in and the vacuum fields−1ṽout and ṽl with the associated transfer functions Mout (iΩI − Mc )I, and Mout (iΩI − Mc )−1

Min , Mout (iΩI − Mc )−1

Mout −

Ml .

Following the amplitude and phase quadrature fields in Eqs. (2.10) and (2.11), the quadratureinfield amplitudes of the input field (δ Ãin
1 and Ã2 ) are defined in the frequency domain (the same asin the time domain) byδ Ãin1 (Ω)δ Ãin2 (Ω)

22ω + Ω inω − Ω in†≡
δ Ãs (ω + Ω) +
δ Ãi (ω − Ω),
ω
ω
@
?2
2
ω + Ω inω − Ω in†
δ Ãs (ω + Ω) −
δ Ãi (ω − Ω) ,
≡ −i
ω
ω
67

(2.50)(2.51)

out(similarly for δṽ1out , δṽ2out , δṽ1l , δṽ2l , δ Ãout
1 , and δ Ã2 ). Correspondingly, Eqs. (2.31) and (2.32) imply

the commutation relations for the quadrature field amplitudes in the frequency domain,
=

>
=
>
"
"
"
in†
in†
in
δ Ãin1 (Ω), δ Ã2 (Ω ) = − δ Ã2 (Ω), δ Ã1 (Ω ) = −2iδ(Ω − Ω ),(2.52)(similarly for δṽ1out , δṽ2out , δṽ1l , and δṽ2l ) and all others vanish.
For simplicity, we assume the frequency of interest is much smaller than the linewidth of the
OPO cavity at each wavelength such that Ω + γsj , γij for j = in, out, and l. It is also reasonable
to assume |ā|δb + |b̄|δa in the case of the undepleted pump since |ā| +| b̄| and the fluctuations in
the pump and seed fields are on the same order. In addition, we assume γsj % γij for j = in, out,
and l and thus drop the subscript by denoting γaj ≡ γsj % γij for j = in, out, and l. Under theseassumptions, the quadrature field amplitudes of the output field are found to beδ Ãout1 (Ω)δ Ãout
2 (Ω)

A2
B@
?
2
.
ω + Ω iφω − Ω −iφDin
2γa C +e +e
δ Ãin
=
1 (Ω)
2
ω−Ω
ω+Ω
C
?
A2
B@
D
2
.
D
ω
+
Ω
ω
−
Ω
+
2γaout C +
eiφ +
e−iφ
− 1 δṽ1out (Ω)
2
ω−Ω
ω+Ω
?
A2
B@
2
/
D
ω + Ω iφω − Ω −iφl
+ 2γa C +e +eδṽ1l (Ω)
2
ω−Ω
ω+Ω
A2
B
2
.
D
ω
+
Ω
ω
−
Ω
−i 2γaineiφ −e−iφ δ Ãin
2 (Ω)
2
ω−Ω
ω+Ω
A2
B
2
.
D
ω + Ω iφω − Ω −iφout
δṽ2out (Ω)−i 2γa
e −
e
2
ω−Ω
ω+Ω
A2
B
2
/
D
ω
+
Ω
ω
−
Ω
iφ
−iφ
−i 2γale −eδṽ2l (Ω),(2.53)
2
ω−Ω
ω+Ω
?
A2
B@
2
.
D
ω + Ω iφω − Ω −iφin=
2γa C −e +e
δ Ãin
2 (Ω)
2
ω−Ω
ω+Ω
C
?
A2
B@
D
2
.
D
ω
+
Ω
ω
−
Ω
iφ
−iφ
+
2γaout C −e +e− 1 δṽ2out (Ω)
2
ω−Ω
ω+Ω
?
A2
B@
2
/
D
ω + Ω iφω − Ω −iφl
+ 2γa C −e +eδṽ2l (Ω)
2
ω−Ω
ω+Ω
A2
B
2
.
D
ω
+
Ω
ω
−
Ω
iφ
−iφ
−i 2γaine −eδ Ãin
1 (Ω)
2
ω−Ω
ω+Ω
A2
B
2
.
D
ω + Ω iφω − Ω −iφout
−i 2γae −eδṽ1out (Ω)
2
ω−Ω
ω+Ω
A2
B
2
/
D
ω
+
Ω
ω
−
Ω
iφ
−iφ
−i 2γale −eδṽ1l (Ω),(2.54)2ω−Ωω+Ω

where φ is the phase of the pump field (or equivalently the squeeze angle) such that B̄in = |B̄in |eiφ
68

and
C≡

.

2γaout 1,
γatot 1 − x2D≡−.

2γaout x.
γatot 1 − x2

(2.55)

Here x is the coupling of the nonlinear interaction normalized to the OPO threshold given by
x=

+|b̄|
2γatot(2.56)and should be less than unity not to lase.
Reduction to the Degenerate CaseIn most cases, Ω + ω, and therefore, the quadrature field amplitudes in NDOPO are approximately
equal to those in DOPO. In the DOPO limit (Ω/ω → 0), the two modes coalesce into a single mode
at frequency ω, such that22
ω + Ω iφω − Ω −iφ
e +
e
ω−Ω
ω+Ω
2
2
ω + Ω iφω − Ω −iφ
e −
e
ω−Ω
ω+Ω

→

2 cos φ,(2.57)→

2i sin φ.

(2.58)out
The quadrature field amplitudes, δ Ãout
1 (Ω) and δ Ã2 (Ω), then reduce toδ Ãout1 (Ω)δ Ãout
2 (Ω)

? .
@
, out
2 γain γaout 1 − x cos φ2γa 1 − x cos φ
in
=
δ
Ã
(Ω)
+
−
1
δṽ1out (Ω)
1γatot
1 − x2γatot
1 − x2@? .
@? .
in γ out x sin φ
2γ
2 γal γaout 1 − x cos φa aδṽ1l (Ω) −δ Ãin
+
2 (Ω)
γatot
1 − x2γatot
1 − x2? .
@, out
2 γal γaout x sin φ
2γa x sin φout−δṽ2 (Ω) −δṽ2l (Ω),(2.59)
γatot 1 − x2
γatot
1 − x2? .
@, out
2 γain γaout 1 + x cos φ2γa 1 + x cos φin=δ Ã2 (Ω) +− 1 δṽ2out (Ω)γatot
1 − x2γatot
1 − x2? .
@? .
@2 γain γaout x sin φ
2 γal γaout 1 + x cos φl+δṽ2 (Ω) −δ Ãin1 (Ω)γatot
1 − x2γatot
1 − x2? .
@, out
2 γal γaout x sin φ
2γa x sin φout−δṽ1 (Ω) −δṽ1l (Ω),(2.60)
γatot 1 − x2
γatot
1 − x2

which agree with the quadrature field amplitudes in the degenerate case [131].
Quadrature Variances
In squeezing experiments, spectrum analyzers are often used to obtain squeezed and anti-squeezed
quadrature variances after measuring squeezed fields with photodetectors. The detection of squeezed
69states will be described in Sec. 2.6. As defined by Eq. (1.44), the normalized quadrature variancesof the output field [ṼAout(Ω) and ṼAout(Ω)] in the frequency domain for quadrature fields [δ Ãout
and
1
1
2
δ Ãout
2 ] are given by [131, 137]
*E
E2 +
3
4E2 + *E
E out
E
E out
E
=
Ã
(Ω)
EÃ1 (Ω) − Ãout
E
Eδ
E
1
1
*E
E2 +
3
4E2 + *E
E out
E out
E
out E
ṼAout
(Ω)
≡
=
δ
Ã
(Ω)
Ã
(Ω)
−
Ã
E
E
E
E
2
2
2
2

ṼAout(Ω) ≡1

Amplitude Variance, (2.61)
Phase Variance,(2.62)(similarly for all the input modes). From Eqs. (2.59) and (2.60), the amplitude and phase quadrature
variances of the output field are, respectively, found to be

ṼAout(Ω)
1

ṼAout
(Ω)
2

@2
? .
, out
-2
2 γain γaout 1 − x cos φ2γa 1 − x cos φ
in
Ṽ
(Ω)
+
−
1
Ṽvout
(Ω)
=
A1
1
γatot
1 − x2γatot
1 − x2
? .
@2
? .
@2
2 γain γaout x sin φ
2 γal γaout 1 − x cos φl+Ṽv1 (Ω) +ṼAin2 (Ω)γatot
1 − x2γatot
1 − x2
? .
@2
, out
-2
2 γal γaout x sin φ
2γa x sin φout+Ṽv2 (Ω) +Ṽvl2 (Ω),γatot 1 − x2
γatot
1 − x2
? .
@2
, out
-2
2 γain γaout 1 + x cos φ2γa 1 + x cos φ
in
=
Ṽ
(Ω)
+
−
1
Ṽvout
(Ω)
A2
2
γatot
1 − x2γatot
1 − x2
? .
@2
? .
@2
2 γain γaout x sin φ
2 γal γaout 1 + x cos φl+Ṽv2 (Ω) +ṼAin1 (Ω)γatot
1 − x2γatot
1 − x2
? .
@2
, out
-2
2 γal γaout x sin φ
2γa x sin φout+Ṽv1 (Ω) +Ṽvl1 (Ω),γatot 1 − x2
γatot
1 − x2(2.63)(2.64)

Since all the input vacuum modes are in the coherent state, from Eq. (1.50), Ṽ (Ω) = 1 for all thevacuum modes, such that Eqs. (2.63) and (2.64) become

ṼAout(Ω)
1

ṼAout
(Ω)
2

? .
@2
? .
@2
2 γain γaout 1 − x cos φ
2 γain γaout x sin φin=ṼA1 (Ω) +ṼAin2 (Ω)γatot
1 − x2γatot
1 − x2, out-2 , out-2
2γa 1 − x cos φ
2γa x sin φ
+
−1 +
γatot
1 − x2γatot 1 − x2
? .
@2 ? .
@22 γal γaout 1 − x cos φ2 γal γaout x sin φ
+
+
,
γatot
1 − x2γatot
1 − x2
? .
@2
? .
@2
2 γain γaout 1 + x cos φ
2 γain γaout x sin φin=ṼA2 (Ω) +ṼAin1 (Ω)γatot
1 − x2γatot
1 − x2, out-2 , out-2
2γa 1 + x cos φ
2γa x sin φ
+
−
1
+
γatot
1 − x2γatot 1 − x2
? .
@2 ? .
@22 γal γaout 1 + x cos φ2 γal γaout x sin φ
+
+
,
γatot
1 − x2γatot
1 − x270(2.65)(2.66)

and therefore, the quadrature variances of the output field are linear functions of the quadrature
variances of the input or seed field, ṼAin1 (Ω) and ṼAin2 (Ω). Since all the coefficients of the input
quadrature variances are frequency-independent, the frequency-dependency of the output quadrature
variances is completely determined by the quadrature variances of the input field at low frequencies.
At high frequencies, although we have omitted the high frequency cutoff determined by the linewidthof the OPO cavity since we have assumed Ω + γain , γaout , γal , it appears in the 1 − x2 denominatorand therefore, all the coefficients containing 1 − x2 in the denominators decay to zero at frequencies
where Ω > γain , γaout , γal .

In the case of generating a squeezed state of light, the seed field is a bright field which istypically noisy at frequencies below 1 MHz due to its classical laser noise and therefore is not shotnoise-limited. In other words, ṼAin1 (Ω) and ṼAin2 (Ω) are much greater than unity at low frequencies
including the GW detection band. On the other hand, in the case of generating a squeezed state of
vacuum, the seed field is a vacuum field in the coherent state, and therefore, ṼAin1 (Ω) = ṼAin2 (Ω) = 1
at all frequencies, such that Eqs. (2.65) and (2.66) become
ṼAout(Ω)
1=

ṼAout(Ω)
2=2x2 − x cos φ − x2 cos φ,(1 − x2 )2
2x2 + x cos φ + x2 cos φ
1 + 4ηout,(1 − x2 )2
1 + 4ηout(2.67)(2.68)

where ηout ≡ γaout /γatot is the escape efficiency. As defined in Sec. 1.5.2, φ is the squeeze angle that
determines which quadrature to squeeze. The squeezed and anti-squeezed quadrature variances arefound from Eqs. (2.67) and (2.68) to be
E
EṼAout
E
1
φ=0
E
out E
ṼA2 E
φ=0

=
=

E
x
E
ṼAout,= 1 − 4ηout
E
2(1 + x)2
φ=πE
xEṼAout= 1 + 4ηout.
E
1(1 − x)2
φ=π(2.69)(2.70)

Therefore, the generated level of squeezing5 is completely determined by the escape efficiency ηoutand the nonlinear coupling efficiency x. These equations are often seen in many publications [138,139, 140, 141, 18, 142, 143]. Again, it is important to note that they are in the limiting case wehave assumed. Throughout the rest of the thesis, we will continue to assume that Ω + ω so thatany term that contains Ω/ω will be ignored.

2.3.4

The Ball-on-Stick Picture Revisited

In the previous section, the generation of squeezed states in an OPO has been mathematicallydescribed using the two-photon formalism. Although the two-photon description of the squeeze
5 The detected level of squeezing is always lower than the generated level of squeezing since the detection efficiency
of a detector has to be taken into account.

71

state production is rigorous and precise, its mathematical complexity does not appeal to an intuitive
understanding. In this section, to facilitate the two-photon description of the squeezing generation,
we revisit the ball-on-stick picture to visualize how the OPO produces a squeezed state.

(a)

(b)
OPO

Seed

Correlated
Signal ω+Ω
Idler ω- Ωεω+ΩSignal ω+Ω

Pum p
2ω

Idler ω- Ω

Pum p
2ω
(c)

u2

u2

SignalSignal

+Ω
-Ω

Idler

Idler

ωt

ωt
u1

u1

Figure 2-5: The production of a squeezed state in the ball-on-stick picture: (a) difference frequency
generation in nondegenerate optical parametric oscillation, (b) energy level diagram for the interaction, (c) a phase-squeezed state created by the sum of correlated signal and idler photons.
Fig. 2-5 shows the production of a squeezed state in the ball-on-stick picture. Suppose that two
coherent photons at ω + Ω are injected as a seed into a second-order nonlinear medium +, and then,an idler photon at ω − Ω is created due to the difference frequency generation in the NDOPO excitedby one of the seed photons at ω + Ω in the presence of a pump photon at 2ω, leaving the other seedphoton (signal) transmitted, as shown in Fig. 2-5(a). The transmitted signal and generated idler
photons are, therefore, correlated. The energy level diagram for the interaction is shown in Fig.
2-5(b).
Suppose that the signal and idler photons beat with a coherent field (carrier) at ω. Assumingthat the signal and idler photons have an equal amplitude for simplicity6 , they rotate with respect
6 The equal amplitude is assumed since we want modulation to occur only along the amplitude or phase axis of
the carrier. If the signal and idler amplitudes are not equal, the modulation axis is tilted relative to the amplitude orphase axis of the carrier, but does not lose generality.

72

to the carrier axis at ω counterclockwise at +Ω and clockwise at −Ω respectively in the ball-on-stick
picture as shown in Fig. 2-5(c). The sum of these signal and idler vectors travels only along theamplitude axis of the carrier, modulating the amplitude of the carrier at Ω. This is analogous toclassical amplitude modulation (AM). If the idler has an initial phase of π so that the signal rotatescounterclockwise at Ωt and the idler rotates clockwise at Ωt + π, the sum of these signal and idler
vectors travels only along the phase axis of the carrier, modulating the phase (frequency) of thecarrier at Ω. This is analogous to classical frequency modulation (FM).
Squeezing is just like AM and FM, but quantum-mechanically. Assume that two vacuum photons(vacuum fluctuations) at ω + Ω enter the second-order nonlinear medium. Due to the difference
frequency generation in the NDOPO excited by the vacuum photons at ω + Ω, the transmitted
signal and generated idler vacuum photons at ω ± Ω are correlated. They rotate with respect to thecarrier just like the coherent signal and idler photons in the ball-on-stick picture. Assuming that theinitial phase difference between the two is 0 or π, the sum of these signal and idler vacuum photons
travels only along either the amplitude or phase axis of the carrier, yielding quantum-mechanical
AM and FM. In practice, when a number of photons enter the OPO, not all of them interact withthe pump fields, and thus, the signal-idler photon correlations are not perfect. Then, the sum of all
the vectors becomes an ellipse – or a squeezed state. The probabilistic distribution of the correlated
photons is given by the Wigner quasi-probability distribution function in Eq. (2.1). Fig. 2-6 shows
the Wigner functions of a coherent state and amplitude and phase squeezed states.

Figure 2-6: The Wigner functions given by Eq. (2.1) for three common states. Left: A coherent state.
Center : An amplitude-squeezed state with ∆a1 = e−1/2 and ∆a2 = e1/2 . Right: A phase-squeezed
state with ∆a1 = e1/2 and ∆a2 = e−1/2 .

73

2.4

Second-Order Nonlinear Optical Processes for Squeezed
State Production

2.4.1

Overview

In the previous section, we have assumed that the nonlinearity + is a given quantity. However, tounderstand its origin, we need to take a close look at the physics of nonlinear optical processes. In
the following sections, requirements for optimum second-order nonlinear optical processes necessary
for the efficient generation of squeezed states are described.

2.4.2

Atomic Polarization of a Dielectric Medium

When an electromagnetic field enters a dielectric medium under the condition that its frequency is
far from the atomic resonances of the medium, the atomic polarization of the medium is given by;<
P = +0 χ(1) E + χ(2) E 2 + χ(3) E 3 + ... ,(2.71)

where +0 is the permittivity of vacuum, E is the applied electric field, and χ(1) , χ(2) , and χ(3) are thelinear or first-order, second-order, and third-order susceptibilities of the medium, respectively. The
atomic polarization is the overall effect of the dipole moments of each of the atoms that compose
the medium. When the applied electric field propagates through the medium, it excites the dipole
moments and induces the atomic polarization. Depending on the strength of the applied electricfield, the polarization oscillates at the frequency of the field if it is linear, at the second-harmonicfrequency of the field if it is second-order, at the third-harmonic frequency of the field if it is thirdorder, and so on, and then, re-radiates fields at those frequencies. When the applied electric field
is weak, the linear effect is dominant. When the field is strong, it excites the second-order effects
such as SHG and OPO, and if it is even stronger, it excites the third-order effects such as Kerrand Raman effects. Materials with somewhat strong second- or third-order susceptibilities haveχ(2) ∼ 10−10 − 10−13 m/V or χ(3) ∼ 10−18 − 10−23 m2 /V2 .
Nonlinear Medium

Pump Wavelength

1 (2)2 χijk (pm/V)

BPM Type

Seed-Pump Angle

Ba2 NaNb5 O15
MgO:LiNbO3
KNbO3 (KN)KH2 PO4 (KDP)
LiIO3KTiOPO4 (KTP)

532 nm532 nm532 nm488 nm351 nm / 413 nm
532 nm(zxx) 11.7(zxx) 5.9(zyy) 11.7(zxy) 0.5(zxx) 7.1(zyy) 4.5

I
I
I
I
I
II

0◦
90◦
90◦
37◦
30◦
90◦

Table 2.1: Nonlinear crystals often used in optical parametric oscillation. BPM: birefringent phasematching. The values are from Refs. [105, 144].
74

Although the third- or higher-order susceptibility can also generate squeezed states, we focus on
the second-order susceptibility throughout this thesis. Nonlinear crystals often used in OPOs areshown in Table 2.1. The type of nonlinear crystals is chosen for an OPO based on the transparency
window (pump wavelength), nonlinearity, phase-matching type, and phase-matching temperature.
Phase-matching types will be described in Sec. 2.4.4.

2.4.3

Conservation Laws

Both energy and momentum must be conserved in all physical processes including second-ordernonlinear optical processes such as up-conversion and down-conversion processes. The energy andmomentum conservation laws are respectively given by
7

h̄ωi

=

i

7

7

h̄ωj

Energy Conservation,(2.72)

h̄kj

Momentum Conservation,(2.73)

j

h̄ki

=

i

7
j

where the left/right hand sides are the inputs/outputs and i/j are the indices of input/output
photons. The photon energies are scalars while the photon momenta are vectors.
In the case of SHG, two photons at frequency ω get absorbed into the medium with the atomic polarization and it radiates a single photon at frequency 2ω, and therefore, the energy and momentum
conservation laws imposeh̄ω + h̄ωh̄nω ω h̄nω ω+cc= h̄(2ω),h̄n2ω (2ω)=.
c(2.74)(2.75)

The condition for the energy conservation is already satisfied while the condition for the momentum
conservation, also called the phase-matching condition, is satisfied when nω = n2ω .
In the case of NDOPO, when the medium is pumped by a photon at frequency 2ω, the medium
absorbs the photon and generates signal and idler photons. The energy and momentum conservation
equations in this nonlinear interaction are given byh̄(2ω)
h̄n2ω (2ω)
c= h̄(ω + Ω) + h̄(ω − Ω),h̄nω+Ω (ω + Ω) h̄nω−Ω (ω − Ω)
=
+
.
c
c

(2.76)(2.77)

In practice, the sideband frequency of interest Ω is much smaller than the frequency of the field since
the linewidth of a typical OPO cavity is ∼100 MHz and the frequency of a typical near-infrared laseris ∼100 THz. Thus, it is fair to assume nω+Ω % nω−Ω , which leads to the phase-matching conditionn2ω = nω . This is the same as the phase-matching condition for SHG.
75

2.4.4Phase Matching Types

Overview
Since the SHG and OPO cavities select photons with energies that satisfy the energy conservation
law, the only fundamental condition that needs to be satisfied for nonlinear optical processes is thephase-matching condition. However, this is not easily achievable since dielectric media are dispersive,
meaning that the refractive index n of a dielectric medium is usually an increasing function of
frequency ω, provided that the frequency of the applied electric field is far detuned from the atomic
resonances of the medium. There are three methods commonly used to achieve phase-matching: (1)type I phase-matching, (2) type II phase-matching, and (3) quasi-phase-matching (QPM). In thissection, these phase-matching types are described in detail.

Normalized SHG Conversion Efficiency1
z = 3 mmz = 6 mmz = 9 mm

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
103

104

105

106
107
108
Temperature (°C)

109

110

111

Figure 2-7: The effect of the phase mismatch on the SHG conversion efficiency in the case of type I
phase-matching for various crystal lengths. The shorter the nonlinear interaction length, the wider
the effective nonlinear interaction temperature range. The temperature stability of a short crystal
is high at the expense of decrease in the nonlinear interaction length, or vice versa.

Type I Phase-Matching
In type I phase-matching, the birefringent property of nonlinear media with both fast and slow axes,
depending on the polarization, is used to match the refractive indices at two different frequencies.
The fast and slow axes are often called the ordinary and extraordinary axes. The most popular
nonlinear medium for type I phase-matching is lithium niobate (LiNbO3 ) which is typically doped by76

magnesium oxide (MgO) which increases its optical damage threshold and reduces the effect of greeninduced infrared absorption (GRIIRA) [145]. MgO:LiNbO3 has a bulk second-order nonlinearity of
about 4.3 pm/V and an intra-crystal loss of about 0.1 - 1.0 %/cm.
The refractive index of this material strongly depends on the temperature (dno /dT = 3.3 ×

10−6 K−1 in the ordinary axis and dne /dT = 37 × 10−6 K−1 in the extraordinary axis, bothat λ = 1.0 - 4.0 µm), and therefore, tuning the temperature of the crystal is an ideal method
for achieving type I phase-matching. The phase mismatch parameter ∆k that depends on thetemperature and wavelength for type I phase-matching using MgO:LiNbO3 is given by the Sellmeierequation [146, 147],'

λ0∆k = k2ω − 2kω = A 1 −λ(+ B (T − T0 ) ,(2.78)

which has been approximated as a linear function near λ = λ0 and T = T0 . Here A = −8666 cm−1 ,B = 7.49 cm−1 K−1 , λ0 = 1064 nm, and T0 = 107 ◦ C. The conversion efficiency (in power) of ideal
single-pass SHG7 is given by [130]ηSHG = sinc2

'

∆kz
2

(

,

(2.79)

where z is the length of the nonlinear interaction. The effect of the phase mismatch on the singlepass SHG conversion efficiency (not within a cavity) for various crystal lengths is plotted in Fig.
2-7.

Type II Phase-Matching
In type II phase-matching, the correlated signal and idler fields are generated in orthogonal polarizations. This technique is often used for spin squeezing and two-photon correlation measurements.
However, this technique is not useful for quadrature squeezing in GW detectors since the signal andidler quantum sidebands are orthogonally polarized, and thus will not be discussed in this thesis.

Quasi-Phase-Matching
QPM is a relatively new technique, but is getting more and more popular nowadays because quasiphase-matched crystals have access to the largest nonlinear coefficient for nonlinear optical processes
such as SHG and OPO. Since this phase-matching technique is used in Chaps. 3 and 6, the details
of QPM will be fully described in the next section.
7 This will be defined in Eq. (2.101).

77

2.4.5

Quasi-Phase-Matching with Periodically Poled Materials

Comparison with Birefringent Phase-Matching
Birefringent phase matching (BPM) has been traditionally used in many nonlinear optical processes
such as SHG and OPO. Although BPM is convenient, it has a few severe disadvantages. The
phase matching range is restricted by the birefringent and dispersive properties of the nonlinear
material. Moreover, the effective nonlinear coefficients vary, depending on both the propagationand polarization directions of the interacting beams. It is worth noting that the largest nonlinear
coefficient for two-wave mixing, d33 , cannot be used in BPM. One way to go around the problems
with BPM is to use QPM.
Periodically poled nonlinear crystals utilize QPM in which a crystal is fashioned in such a waythat the direction of one of its principal axes is reversed periodically as shown in Fig. 2-8 [148, 149].
In a properly chosen crystal orientation and polarization directions of the participating optical fields,
this results in a periodic modulation of the nonlinear coefficient tensor element dij responsible forthe interaction. QPM offers advantages over traditional BPM in nonlinear optical processes. QPM
permits access to the highest effective nonlinear coefficients of materials, thus providing greater
conversion efficiency. Such materials have recently had a considerable effect in the field of nonlinear
optics. In addition, periodically poled materials offer the flexibility of grating-engineered phasematching, allowing any wavelength combination, within the transparency range of the material at
any temperature by a correct choice of the period of the domain inversion to be phase-matched ina noncritical geometry [150].

Λ

Figure 2-8: Left: A homogeneous single crystal. Right: A periodically poled material in which thepositive crystalline axis alternates in orientation with period Λ. An inversion in the direction of
the axis has the consequence of inverting the sign of the effective nonlinear coefficient, deff . This
periodic alternation of the sign of deff can compensate for a nonzero wavevector mismatch ∆k. Eachtime the field amplitude of the generated wave is about to begin to decrease as a consequence of the
wavevector mismatch, a reversal of the sign occurs, which allows the field amplitude to continue togrow monotonically.
Although the QPM interaction reduces the effective nonlinear coefficient by a factor of 2/π asshown in Sec. 2.4.5, the overall conversion efficiency determined by d33 can still be several times
larger than that given by the off-diagonal elements, d24 or d31 , with BPM. d33 can be accessed
only if all the interacting waves are polarized in the same direction, which is impossible in BPM.
78

Furthermore, QPM is well suited for noncritical phase matching and thereby, spatial walk-off can
be avoided, which leads to better beam quality [150].

Mathematical Framework for Quasi-Phase-Matching

When fundamental and second-harmonic fields travel through a nonlinear crystal in which theyinteract, the coupled equations that describe DOPO and SHG given by [148, 149, 130]
dE1
dz
dE2
dz

σ1 µ0 ciω1 µ0 c
E1 −deff E2 E1∗ e−i(k2 −2k1 )z ,2n1
2n1
σ2 µ0 ciω2 µ0 c ∗ 2 i(k2 −2k1 )z
= −
E2 −
d E e
,
2n2
2n2 eff 1= −(2.80)(2.81)

where E1 and E2 are the fundamental and second-harmonic fields respectively, z is the nonlinear
interaction length, c is the speed of light in vacuum, µ0 is the permeability of vacuum, deff is theeffective nonlinear coupling coefficient, σ1 and σ2 are the electrical conductivities of the medium
at the fundamental and second-harmonic frequencies (ω1 and ω2 ), and n1 and n2 are the refractive
indices at the fundamental and second-harmonic frequencies respectively, remain unchanged in QPM
except that deff is replaced by a periodic function d(z), which can be expanded in a Fourier seriesd(z) = dbulk

∞
7

2π

am eim Λ z ,(2.82)

d(z) −im 2π z
Λ dz.
e
dbulk

(2.83)m=−∞where Λ is the period of d(z) and
1
am =
Λ

) Λ
0

Substituting Eq. (2.82) into Eqs. (2.80) and (2.81) yields
dE1
dz
dE2
dz

∞
7
σ1 µ0 ciω1 µ0 c∗= −E1 −dbulk E2 E1
am ei∆kz ,
2n1
2n1
m=−∞

= −

∞
7
σ2 µ0 ciω2 µ0 c
E2 −dbulk E12am e−i∆kz ,
2n2
2n2
m=−∞

(2.84)(2.85)

where ∆k = m 2π
Λ − (k2 − 2k1 ). Phase-matching is obtained if the following condition is satisfied:∆k = m2π− (k2 − 2k1 ) = 0
Λ

for some integer m.

(2.86)

A nonlinear crystal for QPM is typically fabricated such that d(z) is spatially periodic with d(z)79Second−Harmonic Power (a.u.)

2.5

2

BPM (Δk = 0)
BPM (Δk ≠ 0)
QPM (Δk = 0 for m = 1)

1.5

1

0.5

0
0

2

4
6
8
10
12
14
16
Distance Traveled through a Nonlinear Crystal (µm)

18

20

18

20

40

Second−Harmonic Power (a.u.)35

BPM (Δk = 0)
BPM (Δk ≠ 0)
QPM (Δk = 0 for m = 1)

30
25
20
15
10
5
0
0

2

4
6
8
10
12
14
16
Distance Traveled through a Nonlinear Crystal (µm)

Figure 2-9: Upper : The evolution of a second-harmonic field in (a) phase-matched BPM (∆k = 0),(b) non-phase-matched BPM (∆k 0= 0), and (c) phase-matched QPM (∆k = 0 for m = 1). The
BPM and QPM crystal nonlinearities used in the figure are comparable. The vertical lines indicate
the grating period (9µm). Lower : The evolution of the second-harmonic field in (a) phase-matched
BPM (∆k = 0) using MgO:LiNbO3 , (b) non-phase-matched BPM (∆k 0= 0) using MgO:LiNbO3 ,and (c) phase-matched QPM (∆k = 0 for m = 1) using PPKTP. The BPM and QPM crystal
nonlinearities used in the figure are given in Table 4.1. The vertical lines indicate the grating period(9µm).

80

that switches from dbulk to −dbulk every Λ/2. In this caseam = −i'1 − cos mπmπ(

for m 0= 0(2.87)

such that, choosing m = 1, the effective nonlinear constant isdeff = am dbulk =2
dbulk .
π(2.88)

According to Eqs. (2.84) and (2.85), in principle, quasi-phase-matched configurations can give riseto the same conversion efficiency as in the ideal ∆k = 0 phase-matched case, except that a longer
interaction path is required to achieve it. An inversion in the direction of the axis has the consequenceof inverting the sign of the effective nonlinear coefficient, deff . This periodic alternation of the signof deff can compensate for a nonzero wavevector mismatch ∆k. Each time the field amplitude of
the generated wave is about to begin to decrease as a consequence of the wavevector mismatch, areversal of the sign occurs, which allows the field amplitude to continue to grow monotonically.
In SHG with no second-harmonic input, the generated second-harmonic field amplitude after adistance of nonlinearity z = L is given by
E2 (L)= −)∞7
ω2 µ0 c1 − cos mπ L −i∆kzdbulk E12
e
dz
2n2
mπ
0
m=1

∞
7
ω2 µ0 c1 − cos mπ −i ∆kL sin ∆kL/2
22= −dbulk E1 L
e
.
2n2
mπ
∆kL/2
m=1

(2.89)The generated second-harmonic power is proportional to |E2 (L)|2 and its evolution over the distanceis shown in Fig. 2-9.
Grating Period
Assuming that all the interacting waves have the same polarization, when the fundamental andsecond-harmonic waves are quasi-phase-matched, i.e. ∆k = 0, the grating period can be deduced as
Λ=

mλ1
2πm
=
.
k2 − 2k1
2(n2 − n1 )(2.90)

Here, the refractive indices n1 and n2 are given by the one-pole Sellmeier equation with an IR
correction of the form [151]n2 = A +B− Dλ2 ,
1 − Cλ−2(2.91)

where λ is the vacuum wavelength in units of µm, and A, B, C, and D are constants as shown in
Table 2.2. For KTiOPO4 (KTP), the grating period Λ for m = 1 at 1064 nm is typically ∼ 9µm.
81

nx
ny
nz

A
2.16747
2.192292.25411B
0.83733
0.83547
1.06543C
0.046110.049700.05486D
0.017130.016210.02140

Table 2.2: Sellmeier equation coefficients for flux-grown KTP [151].

Types of Quasi-Phase-Matching
There are two types of QPM in SHG and OPO, depending on the two possible orientations forthe linear polarization vectors of the incident beams [150]. In the type I process, both polarization
vectors are parallel whereas in the type II process, the polarization vectors are orthogonal.
In the type I QPM process, the largest nonlinear coefficient, d33 for KTP and its isomorphs suchas KTiOAsO4 (KTA) and RbTiOAsO4 , can be used when the polarization of the fundamental field ischosen to be parallel to the direction of the polar axis of the crystal. Consequently, the polarizationof the second-harmonic wave is also parallel to the polar axis (z-direction). The grating period forthis type is given by
ΛI =

mλ1
,
2(nz2 − nz1 )(2.92)

where nz1 and nz2 are the refractive indices along the polar axis of the crystal for the fundamental
and second-harmonic waves.
The type II QPM process is realized by employing the d24 nonlinear coefficient for a z-cut,
x-propagation direction. Here, two orthogonal fundamental components are used to generate asecond-harmonic wave that is polarized perpendicular to the polar axis (y-axis) in SHG. In thiscase, the grating period is found by replacing nz1 with (nz1 + ny2 )/2, such that
ΛII =

mλ1
,
y
2n2 − nz1 − ny1(2.93)

where ny1 and ny2 are the refractive indices along the y axis of the crystal for the fundamental andsecond-harmonic waves. However, the type II QPM is a much less efficient process compared to thetype I QPM since the d24 nonlinear coefficient is several times smaller than d33 . For KTP, d24 = 4.5
pm/V and d33 = 16.9 pm/V as shown in Table 4.1.

Quasi-Phase-Matching against Thermal Expansion
Since QPM does not use the dn/dT effect to satisfy the phase-matching condition, the only parameterthat needs to be stabilized is the grating period Λ against the thermal expansion of the medium in82Normalized SHG Conversion Efficiency

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
28

30

32

34
36
Temperature (°C)

38

40

42

Figure 2-10: The SHG conversion efficiency of PPKTP against the thermal expansion of the medium
in the light propagation direction. The parameters used for the figure are L = 10 mm, α = 0.2×10−6
K−1 , and T0 = 35◦ C.
the light propagation axis. From Eq. (2.86), the phase mismatch is given by∆k(T ) =2π
2π−,
Λ(T ) Λ0(2.94)

where Λ0 is the optimum grating period, using λ2 = λ1 /2, andΛ(T ) = Λ0 + αL(T − T0 )(2.95)

is the grating period as a function of the temperature of the medium T . Here α is the thermal
expansion coefficient in the light propagation direction and T0 is the optimum temperature. Fig.
2-10 shows the normalized SHG conversion efficiency of periodically poled KTP (PPKTP) against
its thermal expansion. The effective FWHM temperature range of PPKTP is much larger than thatof LiNbO3 with a typical crystal length of z = 6 mm (shown in Fig. 2-7).

2.4.6

Second-Harmonic Generation

To pump an OPO cavity, an energetic second-harmonic field that is typically generated by theprocess of SHG is needed. Although commercial SHG lasers are available, frequency-doubling thefundamental field is desirable to pump the OPO and generate a squeezed field at the carrier frequency
83unless the fundamental pump of an independent SHG laser is phase-locked to the carrier that is usedin an interferometer whose quantum noise needs to be squeezed.
δval δvbl
γal γblab
γain γbin

γaout γboutε

A in

A out B out
δvaout δvbout

Figure 2-11: A model of a SHG cavity that consists of a second-order nonlinear medium and two
coupling mirrors. A pump field at the fundamental frequency Ain (ω) enters the SHG cavity and asecond-harmonic field B out (2ω) is generated due to the nonlinear interaction in the crystal.

A model of a SHG cavity is shown in Fig. 2-11. Consider that a pump field at the fundamental
frequency enters the SHG cavity composed of a second-order nonlinear crystal and two coupling
mirrors and a second-harmonic field is generated due to the nonlinear interaction in the crystal. The
Hamiltonian for the nonlinear interaction is given byHint =

011ih̄+ a†2 b − a2 b† ,2(2.96)and therefore, the quantum Langevin equations for a SHG or DOPO cavity in steady state are foundfrom Eqs. (2.21), (2.22), and (2.23) to be
0
0

.
0
1
= − γatot − iωad ā + +ā∗ b̄ + 2γain Āin ,0
1
1= − γbtot − iωbd b̄ − +ā2 ,2(2.97)(2.98)

where wad = w − wac and wbd = 2ω − wbc are the cavity detunings at the fundamental and secondharmonic frequencies, respectively. Solving Eq. (2.98) for b̄ and substituting it into Eq. (2.97)
yield
0
1
0 = − γatot − iωad ā −

.
+2
0 tot
1 |ā|2 ā + 2γain Āin .
d
2 γb − iωb(2.99)The second term of the equation represents the increased cavity decay rate due to the nonlinear
84

interaction. This equation also indicates that the phase of Āin is directly translated into the phaseof ā. The output of the second-harmonic field is given by
B̄

out

. out
/
2γ
out
= 2γb b̄ = −+ 0 tot b d 1 ā2 ,2 γb − iωb(2.100)where ā is found by solving the nonlinear equation in Eq. (2.99) numerically. As is often the case,the cavities are tuned to resonance so that ωad = ωbd = 0.

1
93%
95%
97%
99%

0.9
SHG Conversion Efficiency

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 −3
10

−2

10

−1

0

10
10
Fundamental Pump Power (W)

1

10

2

10

Figure 2-12: The SHG conversion efficiency of a SHG cavity for various input coupler reflectivities
assuming that the output coupler is HR coated. The parameters used are as follows: the refractive
index of the crystal is 2.233, the axial length of the crystal is 7.5 mm, the distance between the crystal
surface and coupling mirror is 23.1 mm, + = 6000 s−1/2 , the intra-crystal loss at the fundamental
frequency is 0.1 %/cm, the intra-crystal loss at the second-harmonic frequency is 0 %/cm, and theoutput coupler reflectivity is 99.95 %.

The SHG conversion efficiency is given by the optical power of the generated second-harmonic
field divided by the optical power of the input fundamental field,
Efficiency =h̄(2ω)|B̄ out |2
2|B̄ out |2=,h̄ω|Āin |2|Āin |2

(2.101)

which cannot be expressed analytically due to the nonlinear equation in Eq. (2.99) and is, therefore,
typically solved numerically. The SHG conversion efficiency for various input/output coupling mirror
reflectivities is plotted in Fig. 2-12. The optical power of the generated second-harmonic fielddepends on the available optical power of the fundamental field. In practice, the mode-matching85

efficiency of the mode of the fundamental field to the cavity mode needs to be taken into consideration
to estimate the right optical power coupled to the SHG cavity.

2.4.7

Classical Optical Parametric Oscillation

Before generating squeezed states, it is convenient to examine the classical behavior of an OPO to
optimize the coupling of the pump field to the vacuum seed. This is usually done by optimizing
the parametric amplification of a bright seed when pumping the OPO. In a DOPO cavity, thecoherent intra-cavity field at the fundamental frequency is obtained from Eq. (2.97) and its complex
conjugate,



0
0





=

−γatot+b̄∗+b̄−γatot




ā
ā∗



+

.



2γain 

Āin
Āin



,

(2.102)where we have assumed that there is no cavity detuning and the classical amplitude Āin is real
without the loss of generality. Solving the equations for ā and using the transmitted field Āout =.
2γaout ā yield
1 + xeiφ in√Ā ,Āout = 2 ηin ηout
1 − x2(2.103)where ηin ≡ γain /γatot . The parametric gain is defined byG=|Āout (x)|2
1 + x2 + 2x cos φ
=
,
out
2
(1 − x2 )2|Ā (0)|(2.104)

which can be greater or smaller than 1, depending on the phase φ. If φ = 0, then G = 1/(1 − x)2 ,which indicates parametric amplification. If φ = π, then G = 1/(1 + x)2 , which indicates parametricdeamplication. The parametric gain as a function of the pump power is plotted in Fig. 2-13.

2.4.8

Designing Stable Resonators with Nonlinear Media

Cavities with High Refractive Index MediaThe resonator stability condition for an empty cavity with two mirrors as described in Refs. [152, 153]
cannot be applied to a cavity with a crystal due to its high refractive index that slows down the
propagation of light in the material. In this section, a more generalized resonator stability condition
for such cavities is derived. It will be shown that in the limit of n = 1, the condition reduces to thecondition for empty cavities as expected.
A cavity composed of two curved mirrors and a crystal in between them is schematically shownin Fig. 2-14. For a stable resonator, the wavefront R(z) of a Gaussian beam must match the mirror
86

6

10

5

10

4

Parametric Gain

10

3

10

2

10

Parametric Amplification

1

10

0

10

Parametric Deamplification

−1

10

0

0.1

0.2

0.3

0.4

0.5
x

0.6

0.7

0.8

0.9

1

Figure 2-13: The parametric gain obtained through an OPO cavity. If φ = 0, parametric amplification occurs. If φ = π, parametric deamplification occurs.

curvature at each mirror. Assume that the beam waist is located exactly at the middle point of thecrystal in order to maximize the nonlinear interaction in the case of a nonlinear crystal as a highrefractive index material.
R1

R2

d1

d2
n
l

Figure 2-14: An optical cavity composed of two external mirrors and a crystal in between them.
The refractive index and length of the crystal are n and l. The radii of curvature of the mirrors are
R1 and R2 , respectively.

Starting with waist size ω0 at the center of the crystal and propagating toward the mirror R1 ,the ABCD matrices for free space with index n and distance l/2, the interface from index n to index
1, and free space with index 1 and distance d1 are respectively given by [153, 154, 152]

M1 = 

1

l
2n

0

1



,



M2 = 

1

0

0 n
87



,



M3 = 

1

d1

0

1



.

(2.105)

The combined ABCD matrix is therefore found to be

M = M3 M2 M1 = 

1 d1
0

1




1

0

0 n




1

l
2n

0

1





=

1

l
2n + nd1

0

n



.

(2.106)The spot size ω1 and the radius of curvature R1 of the beam at d1 are given byq̃1 =Aq̃0 + B,
C q̃0 + D

(2.107)

where
1
λ
= −i
,
q̃0
nπω02

1
1
λ
=
− i 2.
q̃1R1πω1(2.108)The real and imaginary parts in Eq. (2.107) yieldlnπ 2 ω 4 R1
+ nd1 = 2 4 1 2 2 ,2nπ ω1 + λ R 1nπλω 2 R2nπω02= 2 4 1 21 2 .
λπ ω1 + λ R 1(2.109)The corresponding equations for the spot size ω2 and the radius of curvature R2 of the beam at d2
are similarly given bynπ 2 ω 4 R2l
+ nd2 = 2 4 2 2 2 ,2nπ ω2 + λ R 2nπω02nπλω 2 R2= 2 4 2 22 2 .
λπ ω2 + λ R 2(2.110)Using the definitions
L1 ≡l+ d1 ,
2n2

L2 ≡l+ d2 ,
2n2(2.111)

which we call the “effective” distances that are not equal to the physical distances, the waist size ω0
is found to be
ω04 =

λ2
λ2
L
(R
−
L
)
=
L2 (R2 − L2 ).
1
1
1
π2
π2

(2.112)

If we also define the “effective” total distance between the mirrors by
Leff ≡ L1 + L2 =l+ d1 + d2 ,n2

(2.113)

L1 and L2 can be rewritten in terms of Leff , R1 , and R2 as
L1 =R2 − Leff
Leff ,
R1 + R2 − 2Leff

L2 =R1 − Leff
Leff .
R1 + R2 − 2Leff(2.114)

It is important to note that n2 in Eq. (2.113) comes from two effects: one factor of n is due to the88slow propagation and the other factor of n is due to the refraction at the interface from index n toindex 1. As n → 1, the effective distance reduces to the physical distance L.

Resonator Stability Condition
Likewise, if we define the “effective” g parameters by
g1eff ≡ 1 −

Leff
,
R1

g2eff ≡ 1 −

Leff
,
R2

(2.115)the waist size ω0 and the spot sizes, ω1 and ω2 , are respectively found in terms of the effective gparameters and the effective total distance Leff (similar to the equations in Refs. [152, 153]) to be
Leff λω02 =π:g1eff g2eff (1 − g1eff g2eff )(g1eff + g2eff − 2g1eff g2eff )2(2.116):(2.117)

andLeff λ
ω12 =
π

:

g2eff
,
eff
g1 (1 − g1eff g2eff )Leff λ
ω22 =
π

g1eff
.
eff
g2 (1 − g1eff g2eff )It is obvious from Eqs. (2.116) and (2.117) that real and finite solutions for the Gaussian beam
parameters and spot sizes can exist only if the effective g parameters are confined to a stability range
defined by
0 < g1eff g2eff < 1.

(2.118)This is the resonator stability condition for cavities with high refractive index media such as nonlinear
crystals.
For nonlinear interactions through nonlinear crystals contained inside cavities such as OPO and
SHG, the waist size ω0 needs to be considered as well as the cavity stability because the strengthof the interaction is inversely proportional to the waist size ω0 . In general, it is difficult to satisfy
both requirements. If the waist size is made small to maximize the nonlinear interaction, the cavity
becomes near-concentric, and therefore, unstable since g1eff g2eff is very close to 1. If the cavity ismade stable, the waist size needs to be large, resulting in decrease in the strength of the nonlinear
interaction which is proportional to the intensity.
The generalized resonator stability condition and waist size allow us to visualize the requirements.
They are useful since the design of an OPO or SHG cavity is a delicate issue. Fig. 2-15 shows anexample of a symmetric cavity with a nonlinear crystal in between two cavity mirrors.
89

2

geff
geff
1 2

1.5
1
0.5
0
−0.5
1

2

3

4

5

6

7

8

9

10

2

3

4

5
6
d1 (mm)

7

8

9

10

ω0 (µm)

60
40
20
0
1

Figure 2-15: g1eff g2eff and the waist size ω0 as functions of the distance between the crystal face andthe end mirror d1 in the case of a symmetric cavity (d1 = d2 ). The radii of curvature of the mirrors
are R1 = R2 = 10 mm. The refractive index of the material is 1.8. To satisfy both requirementsthat the 0 < g1eff g2eff < 1 and the waist size is small, it is optimal to choose 4 - 8 mm for d1 , taking
into account that d1 < 3 mm is too close to the crystal face to be practical.

2.5

Degradation of Squeezed States

A squeezed state is a manifestation of the sum of correlated quantum sidebands centered around thefrequency of a carrier field, as described in Sec. 2.3. The quantum correlation, however, degrades ordecoherence occurs when it interacts with the universe through losses, assuming that the universe
consists of random vacuum fields. The beamsplitter model of optical losses or decoherence is often
used to understand the interaction of squeezed states with the universe [134, 137, 142]. In thissection, the beamsplitter model is reviewed to see how the level of squeezing decreases.
Loss
ηloss
Ain

Aout
Beamsplitter
δv

Figure 2-16: A beamsplitter model of squeezing degradation. Ain , Aout , and δv are the input field,
output field, and vacuum field that couples in at the unused port of the beamsplitter when an opticalloss occurs. ηloss is the power transmissivity through the beamsplitter.

90

As shown in Fig. 2-16, consider that field Ain enters a beamsplitter with power transmissivity
ηloss and the output is Aout . Any loss mechanism can be interpreted as a beamsplitter and therefore
treated in this beamsplitter model. When an optical loss occurs, a random vacuum field couples
into the incoming field at the unused port of the beamsplitter and replaces the portion of the fieldthat is lost by the beamsplitter. When the vacuum field is δv, the output field Aout is given by
Aout (t) =.
√ηloss Ain (t) + i 1 − ηloss δv(t).

(2.119)In the frequency domain, Eq. (2.119) becomesÃout (Ω) =.
√ηloss Ãin (Ω) + i 1 − ηloss δṽ(Ω).

(2.120)Linearizing the operators [136] and taking the fluctuation components, we find
δ Ãout (Ω) =.
√ηloss δ Ãin (Ω) + i 1 − ηloss δṽ(Ω),(2.121)and therefore, the amplitude and phase quadrature variances of the output field, respectively, arefound to beṼ1out= ηloss Ṽ1in + 1 − ηloss ,(2.122)Ṽ2out= ηloss Ṽ2in + 1 − ηloss .

(2.123)

When the transmission is 100% or ηloss = 1, the output squeezing level is retained. When thetransmission is 0% or ηloss = 0, squeezing completely disappears as expected.

2.6

Detection of Squeezed States

2.6.1

Overview

In the previous sections, the generation and degradation of squeezed states have been discussed. In
the following sections, the detection of squeezed states with one or two photodiodes and the conversion of the induced photocurrent(s) into frequency spectra are described. In squeezing experiments,
spectrum analyzers are often used to obtain quadrature variances. The detailed operation of spectrum analyzers to obtain quadrature variances is well-described in Ref. [137]. Two methods of detecting squeezed states are introduced: (1) balanced homodyne detection [140, 141, 17, 155, 16, 156, 139]and (2) unbalanced homodyne detection [15, 13, 14]. The balanced homodyne detection is usefulin situations where both the amplitude and phase quadrature variances of an optical field in asqueezed state need to be studied. The unbalanced homodyne detection is the scheme used to ob91

serve quantum-enhancement in the amplitude quadrature variance of an optical system such as aninterferometer. Other methods of detecting squeezed states can be found in Refs. [140, 141].

2.6.2

Balanced Homodyne Detection

Detector Description
The balanced homodyne detection technique is the most commonly used technique in squeezing
experiments [140, 141, 17, 155, 16, 156, 139]. As shown in Fig. 2-17, a balanced homodyne detector
is composed of a 50/50 beamsplitter, a coherent local oscillator (LO) field, and a pair of photodiodes
or a balanced photodetector. The input field is a squeezed state of light. The wavelength of the LO
is the same as the wavelength of the squeezed light. The balanced photodetector is interchangablewith a pair of photodetectors with a subtractor that subtracts one signal from the other. Assume
that the two input fields have the same initial phase. The balanced homodyne detector forms an
opto-electrical Mach-Zehnder interferometer to subtract uncorrelated noise on the input field, leaving
its correlated noise, that is, squeezed quantum noise.

PD2
I(t)
PD1
Squeezed
Vacuuma(t)+V
Balanced
Homodyne
Beamsplitter

d(t)

PD2
I(t)

Phase
Modulator

c(t)

PD1
-V

b(t)

Balanced
Photodetector

LO

Figure 2-17: A balanced homodyne detector composed of a 50/50 beamsplitter, a coherent local
oscillator (LO) field, and a pair of photodiodes or a balanced photodetector. The input field is asqueezed state of light or vacuum.

Consider that the input squeezed state, a(t), and the LO, b(t), enter the 50/50 beamsplitter.
92

The output fields, c(t) and d(t), are given by





1 i
a(t)
1

= √ 

.
2
d(t)
i 1
b(t)
c(t)



(2.124)The photocurrents induced by the two photodiodes, PD1 and PD2, are given by
Ic (t)=Id (t)

=

E2 †
1 2 †
E c c=(a − ib† )(a + ib),
2
4
1 2 †
E2
E d d=(−ia† + b† )(ia + b),2
4(2.125)(2.126)

where E is given by Eq. (2.4). As in Sec. 2.3.3, we linearize the operators, a(t) and b(t), such thata(t) → ā + δa(t) and b(t) → b̄ + δb(t). In addition, we further assume that b(t) has variable phaseθ(t) relative to a(t), such that b = (b̄ + δb)eiθ . Then, ignoring second-order fluctuation terms suchas δa† δb, δaδb† , δb† δa, and δbδa† , Eqs. (2.125) and (2.126) become
Ic (t)Id (t)

E2 2[ā + b̄2 − 2āb̄ sin θ + āδa1 (t) + b̄δb1 (t)4−ā (δb1 (t) sin θ + δb2 (t) cos θ) − b̄ (δa1 (t) sin θ − δa2 (t) cos θ)],E2 2=
[ā + b̄2 + 2āb̄ sin θ + āδa1 (t) + b̄δb1 (t)4+ā (δb1 (t) sin θ + δb2 (t) cos θ) + b̄ (δa1 (t) sin θ − δa2 (t) cos θ)],=(2.127)(2.128)where δa1 = δa + δa† , δa2 = −i(δa − δa† ), δb1 = δb + δb† , and δb2 = −i(δb − δb† ) in the limitingcase of Ω + ω, as defined in Eqs. (2.50) and (2.51). Therefore, the difference photovoltage obtained
from the difference photocurrent multiplied by current-to-voltage conversion coefficient R is givenbyVc−d= R(Ic − Id )6
RE 2 5= −2āb̄ sin θ + ā (δb1 sin θ + δb2 cos θ) + b̄ (δa1 sin θ − δa2 cos θ) ,2

(2.129)which can be decomposed into the time-independent DC component (first term) and the timedependent fluctuation component (terms containing δa1 , δa2 , δb1 , and δb2 ). In the frequency domain,
the fluctuation component is given byṼc−d (Ω) = −<>RE 2 = ;ā δ b̃1 sin θ + δ b̃2 cos θ + b̄ (δã1 sin θ − δã2 cos θ) ,2

93

(2.130)and therefore, the variance of the photovoltage is found to beÑc−d (Ω)

=
=

*E
E2 +
E
E
EṼc−d (Ω)E<;<>
R2 E 4 = 2 ;ā Ṽb1 sin2 θ + Ṽb2 cos2 θ + b̄2 Ṽa1 sin2 θ + Ṽa2 cos2 θ .
4(2.131)

In most cases, the amplitude of the squeezed field is much smaller than the amplitude of the LO(ā + b̄). In the case of a squeezed vacuum field, it does not contain any coherent amplitude (ā = 0).
Thus, Eq. (2.131) becomesÑc−d (Ω) %<R2 E 4 b̄2 ;Ṽa1 (Ω) sin2 θ + Ṽa2 (Ω) cos2 θ .
4(2.132)Since Ṽa1 (Ω) = Ṽa2 (Ω) = 1 at all frequencies when the input field is blocked so that a random
vacuum field enters the balanced homodyne detector, the normalized variance of the input squeezed
field can be obtained by dividing the variance of the photovoltage when the input field is unblocked
by the variance when it is blocked, such thatṼaθ (Ω) ≡EEÑc−d (Ω)E

UnblockedE= Ṽa1 (Ω) sin2 θ + Ṽa2 (Ω) cos2 θ.
EÑc−d (Ω)E

(2.133)BlockedAs shown in the derivation, the biggest advantages of the balanced homodyne detector are thecancellation of classical noise on the LO and the measurement of the linear response of the input
squeezed field so that both amplitude and phase quadratures can be observed. If the beamsplitter is
not perfectly balanced, the subtraction of the classical noise becomes imperfect, leaving some excess
noise in the squeezing spectrum, especially at low frequencies.

Detection Efficiency
So far we have assumed that the mode overlap between the input squeezed light and LO is perfect,
but this is not often the case experimentally. The homodyne efficiency is the degree of the mode
overlap between the two fields given byηhom ='

Pmax − Pmin
Pmax + Pmin

(2,

(2.134)

where Pmax and Pmin are the maximum and minimum values of the interference fringe measured
by the photodetectors (assuming that both photodetectors measure the same homodyne visibility)[137, 141].
In general, photodiodes do not convert all incident photons to electrons in the conduction band.
The quantum efficiency of photodiodes is the efficiency of the energy conversion from photons to94

electrons, defined by8
ηPD =

hc I
,
λe P

(2.135)where c is the speed of light in vacuum, h is Planck’s constant, e is the electron charge, λ is the laser
wavelength, I is the induced photocurrent, and P is the incident optical power.
Taking into account these two effects and a possible optical loss in the path between the OPO andhomodyne detector (characterized by ηloss and following Sec. 2.5, the quadrature variance detectedby the imperfect homodyne detector is given by making the substitution Ṽaθ → ηloss ηhom ηPD Ṽaθ +1−
ηloss ηhom ηPD , such that when the OPO-generated squeezed field characterized by Eqs. (2.69) and(2.70) is detected by the imperfect homodyne detector, the detected quadrature variance is given
from Eq. (2.133) byṼaθ (Ω)=>= ηloss ηhom ηPD Ṽa1 (Ω) sin2 θ + Ṽa2 (Ω) cos2 θ + 1 − ηloss ηhom ηPD

sin2 θ + Ṽadet
cos2 θ,= Ṽadet2
1(2.136)

where the squeezed and anti-squeezed quadrature variances measured by the detector (assumingthat the amplitude quadrature is initially squeezed) are, respectively, given byṼadet≡ 1 − 4ηloss ηout ηhom ηPD
1x,(1 + x)2Ṽadet≡ 1 + 4ηloss ηout ηhom ηPD
2x.
(1 − x)2(2.137)Note that the squeezing level is completely determined by the detection efficiencies (ηhom and ηPD ),the escape efficiency ηout , the optical loss 1 − ηloss , and the nonlinear coupling efficiency x. Moredetails about the detection efficiencies can be found in Ref. [137].

2.6.3

Unbalanced Homodyne Detection

Detector Description
Although squeezed states can be measured by a balanced homodyne detector, it is useful only forstand-alone squeezed state generators. When a squeezed state of light is used to reduce quantum
noise in an optical system where its field is the quantity of interest to measure, the balanced homodyne detector may not be practical. This is analogous to the situation in which a squeezed stateof light is injected into the antisymmetric port of a Michelson interferometer with an offset from adark fringe that leaks out some amount of the carrier light that a possible GW signal beats against.
This scheme called the DC readout scheme [7, 95, 157] is planned to be used in Advanced LIGO. In
this case, another detection method is needed for this type of detection. In this section, unbalanced
8 More rigorously, quantum efficiency is a quantity defined for a photosensitive device as the percentage of photons
hitting the photoreactive surface that will produce an electron-hole pair.

95homodyne detection is described for the application.
As shown in Fig. 2-18, an unbalanced homodyne detector can be modeled as a beam sampler
or an unbalanced homodyne beamsplitter with power transmissivity ηubs , a coherent LO field, anda single photodiode. If ηubs < 1, the unbalanced beamsplitter is equivalent to an imperfect mirror
with power transmissivity ηubs and hence, acts like a Michelson interferometer with a small offset
from a dark fringe. The input field a(t) is a squeezed state of light or vacuum. The wavelength of
the LO is the same as the wavelength of the squeezed light. Assume that the two input fields havethe same initial phase, but b(t) has variable phase θ(t) relative to a(t), such that b = (b̄ + δb)eiθ .
Similar to the calculation in Sec. 2.6.2, consider that the input squeezed state, a(t), and the LO,b(t), enter the unbalanced beamsplitter. The output field, c(t), is given byc=√.
ηubs b + i 1 − ηubs a.

(2.138)Linearizing the operators of the input fields and LO and ignoring any second-order fluctuation term,
the photocurrent induced by the photodiode is found to be
I(t)

=
=

E2 †
c c
2
.
E2 =
ηubs b̄2 + (1 − ηubs )ā2 + 2 ηubs (1 − ηubs )āb̄ sin θ + ηubs b̄δb1 + (1 − ηubs )āδa1
2>.
.
+ ηubs (1 − ηubs )ā (δb1 sin θ + δb2 cos θ) + ηubs (1 − ηubs )b̄ (δa1 sin θ − δa2 cos θ) .

(2.139)Separating the time-independent and fluctuation components of the equation, the variance of thedetected photovoltage in the frequency domain is given byÑ (Ω)=

R2 E 4 = 2 2
ηubs b̄ Ṽb1 (Ω) + (1 − ηubs )2 ā2 Ṽa1 (Ω)
4;<+ηubs (1 − ηubs )ā2 Ṽb1 (Ω) sin2 θ + Ṽb2 (Ω) cos2 θ;<>+ηubs (1 − ηubs )b̄2 Ṽa1 (Ω) sin2 θ + Ṽa2 (Ω) cos2 θ .

(2.140)If ηubs = 0, the beamsplitter acts as a perfect mirror and the detector measures only the amplitude quadrature variance of the input squeezed field. Since ā + b̄ in most cases, Eq. (2.140) is
approximately equal toÑ (Ω) %;<>R2 E 4 b̄2 = 2
ηubs Ṽb1 (Ω) + ηubs (1 − ηubs ) Ṽa1 (Ω) sin2 θ + Ṽa2 (Ω) cos2 θ .
4(2.141)

However, since the first term and the terms containing Ṽa1 and Ṽa2 scale identically, if the LO is
not shot-noise limited in the amplitude quadrature, the effect of squeezing that beats with the LO96

is contaminated by the dominant classical noise of the LO.
d(t)Unbalanced
Homodyne
Beamsplitter

Phase
Modulator
b(t)

+V

c(t)

LO

I(t)

a(t)

Figure 2-18: An unbalanced homodyne detector composed of a beam sampler or an unbalanced
homodyne beamsplitter, a coherent LO field, and a single photodiode. The input field is a squeezed
state of light or vacuum.
Similar to the balanced homodyne detection, since Ṽa1 (Ω) = Ṽa2 (Ω) = 1 at all frequencies when
the input field is blocked so that a random vacuum field enters the unbalanced homodyne detector,
the normalized variance of the input squeezed field can be obtained by dividing the variance of thephotovoltage when the input field is unblocked by the variance when it is blocked, such that at
frequencies where Ṽb1 (Ω) = 1,Ṽaθ (Ω) ≡EEÑ (Ω)E

Unblocked
EEÑ (Ω)E

Blocked=>= (1 − ηubs ) Ṽa1 (Ω) sin2 θ + Ṽa2 (Ω) cos2 θ + ηubs .

(2.142)

If Ṽb1 ) 1, assuming that the generated squeezing level is moderate, then Ṽaθ % 1, and thus, theeffect of squeezing cannot be observed.
Detection Efficiency
Similar to the balanced homodyne detection in the previous section, the detection efficiency isdetermined by the mode overlap between the LO and the input squeezed field and the quantum
efficiency of the photodiode.

2.7

Control of Squeezed States

2.7.1

Overview

Squeezed states are a much utilized resource required to perform quantum non-demolition readout[158] and to enhance the sensitivity of quantum noise limited applications [10]. The application of97

squeezed states requires, firstly, the generation of quality squeezed states and, secondly, the readout
and control of the phase of the states, such that the appropriate quadrature is measured. When
we generate a squeezed state in a sub-threshold OPO and detect it with a homodyne detector, weneed to control the so-called squeeze angle which is defined by the difference between the semiminor
axis of the squeezed noise ball (ellipse) and the amplitude axis of a coherent field that the squeezed
field beats against. For the readout and control of squeezed states of light, information about thesqueeze angle can be obtained from the parametric amplification or deamplification of a coherent
seed field such that the standard RF modulation/demodulation or DC readout techniques can beused to stabilize the phase [159, 137]. For quantum states which do not have a coherent amplitude
at the carrier frequency such as a squeezed state of vacuum, the standard phase-locking techniques
are not available.
In this situation, there are two available control schemes: (1) quantum noise-locking [160, 17]
and (2) coherent control of squeezing [161, 18]. In the former scheme, the squeeze angle is modulated
and the squeezing spectrum is demodulated to obtain an error signal for feedback control. In thiscase, the field that carries the squeezed state can be completely devoid of a coherent amplitude. In
the latter scheme, the parametric amplification or deamplification of a coherent sideband along witha squeezed vacuum field is measured to obtain an error signal for feedback control.
Throughout this thesis, the quantum noise-locking technique is experimentally used because it
has some advantage over the the coherent control in squeezing-enhanced laser interferometric GW
detectors. In the following sections, the two methods of controlling squeezed states of vacuum aredescribed.

2.7.2

Quantum Noise Locking

Introduction
An error signal for the quantum noise locking technique can be obtained for the case of locking
the squeezed vacuum phase on a homodyne detector in an analogous way to coherent modulated
techniques [162]. The squeezed vacuum phase is modulated at some frequency, and then, the noise
power detected by the homodyne detector is demodulated. This produces an error signal which has
zero crossings at both the minimum and maximum variance points. This technique, which we referto as quantum noise-locking or noise-locking [160], has application in the locking of the phase ofquantum states that have non-polar symmetric phase space distribution functions. This technique
has been used already in previous squeezed vacuum experiments [163, 164, 17, 155].
Moreover, in applications where the properties of squeezed vacuum states are desirable, thenoise-locking technique is a good choice for phase control. One could conceive a scheme where avacuum squeezed state is generated with coherent phase modulation sidebands without a coherent98carrier amplitude imposed on it [18]. One such application may be in laser-interferometric GW
detectors where low-frequency squeezing at the optimum phase is required. The relative immunityof squeezed vacuum states to classical noise sources can result in squeezing being produced in the
GW detection band [17, 18, 134]. For this reason, the quantum noise-locking technique will be usedin the experiments in Chaps. 3, 4, and 6.

Noise-Locking Error Signal
In this section, the error signal of the quantum noise-locking technique is derived theoretically.
Much of the rigorous derivation is described in Ref. [160]. Fig. 2-19 shows two input fields, a andb, with relative phase denoted θ that interfere on a balanced beamsplitter. These operators satisfy
the standard commutation relations in Eq. (1.45). The output fields, c and d, are incident on thephotodetectors (PD1 and PD2). In the case of noise-locking the phase of a squeezed vacuum fieldon the homodyne detector, a is the squeezed state and b is the LO. The output of the balanced
homodyne detector is sent to a band-pass filter (BPF). The band-pass-filtered output is sent to an
envelope detector which gives an output proportional to the real envelope of the input. This signal
is then demodulated at the modulation frequency and low-pass filtered to give the error signal.
The generation of noise-locking error signals requires modulation of the relative phase between
the two input fields. The phase of the LO is modulated at frequency Ωm so that θ = θ0 + θ1 sin Ωm t,
where θ0 is the average DC phase and θ1 is the modulation depth. For small modulation depth(θ1 + θ0 ), we make the approximation eiθ1 sin Ωm t % J0 (θ1 )+J1 (θ1 )eiΩm t −J1 (θ1 )e−iΩm t . If the input
squeezed field has a coherent amplitude, the DC component in Eq. (2.129) can be modulated andthe photodetector signal can be demodulated to obtain an error signal. However, in the absence of acoherent amplitude in the squeezed vacuum field, the DC term vanishes and therefore, the correlated
noise obtained by band-pass-filtering and envelope-detecting the signal needs to be demodulated toacquire an error signal. When the phase of the LO is modulated in the case of ā = 0, the difference
photovoltage in Eq. (2.129) becomes
Vc−d = −RE 2 b̄[δa1 (J0 sin θ0 + 2J1 cos θ0 sin Ωm t) − δa2 (J0 cos θ0 − 2J1 sin θ0 sin Ωm t)] . (2.143)
2

In the frequency domain, neglecting J12 (θ0 ) terms, the variance of the photovoltage after band-passfiltering the photovoltage is given byÑc−d (Ω)%<
R2 E 4 b̄2 = 2 ;J0 Ṽa1 sin2 θ0 + Ṽa2 cos2 θ0
4;<>+2J0 J1 Ṽa1 − Ṽa2 sin 2θ0 sin Ωm t ∆Ω,(2.144)

where ∆Ω is the detection bandwidth. Here we have assumed a perfect band-pass filter with hard
99

PD2

PD1

Monitor

Squeezed
Vacuuma(t)+Vd(t)
Homodyne
Beamsplitter
Phase
Modulator

c(t)

PD2

Bandpass
Filter

PD1

Envelope
Detector-Vb(t)Balanced
Photodetector

LO

Mixer

Low-Pass
FilterAdder

Oscillator

Figure 2-19: A balanced homodyne detector with a noise-locking servo. Two input fields, a(t) andb(t), interfere with relative phase, θ, on a balanced 50/50 beamsplitter. a(t) is a squeezed vacuum
field and b(t) is a coherent local oscillator (LO) field. Here the LO beam b(t) passes through a phasemodulator with an applied sinusoidal modulation signal at frequency Ωm /2π. The output fields,c(t) and d(t), are incident on the photodiodes, PD1 and PD2, that form a balanced photodetector
when PD1 and PD2 are respectively biased negatively and positively. The balanced photodetector
can be replaced with a pair of photodetectors with a differencer that subtracts one signal from theother. To derive the noise-locking error signal, the output of the homodyne is band-pass-filtered(BPF), and then envelope-detected (ED). The output of the envelope detector is demodulated at
the modulation frequency, and then low-pass filtered (LPF), yielding a control signal that is fed backto the phase modulator.

edges although experimentally this is not the case. Demodulating Eq. (2.144) at Ωm and low-passfiltering its output to remove sin Ωm t and sin 2Ωm t terms yields the noise-locking error signal,κ=<J0 (θ1 )J1 (θ1 )R2 E 4 b̄2 ;Ṽa1 − Ṽa2 ∆Ω sin 2θ0 ,2(2.145)which has zero crossings at θ0 = 0 and π/2. The error signal amplitude depends on the quadrature
variances and their asymmetry. Thus, if there is no asymmetry in the quadrature variances (Ṽa1 =Ṽa2 = 1), the error signal vanishes. The error signal compared with the quadrature variance of asqueezed vacuum field is shown in Fig. 2-20.
Stability of Noise-Locking
Since it is the variance or noise power of the detected squeezed vacuum field which is used to derive
the error signal, the noise performance of the squeeze angle control depends on the variance of the100

Quadrature Variance (dB)

10
5
0
−5
−10
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1
θ (π)

1.2

1.4

1.6

1.8

2

Error Signal (a.u.)

2
1
0
−1
−2
0

Figure 2-20: The quadrature variance of a phase squeezed vacuum field relative to the noise-locking
error signal as the relative phase to the LO is varied. The zero crossing points of the error signal
indicate that the homodyne angle can be locked to both the squeezed and anti-squeezed quadratures.
The scale of the error signal is arbitrary.

variance, or the noise on the noise of the field. This can be found by taking the kurtosis, which welabel ∆Ṽa1 and ∆Ṽa2 . The kurtosis in the amplitude and phase quadratures, respectively, is given
by
∆Ṽa1
∆Ṽa2

=
=

23

4

(δã1 − -δã1 .)23

4(δã2 − -δã2 .)It is important that the kurtosis is a factor of

4
4

3
42 √
2
− (δã1 − -δã1 .)= 2Ṽa1 ,3
42 √2− (δã2 − -δã2 .)= 2Ṽa2 .

(2.146)(2.147)√2 larger than the variance. As a measure of locking

stability, we express the kurtosis of the normalized variance of the photovoltage ∆Ṽc−d in terms ofphase fluctuations ∆θ.

From Eq. (2.133), equating its kurtosis with its variance due to phase fluctuation and using a
Taylor expansion of Ṽaθ to second-order around θ = θ0 , we findE
E
dṼ EE1 d2 Ṽ EE∆Ṽ (θ0 ) %∆θ +(∆θ)2 .
dθ Eθ02 dθ2 Eθ0
θ101(2.148)

Expanding both sides, the equation becomes
E
E
√
E
E
2(Ṽa1 sin2 θ0 + Ṽa2 cos2 θ0 ) = E(Ṽa1 − Ṽa2 ) sin 2θ0 ∆θ + (Ṽa1 − Ṽa2 ) cos 2θ0 (∆θ)2 E ,(2.149)which when solved for ∆θ at the two locking points (θ0 = 0, π/2) yields∆θ|θ0 =π/2
∆θ|θ0 =0

: √
2Ṽa1
=
,
Ṽa2 − Ṽa1
: √
2Ṽa2
=
.
Ṽa2 − Ṽa1

(2.150)(2.151)In the case of perfect squeezing in the amplitude quadrature with a squeeze factor of r, taking into√
account that the variance is proportional to ∆Ω and the kurtosis is proportional to ∆Ω, Eqs.
(2.150) and (2.151) become∆θ|θ0 =π/2
∆θ|θ0 =0

(1/4
2
∝
,
∆Ω
2
'
(1/4
1
2
∝
.
1 − e−4r ∆Ω21
e4r − 1'(2.152)(2.153)

If the detection efficiency ηdet is taken into account, the amplitude and phase quadrature variances
are given by Ṽa1 = ηdet e−2r + 1 − ηdet and Ṽa2 = ηdet e2r + 1 − ηdet , and then, Eqs. (2.152) and(2.153) become∆θ|θ0 =π/2
∆θ|θ0 =0:'
(1/4
ηdet e−2r + 1 − ηdet
2∝,
ηdet (e2r − e−2r )
∆Ω
:
'
(1/4
ηdet e2r + 1 − ηdet
2∝.
ηdet (e2r − e−2r ) ∆Ω

(2.154)(2.155)The stability of the two locking points is plotted as a function of squeeze factor r in Fig. 2-21.
For both the squeezed and anti-squeezed quadratures, the stability of the noise-locking improves as
the squeeze factor is increased. This is not surprising since it is from the quadrature asymmetrythat the error signal is derived. Eqs. (2.152) and (2.153) show that the locking stability for both
quadratures improves as the detection bandwidth is increased, albeit with a weak dependence.

2.7.3

Coherent Control of Squeezing

Introduction
Although the noise-locking technique is powerful in that it does not require any coherent fields, its
stability is limited by the kurtosis of the squeezed vacuum field. The coherent control technique is
102

1

10

Squeezed
Antisqueezed0

Dq (a.u.)

10

10

1

2

10

0

0.2

0.4

0.6

0.8

1
r

1.2

1.4

1.6

1.8

2

Figure 2-21: The stability of the two locking points as a function of squeeze factor r. As r increases,
the level of fluctuations in θ decreases for the squeezed quadrature and converges to unity for theanti-squeezed quadrature.
another useful method of locking the squeeze angle of a squeezed vacuum field that beats with acoherent LO field [161]. A subcarrier field which is frequency-shifted by Ωs from the carrier frequency
is injected into an OPO cavity within its linewidth. Due to the nonlinear interaction between thesubcarrier as a signal field at frequency ω + Ωs and the pump field at frequency 2ω, the difference
frequency generation takes place in the NDOPO, yielding the idler field at frequency ω − Ωs that iscorrelated with the signal. Since squeezing is a representation of the quantum correlation between
upper and lower quantum sidebands centered around the carrier frequency, the coherent correlation
between the signal and idler fields can also be used to obtain information about the squeeze angle.
Once the squeezed vacuum is generated by the NDOPO along with the coherent signal and idler
fields, when it is detected by a homodyne detector with a LO field, the beat between the LO andsignal/idler can be used to obtain the homodyne angle.

Error Signal for Squeeze Angle Control
Assuming that γsin % γiin , γsout % γiout , and the amplitude of the input signal field is real without theloss of generality, from Eqs. (2.37) and (2.45), the amplitudes of the output fields are given by



Āout
s
Āout∗i.
 = 2 γain γaout γatot− 12 E b̄∗103− 12 E b̄
γatot

−1 




Āin
s
0



,

(2.156)and found to be
Āout
=
s

√
2 ηin ηout inĀs ,1 − x2Āout∗=i√2 ηin ηout xe−iφ inĀs .
1 − x2(2.157)

Transforming back from the rotating frame of each field to the stationary frame, the output field isthe sum of the signal and idler fields,i(ω+Ωs )ti(ω−Ωs )tĀtot = Āout+ Āout.
s e
i e(2.158)When the field is detected by a photodetector, the measured intensity is given by|Ātot |2 = 4ηin ηout

1 + x2xĀin2 + 8ηin ηoutĀin2 cos(2Ωs t − φ).
(1 − x2 )2 s
(1 − x2 )2 s

(2.159)Demodulating Eq. (2.159) at frequency 2Ωs and low-pass filtering the demodulator output yield anerror signal for the coherent control of the squeeze angle,
κ = 8ηin ηoutxĀin2 cos φ.
(1 − x2 )2 s

(2.160)The error signal crosses zeros when φ = π/2 and φ = 3π/2 as expected. If x = 0, the parametricdown-conversion process does not occur and, therefore, the error signal vanishes.

Error Signal for Homodyne Angle Control
Similarly, an error signal for locking the homodyne angle or the relative phase between the squeezed
vacuum field and the LO field for homodyne detection can be obtained from the beat between the
LO field and the signal/idler.
Suppose that the squeezed vacuum field and the LO field B̄eiωt with phase θ relative to thesqueezed vacuum field enter a homodyne beamsplitter, as shown in Fig. 2-17. The two outputs of
the homodyne beamsplitter are given by



C





1

 = √1 
2
i
D

i
1



i(ω+Ωs )ti(ω−Ωs )tĀout+ Āout
s e
i e



B̄ei(ωt+θ),(2.161)and therefore, the difference photovoltage obtained from the difference photocurrent multiplied bycurrent-to-voltage coefficient R is given by
VC−D

E 2R †(C C − D† D)= R(IC − ID ) =2√
2 ηin ηout 2 in=
E Ās B̄ [sin(θ − Ωs t) + x sin(θ + Ωs t + φ)] .
1 − x2104(2.162)

Demodulating Eq. (2.162) at frequency Ωs and low-pass filtering the demodulator output yield anerror signal,√
2 ηin ηout 2 inκ=E Ās B̄ [sin θ + x sin(θ + φ)] .
1 − x2(2.163)The combination of both error signals in Eqs. (2.160) and (2.162) gives an error signal for controllingthe homodyne angle, θ.

105

106

Chapter 3

Experimental Generation,
Detection, and Control ofSqueezed States
3.1

Overview

In the previous chapter, the generation, degradation, detection, and control of squeezed statesusing a sub-threshold optical parametric oscillator (OPO) with a second-order nonlinear medium
and a homodyne detector have been theoretically introduced. In this chapter, the experimental
demonstration of the generation, detection, and control of squeezed states is presented. We focus
on the generation of squeezed states using a periodically-poled KTiOPO4 (PPKTP) crystal in aquasi-phase-matched OPO in a linear traveling-wave cavity configuration. The reasons for choosing
PPKTP as a nonlinear medium and the optical configuration will be explained in Chap. 4. In Sec.
3.2, the apparatus of a squeezed vacuum generator built in the Caltech 40m lab [165] is described1 .
In Sec. 3.3, experimental results of squeezing are presented.

3.2

Experimental Apparatus

3.2.1

Overview

As schematically shown in Fig. 3-1, the squeezed vacuum generator – or squeezer – consists of (1)a laser source, (2) a second-harmonic generator (SHG), (3) an optical parametric oscillator (OPO),
1 Although a few different squeezers (mainly with different crystals) have been built during my Ph.D., the latest
squeezer built at Caltech is mainly discussed in this chapter. Some results presented in this chapter come from theprevious squeezers, but the overall performance of squeezing is intact.

107

Nd:YAG
MOPA LASER

Spectrum
A nalyzer

Faraday
Isolator PO

Beam Dump
Subtractor
PZT4
Fiber Mode Cleaner

HPD2

Fiber Output
CouplerBS

PO
Faraday λ/2Isolator

Fiber Input
Coupler

S

PZT3

HPD1

S

SHG
PO

A uxiliary
Nd:Y A G LA SER

PM1

PD1

DBS
PZT2

Faraday
Isolator

DBS
BS

PBS

PD3

PZT1

S

PM2

λ/2
BS

OPO

PD2

Beam Dump

Figure 3-1: A schematic of the squeezer. SHG: second-harmonic generator, OPO: optical parametric oscillator, PM1 and PM2: phase modulators, PZT1-4: piezo-electric transducers, DBS: dichroic
beamsplitter, λ/2: half-wave plate, PBS: polarizing beamsplitter, BS: 50/50 beamsplitter, PO: pickoff beam sampler, PD1-3: photodetectors, HPD1 and HPD2: homodyne photodetectors. Spectrum
analyzers HP8591E (Hewlett-Packard) and SR785 (Stanford Research Systems) are used to measure
the response of the homodyne detector at MHz and lower frequencies respectively.

(4) subcarrier optics, (5) a homodyne detector with a mode-cleaned local oscillator (LO) field, and(6) feedback control systems to lock the SHG and OPO cavities and the squeeze angle. Simply put,
the OPO pumped by a second-harmonic field generated by the SHG produces a squeezed vacuum
field, which is then detected by the homodyne detector with the LO. In the following sections, theroles and properties of each major component are described.

3.2.2

Laser

The main laser source that powers the squeezer is a single-frequency Nd:YAG Master Oscillator
Power Amplifier (MOPA) laser in which a Lightwave 126 (Lightwave Inc.) master laser is amplified
by a series of laser amplifiers. The laser wavelength is 1064 nm. The output of the MOPA laser isabout 6 W, and about 1.5 W of it is utilized for the SHG, OPO, and other optics in the squeezer.
Although a pre-stabilized laser system [166] is available at the Caltech 40m lab [165], it is notnecessary for the generation of squeezed states, and thus, it is not discussed here.
108

Differential
AmplifierSecondHarmonic
Field

Setting
Temperature

Thermistor
Phase
Modulator

HeaterPZT
MgO:LiNbO 3

Fundamental
Field

Low-Pass
FilterDichroic
MirrorOutputCoupler

Photodetector

Oven

Mixer
High Voltage
Amplifier

Low-Pass
FilterOscillator

Figure 3-2: A detailed schematic of the SHG that consists of a 5% MgO:LiNbO3 crystal and an
external coupling mirror. The reflectivity of the mirror is 95.0% at 1064 nm and 4.0% at 532 nm.
The reflectivity of the curved surface of the crystal is 99.95% at both 1064 nm and 532 nm. Thus,
the SHG forms a cavity at 1064 nm, but not at 532 nm. The SHG cavity is locked on resonance by
the transmission PDH locking technique. The temperature of the crystal is also controlled by the
feedback control system using a PID controller.

3.2.3

Second-Harmonic Generator

The role of the SHG is to generate a second-harmonic field to pump the OPO. A detailed schematicof the SHG is shown in Fig. 3-2. The SHG is a cavity composed of a 5% MgO:LiNbO3 hemilithic
crystal (Photon LaserOptik Inc.) and an external coupling mirror. One surface of the crystal is flatand AR coated. The other surface is curved with a radius-of-curvature of 8 mm and its reflectivity is
99.95% at both 1064 nm and 532 nm. The coupling mirror has a radius-of-curvature of 50 mm andis located at 48 mm from the center of the crystal. Its reflectivities are 95.0% at 1064 nm and 4.0%
at 532 nm. It is mounted on a piezo-electric transducer (PZT) as an actuator that adjusts the cavity
length to resonance at 1064 nm. The SHG cavity is locked by the transmission Pound-Drever-Hall(PDH) technique in which the 1064 nm transmission through the SHG cavity is demodulated at thefrequency at which the phase of the 1064 nm pump is modulated and the mixer output is low-passfiltered and fed back to the PZT after some amplification. The crystal is enclosed in an oven heatedby a current-driven NiCr wire and is maintained at 114.05 ◦ C to optimize the SHG conversion
efficiency for type I phase-matching in which the fundamental field is polarized in the ordinary axisof the crystal while the second-harmonic field is polarized in the extraordinary axis. When thetemperature is optimized, their matched refractive index is 2.23. The measured dependence of the109

SHG conversion efficiency on the temperature of the crystal is shown in Fig. 3-3. As compared tothe single-pass SHG conversion efficiency in Fig. 2-7, the measured SHG conversion efficiency does
not show a clear sinc function that has been predicted in Sec. 2.4.4. This is due to the dependenceof the cavity-SHG conversion efficiency on the fundamental pump power as shown in Fig. 2-12.

9Second Harmonic Power (a.u.)

8
7
6
5
4
3
2
1
0
104

106

108

110
112
Temperature (°C)

114

116

118

Figure 3-3: The measured dependence of the SHG conversion efficiency on the temperature of thecrystal. As compared to the single-pass SHG conversion efficiency in Fig. 2-7, the measured SHG
conversion efficiency does not show a clear sinc function that has been predicted in Sec. 2.4.4. Thisis due to the dependence of the cavity-SHG conversion efficiency on the fundamental pump power
as shown in Fig. 2-12.
The SHG cavity is pumped by 1.2 W of the MOPA laser at 1064 nm and generates 350 mW at 532
nm, which is then used to pump the OPO cavity. The cavity size of 20 mm - 60 mm is often chosenbecause both the fundamental and second-harmonic fields diverge significantly at large distances dueto their small waist sizes. The mode structure of the SHG cavity using the fundamental field andthe corresponding modes of the second-harmonic field are shown in Fig. 3-4. The mode structureof the generated second-harmonic field is also shown.

3.2.4

Optical Parametric Oscillator

The role of the OPO is to produce squeezed states of vacuum, and therefore, the OPO is thecore of the experiment. A detailed schematic of the OPO is shown in Fig. 3-5. The OPO is acavity composed of a 10 mm long, quasi-phase-matched, domain-engineered PPKTP crystal (RaicolInc.) with AR coated flat surfaces and two external coupling mirrors with radii-of-curvature of 10
110

Pump Transmission (a.u.)
Second Harmonic Power (a.u.)

15
10
5
0
−5

−6

−4

−2

0

2

4

6

−6

−4

−2

0
2
Cavity Length (a.u.)

4

6

0.6
0.4
0.2
0
−0.2

Figure 3-4: The mode structure of the SHG cavity using the fundamental transmission and thecorresponding modes of the second-harmonic field. The evident modes are TEM00 modes. Higher
modes of the cavity are suppressed by optimizing the transmission of the TEM00 modes. The reason
why a few higher modes are present in the pump transmission, but no corresponding higher modes
are present in the second-harmonic power is because the SHG cavity mirror reflectivities are chosen
to optimize the SHG conversion efficiency of the fundamental pump field in the optimized TEM00
transmission.

mm. The reflectivities of the input and output coupling mirrors are 99.95% at both 1064 nm and532 nm and 92.0% at 1064 nm and 4.0% at 532 nm respectively. The OPO cavity length is 2.3
mm. The crystal is mounted on a heater that consists of a current-driven NiCr wire in an ovenand maintained at 33.5 ◦ C to optimize the 1064/532 parametric down-conversion. To optimize
the temperature, the parametric gain of a bright seed field is usually examined and maximized bytuning the temperature. The measured dependence of the nonlinear interaction strength in the OPO
cavity on the temperature of the crystal is shown in Fig. 3-6. The OPO cavity is locked by thetransmission PDH locking technique with a frequency-shifted subcarrier which will be described in
the next section. One of the coupling mirrors is mounted on a PZT as an actuator used to keep
the cavity on resonance at 1064 nm. The pump is aligned and mode-matched to the OPO cavity
by optimizing the parametric gain. The measured dependence of the OPO parametirc gain on thepump power is shown in Fig. 3-7.
When pumped by the second-harmonic pump at frequency 2ω, the OPO correlates upper andlower quantum sidebands centered around the carrier frequency ω. The sum of the correlated
quantum sidebands appears as a squeezed vacuum field, as discussed in Chap. 2. In the case of111

Oscillator

Squeezed
VacuumPolarizer
High Voltage
Amplifier

Mixer

Low-Pass
FilterOven
Subcarrier
Field

Photodetector
OutputCoupler

Dichroic
Mirror

PZT
PPKTP

Phase
Modulator

SecondHarmonic
Field

Thermistor
Input
Coupler

HeaterSetting
Temperature

Differential
Amplifier

Low-Pass
FilterFigure 3-5: A detailed schematic of the OPO that consists of a periodically-poled KTiOPO4 (PPKTP) crystal and two coupling mirrors. Both surfaces of the crystal are AR coated. The reflectivitiesof the input and output coupler are 99.95% at both 1064 nm and 532 nm and 92.0% at 1064 nm
and 4.0% at 532 nm respectively, and therefore, the OPO forms a cavity at 1064 nm, but not at
532 nm. The subcarrier field is used to lock the OPO cavity using the transmission PDH locking
technique. The polarization of the subcarrier is orthogonal to that of the squeezed vacuum field.
The subcarrier is frequency-shifted relative to the fundamental field so that the vacuum field at
the fundamental frequency can resonate and interact with the second-harmonic pump in the OPO
cavity. The temperature of the PPKTP crystal is maintained stable by the feedback control systemusing a PID controller.
generating squeezed vacuum as opposed to squeezed light, the seed field that couples into the cavity
from the input coupler contains no coherent amplitude, and therefore the squeeze angle which isdefined by the angle between the amplitude axis and the squeeze axis or the semiminor axis of thenoise ellipse does not exist unless the squeezed vacuum beats with a coherent field.

3.2.5

Subcarrier Optics

The role of the subcarrier is to lock the OPO cavity on resonance for the stable generation of squeezed
states. A coherent seed at the carrier frequency cannot be used as a locking probe field because
it contaminates squeezing at low frequencies due to its laser excess noise. Instead, a frequencyshifted subcarrier field which is orthogonally polarized to the squeezed vacuum can be used as alocking probe. There are two possible ways to generate such a subcarrier field: (1) an acousto-optic
modulator (AOM), and (2) a second laser. The first scheme is useful if the closest usable mode is near
112

40

Second Harmonic Power (a.u.)

35
30
25
20
15
10
5
0
20

25

30
35
Temperature (°C)

4045Figure 3-6: The measured dependence of the nonlinear interaction strength in the OPO cavity on thetemperature of the crystal. This is obtained by using the OPO cavity as a SHG and measuring thegenerated second-harmonic power. As discussed in the previous section, the measured temperaturedependence does not show a clear sinc function due to the dependence of the cavity-SHG (notsingle-pass SHG) conversion efficiency on the fundamental pump power. The nonlinear interaction
temperature range of PPKTP is much larger than that of MgO:LiNbO3 .

the carrier frequency. In general, since the OPO cavity length is short, the free spectral range (FSR)
is about 5 - 10 GHz, and therefore, the usable modes tend to be away from the carrier frequency byseveral GHz or so. If an AOM for such a high frequency-shift is available, the first scheme can beused. Double-passing a beam through an AOM is often used to obtain a high frequency-shift, but
its low diffraction efficiency at GHz frequencies might lead to a large optical loss in the frequencyshifted subcarrier. On the other hand, the second scheme does not have this problem although thefrequency ranges of the two lasers have to overlap. However, this scheme requires the second laser
to be phase-locked to the main laser to maintain the constant frequency-shift. In this experiment,
the closest TEM00 mode in the orthogonal polarization to the squeezed vacuum is 642 MHz awayfrom the carrier frequency. Since such a high frequency-shift AOM is costly, the second scheme isemployed.
The second laser is also a Lightwave 126 (Lightwave Inc.) laser with a throughput of 650 mW.
About 150 mW of it is injected into the OPO cavity in the P polarization. A random vacuum
field centered around the carrier frequency enters the OPO cavity in the S polarization so that thegenerated squeezed vacuum comes out of the cavity in the S polarization. A vacuum field in the
P polarization does not participate in the nonlinear optical process in the OPO cavity due to its
113

6

10

Parametirc Amplification
Parametric Deamplification
No Pump5

10

4

Parametric Gain

10

3

10

2

10

1

10

0

10

1

10

0

50

100

150
200
250
300
Second Harmonic Pump Power (mW)

350

400

Figure 3-7: The measured dependence of the OPO parametric gain on the pump power. When thepump and seed are in phase in the crystal, the seed is parametrically amplified. When they are outof phase in the crystal, the seed is parametrically deamplified.
non-interacting polarization with the pump field. The OPO cavity is locked using the transmission
PDH technique in which the subcarrier is phase-modulated by an electro-optic modulator (EOM) asa phase modulator (PM), the transmission is demodulated at the modulation frequency, yielding anerror signal, and then, the control signal is fed back to the PZT-actuated mirror of the OPO cavity.
To maintain the 642 MHz frequency-shift between the two lasers, the second laser is phase-locked
to the MOPA laser. The beat of the two lasers is detected by a PD1611 photodetector (New Focus
Inc.) with a bandwidth of 1 GHz. To maximize the beat signal with respect to the DC power of thetwo lasers, their optical powers are set to be equal. The beat signal is demodulated at the frequencyof 642 MHz given by an external oscillator, yielding an error signal, and then, the produced control
signal is fed back to the piezo and thermal actuators of the second laser.

3.2.6

Homodyne Detector

The role of the homodyne detector is to measure the quadrature variance of an input squeezed field.
The homodyne detector is composed of a 50/50 beamsplitter, a pair of photodetectors (HPD1 and
HPD2), and a LO field. The difference between the two optical responses measured by the HPD1 and
HPD2 with a LO field is analyzed by spectrum analyzers HP8591E (Hewlett-Packard) and SR785(Stanford Research Systems) at MHz and lower frequencies respectively. The homodyne beamsplitterused in this experiment is a polarization-sensitive 50/50 beamsplitter so that the splitting ratio of
114

the beamsplitter can be changed by changing the polarizations of the input fields to optimize thehomodyne efficiency. When the squeezed vacuum and LO fields enter the homodyne beamsplitter,the detector forms an opto-electrical Mach-Zehnder interferometer, giving the linear response to thelevel of squeezing beated against the LO.
To ensure a high homodyne efficiency at the homodyne detector, the LO field needs to be modecleaned. Mode-cleaning can be done by either (1) an optical cavity, or (2) an optical fiber. In thisexperiment, the second method is adopted because it is less costly than the first method and also
convenient to use in the optical layout. The squeezed vacuum mode is also clean due to the OPO
cavity. Therefore, when the two fields beat at the homodyne beamsplitter after some alignment
and mode-matching work, a high homodyne efficiency can be obtained. In this setup, a homodyne
visibility of 98.0 % is achieved.

3.2.7

Quantum Noise Locking

The squeeze angle is locked to either the amplitude or phase quadrature by use of the quantum noiselocking technique which has been described in Sec. 2.7.2. AD8361 (Analog Devices) together witha high-pass filter at 100 kHz and a low-pass filter at 100 MHz is used as an envelope detector. The
measured stability of the noise locking is shown in Fig. 3-8. The difference in stability between thesqueezing and anti-squeezing has been predicted in Sec. 2.7.2. The measured difference in stability2
is in good agreement with the theoretical model given by Eqs. (2.154) and (2.155).

3.3

Experimental Results

3.3.1

Overview

In most stand-alone vacuum-squeezing experiments, the squeezed variance is analyzed by a highfrequency spectrum analyzer when the squeeze angle is scanned or locked and then by a low-frequency
spectrum analyzer when the squeeze angle is locked. In the next two sections, the spectra of scanned
and locked squeezed shot noise are experimentally measured and presented. In typical low-frequency
squeezing experiments, the high-frequency spectrum analyzer is used to diagnose the level of squeezing. It is important to note that optimizing the squeezing level at high frequencies by balancing
the homodyne photodetector responses does not always mean optimizing the squeezing level at low
frequencies since it is possible that the two photodetectors have different frequency responses. In thelast section, the quantum correlation between upper and lower quantum sidebands centered aroundthe fundamental (carrier) frequency is verified by detuning the OPO cavity.
2 The noise-locking stability was measured using a different squeezer, not the one described in this chapter, andtherefore, the parameter values for the fit are different. However, the noise-locking performance of the squeezer inthis chapter is expected to be similar.

115

20
Squeezed
Anti squeezed

Error Signal Spectrum (dBm)

25
30
35
40
45
50
55
60
65 2
10

3

10
Frequency (Hz)4

10

Figure 3-8: The stability of the noise locking technique. The noise spectrum of the noise locking
error signal in the case of locking to the squeezed quadrature is smaller than in the case of locking
to the anti-squeezed quadrature. The theoretical model given by Eqs. (2.154) and (2.155) withthe measured efficiency of η = 0.55 and the initial squeeze factor of r = 0.9 that yields a stability
separation of 4.2 dB is in good agreement with the measured stability difference.

3.3.2

Spectrum of Scanned Squeezed Shot Noise

When the squeezed field has no coherent amplitude, the homodyne angle is equivalent to the squeeze
angle. The squeeze angle can be scanned by scanning either the phase of the pump field relative to
the OPO cavity (PZT3) or the phase of the LO relative to the squeezed vacuum (PZT4). In thisexperiment, the second scheme (PZT4) is used because in the first scheme, the PZT could be noisy
enough to excite either the SHG or OPO cavity if it is in their vicinity.
Fig. 3-9 shows the fixed-frequency, zero-span spectra of the shot noise and squeezed/antisqueezed shot noise when scanning the squeeze angle at 10 Hz. The periodic oscillation of thenoise power with respect to the squeeze angle is evident in the figure. All the measurements aredone at 900 kHz with zero frequency span. The resolution bandwidth (RBW) is 100 kHz and thevideo bandwidth (VBW) is 3 kHz. The squeeze angle is scanned by the PZT4 at 10 Hz, but calibrated to θ in the figure. The electronic noise, which consists of the electronic noise of the HPD1
and HPD2, a SR560 low-noise pre-amplifier (Stanford Research Systems) used as a subtracter witha gain of 1000, and the SR785 spectrum analyzer, was measured to be 17.4 dB below the shot noiselevel. This large separation between the shot noise level and electronic noise is sufficient to observe
squeezing without subtracting the electronic noise. The measured squeezed/anti-squeezed noise
116

Noise Power Relative to Shot Noise [dB]

20
15
10
Squeezed/Antisqueezed
Shot Noise

50

Shot Noise

−5
−10
0

0.5

1
θ [π radian]

1.5

2Figure 3-9: The fixed-frequency spectra of shot noise and squeezed/anti-squeezed noise power when
the squeeze angle was scanned. The measurements were done at 900 kHz with zero frequency span.
The resolution bandwidth (RBW) is 100 kHz and the video bandwidth (VBW) is 3 kHz. The squeeze
angle was scanned by the PZT4 with a ramp function at 10 Hz. The electronic noise was 17.4 dB
below the shot noise level. The measured squeezed/anti-squeezed noise power is in good agreementwith the theoretical fit (solid curve) given by Eqs. (2.132), (2.69), and (2.70) based on the measured
composite efficiency of η = ηout ηhom ηPD ηloss = 0.85. The nonlinear coupling efficiency is found fromthe fit to be x = 0.75.
power is in good agreement with the theoretical fit (solid curve) given by Eqs. (2.132), (2.69), and(2.70) based on the measured composite efficiency of η = ηout ηhom ηPD ηloss = 0.85. The nonlinear
coupling efficiency is found from the fit to be x = 0.75.
There are a few important things to note: (1) the imbalance in level between the squeezing
and anti-squeezing spectra, and (2) fitting to estimate the generated level of squeezing in the OPO.
Ideally, the squeezing and anti-squeezing levels are equal in the OPO, such that ∆a1 ∆a2 = 1.
However, since squeezing is more susceptible to optical losses than anti-squeezing as discussed inSec. 2.5, the product of the amplitude and phase quadrature variances becomes greater than unity,
such that ∆a1 ∆a2 > 1. As for the second point, one could try to fit Eq. (2.132) with Eqs. (2.69)
and (2.70) as input quadrature variances to the squeezing/anti-squeezing spectrum. However, thisis not an easy task since as losses increase, the asymmetry between the squeezing and anti-squeezing
levels increases, making it more likely to see anti-squeezing than squeezing. If equal weight is oneach data point when fitting, it is often the case to incorrectly estimate the initial level of squeezingin the OPO. In addition, as the squeezing level increases, there are fewer data points in the squeeze
angle range of squeezing. This makes it difficult to fit the right function to the data. For these117

Noise Power [dBm]−70
Shot Noise

−75
−80

7.4 dB−85

Squeezed Shot Noise−90
Electronic Noise
−95

3

10

4

10
Frequency [Hz]5

10

Figure 3-10: The broadband spectra of shot noise, squeezed shot noise, and electronic noise. The
spectra are averaged 2000 times. The squeeze angle is noise-locked without any coherent light.
The spikes at 13.5 kHz and 15.7 kHz are due to the modulation of the PZT4 for the noise-locking
technique. The shot noise and squeezed shot noise spectra are in good agreement with the theoreticalpredictions (solid lines) based on the measured optical power and losses. The noise increase at
frequencies below 2 kHz in both the shot noise and squeezed shot noise spectra is due to theinadequate isolation of scattered photons (mainly from the homodyne detector) and the imperfectbalancing of the homodyne photodetector responses.
reasons, if one wants to measure the exact level of squeezing, one should lock the squeeze angle andtake the average of the squeezing data points over some period of time.

3.3.3

Spectrum of Locked Squeezed Shot Noise

When the squeeze angle is locked, a broadband squeezing spectrum can be obtained. The noiselocking technique is used, in which the squeeze angle is locked by dithering the PZT4 to obtain anerror signal and correcting the phase of the LO with the same PZT. Fig. 3-10 shows the noise powerof the squeezed and anti-squeezed shot noise when locking the squeeze angle as well as the electronicnoise of the detection system. A broadband squeezing of 7.4 ± 0.1 dB at frequencies above 3 kHz
and a cutoff frequency for squeezing at 700 Hz are evident. The spikes at 13.5 kHz and 15.7 kHz are
due to the modulation of the PZT4 for the noise-locking technique. Ignoring the low frequency noise
increase, the shot noise and squeezed shot noise spectra are in good agreement with the theoreticalpredictions based on the measured optical power and losses.
The noise increase at frequencies below 2 kHz in both the shot noise and squeezed shot noise
spectra is due to the inadequate isolation of scattered photons (mainly from the homodyne detector)
118

Noise Power Relative to Shot Noise (dB)10Squeezed
Antisqueezed

8
6
4
2
0
−2
−4
500

550

600
650
700
Frequency (MHz)

750

800

Figure 3-11: The degradation of squeezing as a function of the detuning of the subcarrier frequencyshift. As the detuning increases, the destruction of the correlation between upper and lower quantum
sidebands around the fundamental frequency leads to decrease in the squeezing level. The measured
data are compared with the theoretical fits that yield the resonance frequency to be 642.9 MHz.
and the imperfect balancing of the homodyne photodetector responses. To observe squeezing at
lower frequencies requires careful shielding of the LO or ambient light induced scattered photons
from coupling to the OPO cavity. This was demonstrated very well in Refs. [17] and [18], and is
not part of the initial goals of the present experimental demonstration.

3.3.4

Verification of Quantum Correlations

By detuning the frequency-shift of the subcarrier, the OPO cavity can be detuned with respect
to the fundamental frequency. This destroys the quantum correlation between upper and lower
quantum sidebands centered around the fundamental frequency. As discussed in Sec. 2.3.3, when
the OPO cavity is detuned continuously, the quantum correlation is weakened gradually, degrading
the overall level of squeezing in a frequency-dependent manner. The measured destruction of thequantum correlation as a function of the detuning is shown in Fig. 3-11. The figure also showstheoretical fits to the measured data points3 . The resonance frequency is found to be 642.9 MHz
from the fits. As the detuning keeps increasing, the next TEM00 mode at one FSR away from theresonance frequency hits resonance, and squeezing can be observed again.
3 The measured maximum level of squeezing seems to be lower than the locked level of squeezing in Fig. 3-10
because this measurement was taken at a different time when the squeezer was not optimized. In other words, thismeasurement was necessary before optimizing the squeezer, in order to find out the subcarrier resonance frequency.

119

120

Chapter 4

Requirements for Quantum
Enhanced Gravitational Wave
Detectors
4.1

Overview

The basic generation, degradation, detection, and control of squeezed states have been theoretically described in Chap. 2 and experimentally demonstrated in Chap. 3. However, although thesetechniques are useful for stand-alone squeezers and quantum-enhancement in table-top scale interferometers, they are not directly applicable to quantum-enhanced laser-interferometric gravitational
wave (GW) detectors. Key requirements toward the actual implementation of quantum-enhancementin GW detectors, such as Advanced LIGO, include (1) squeezing in the GW detection band (10 Hz- 10 kHz), (2) a high level of squeezing (ideally 10 dB or higher)1 , (3) long-term stability, and (4)frequency-dependent squeezing. In this chapter, these four requirements are studied and solutions
to them are also presented.

4.2

Squeezing in the Gravitational Wave Band

4.2.1

Overview

The first requirement is to generate squeezed states in the GW detection band. However, this is notan easy task because the frequency of squeezing is limited by various noise sources that couple into
an optical parametric oscillator (OPO) that generates squeezed states. The limiting noise sources
1 Although this is not exactly a requirement, generating a high level of squeezing is necessary since losses in theinterferometer and injection optics attenuate some amount of squeezing.

121

are (1) seed noise [134, 17, 142], (2) pump noise [134, 167, 168, 169], (3) photothermal noise [134],and (4) scattered photon noise [18, 17]. In the following sections, these noise sources are discussed
in detail.

4.2.2

Seed NoiseThe seed noise is represented by ṼAin1 (Ω) and ṼAin2 (Ω) in Eqs. (2.65) and (2.66). When the seed isshot-noise-limited in some frequency band so that ṼAin1 (Ω) = ṼAin2 (Ω) = 1, then squeezing is achievablein the band. If the seed is noisy so that ṼAin1 (Ω) > 1 or ṼAin2 (Ω) > 1, the quadrature of the output
field where the input seed is noisy cannot be squeezed. Typically, the seed is not shot noise limited
at frequencies below 1 MHz due to its laser classical noise (laser amplitude and phase noise). In
most cases, this is the dominant noise that limits the frequency of squeezing. For this reason, thegeneration of squeezed states has been traditionally done in the MHz regime [17, 18].
To solve this issue, a few schemes have been proposed and experimentally demonstrated. Bowenet al. used two OPOs to cancel classical noise on a pair of squeezed light fields generated by the two
OPOs with two seeds that originate from a single laser source and share the same classical noise,
and achieved broadband squeezing at frequencies down to 220 kHz [170]. Another scheme is to use
a vacuum field at the fundamental frequency as a seed to the OPO cavity since it is in the coherent
state at all frequencies [17, 16, 18]. However, this leads to two issues: (1) the OPO cavity needs tobe controlled without coherent light in the case of a singly-resonant OPO cavity which is often used
as a squeezed state generator [13, 17, 155, 138, 139, 142], and (2) the generated squeezed vacuum
does not contain a coherent amplitude that is often used as a probe to lock the squeeze angle whenmixed with a local oscillator (LO) field.
Squeezing in the audio frequency band was first observed by McKenzie et al. [17]. They seeded a
sub-threshold OPO with a vacuum field while locking the OPO cavity manually and controlled thehomodyne angle using the quantum noise-locking technique [160]. Later, Vahlbruch et al. achieved
the generation of squeezing in the full GW detection band, using a frequency-shifted bright subcarrier
field along with the squeezed vacuum field and detecting photons generated in a nondegenerate OPO(NDOPO) to obtain the information of the squeeze angle [18]. They used another frequency-shifted
subcarrier to lock the OPO cavity to the mode which is one free-spectral-range (FSR) away from thecarrier mode such that the vacuum field at the carrier frequency can resonate in the OPO cavity in a
TEM00 mode. Recently, a doubly-resonant OPO cavity for the production of low-frequency squeezed
states has been proposed and demonstrated [115]. This scheme does not require a subcarrier field
to lock the OPO cavity since it is done by the pump field with a phase compensation plate.
122

4.2.3

Pump NoiseThe pump noise is represented by ṼBin1 (Ω) and ṼBin2 (Ω), but they do not appear in Eqs. (2.65) and(2.66) because we have assumed |ā|δb + |b̄|δa in the case of the undepleted pump. This assumption
is valid in most cases since typically |ā| +| b̄| and the fluctuations in the pump and seed fields areon the same order. However, if this is not the case, the pump noise couples into Eqs. (2.65) and(2.66). For example, if a second-harmonic generator (SHG) is noisy and adds noise to the generated
second-harmonic field, the assumption |ā|δb + |b̄|δa is no longer valid. The details of this case canbe found in Ref. [134].

4.2.4

Photothermal NoiseThe photothermal noise arises from changes in temperature of the optical medium due to absorbed
optical power [79, 171, 172]. By nature, photothermal effects are significant at low frequencies since
thermal effects are typically slow. This effect may be significant in most nonlinear crystals since many
have relatively high absorption rates. For example, a MgO:LiNbO3 crystal has a linear absorption
rate of about 4 %/cm at 532 nm [142]. High absorption rates coupled with the high circulating power
required for strong nonlinear interaction result in a large amount of optical power absorbed into the
crystal, which may cause a significant temperature change. The average temperature change dueto the power absorbed in the crystal can be compensated for by using a temperature controller anddoes not pose significant problems for most experimental systems.
Instead, we focus on the effect of photothermally induced temperature fluctuations, caused byfluctuations in the circulating power in an OPO cavity. The circulating power fluctuations could haveboth classical and quantum origins, or in the case of a shot noise limited system, only the quantum
mechanical origin. The photothermal noise caused by thermal-expansive noise and thermal-refractive
noise has two degrading effects on the production of squeezed light in the OPO. The first effect is
via fluctuations in the nonlinearity. This arises as the nonlinear strength is temperature-dependent
due to the phase-matching condition. It is a significant noise source in birefringent phase-matchedcrystals since they have strong temperature dependence. The second effect is via optical path length
fluctuations. The temperature fluctuations cause the optical path length to change, potentiallycausing a detuning of the optical cavity. These effects appear as 1/(Ω2 + Ω2T ) in variance, where Ωis the sideband (measurement) frequency and ΩT is the thermal relaxation cutoff frequency of thenonlinear medium, and are, therefore, primarily significant at low frequencies. The full derivationof the photothermal-noise-coupled quadrature variances can be found in Ref. [134].
However, the photothermal noise can completely vanish in the absence of the bright seed since
the coupling of the photothermal noise is proportional to the seed power. In addition, by use ofquasi-phase-matched crystals instead of type I phase-matched crystals, the effect of the photothermal
123noise can be reduced significantly. For these reasons, the photothermal noise is not as significant as
the seed noise.

4.2.5

Scattered Photon NoiseThe scattered photon noise appears at low frequencies and often shows 1/f spectrum. It is caused
by back-scattered photons mainly from a homodyne detector that return to the OPO, behave likea seed, and degrade squeezing at low frequencies. This noise is analogous to the noise induced byback-scattered light in the LIGO interferometers at low frequencies [91].
To prevent the back-scattered photons that come from the homodyne detector and all otheroptics as reflections, a few different techniques can be used. One is to install a Faraday isolator
between the OPO and homodyne detector. It prohibits the back-scattered photons from retuning to
the OPO cavity to some degree at the expense of some loss in the level of squeezing [17]. Anothermethod is to protect the homodyne detector from the environment using an enclosure and beam
dumps [18]. Removal of scattered light is the key factor that allows measurement of squeezing at
frequencies below 100 Hz [18]. One of the best solutions is to use a bow-tie OPO cavity configuration
since it geometrically isolates back-scattered photons from the OPO output field and does not requirea Faraday isolator [115].

4.3

High Level of Squeezing

4.3.1

Overview

The second requirement is a high level of squeezing. Although this is not exactly a requirement, it
is required to generate a high level of squeezing so that losses in interferometers and injection optics
do not completely kill the effect of squeezing. The typically achieved level of squeezing is about 3 to
6 dB [16, 13, 17, 170, 115, 142, 138, 139, 112, 18]. Fig. 4-1 shows the history of observed squeezing
levels between 1985 and 2007 [173, 11, 120, 17, 16, 15] since the first observation of squeezing by
Slusher et al. [12]. It is evident that the squeezing level has grown gradually2 .
Reflecting this trend, 10 dB of squeezing has been the world-wide goal for the past several
years [156, 16], not just for GW detectors at 1064 nm, but also for quantum optics research suchas quantum teleportation [112, 138] and quantum imaging [116] at other wavelengths. For GW
detectors, 10 dB of squeezing or a factor of 3.16 increase in strain sensitivity has been the researchtarget in the GW quantum research consortium because it is a reasonably achievable goal for the
past and next several years and in addition, even if more than 10 dB of squeezing was available,
2 This figure contains squeezing levels at wavelengths other than 1064 nm. High levels of squeezing at somewavelengths can be achieved more easily than at 1064 nm since the quantum efficiencies of both Si and InGaAs
photodiodes at 1064 nm are not as high as at some particular wavelengths.

124interferometer losses would attenuate it down to 10 dB [82, 103].

12

Squeezing Level (dB)10Squeezing in Four−Wave Mixing
MHz Squeezing in OPO
Audio−Frequency Squeezing in OPO
Squeezing−Enhancement in MHz Interferometry
Squeezing−Enhancement in Prototype GW Detectors

Vahlbruch et al.

Goda et al.

8
Polzik et al.
6

4

McKenzie et al.

Wu et al.

Goda et al.

Xiao et al.

2

Slusher et al.
0
1980

1985

1990

1995
Year

2000

2005

2010

Figure 4-1: The history of observed squeezing levels between 1985 and 2007 [173, 11, 120, 17, 16, 15]since the first observation of squeezing by Slusher et al. [12].
From Eqs. (1.61) and (1.66), 10 dB of squeezing in GW detectors corresponds to a factor of 3.16
increase in detector sensitivity or equivalently a factor of 3.163 = 31.6 increase in GW detection
rate for isotropically distributed GW sources3 . It is also equivalent to a factor of 3.16 increase ininterferometer arm lengths or a factor of 3.162 = 10 increase in laser power (in the case of reducing
shot noise), or equivalently a factor of 3.16 decrease in interferometer arm lengths or a factor of 10
decrease in laser power for the same shot noise when the interferometer is squeezing-enhanced. In
LIGO, the length of the arms is 4 km and it is nearly impossible to increase it by a factor of 3.16
because it is costly. Moreover, a factor of 10 increase in laser power is technically very difficult toachieve. Therefore, squeezing is a very powerful technique if the injection of it into GW detectors
does not add any noise (which will be shown in Chap. 6). Very recently, the generation of 10
dB of squeezing has been reported by Vahlbruch et al. [174] and it is not far to realize 10 dB ofsqueezing-enhancement in GW detectors.
The requirements for high levels of squeezing are (1) low optical losses (both intra-cavity andextra-cavity), (2) high nonlinearity, (3) high pump power, and (4) high stability of squeezing control.
Although these requirements mutually influence each other, satisfying any one of the requirements
helps increase the level of squeezing. For example, increasing the escape efficiency of the OPO cavity
by decreasing the reflectivity of the output coupler compensates for a high intra-cavity optical loss
3 Appendix D will show this conversion in detail.

125

although it requires a high pump power. A low value of nonlinearity can also be compensated forby a high pump power, but a high pump power often leads to photothermal problems because anonlinear crystal absorbs some mount of the pump power, leading to an increase in its temperatureand making the OPO cavity unstable.
In the following sections, the first three requirements are discussed since they are strongly related
to each other. The last requirement will be described in Sec. 4.4 since it is directly related to thethird requirement for quantum-enhancement (long-term stability).

A
B
C
D
E
F
G
H
I

MgO:LiNbO3
4.3 pm/V
BPM type I
60 - 120 ◦ C
0.1 - 1.0 %/cm
a few %/cm
0.7 %/cm
250 MW/cm2
50 MW/cm2
strong GRIIRA
doping inhomogeneity

KTP
4.5 pm/V
BPM type II
60 - 100 ◦ C
0.1 - 0.3 %/cm
1 %/cmnegligible
500 MW/cm2
100 MW/cm2
grey-tracking
type II QPM

PPLN
16.5 pm/V
QPM
above 200 ◦ Ca few %/cm
high
0.7 %/cm
300 MW/cm2
60 MW/cm2
photorefractive effecthigh optical loss

PPKTP
16.9 pm/V
QPM
around 35 ◦ C
0.1 - 0.3 %/cm
1 %/cmnegligible
500 MW/cm2
100 MW/cm2
grey-tracking

Table 4.1: Comparison of the optical properties of different nonlinear media [175, 176, 177, 178, 179,
180, 181, 150, 182, 183, 138, 184]. A: nonlinearity (bulk), B: phase-matching type, C: phase-matchingtemperature, D: optical loss at 1064 nm, E: optical loss at 532 nm, F: GRIIRA, G: damage thresholdat 1064 nm (10 ns), H: damage threshold at 532 nm (10 ns), I: disadvantages. Since different papers
report different parameter values, their averages are taken into account. KTP and PPKTP do notshow the sign of GRIIRA [138]. PPLN is lossy because the poles cause scattering, while PPKTP is
not lossy because the poling technique for PPKTP is much easier than for PPLN.

4.3.2

Low Optical Losses

Overview
In Sec. 2.5, the degradation of squeezed states using the beamsplitter model in the quantum optical
picture has been introduced. This model can be applied to any lossy mechanism including intracavity losses in an OPO cavity, losses in the path of a squeezed vacuum field between the OPO and ahomodyne detector, and losses at the homodyne detector. These losses are major losses of squeezing
and thus discussed in this section.
OPO Intra-Cavity Losses
As discussed in Sec. 3.2.4, a typical OPO cavity is composed of either an AR/AR coated nonlinear
crystal and two coupling mirrors [16, 112, 138, 115] or a hemilithic crystal with a HR coated surface
and an external coupling mirror [13, 17, 170, 18]. The intra-cavity losses mainly come from theintra-crystal loss, the imperfect AR coating of one or two crystal surface(s), and the mirror internal126

loss. Table 4.1 shows the intra-crystal losses of various nonlinear media at both 1064 nm and 532
nm. The imperfect AR coating loss depends on how well coating manufacturers can do their job
and how much money can be afforded, but R < 0.2% is an easily achievable AR coating. The
mirror internal loss is negligible compared to the intra-crystal and AR coating losses since they are
typically less than 0.1%. Comparing these losses, the biggest intra-cavity loss typically comes fromthe AR coating, assuming that low-loss crystals are chosen. If one designs an OPO in a bow-tie

Quadrature Variance Relative to Shot Noise (dB)cavity configuration, its non-coupling mirror losses need to be small as well.

−5.5
−6

99.0%

−6.5
−7
99.5%

−7.5
−8
−8.5
−9

99.95%
−9.5
−10
0

2

4

6

8
10
12
Number of Optics

14

16

18

20

Figure 4-2: The degradation of squeezing for various transmissivities or reflectivities as the numberof optical elements increases, assuming that mirrors with the same reflectivity are used consecutively
for each reflectivity. 99.95% is the reflectivity of typical supermirrors, 99.5% is the reflectivity of
most standard mirrors, and 99.0% is the reflectivity of cheap mirrors. The initial level of squeezing
is assumed to be 10 dB.

Losses in Propagation of Squeezed States
Once a squeezed vacuum field is produced in a well-designed OPO cavity, it is also important topropagate the field to a GW detector and length sensing photodetector. One should minimize
the number of optics used in the path between the OPO and photodetector because every opticalelement has some finite loss. As often used in squeezing experiments [138, 112], one could utilize
supermirrors. Fig. 4-2 shows the degradation of squeezing as a function of the number of opticalelements for various transmissivities or reflectivities, as the degree of the squeezing degradation has
been given by Eq. (2.122). If the optical elements are reflective optics such as mirrors, the values
are their reflectivities. If they are transmissive optics such as lenses and waveplates, the values are127their transmissivities. Since about twenty optics need to be used to guide, align, and mode-matchthe squeezed vacuum to the interferometer as discussed in Chap. 6, the reflectivity or transmissivityof each optical element is important although each loss may not be significant, or “a little at a time
adds up to a lot.”
Losses in Homodyne Detection
The homodyne detection is another big loss factor. It consists of two different losses/efficiencies:(1) the homodyne efficiency and (2) the quantum efficiency of each photodetector. They have been
defined in Sec. 2.6.2.
The measurement of the interference fringe visibility is possible only in stand-alone squeezers,
not in the case of injecting the squeezed vacuum field into the interferometer since the squeezed
vacuum and interferometer fields are counter-propagating. This issue will be explained and resolvedin Sec. 6.3.5.
The quantum efficiency of photodiodes at 1064 nm is in general not very high. The bandgap ofSi photodiodes is too large for photons with a wavelength of 1064 nm to excite electrons into the
conduction level. In addition, most commercial InGaAs photodiodes are designed and optimized forthe telecommunication wavelength of 1.5 µm and do not have a high responsivity at 1064 nm. Afterall, typical InGaAs photodiodes have a quantum efficiency of 75 - 85% and sometimes about 93% at
most. This is, unfortuantely, an unavoidable loss of squeezing. Imperfect AR coatings on windowsthat protect photodiodes also lead to some loss. However, it is desirable not to remove the windows
since photodiodes are susceptible to humidity and their performance degrades with time.

4.3.3

Recent Progress in Crystal Development

Choosing the right kind of nonlinear media for the production of squeezed states is critical since it
defines many important parameters such as nonlinearity, phase-matching type, pump induced seedabsorption, damage threshold, and absorption loss. Choosing a nonlinear material with the highest
nonlinearity does not mean choosing the right medium since its applicability to the generation ofsqueezed states needs to be well considered [16]. In this section, updates on crystal development andapplications are summarized. Moreover, the optimal choice of nonlinear media is proposed toward
10 dB of squeezing at 1064 nm.
LiNbO3 is the most commonly used nonlinear crystal with unique electro-optical, piezoelectric, photoelastic, and nonlinear optical properties. Its applications include laser frequency doubling, Pockels cells, optical parametric processes, Q-switching devices, acousto-optic devices, opticalswitches for GHz frequencies, and optical waveguides. LiNbO3 is usually doped by magnesium oxide(MgO) which increases its optical damage threshold and reduces the effect of green-induced infraredabsorption (GRIIRA) [145]. However, the doping increases inhomogeneity in the crystal and there128

Quadrature Variance Relative to Shot Noise (dB)

0
−2
−4
−6
−8
−10
−12

MgO:LiNbO3 with an AR Coating Loss of 0.05%−14

PPKTP with an AR Coating Loss of 0.05%
PPKTP with an AR Coating Loss of 0.5%

MgO:LiNbO with an AR Coating Loss of 0.5%
3−16 −4
10

−3

−2

−1

10
10
10
Single−Pass Pump Power in a Seed−Resonant OPO Cavity (W)010Figure 4-3: Comparisons in the generated level of squeezing as a function of single-pass pump power
in a seed-resonant OPO cavity between MgO:LiNbO3 and PPKTP with the same conditions (thecrystal length, the cavity length, the reflectivity of the input coupler, the intra-crystal loss, no
GRIIRA, etc.) except for the nonlinearity quoted from Table 4.1 and the AR coating loss of thecrystal, assuming any reflectivity is available for the output coupler to maximize the squeezing level.
The intra-crystal loss is 0.1 %/cm for all the curves.

fore increases the intra-crystal optical loss. Type I phase-matching is required for LiNbO3 withtemperature tuning of the refractive indices in the ordinary and extraordinary axes. The typical
phase-matching temperature is 60 - 120◦ C.
On the other hand, KTiOPO4 (KTP) has the advantages of a higher laser damage threshold, higher resistance to photorefractive damage, and lower susceptibility to thermal lensing than
LiNbO3 . This is especially important for OPOs pumped by CW visible sources, in which the lower
thermal lensing and higher damage threshold of the material are particularly important [151, 150].
In addition, the effective nonlinear coefficient of KTP at 1064 nm is more than 1.5 times that ofb-BaB2 O4 (BBO). KTP has been shown to generate relatively high levels of quantum correlations[185, 186]. However, it is limited to moderate to low-power applications due to its grey-tracking
problem [187, 188, 189]. Another disadvantage of KTP from the viewpoint of squeezing is that
although the indices of refraction allow both type I and II phase matchings, deff is much smaller for
type I than for type II, but type II is not useful for squeezing since the signal and idler (upper andlower sidebands) are orthogonally polarized.
Nonlinear frequency conversion using quasi-phase-matched, domain-engineered ferroelectric crystals has become a relatively mature technique for efficient OPO. Periodically poled LiNbO3 (PPLN)
129

and LiTaO3 (PPLT) have been the main focus of research for the past several years [175, 180].
However, when pumped by a visible light source, CW OPOs using these materials are significantly
affected by thermal effects originating from the relatively high absorption of the pump. Furthermore,
PPLN and PPLT show strong photorefractive effects that prevent the stable operation of CW OPOs
near room temperature. To reduce these effects, the nonlinear crystals have to be operated at temperatures near above 140 ◦ C, which is inconvenient and significantly reduces the wavelength tuning
range obtainable with a single grating period. Therefore, the development and characterization of
new periodically poled materials with low visible absorption and negligible photorefractive effects isof great relevance.
Recent progress in the electric field poling of flux-grown KTP has made periodically poled KTP(PPKTP) a very promising candidate. With its high nonlinearity of |deff | % 10.8 pm/V, PPKTP isan interesting alternative to PPLN and PPLT. The properties of these different kinds of nonlinear
crystals are compared in Table 4.1. PPKTP has already been used in CW resonant second-harmonic
generation (SHG) [180, 181, 150] and OPO [175, 176, 177, 178, 179] at 1064 nm. These experiments
showed high second-harmonic conversion efficiencies and parametric gains. Very recently, PPKTP
has been used in the generation of entangled photons at 1064 nm [182] and bright squeezed light froma CW SHG also at 1064 nm [183]. In the latter experiment, Andersen and Buchhave observed a noise
reduction of 13 % below the shot noise limit in the generated second-harmonic field. They reported
that this noise reduction is greater than what could be expected using a normal birefringence phasematched KTP crystal with the same experimental parameters. Aoki et al. reported the generationof CW squeezed vacuum at 946 nm using a PPKTP crystal in an OPO and observed a squeezing
level of 5.6±0.1 dB and an anti-squeezing level of 12.7±0.1 dB [138]. Very recently, Suzuki etal. have observed 7.2±0.2 dB of CW squeezing at 860 nm using a PPKTP crystal in a bow-tie
cavity [139]. Hirano et al. also observed the generation of pulsed squeezed light from a single-passdegenerate optical parametric amplifier (OPA) pumped by a CW second-harmonic wave using a
PPKTP crystal and reported a squeezing level of 3.2 dB and an anti-squeezing level of 6.0 dB at
1064 nm [184]. These demonstrations of high level squeezing agree with the theoretical predictionthat a considerable amount of squeezing can be generated with large input pump powers and wellphase-matched interactions [190]. To see the squeezing level gain, Fig. 4-3 compares MgO:LiNbO3and PPKTP in the generated level of squeezing as a funtion of single-pass pump power in a seedresonant OPO cavity with the same conditions (the crystal length, the cavity length, the reflectivityof the input coupler, no GRIIRA, etc.) except for the nonlinearity quoted from Table 4.1, theintra-crystal loss, and the AR coating loss of the crystal, assuming any reflectivity is available forthe output coupler to maximize the squeezing level. In the figure, we can learn that the separationin the level of squeezing between MgO:LiNbO3 and PPKTP for different losses changes weakly, andtherefore, even if the crystal or the AR coating on its surfaces is lossy, PPKTP in an OPO cavity
130gives a gain of about 4 dB compared with MgO:LiNbO3 in the same OPO cavity.
Recent experiments on quasi-phase-matching in gallium arsenide (GaAs), which has a very highnonlinear coefficient of 69 - 170 pm/V (several to ten times larger than that of PPKTP and PPLN)
and a wide transparency range of 0.7 - 17 µm, make this material very attractive for SHG and OPO
as well as squeezing in the mid-IR range [191, 192]. However, the growth of periodically poled GaAs
structures with sufficient quality and thickness is a delicate issue.

4.3.4

Cavity Configurations

As discussed in Chap. 2, the optical configuration of an OPO cavity needs to be well-designed
to optimize the level of squeezing generated in the OPO. From Eq. (2.69), the generated level ofsqueezing is completely determined by the nonlinear coupling efficiency x and the escape efficiency
ηout . Typically, the input coupler is HR-coated. Fig. 4-4 shows the theoretically achievable levelsof squeezing as a function of pump power for various output coupler reflectivities. An input coupler

Quadrature Variance Relative to Shot Noise (dB)with a reflectivity of 99.5% and an intra-cavity loss of 0.2% are assumed for the figure.

0
−2
−4

98%
94%
90%
86%

−6
−8
−10
−12
−14
−16 −3
10

−2

10

−1

10
P/Pth

0

10

1

10

Figure 4-4: The theoretically achievable levels of squeezing as a function of pump power normalized
to the OPO threshold in the case of an output coupling reflectivity of 86% for various output
coupler reflectivities. The reflectivity of the input coupler and the intra-cavity loss are 99.8% and0.2% respectively.
What we can read from this figure is that if a sufficiently high pump power is available, a
low-reflectivity output coupler should be used to obtain a high level of squeezing. However, if
it is not available, an output coupler with a higher reflectivity should be used since the level of131squeezing decays more quickly. In addition, when a high pump power is used, it triggers undesirable
photothermal effects [134], degrading the level of squeezing, although it is not shown in the figure.
This situation also requires a high mode-matching efficiency between the pump and the cavity modein the case of a singly-resonant OPO cavity since otherwise, it would cause unnecessary photothermal
noise [134].

4.4

Long-Term Stability

4.4.1

Overview

The third requirement is long-term stability for the practical use of squeezing in the GW detectors
such as LIGO, VIRGO, GEO600, and TAMA300 that typically run for a few years continuously
although there are periods for maintenance from time to time during their science runs. First ofall, the OPO cavity needs to be kept stably locked on resonance to maintain a constant level ofsqueezing. In the case of an empty cavity, a bright coherent field is typically used to lock thecavity on resonance. However, as discussed in Sec. 4.2.2, in the case of the OPO cavity, a brightfield cannot be used as a seed to produce squeezing in the GW detection band, and therefore, analternative method is required to maintain vacuum at the fundamental frequency on resonance in
the OPO cavity. Secondly, the phase-matching condition of the OPO needs to remain satisfied. In
type I phase-matching, temperature tuning of the nonlinear crystal via change in refractive index
with temperature, or the so-called dn/dT effect, is used to match the two refractive indices in theordinary and extraordinary axes of the nonlinear crystal. In the case of quasi-phase-matching in aperiodically poled nonlinear crystal, temperature tuning of the crystal is also required, but against
thermal expansion. Thus, in all cases, stable temperature control of the crystal is required, whichindicates that a temperature control servo and an oven for the crystal need to be well-designed.
Finally, the interface of squeezing to the interferometer also needs to be stabilized. Otherwise, thecoupling of the anti-squeezed noise that comes with the squeezed noise would degrade the strain
sensitivity of the GW detector. However, since the injected squeezed vacuum field does not containany coherent amplitude, the standard scheme to read out the relative phase between the squeezed
vacuum and the interferometer field is not applicable, and therefore, an alternative readout scheme
is necessary.
In summary, the following three requirements need to be satisfied for the stable operation ofsqueezing in the GW detectors: (1) stable control of the OPO cavity resonance, (2) stable controlof OPO phase-matching, and (3) stable control of squeezing (the squeeze angle). In the following
sections, these requirements are discussed and a few different techniques are compared. In Chap. 7,
a few ideas about increased stability by cavity-less generation of squeezed states will be presented.
132

4.4.2Control of OPO Cavity ResonanceControl with a Subcarrier
In many cases, the OPO cavity is a singly resonant cavity in which the cavity mirror reflectivities
form a cavity at the fundamental frequency, but not at the second-harmonic frequency. The secondharmonic field typically singly or doubly passes through the OPO. In this case, a frequency-shifted
subcarrier can be used to lock the OPO cavity such that a vacuum field at the fundamental frequency
resonates in the OPO cavity. The frequency of the subcarrier can be either one FSR away from the
TEM00 mode in which the vacuum at the fundamental frequency resonates or one of the TEM00
modes near the fundamental vacuum resonance frequency in the orthogonal polarization. The difference in optical length between the two polarizations is attributed to the birefringent property
of nonlinear crystals (certainly in type I phase-matched crystals, but even in quasi-phase-matchedperiodically poled crystals). In the former case, the FSR of a typical OPO cavity is on the order ofGHz due to its short cavity distance, and therefore, it is most likely that a second laser is used tosupply the subcarrier unless a GHz acousto-optic modulator (AOM) is available. In the latter case,
if a TEM00 near the fundamental vacuum resonance frequency is available and its frequency shift ison the order of 100 MHz, passing some portion of the fundamental field through an AOM singly or
doubly is sufficient to frequency-shift it.
If locking the OPO cavity with a subcarrier is only of interest, both methods work well. However,if the generated squeezed vacuum field is injected into the interferometer, the second method works
better than the first method in that in the first method, since the frequency-shifted subcarrier is in
the same polarization as the squeezed vacuum field, it is not an easy task to isolate the squeezed
vacuum field from the composite field without attenuating the level of squeezing. One could suggest
an optical cavity to filter out the subcarrier, but mode-matching the squeezed vacuum field to sucha cavity is typically 80% in practice, meaning that 20% of squeezing is lost in the filtering process.
Without filtering, the subcarrier travels along with the squeezed vacuum to the interferometer andmight have an undesirable effect on the interferometer although it could be rejected by the signalrecycling cavity (SRC) or output mode-cleaner (OMC) in Advanced LIGO, depending on the designof the cavity resonance frequencies. On the other hand, in the second method, a polarizer can easily
separate the squeezed vacuum from the subcarrier without any significant optical loss and hence,locking the OPO cavity can be done independently from the interferometer.

Control with a Second-Harmonic Pump
A doubly resonant OPO cavity at both the fundamental and second-harmonic wavelengths has a
few advantages over a singly resonant OPO cavity [115]. One advantage is that the pump power
can be increased by a factor of the cavity finesse, and thus, when the power of the incident second133

harmonic field is insufficient, this is a good strategy. As discussed in Sec. 4.3.4, in principle, themore pump power is available, the higher the achievable level of squeezing is. However, since thenonlinear crystal absorbs some amount of the pump power, the pump with a higher power makes
the cavity thermally unstable. Another advantage is that the mode coupling of the pump to the seedis perfect in the doubly resonant OPO cavity compared to the singly resonant OPO cavity because
the cavity selects the resonating mode. This is an efficient pumping process in that no pump poweris wasted in the OPO cavity. One disadvantage is that vacuum at the fundamental frequency might
not resonate exactly due to the difference in optical travel distance between the fundamental andsecond-harmonic frequencies. In type I phase-matching, their refractive indices are equivalent under
the condition that the phase-matching condition is satisfied. In quasi-phase-matching, a tunable
phase compensation plate that compensates for the phase difference between the two frequencies isrequired, but it might be lossy in the cavity.

4.4.3

Control of OPO Phase Matching

In type I phase matching, the dependence of the refractive index of a nonlinear crystal on its
temperature or the so-called dn/dT effect is used to match the refractive indices of the material in
the ordinary and extraordinary axes. As mentioned in Sec. 2.4.4, in the case of LiNbO3 , dno /dT =3.3 × 10−6 K−1 in the ordinary axis and dne /dT = 37 × 10−6 K−1 in the extraordinary axis, bothat λ = 1.0 - 4.0 µm. These are fairly large values for temperature change on the order of 10 mK.
As shown in Fig. 2-7, the strength of nonlinear interaction in both SHG and OPO cavities strongly
depends on the temperature of the nonlinear crystals, and therefore, temperature fluctuations in the
crystals are directly coupled to nonlinearity fluctuations, that is, fluctuations in the generated levelof squeezing. In addition, as the size of the crystal increases, the effective temperature range gets
narrower. The typical size of nonlinear crystals is about 6 mm, and therefore, the effective FWHM
temperature range is 0.5 K.
On the other hand, periodically poled nonlinear crystals do not require temperature-tuning of
their refractive indices for phase-matching. The only requirement for temperature control is tomaintain the grating period against thermal expansion. The typical FWHM temperature range of
PPKTP at 1064 nm is 5 K, which is about 10 times larger than that of LiNbO3 in type I phasematching. Another advantage of the large temperature range of PPKTP is that it significantly
reduces difficulties in fabrication of ovens for temperature control that would be needed for type I
phase-matching. In this respect, periodically poled nonlinear crystals such as PPKTP are better instability against fluctuations in phase-matching than nonlinear crystals that require type I phasematching such as LiNbO3 .
134

4.4.4Control of Squeezing

Overview
In Sec. 2.7, two methods of controlling squeezed vacuum fields without a carrier field have been
described. However, while these techniques are useful in stand-alone squeezers, the validity of thetechniques for the injection and control of a squeezed vacuum field in a laser-interferometric GW
detector needs to be investigated. In this section, practical issues and solutions are discussed.
Quantum Noise Locking
The quantum noise locking technique is useful in any optical system since it does not require any
coherent sidebands (although there are vacuum sidebands that are generated by modulating thesqueeze angle of the squeezed vacuum field). The only requirement is that the length sensing
detector of an interferometer is required to detect its quantum noise that is demodulated to obtainan error signal for noise-locking the squeeze angle. Because of the absence of any coherent sidebands,
the level of shot noise that is equal to the quadrature sum of all the fields incident on the length
sensing photodetector does not change. One disadvantage is its low stability (both long-term andshort-term). Since the noise-locking error signal relies on the noise on the noise or kurtosis, itsvariance strongly couples to the stability of the noise-locking. Therefore, as shown in Eqs. (2.152)
and (2.153), unless the generated level of squeezing is very large, the stability of the noise-locking
cannot be better than the stability of the coherent control method.
Advantages• Does not require any coherent sidebands• Does not require any frequency filters or photodetectors• Requires only the length sensing detector to measure the quantum noise of an interferometer
Disadvantages• Low stability• Lower achievable level of squeezing
Table 4.2: Advantages and disadvantages of the quantum noise locking technique.

Coherent Control of Squeezing
The coherent control method has the obvious advantage over the quantum noise locking methodthat its stability is much higher, indicating that the maximum achievable level of squeezing is higher[138]. On the other hand, it has one serious disadvantage that the coherent signal and idler fields
co-propagate with the squeezed vacuum field. In the DC readout scheme of Advanced LIGO as
shown in Fig. 4-5, all RF sidebands are removed by the OMC before measuring the carrier fieldwith the length sensing photodetector. The purpose of the OMC is to remove RF sidebands that135

are not filtered by the arm cavities of the interferometer and clean the mode of the interferometer
carrier field, and therefore, the signal and idler fields should not transmit through the OMC since
they would otherwise contaminate the carrier field that possible GW signals would beat with. In
this respect, an error signal for the coherent control needs to be obtained from the beat between
the signal/idler and interferometer RF sidebands (not the interferometer carrier) at the rejection
port of the OMC. One could suggest putting a pickoff mirror in between the optical circulator and
OMC, but it should be suspended not to pick up any ground and mechanical noise, assuming that
the OMC is suspended for the same reason.
If these issues can be overcome, the coherent control technique should be used in the implementation of squeezing in next generation GW detectors such as Advanced LIGO since its control
stability can simply be enhanced by increasing the gain of the servo.

Input
Mode
Cleaner

Fabry-Perot
Michelson
Power
Recycling
MirrorLASER
Signal Recycling
Mirror
Squeezed
Vacuum

OpticalCirculator

Photodetector

OutputMode
Cleaner

Photodetector

Figure 4-5: The optical configuration and readout scheme of Advanced LIGO. Another filter cavity
called the output mode cleaner (OMC) is planned to be added to the interferometer output toremove any RF sidebands and clean the mode of the carrier field.

4.5

Frequency-Dependent Squeezing

4.5.1

Introduction

The fourth requirement is frequency-dependent squeezing. In Chap. 1, two limiting quantum noise
sources in GW detectors have been addressed. They are photon shot noise at high frequencies
136

Advantages• High stability• Higher achievable level of squeezing• Easy to control
Disadvantages• Requires RF sidebands• Requires a complicated optical setup to generate and detect correlated signal and idler fields• Requires a filter cavity that filters out the RF sidebands before the length sensing
photodetector• Requires the beat between the signal/idler and RF sidebands (not the carrier field) from theinterferometer to obtain an error signal for the coherent control
Table 4.3: Advantages and disadvantages of the coherent control method.

that arises from uncertainty due to quantum mechanical fluctuations in the number of photons
at the interferometer output and quantum radiation pressure noise at low frequencies that arises
from mirror displacements induced by quantum radiation pressure fluctuations [82, 83]. As in Eqs.
(1.61) and (1.66), in a simple Michelson interferometer, the strain associated with the shot noise is√
proportional to 1/ Pin while the strain associated with the radiation pressure noise is proportional√
to Pin . Thus, increasing the laser power improves the strain sensitivity by reducing the shot noise
while degrading it by increasing the radiation pressure noise, or vice versa.
The effect of the radiation pressure noise is to ponderomotively squeeze the optical fields with afrequency dependent squeeze angle due to the frequency dependent response of the test masses [82].
This presents difficulty in injecting squeezed states into the interferometer because the squeeze angleof the squeezed state source must be matched to the ponderomotive squeeze angle. Squeezing with afrequency-independent or constant squeeze angle improves the sensitivity at some frequencies in the
GW detection band, but degrades it at other frequencies due to the orthogonality of amplitude andphase quadratures corresponding to radiation pressure noise and shot noise in the GW detectors.
For a conventional GW detector with arm lengths L and mirror masses m, the quantum noise in
the GW strain signal when a squeezed field a with squeeze factor r and squeeze angle θ is injected
into the dark port of the interferometer is given by [82, 193]h̃(Ω) = h̃SQL

2>
1 + K2 =Ṽa1 sin2 (θ + Φ) + Ṽa2 cos2 (θ + Φ) ,
2K

(4.1)

where h̃SQL is given by Eq. (1.70), Ω is the sideband (measurement) frequency, K is the effective
coupling constant given by Eq. (1.71), Ṽa1 and Ṽa2 are the amplitude and phase quadrature variancesof the input squeezed field, and Φ = cot−1 K is the phase gained by the sideband frequency Ω while

traveling one way in the SRC4 . Fig. 4-6 shows quantum noise and squeezed quantum noise with
4 In other words, it is the effective ponderomotive squeeze angle of the interferometer.

137

−21

10

Unsqueezed Quantum Noise
Phase−Squeezed Quantum Noise
Amplitude−Squeezed Quantum Noise

Strain (Hz−1/2)

−22

10

−23

10

−24

10

1

10

2

10

3

Frequency (Hz)

10

4

10

Figure 4-6: The comparison of the unsqueezed quantum noise and the squeezed quantum noise withfrequency-independent squeeze angles in a conventional GW detector with parameters m = 30 kg,
L = 4 km, λ = 1064 nm, γ/(2π) = 100 Hz, P = PSQL , and a squeeze factor of r = 2. Squeezing
with a constant squeeze angle reduces the quantum noise at high frequencies while increasing it at
low frequencies, or vice versa. Phase-squeezing and amplitude-squeezing correspond to θ = 0 andθ = π/2 respectively.
a constant (frequency-independent) squeeze angle in a conventional GW detector. Squeezing witha constant squeeze angle reduces the quantum noise at high frequencies while increasing it at low
frequencies, or vice versa. Squeezing with squeeze angle θ = −Φ reduces the quantum noise at all
frequencies.
Squeezed states generated by OPOs are generally frequency-independent, but the desired angle
may be produced by using optical filters placed between the squeezed state source and the interferometer so that broadband squeezing can be achieved in the GW detection band. A few different
filtering schemes have been proposed: (1) optical cavities with narrow linewidths comparable to the
bandwidth of the interferometer (100 Hz) [82, 104], (2) filters by use of electromagnetically induced
transparency (EIT) [193], and (3) filters for squeeze amplitude attenuation [194]. The properties
and practicality of these filters are discussed in the following sections.

4.5.2

Cavities with Narrow Linewidths

A scheme that uses a pair of successive lossless Fabry-Perot filter cavities as a frequency-dependent
squeeze angle filter has been proposed [82], and the proof of principle has been experimentallydemonstrated at MHz frequencies [159]. When an input squeezed field is incident on a cavity which138

is not impedance-matched and the carrier frequency and cavity resonance frequency are equal, its
sidebands obtain frequency-dependent phase shifts when the field reflects back. Since squeezing is
the sum of the quantum sidebands centered around the carrier frequency, the sum of those quantum
sidebands with frequency-dependent phase shifts yields frequency-dependent squeezing.
The two filter cavities (denoted I and II) produce phase shifts θI± and θII± on the sidebands at
frequencies ±Ω. The net phase shifts acquired by the two cavities are given by θ± = θI± + θII± .
Assuming that the carrier frequency of the squeezed field is equal to the resonant frequency of thecavities, the phase shifts are given by
θI± = tan

'

±Ω
γI

(

,

θII± = tan

'

±Ω
γII

(

,

(4.2)

where γI and γII are the HWHM linewidths of the cavities. The detail calculations of the squeeze
angle rotation will be given in Sec. 5.2.2.
The biggest problem of this type of filter cavities is that the cavity filters, while producing thedesired squeeze angle, are difficult to build because they require cavities with narrow linewidths
comparable to the bandwidth of the interferometer (100 Hz), which require either km-scale cavities
or high-finesse cavities. For squeeze-angle rotation, high-finesse cavities are somewhat undesirable
due to high optical losses. For example, assuming a round-trip optical loss of 20 × 10−6 , a cavity
would have to be nearly 100 m to maintain an effective loss of 10% or less.

4.5.3

Filters by Use of Electromagnetically Induced Transparency

Overview
Alternative filter designs based on EIT media have been proposed [193]. Several properties of
EIT media make them a potentially advantageous alternative to ultra low-loss or km-scale opticalcavities. First, EIT media typically have narrow transmission resonance linewidths. The narrowest
linewidth reported is 1.3 Hz in a paraffin coated cell with 85 Rb as the EIT medium [195]. The
EIT media can also be very compact, with typical lengths of ∼ 10 cm. Second, the resonancelinewidth may be adjusted in non-invasive ways, by changing the atomic density [196] or drive field
intensity [197], to optimize the filtering. Altering the linewidth of an optical cavity, on the other
hand, usually requires changing the transmission of the mirrors or the macroscopic cavity length.
Other variable reflectivity techniques such as three-mirror cavities are possible, but issues of fringe
control, scattered light and mode mismatch are compounded in such systems. Third, EIT media
can be nearly 100% transmissive for the probe field [198, 199]. Fourth, mode matching of the spatial
mode of the squeezed (probe) beam to that of the EIT is not critical, as it is for an optical cavity,
where mode mismatch can be a significant source of optical loss [198]. Fifth, EIT media have also
been shown to preserve squeezed states in transmission [200, 201]. These reasons lead naturally to139

Michelson
Interferometer

Ω

δ
∆

Ω
ω+Ω
ω

|a〉

Pickoff
LASER
SQZ

ωd

EIT

ω-Ω

OpticalCirculator

Squeezed
Vacuumωbc |c〉Photodetector|b〉

Figure 4-7: Left: A generic three-level EIT system. The susceptibility of the EIT medium, χ, isa function of the frequency of the probe field, ω, the sideband frequency, Ω (relative to the carrier
frequency of the probe, ω), the frequency of the ground state splitting, ωbc , the two-photon detuningof the probe field, δ = ω − ωd − ωbc , the one-photon detuning of the drive field, ∆, and the lengthof the EIT medium, l. Right: A proposed configuration for a squeeze EIT filter and GW detector,
where SQZ: squeezed field generator, EIT: EIT filter.
consideration of using EIT for squeezed state filters.

Theory of Squeeze Amplitude Attenuation and Squeeze Angle Rotation by EIT
The energy level scheme for one photon resonant EIT is shown in Fig. 4-7. Using the parameters
defined in Fig. 4-7, the EIT transmission as a function of the sideband frequency, Ω, is given byA )T (ω + Ω) = exp ilkeit (ω + Ω, z) dz

0

B

,

(4.3)

where keit is the wavevector of the probe field through the EIT medium. For most cases, it isreasonable to assume that the susceptibility of the medium |χ(ω + Ω, z)| + 1 and Ω + ω, then the
wavevector can be written as
keit (ω + Ω, z) =ω + Ω.
ω
1 + χ(ω + Ω, z) %
c
c

'

1+

χ(Ω, z)
2

(

.

(4.4)

In the frequency domain defined by Eq. (1.60), substituting Eq. (4.4) into Eq. (4.3) and moving
into the rotating frame of the probe field a with frequency ω [131], we find the EIT transmission at
sideband frequencies ±Ω,T (±Ω) = T± ei Θ± ,
140

(4.5)

whereAB) lω≡ exp −
χ2 (±Ω, z) dz ,
2c 0
)
ω l
≡
χ1 (±Ω, z)dz.
2c 0

T±
Θ±

(4.6)
(4.7)

Here χ1 (Ω, z) and χ2 (Ω, z) are the real and imaginary parts of χ(Ω, z), respectively. χ1 (Ω, z) is
responsible for phase shift and χ2 (Ω, z) is responsible for signal strength attenuation.
When the input field ã is injected into an EIT medium, the output field b̃ and its adjoint b̃† aregiven in terms of ã and its adjoint ã† by= T (Ω)ã(Ω) + L+ (Ω)ṽ(Ω),b̃(Ω)
b̃† (−Ω)=(4.8)

T ∗ (−Ω)ã† (−Ω) + L∗− (−Ω)ṽ † (−Ω),(4.9)

where L± is the EIT absorption coefficient such thatL± =/1 − T±2(4.10)and ṽ is a vacuum field coupled in by the absorption loss. ã(Ω) and ã† (−Ω) satisfy the commutation
relations56"ã(±Ω), ã† (±Ω) ) = 2πδ(Ω − Ω ),(4.11)

and all others vanish [similarly for v(Ω) and v † (−Ω)]. Using the definitions of the amplitude andphase quadratures in Eqs. (2.10) and (2.11), we find the quadrature field amplitudes of the output,
in a compact matrix form, to beb̃ = Mã +/0
1
1 − A2+ + A2− ṽs ,(4.12)

where we use the two-photon matrix representation

ã



≡ 

ã1
ã2



(4.13)

for the operator, a (and similarly for b and v), and

M = eiϕ− cos ϕ+− sin ϕ+sin ϕ+

cos ϕ+




A+

iA−

−iA−

A+




(4.14)

is a matrix representing propagation through the EIT medium. M comprises an overall phase shift,141ϕ− , rotation by angle, ϕ+ , and attenuation by a factor, A+ . Here we have definedϕ± ≡1(Θ+ ± Θ− ) ,
2

A± ≡

1
(T+ ± T− ) ,2(4.15)and performed a unitary transformation on v, such that/
L+ + L−1
1 − A2+ + A− ṽs = 2−i (L+ − L− )120i (L+ − L− )L+ + L−

 ṽ

(4.16)and vs behaves as ordinary unsqueezed vacuum. For symmetrical lineshapes with respect to thecarrier, χ1 (Ω) = −χ1 (−Ω), and ϕ+ therefore vanishes, giving no quadrature angle rotation, butattenuating the signal strength. For asymmetrical lineshapes, nonzero ϕ+ gives quadrature angle
rotation.
Application to GW Detectors
Assuming no losses other than those associated with the EIT filter, a squeezed field b with squeeze
factor r and squeeze angle θ = θ0 + ϕ+ is injected into the antisymmetric port of the GW detector
through the EIT filter as shown in Fig. 4-7. The input angle θ0 may be arbitrarily chosen by
microscopic variations in the distance between the squeezed state source and the interferometer.
Following Eq. (4.1), the spectral density of the noise at the output of the GW detector is given byS̃(Ω) =

h2SQL
2

'

K+

1
K

(=

>
Ṽ+ sin2 (θ + Φ) + Ṽ− cos2 (θ + Φ) ,(4.17)

where the noise in the anti-squeezed and squeezed quadratures, Ṽ+ and Ṽ− , respectively, is given by



V+
V−





 = 

A2+

A2−







0
1 
1 − A2+ + A2−

+
0
1 .
A2+
e−2r
1 − A2+ + A2−

A2−

e+2r

(4.18)EIT Filter Overview
The following three kinds of EIT filters are proposed: (i) low-pass, band-pass, and high-pass squeeze
amplitude attenuation filters, (ii) an S-shaped filter as a frequency-dependent squeeze angle rotator,and (iii) an intra-cavity EIT filter as a frequency-dependent squeeze angle rotator.
To obtain expressions for the EIT transmission T (±Ω), it is useful to use the formulas for EIT
susceptibility χ [202, 203]
ω
2c
ω
2c

) l

χ1 (±Ω, z) dz= A(∆)

χ2 (±Ω, z)dz

= A(∆)0) l0Γ(δ ± Ω)

Γ2 + (δ ± Ω)2Γ2

Γ2 + (δ ± Ω)2142+ B(∆)+ B(∆)

Γ2
Γ2 + (δ ± Ω)2,

Γ(δ ± Ω)+ C(∆),
Γ2 + (δ ± Ω)2(4.19)(4.20)

10
a1 squeezed

Sh / Sh(Unsqueezed)

a2 squeezed
S resonance, low absorption

S resonance, high absorption

Low pass

1.0High passBand pass

Optimal rotation/intra−cavity EIT

0.1
10

100
Frequency [Hz]1000

Figure 4-8: The noise spectral density of a GW detector, normalized to the noise density with nosqueezed state injection. We show the effect of low-pass (red), band-pass (green), high-pass (cyan),S-shaped (blue for low and brown for high absorption levels), and intra-cavity (black) filters andcompare it to frequency independent schemes. The dashed curves show unfiltered squeezing of thea1 (green) and a2 (red) quadratures.
where Γ is the EIT resonance linewidth and the coefficients A, B and C depend on the intensity anddetuning, ∆, of the drive field. The first term on the right hand side in Eq. (4.19) corresponds toasymmetric phase dependence on the two-photon detuning δ ± Ω and the second term is responsible
for symmetrical Lorentzian dependence. Similarly, the first term in Eq. (4.20) corresponds tosymmetrical dependence on the absorption coefficient of the EIT medium, the second term indicatesasymmetric absorption, and the third term corresponds to broadband absorption in the medium.
Inserting Eqs. (4.19) and (4.20) into Eqs. (4.6) and (4.7) respectively, we get explicit expressions
for T± and θ± , which are used to calculate the noise at the output of the GW interferometer S̃(Ω),
as described in Sec. 4.5.3.

Squeeze Amplitude Attenuation FiltersFlωχ (±Ω, z)dz ≥ 10 for Ω ) Γ and
If we take A = −C, B = 0, C ≥ 10, and δ = 0 [197], then 2c0 2Flω2c 0 χ2 (±Ω, z)dz % 0 for Ω + Γ, and we obtain a symmetric EIT transmission line with A− = 0

and ϕ+ = 0. This can be used as a squeeze amplitude filter that retains squeezing at low frequencies,
but causes ordinary (unsqueezed) vacuum to replace the anti-squeezed noise at higher frequencies
that are outside the EIT bandwidth. This is equivalent to the low-pass filters proposed in Ref.
[194]. Similarly, buffer-gas-induced electromagnetically induced absorption (BGIEIA) [202] may be
used as a high-pass filter. BGIEIA is similar to EIT, with a narrowband absorption resonance143

instead of the transmission resonance of EIT. High-pass filtering with BGIEIA can be realized withFl
ωparameters A ≥ 10, B = 0, C = 0, and δ = 0, such that 2c
χ (±Ω, z)dz % 0 for Ω ) Γ and
0 2
F
l
ω
2c 0 χ2 (±Ω, z)dz ≥ 10 for Ω + Γ. A combination of two EIT resonances, equally detuned from thecarrier (obtained by Zeeman splitting [204]), serves as a band-pass filter. In Fig. 4-8 the effect of low-

pass, band-pass, and high-pass EIT filters on the noise spectral density of a GW detector are shown.
The noise spectra are normalized by the noise spectral density of a conventional interferometer withno squeezed state injection, which corresponds to the unity in Fig. 4-8. In each case, the harmful
effects of squeezing with a constant squeeze angle are reduced.
Squeeze Angle Rotation Filters
In principle, optimum squeeze angle rotation, corresponding to θ % −Φ, can be obtained with anasymmetrical or S-shaped EIT filter, realized under conditions similar to BGIEIA with parameters
A % 0, B = −π/2, and C > |B|. In this case, the noise spectral density can be optimized over mostfrequencies. However, as is evident from the curves labeled S resonance in Fig. 4-8, the improvement
is small because of the high off-resonance losses associated with the coefficient C. Optimization would
require setting C = 0. The noise is decreased at both high and low frequencies, but worsened in
the middle of the band due to two effect. First, the imbalance in absorption (A− 0= 0) between theupper and lower sidebands required to obtain the squeeze angle rotation causes the quadratures tobe mixed and the noise may be higher than shot noise. Second, the imperfect squeeze angle rotation,
θ % −Φ, couples in noise from the anti-squeezed quadrature.
Placing an EIT medium in an optical cavity narrows the cavity linewidth [205, 206]. We consider
a symmetric EIT transmission resonance, such as the one used for our low-pass filter, with a large
E
dχ1 E
Ω
in the frequency band oflinewidth Γ % 10 kHz, such that T± % 1, ϕ+ % 0, and ϕ− % ± ωl
2c
dΩ E
Ω=0

interest (100Hz). This configuration may be understood in terms of the group velocity of the lightthrough the EIT medium

c
vg =
ω
We may then express
ϕ− %

'

E (−1
dχ1 EE
.
dΩ E0

Ωle
l
Ω=
,
2vg
c

(4.21)(4.22)

where le = 2vcg l is the effective cavity length. The EIT medium serves as an additional delay line
inside the cavity. A group velocity as low as 8 m/s in a 12 cm long Rb vapor cell was demonstrated
by Budker et al. [195], giving an effective length on the order of 106 m. To use an intra-cavity EIT
medium as a squeeze angle rotation filter, we must detune the cavity (but not the EIT medium)from the carrier. For this case, the rotation arises from the detuned cavity, similar to the filters
in Ref. [82], and not from the EIT. The EIT acts only to modify the resonant linewidth of the144cavity by increasing its effective length. To achieve the required 100 Hz linewidth for the filter,
we use an EIT with a large effective length, combined with a short cavity. The performance of
this cavity is essentially identical to that of an isolated cavity with no EIT and length equal to le ,and may achieve optimal squeeze angle rotation with a much smaller optical loss than a traditional
cavity of the same linewidth with the same real (not effective) length, giving improved GW detector
sensitivity at all frequencies. We note that T± % 1 is a reasonable approximation because of thenarrowed linewidth of the system (Ti vg /l + Γ, where Ti is the transmission of the cavity input
mirror). Detailed calculations show that the effective loss of the system may even be reduced byplacing the EIT in a shorter cavity for some parameter choices. The broadband reduction of noisefrom such an intra-cavity EIT filter is shown in Fig. 4-8, and corresponds exactly to the much
coveted optimal frequency-dependent squeezing of Ref [82].

4.5.4Filters for Squeeze Amplitude Attenuation

Introduction
Although frequency-dependent squeezing can be performed by an EIT filter that has advantages
over other filter designs in optical losses, compactness, and the tunability of the filter linewidth, an
EIT medium for the laser wavelength used in the existing or future GW detectors such as LIGO andAdvanced LIGO (1064 nm) is unavailable.
To avoid the harmful effects of squeezing with a constant squeeze angle and the need for a low-loss
or high-finesse cavity, another kind of filter cavities that behaves as a high-pass filter for the squeeze
amplitude instead of giving a frequency-dependent squeeze angle has been proposed [194]. Thesefilters operate on the principle that the squeezed state with a frequency-independent squeeze angle
has beneficial effects over some frequency band, but harmful effects over other frequency bands, andthose harmful effects can be mitigated by introducing a frequency-dependent optical loss.
When a squeezed vacuum field with a constant squeeze angle is injected into an impedancematched optical cavity, it is entirely reflected by the filter cavity outside of the cavity linewidth
whereas within the linewidth it is entirely transmitted and replaced by a vacuum field that couples in due to the cavity transmission and intra-cavity losses of the squeezed vacuum field. When
this squeeze amplitude filter is applied to a squeezed vacuum field before being injected into GW
detectors, the interferometer shot noise can be reduced by squeezing at high frequencies while antisqueezing is attenuated at low frequencies, giving a semi-broadband reduction of the quantum noise.
In this section, such a squeeze amplitude high-pass filter is theoretically described and thenexperimentally demonstrated by measuring the reflection of a squeezed vacuum field from the filter
cavity. It is found that cavity detunings play an important role in the performance of the high-passfiltering effect.
145

Theory

Consider a squeezed vacuum field [ã = (ã1 , ã2 )T ] with a constant squeeze angle injected into afilter cavity with input mirror reflectance r1 and transmittance t1 , output mirror reflectance r2 , andintra-cavity mirror reflectance r3 that accounts for intra-cavity losses, as shown in Fig. 4-9. Then,moving into the rotating frame of the carrier field, the reflection from the filter cavity b̃ is given in
the frequency domain defined in Eq. (1.60) byb̃1 = ρ(Ω)ã1 +.
1 − |ρ(Ω)|2 ṽ1 ,b̃2 = ρ(Ω)ã2 +.

1 − |ρ(Ω)|2 ṽ2 ,(4.23)

where ρ is the cavity reflection coefficient given byρ(Ω) = r1 −t21 r2 r3 e−2iΩLAF /c,
1 − r1 r2 r3 e−2iΩLAF /c(4.24)and ṽ = (ṽ1 , ṽ2 )T is the vacuum field that couples with the reflection due to the transmission andintra-cavity loss of the input squeezed vacuum field. Here LAF is the half-length of the amplitudefilter cavity.

Loss-AssociatedVacuum Mode
Intra-Cavity
Lossr 3, t3

r 1, t2

r 2, t2abTransmission-AssociatedVacuum Mode

Figure 4-9: A schematic of a squeeze amplitude filter cavity that consists of input mirror reflectance
r1 and transmittance t1 , output mirror reflectance r2 , and intra-cavity mirror reflectance r3 that
accounts for intra-cavity losses.

The mathematical description of the filter is similar to that of the EIT filter that has been given
in Sec. 4.5.3, except that in the case of the squeeze amplitude filter, the linewidth of the filter cavity
is symmetric and the center frequency of the input squeezed vacuum field is equal to the resonancefrequency of the filter cavity (or not detuned from it). The quadrature variances of the reflection is146then given byṼb1 (Ω) = |ρ(Ω)|2 Ṽa1 + 1 − |ρ(Ω)|2 ,Ṽb2 (Ω) = |ρ(Ω)|2 Ṽa2 + 1 − |ρ(Ω)|2 ,(4.25)

where we have used Ṽv1 = Ṽv2 = 1. At frequencies below the linewidth of the filter, the squeezed
vacuum field transmits through the filter cavity, whereas at frequencies above the linewidth, the fieldreflects back by the cavity, and therefore, the filter cavity behaves as a squeeze amplitude high-passfilter.

Experiment
To demonstrate the squeeze amplitude filtering effect at a wavelength of 1064 nm, a squeezed vacuum
field is prepared using a sub-threshold OPO. Since it is known that squeezed fields generated bylight-seeded OPO cavities have squeezing only at MHz frequencies, a vacuum field is used as aseed to the OPO. The OPO (different from the OPO in Chap. 3) that consists of a hemilithic
7%MgO:LiNbO3 crystal and an output coupling mirror is pumped by 600 mW of a second-harmonic
field at 532 nm. Type I phase-matching is used to satisfy the OPO phase-matching condition. The
FSR and FWHM linewidth of the OPO cavity at 1064 nm are 362.1 MHz and 2.0 MHz respectively.
The second-harmonic pump doubly passes through the OPO5 . The cavity is locked using the PoundDrever-Hall (PDH) locking techcnique with a frequency-shifted subcarrier field. The OPO crystal
is temperature-controlled at the phase-matching temperature of 72.1◦ C to maximize the nonlinear
interaction between the seed and pump fields.
The generated squeezed vacuum field is injected into a test filter cavity that consists of three
mirrors with a FWHM linewidth of 856 kHz and a FSR of 713 MHz. The triangular geometry is
chosen such that the reflected field can be geometrically isolated from the injected field. The filter
cavity is locked with a coherent light injected into the cavity from its back so that the reflected
squeezed vacuum field and control field do not propagate in the same direction. A Faraday isolator
is used to remove the control field from coupling into the OPO cavity. To ensure a higher isolation
efficiency, the control field is frequency-shifted relative to the center frequency of the squeezed
vacuum field. The frequency of the control field is carefully tuned so that the center frequencyof the squeezed vacuum field matches the resonance frequency of the filter cavity. The filteredsqueezed vacuum field is measured by a homodyne detector that consists of a 50/50 beamsplitterand a pair of photodiodes with matched quantum efficiencies of 90%. The difference photocurrentof the photodetectors is measured by an HP4195A spectrum analyzer (Hewlett-Packard) to measure
the effect of the filter cavity on both squeezed and anti-squeezed shot noise spectra.
The experimental result is shown in Fig. 4-10. The cutoff of the squeezed and anti-squeezed
5 Thus, the OPO is not a cavity at 532 nm

147Quadrature Variance Relative to Shot Noise (dB)8
76

Unfiltered Anti−squeezed
Shot Noise543Filtered Anti−squeezed
Shot Noise2
1

Filtered Squeezed Shot Noise

0
−1
0.4

0.6

0.8

Unfiltered Squeezed Shot Noise
1
1.2
1.4
1.6
Frequency (MHz)1.8

2

Figure 4-10: The spectra of squeezed and anti-squeezed quadrature variances which are both filteredand unfiltered by the squeeze amplitude filter. The harmful effect of the anti-squeezing is attenuatedby the filter cavity. The measured spectra are also compared with the theoretical model (solid
curves).
shot noise spectra at about 1 MHz is due to the OPO cavity linewidth. The high-pass filtering
effect of the filter cavity on the squeezed and anti-squeezed shot noise spectra is present in the
mid-band. The peaks at 1.23 MHz are due to the modulation of the OPO crystal for the PDHlocking technique. The cavity resonance frequency is slightly detuned from the center frequency ofthe squeezed vacuum field, such that the cross-coupling of the squeezed and anti-squeezed shot noise
spectra is present (similar to the detuning effect in Sec. 4.5.3). Taking this effect into consideration,
theoretical predictions with the filter cavity mode-matching efficiency ηm and cavity detuning ωd as
free parameters are also shown in Fig. 4-10. They are found to ηm = 0.79 and ωd /(2π) = 320 kHz.

148

Chapter 5

Theory of Quantum Enhanced
Gravitational Wave Detectors
5.1

Overview

In Sec. 1.5, it has been briefly mentioned that the sensitivity of quantum-noise-limited laser interferometers for precision measurements such as gravitational wave (GW) detectors can be improvedby the injection of squeezed states of light [8, 9] into the antisymmetric port of the interferometers[10, 207]. The proposed squeezing injection scheme and the effect of squeezing on the sensitivity of
a Michelson interferometer (with a proper frequency-dependent squeeze angle) have been shown in
Fig. 1-6 and 1-7. The effect of squeezing on Advanced LIGO (with a proper frequency-dependent
squeeze angle) has been shown in Fig. 1-8.
In the quantum optical picture, the injection of an optical field in a squeezed state into theantisymmetric port of the interferometer is equivalent to the replacement of a vacuum (uncorrelated)
field that couples into the port by a correlated field, or in other words, the replacement of an
uncorrelated vacuum field with a correlated (squeezed) field. Since an interferometer is essentiallya two-photon device [82, 103, 96, 104, 102, 20, 19] and a squeezer is a two-photon correlator that
correlates upper and lower quantum sidebands centered around the carrier frequency, the mostproper explanation of the propagation of a squeezed state in the interferometer and its interaction
with the interferometer field is given by the two-photon formalism [19, 20]. In other words, quantumenhancement in the interferometer is equivalent to inserting a two-photon correlator (an OPO in
our case) in the path of two uncorrelated photons (vacuum) that couple into the antisymmetric portof the interferometer as shown in Fig. 5-1.
In this chapter, a basic theory of the propagation of a squeezed vacuum field in an interferometer and its interaction with the interferometer field using the two-photon formalism is described.
149

Michelson

Michelson

LASER

LASER

Uncorrelated
Photons

Uncorrelated
Photons

ω+Ω

ω+Ω

ω+Ω
Two-Photon
Correlator

OpticalCirculator
ω- Ω

ω- Ω
Two-Photon
Correlator

Photodetector

OpticalCirculator

ω- ΩCorrelated
Photons

Photodetector

Figure 5-1: A model of a Michelson interferometer with a two-photon correlator inserted in the pathof two vacuum (uncorrelated) photons that enter the antisymmetric port of the interferometer. The
two-photon correlator or an OPO correlates the two input uncorrelated photons, and the output is
two correlated photons (squeezed vacuum).
A signal-recycled Michelson interferometer (SRMI) is used as a model to understand quantumenhancement since it is a relatively simple interferometer configuration, but similar to the configuration of Advanced LIGO. The SRMI configuration will be used in the next chapter to experimentallydemonstrate quantum-enhancement. The effect of squeezing on more complex interferometers isbest understood using computer simulations [103].

5.2

Theory of a Quantum-Enhanced Signal-Recycled Michelson Interferometer

5.2.1

Equivalent Optical Model

Overview
The interferometer configurations that are used in the currently operational GW detectors such as
LIGO, VIRGO, GEO, and TAMA and envisioned for Advanced LIGO are very complex. Therefore, it
is difficult to understand how a squeezed vacuum field that enters the antisymmetric port propagatesin such a complex interferometer, interacts with the interferometer field, and reduces its quantum
noise. Thevenin’s equivalent optical circuitry is the optical version of Thevenin’s theorem thatany combination of voltage sources, current sources, and resistors with two terminals is electricallyequivalent to a single voltage source and a single series resistor. Using this theorem, the complex
interferometer components which are all linear media can be simplified to simple optical elements.
In this section, the optical equivalent model of a SRMI detector is studied since it is a relatively
150simple interferometer configuration similar to the configuration of Advanced LIGO and also theconfiguration used to experimentally demonstrate quantum-enhancement in this thesis, which will
be discussed in the next chapter. The optical equivalent model of the quantum-enhanced SRMI
detector is also studied in this section.

The Simple Michelson Interferometer(a)Michelson(b)

L2

r m, tm

Ein

Ein

L1

LASER

Etrans

LASER
Erefl

Erefl
EtransMichelson

Photodetector

Figure 5-2: A model of (a) a Michelson interferometer that is equivalent to (b) a mirror withreflectivity rm and transmissivity tm . L1 and L2 are the interferometer arm lengths. For simplicity,
we assume that the beamsplitter is perfectly balanced and lossless. The reflectivity of the two end
test mass mirrors is retm .

As shown in Fig. 5-2, the equivalent optical component of a Michelson interferometer thatconsists of a perfectly balanced, lossless beamsplitter and two end test mass mirrors with equalreflectivity retm is represented by a mirror with reflectivity rm and transmissivity tm given by
rm

=

tm

=

Erefl
φd
= iretm eiφc /2 sin ,Ein
2
Etransφd= iretm eiφc /2 cos ,
Ein
2

(5.1)
(5.2)

where φc and φd are the common and differential phase shifts that the input field acquires during
its round-trips in the interferometer arms, given byφc =2ω(L1 + L2 ),
c

φd =

2ω
(L1 − L2 ).
c

(5.3)

Here we have assumed for simplicity that the end mirrors are perfect and lossless, and the beamsplitter is perfectly balanced and lossless.
151

(a)
Michelson

(b)
v

L2
Loss

a

r l, tl

L1

LASER

SignalRecycling
Mirrorb

SignalRecycling
MirrorMichelsona
LASER

OpticalCirculatorcr m, tm

r s, tsbc
Photodetector

Figure 5-3: A model of (a) a quantum-enhanced SRMI that is equivalent to (b) a three-mirror
ring cavity that consists of a Michelson-interferometer equivalent mirror with reflectivity rm andtransmissivity tm , a signal-recycling mirror (SRM) with reflectivity rs and transmissivity ts , and amirror that accounts for an optical loss in the interferometer with equivalent coupling reflectivity
rl and transmissivity tl . Field a is the main laser that enters the interferometer at the input port.
Field v enters the interferometer when the loss occurs. Field b which is either a random vacuum
field or a squeezed vacuum field enters the antisymmetric port of the interferometer. Field c is theinterferometer output field.
The Quantum-Enhanced Signal-Recycled Michelson Interferometer
An optical equivalent model of a quantum-enhanced SRMI that consists of a balanced beamsplitter,
two end test mass mirrors, a signal-recycling mirror (SRM), and an optical circulator is shown in Fig.
5-3. The SRMI is optically equivalent to a three-mirror ring cavity1 with resonance frequency ωcthat consists of a Michelson-interferometer equivalent mirror with reflectivity rm and transmissivity
tm , the SRM with reflectivity rs and transmissivity ts , and a mirror that accounts for an opticalloss in the interferometer with equivalent coupling reflectivity rl and transmissivity tl . Field a isthe main laser at frequency ω that enters the interferometer at the input port, field v enters theinterferometer when the loss occurs, field b which is either a random vacuum field or a squeezed
vacuum field enters the antisymmetric port of the interferometer, and field c is the interferometer
output field.
This three-mirror triangular cavity model is essentially equivalent to a two-mirror Fabry-Perotcavity model with the only difference that the extra mirror accounts for optical losses inside theinterferometer that can be absorbed into losses in the Michelson-equivalent mirrors or SRM. The
three-mirror cavity model is chosen in this thesis since it is easier to visualize the couplings of thevacuum field and the squeezed vacuum field that enter the signal-recycling cavity (SRC) than in thetwo-mirror cavity model.
1 This is not exactly correct although it is not important here since a three-mirror cavity is polarization-sensitive1525.2.2

Quadrature Field Propagation

To understand the propagation of the squeezed vacuum field in the interferometer requires the twophoton formalism, as we have discussed in Chap. 2. Consider as (ω + Ω), bs (ω + Ω), and cs (ω + Ω) to
be upper quantum sidebands (signals) and a†i (ω − Ω), b†i (ω − Ω), and c†i (ω − Ω) to be lower quantum
sidebands (idlers). Assuming that the input fields to the cavity are perfectly mode-matched, thesignal and idler of the interferometer output are respectively given bycs (ω + Ω)=

tsa (ω + Ω)as (ω + Ω) + rbs (ω + Ω)bs (ω + Ω) + lvs (ω + Ω)vs (ω + Ω),(5.4)

c†i (ω − Ω)

†
†
†
i∗
i∗
= ti∗
a (ω − Ω)ai (ω − Ω) + rb (ω − Ω)bi (ω − Ω) + lv (ω − Ω)vi (ω − Ω),(5.5)

where tsa (ω + Ω) and tia (ω − Ω) are the frequency-dependent cavity transmission coefficients seenfrom the input port of the interferometer, rbs (ω + Ω) and rbi (ω − Ω) are the frequency-dependent
cavity reflection coefficients seen from the output port of the interferometer, and lvs (ω + Ω) andlvi (ω − Ω) are the frequency-dependent cavity transmission coefficients seen from the loss port. They
are respectively given bytsa (ω + Ω)=

rbs (ω + Ω)=

lvs (ω + Ω)=

−tm ts e−i(ωd +Ω)LSRC /c,tia (ω − Ω) = tsa (ω − Ω),
1 − rm rs rl e−2i(ωd +Ω)LSRC /ct2s rl rm e−2i(ωd +Ω)LSRC /crs −,rbi (ω − Ω) = rbs (ω − Ω),
1 − rs rl rm e−2i(ωd +Ω)LSRC /c/
1 − |tsa (ω + Ω)|2 − |rbs (ω + Ω)|2 ,lvi (ω − Ω) = lvs (ω − Ω),

(5.6)
(5.7)
(5.8)

where ωd is the detuning from the cavity resonance given by ωd = ω − ωc and LSRC is the length of
the SRC which is the distance between the SRM and beamsplitter.
For simplicity, we transform into the rotating frame of the carrier frequency ω in the frequency
domain defined by Eq. (1.60), such that Eqs. (5.4) and (5.5) becomec̃s (Ω)
c̃†i (−Ω)= tsa (ωd + Ω)ãs (Ω) + rbs (ωd + Ω)b̃s (Ω) + lvs (ωd + Ω)ṽs (Ω),
†
†
†
i∗
i∗
= ti∗
a (ωd − Ω)ãi (−Ω) + rb (ωd − Ω)b̃i (−Ω) + lv (ωd − Ω)ṽi (−Ω),(5.9)(5.10)

where tsa (ωd +Ω) = tsa (ω +Ω), tia (ωd −Ω) = tia (ω −Ω), rbs (ωd +Ω) = rbs (ω +Ω), rbi (ωd −Ω) = rbi (ω −Ω),lvs (ωd + Ω) = lvs (ω + Ω), and lvi (ωd − Ω) = lvi (ω − Ω). Following the definition of the amplitude andphase quadrature fields in Eqs. (2.10) and (2.11) for fields a, b, and c in the limiting case of Ω + ω

similar to Eqs. (2.15) and (2.16), the amplitude and phase quadrature fields of the output field ccan be written in a compact matrix form,c̃ = Ma ã + Mb b̃ + Mv ṽ

Quadrature Field Propagation,
153(5.11)

where we use the two-photon matrix representationã ≡ 

ã1
ã2





b̃ ≡ 

,

b̃1
b̃2





c̃ ≡ 

,

c̃1
c̃2



,

(5.12)and Ma , Mb , and Mc are the matrices that represent field propagation through the cavity, given by

Ma

Mb

Mv



cos α+− sin α+sin α+

cos α+

cos β+− sin β+sin β+

V+

iV−

−iV−

V+

cos β+
≡ eiα− ≡ eiβ− 


≡ 



A+



iA−

−iA−

A+

B+



iB−

−iB−

B+





,

(5.13)

,(5.14)

,(5.15)

where Ma and Mb comprise overall phase shifts α− and β− and quadrature rotation by angles α+
and β+ . Ma , Mb , and Mv also comprise field attenuation by factors A+ , B+ , and V+ . Here we havedefined
α±

≡

A±

≡

6
6
15
15
arg(tsa ) ± arg(tia ) ,β± ≡arg(rbs ) ± arg(rbi ) ,
2
2
1
1
1
10 s
10 s
10 s
i
|ta | ± |ta | ,B± ≡|rb | ± |rbi | ,V± ≡|lv | ± |lvi | .
2
2
2(5.16)(5.17)si∗
In the case of no carrier detuning (ωd = 0), tsa (ωd + Ω) = ti∗a (ωd − Ω) and rb (ωd + Ω) = rb (ωd − Ω),and thus α+ , β+ , A− , and B− vanish, giving neither quadrature angle rotation nor asymmetrical
amplitude attenuation. In the case of cavity detunings (ωd 0= 0), nonzero α+ and β+ give quadrature
angle rotation.
Eq. (5.11) is the most general system of equations for the quadrature field amplitudes of the
SRMI output which are linear combinations of all the input fields with corresponding frequencydependent coupling transfer functions. Although in this section, only the SRMI is considered, thesame formalism can be extended to any complex interferometer configuration.

5.2.3

Quadrature Variances

The Most General Case
When we detect the interferometer output field with a single homodyne photodetector, just likean unbalanced homodyne detector as discussed in Sec. 2.6.3, the measured amplitude and phase154

1

Quadrature Rotation (π)

0.8

β+(0.3×HWHM)
β+(0.6×HWHM)

0.6
0.4

β (0.9×HWHM)+
β+(1.2×HWHM)

0.2
0α+(0×HWHM), β+(0×HWHM)

α+(0.3×HWHM)
α+(0.6×HWHM)−0.2 α+(0.9×HWHM)
α+(1.2×HWHM)−0.4 5
10

6

10

7

8

10

Frequency (Hz)10

Figure 5-4: Quadrature rotation angles α+ and β+ for various SRC detunings. The HWHM is thehalf linewidth at the half maximum of the SRC.

quadrature variances of the field, given byṼc1 (Ω)
Ṽc2 (Ω)

342|c̃1 (Ω) − -c̃1 .| ,342=
|c̃2 (Ω) − -c̃2 .| ,=(5.18)(5.19)(similarly for Ṽa1 , Ṽa2 , Ṽb1 , and Ṽb2 ), are found in terms of the amplitude and phase quadrature
variances of the input fields a, b, and v to be



Ṽc1
Ṽc2





 = 

cos2 α+sin2 α+



A2+

A2−



Ṽa1






sin2 α+ cos2 α+
A2− A2+
Ṽa2




2
2
cos2 β+ sin2 β+
B+
B−
Ṽb1



+
2
2
sin2 β+ cos2 β+
B−
B+
Ṽb2



2
2
V+ V−
Ṽ
  v1  .
+
2
2
V− V+
Ṽv2

(5.20)

This is the most general system of equations for the SRMI output quadrature variances which
are linear combinations of the quadrature variances of the input fields with frequency-dependent
coupling transfer functions. When all the fields a, b, and v are shot-noise-limited, Ṽa1 = Ṽa2 = Ṽb1 =Ṽb2 = Ṽv2 = Ṽv2 = 1, and thus, Ṽc2 = Ṽc2 = 1 as expected. Although it is previously said that a, b,
155

and c are an unsqueezed field, a squeezed field, and a vacuum field, respectively, the derived model
is general enough to allow any of them to be squeezed fields or noisy fields as one wishes.
If a is shot-noise-limited at all frequencies and v is a vacuum field, not a squeezed vacuum field
to compensate for the decoherence or loss of squeezing, Ṽa1 = Ṽa2 = Ṽv1 = Ṽv2 = 1. Moreover, if b isa squeezed vacuum field with a squeeze factor of r in the amplitude quadrature, Eq. (5.20) becomes



Ṽc1
Ṽc2





 = 

cos2 α+sin2 α+

A2+



A2−

1




 
sin2 α+ cos2 α+
A2− A2+
1




2
2
cos2 β+ sin2 β+
B+
B−
e−2r



+
2
2
sin2 β+ cos2 β+
B−
B+
e2r

 
V+2 V−2
1
 .
+
2
2
V− V+
1

The result of the squeezed quadrature variance for rs =(5.21)√√√
0.93, rl = 0.9995, retm = 0.995, r =Quadrature Variance Relative to Shot Noise (dB)

1, wd /(2π) = 50 kHz, φd = π + π/30, and LSRC = 2 m is plotted in Fig. 5-5.

0
DetunedResonant

−1
−2
−3
−4
−5
−6
−7
−8
−9
−10 3
10

4

10

5

10
Frequency (Hz)

6

10

7

10

Figure 5-5: The
quadrature variance of an interferometer output√ simulation√ results of the squeezed√
field for rs = 0.93, rl = 0.9995, retm = 0.995, r = 2, φd = π + π/30, and L = 2 m in two cases:(a) wd /(2π) = 10 kHz and (b) wd /(2π) = 0. The squeezing level decrease at frequencies below 1
MHz is due to the cavity pole of the SRC.

156

Case of No Cavity Detuning
In the case of no cavity detuning (ωd = 0), α+ = β+ = 0 and A− = B− = V− = 0, and therefore,
there is no cross-coupling between the amplitude and phase quadrature variances of the interferometer input and squeezed vacuum fields. This simplifies Eq. (5.20) to be



Ṽc1
Ṽc2





= A2+ 



Ṽa1
Ṽa2





2 
 + B+

Ṽb1
Ṽb2





 + V+2 

Ṽv1
Ṽv2



,

(5.22)

where A+ , B+ , and V+ are also simplified to be
A+
B+
V+

= |tsa | = .
|rbs | =:= |lvs | =

/

=

tm ts
,
2
2
2
1 + rm rs rl − 2rm rs rl cos(2ΩLSRC /c)(5.23)

2 − 2(r 2 + t2 )r r r cos(2ΩL
rs2 + (rs2 + t2s )2 rl2 rm
SRC /c)ss s l m,2
2
2
1 + rm rs rl − 2rm rs rl cos(2ΩLSRC /c)(5.24)

1 − |tsa |2 − |rbs |2 .

(5.25)

Again, if a is shot-noise-limited at all frequencies and v is a vacuum field, Ṽa1 = Ṽa2 = Ṽv1 =Ṽv2 = 1. In addition, if b is a squeezed vacuum field with a squeeze factor of r in the amplitude
quadrature, Eq. (5.22) becomes



Ṽc1
Ṽc2




=

0

 


−2r
1
1
e
.
1 − |rbs |2   + |rbs |2 
1e2r

A result of the squeezed quadrature variance for rs =(5.26)√√√
0.93, rl = 0.9995, retm = 0.995, r =1, wd /(2π) = 0 Hz, φd = π + π/30, and LSRC = 2 m is also plotted in Fig. 5-5.

5.3

Theory of Quantum-Enhancement in Advanced Interferometer Configurations

Long baseline laser-interferometric GW detectors such as LIGO, VIRGO, GEO600, and TAMA300
are based on a Michelson interferometer with a Fabry-Perot cavity in each arm and a power-recycling
cavity (PRC) formed from a mirror called a power-recycling mirror (PRM) between the laser source
and the Michelson interferometer to increase the optical gain of the displacement measurement
or equivalently increase the signal-to-shot-noise ratio. The next generation GW detectors such as
Advanced LIGO are planned to improve their sensitivities by making use of even higher laser powers[95, 7].
The next generation GW detectors also plan to employ a detuned SRM in order to manipulate
and optimize the frequency response of the detectors to possible GW signals. The SRC is detuned
157

from carrier resonance to recycle a GW signal at some frequencies and resonantly extract it atother frequencies [208, 209]. This configuration is called the resonant sideband extraction (RSE). A
detuned SRC exhibits two optical resonances at frequencies in the GW detection band that can bechosen to optimize the sensitivity in the presence of other noise sources, as shown in Fig. 1-8. The
peak at a higher frequency arises from the unbalanced response of a GW sideband resonating in
the detuned SRC. The peak at a lower frequency arises because the GW sidebands induced by thedifferential displacement of the arms enter the detuned SRC, forming a radiation pressure induced
opto-mechanical spring which enhances the optical response at the resonant frequency of the spring[96, 101]. The configuration allows the possibility of overcoming the standard quantum limit (SQL),
which has been addressed in Sec. 1.4.3.
As discussed in Sec. 1-8, a Michelson interferometer with Fabry-Perot cavity arms is opticallyequivalent to a simple Michelson interferometer with arms that possess cavity frequency responses.
Likewise, a Michelson interferometer with two Fabry-Perot cavity arms and both a PRC and SRC is
optically equivalent to a three-mirror cavity that consists of the PRM, Michelson-equivalent mirror,
and SRM [210]. However, the two-photon transfer function of such an interferometer with unequal
losses and reflectivities in the arms is extremely complex, and therefore the propagation of a squeezed
vacuum field in the interferometer and its interaction with the interferometer field are better understood by use of computer simulations. A few computer programs are available to calculate signal
and noise fields in complex interferometers [103]. Fig. 1-8 shows the result of a computer simulationof quantum-enhancement in Advanced LIGO.

158

Chapter 6

Demonstration of Quantum
Enhancement in a Gravitational
Wave Detector
6.1

Overview

In the previous chapters, the techniques necessary for the implementation of quantum-enhancement
in long baseline laser-interferometric gravitational wave (GW) detectors have been studied. The
generation, degradation, detection, and control of squeezed states using a sub-threshold opticalparametric oscillator (OPO) and a homodyne detector have been theoretically discussed in Chap. 2
and experimentally demonstrated in Chap. 3. The key requirements for the implementation of thequantum-enhancement in GW detectors such as squeezing in the GW detection band, the high levelof squeezing, the long-term stability of squeezing, and frequency-dependent squeezing have been
investigated in Chap. 4. The propagation of a squeezed vacuum field in a GW detector, especiallyin a signal-recycled Michelson interferometer (SRMI) configuration, and its interaction with theinterferometer field have been theoretically formulated in Chap. 5. With these building blocks, anexperimental test of quantum-enhancement in a GW detector is ready to be conducted.
In this chapter, the apparatus and results of the quantum-enhancement experiment in the LIGO
40m prototype interferometer at California Institute of Technology (Caltech) [15] are presented.
In Sec. 6.2, the role of the Caltech 40m LIGO interferometer prototype is described. In Sec.
6.3, the details of the squeezer, the prototype interferometer, the interface of the squeezer to theinterferometer, the length sensing photodetector, and the readout and control schemes are presented.
In Sec. 6.4, the experimental results of broadband quantum-enhancement and an increase in signal159

to-noise ratio (SNR) by squeezing are described.

6.2

Caltech 40m LIGO Interferometer Prototype

The Caltech 40m LIGO Prototype Interferometer plays an important role in Advanced LIGO R&D
as a test bed or staging area for the design, testing, refinement, and staging of advanced GW detector
configurations anticipated for Advanced LIGO, which require a full interferometer for testing. It is
anticipated that Advanced LIGO will operate with both power- and signal-recycling mirrors (PRM
and SRM), operated in the resonant sideband extraction (RSE) configuration. These signal recycling
techniques need to be prototyped with a full interferometer.
The interferometer can also be used as a test bed or staging area for the design, testing, refinement, and staging of other advanced detector elements. With its full control systems similar to
Advanced LIGO’s, the Caltech 40m LIGO Interferometer Prototype is an ideal choice for testing
quantum-enhancement before implementing it in long baseline GW detectors such as LIGO. The
details of the conceptual design of the Caltech 40m LIGO Interferometer Prototype can be foundin Ref. [165]. Although quantum-enhancement is compatible with an output mode cleaner (OMC)
that is planned to be used in Advanced LIGO, the experiment is conducted without an OMC since
it is not ready at the time of the experiment.

6.3

Experimental Apparatus

6.3.1

Overview

As shown in Fig. 6-1, the quantum-enhanced prototype GW detector consists of the following
components: (1) a pre-stabilized laser (PSL) system that is the light source of the interferometerand squeezed vacuum generator that includes a Nd:YAG Master Oscillator Power Amplifier (MOPA)
laser with a throughput of 5 W at 1064 nm and all the optical and electro-optical devices to stabilize
the frequency, amplitude, and mode of the laser, (2) a triangular optical cavity – or mode cleaner –which consists of three free hanging mirrors with a linewidth of 4 kHz, (3) a test interferometer ina signal-recycled Michelson interferometer configuration that comprises a 50/50 beamsplitter, twohigh-reflective end mirrors, and a SRM, all suspended as single loop pendulums, (4) a squeezed
vacuum generator – or squeezer – that consists of a second-harmonic generator (SHG), an opticalparametric oscillator (OPO), a monitor homodyne detector, subcarrier optics to lock the OPO cavity,
and an optical circulator to inject the generated squeezed vacuum field to the antisymmetric portof the interferometer, and (5) a length sensing detector that consists of a high quantum efficiency
photodetector to sense differential motion of the interferometer mirrors. In the following sections,the details of each major component are described.
160

Pre-Stabilized
Laser

Mode
Cleaner

PD3
λ/4

PD4

PBS

λ/4

PO

PZT1

PD6
PM3

PD2

PO

λ/2

PO

BS

PM4

BS

PO
SignalRecycling
Mirror

PD5
FI

Squeezer

PZT4

λ/2

FIC

DBS

FI

PZT5 OPO

BS

DBS

PBS
FM

PZT6

PZT3
FOC

BS
λ/2

FR
PD9

PM5

λ/2

PD10

PBS

SHG
PO

Test
Mass

BS

PD1

PO

PD7

PZT2

PM1

PMC

Auxiliary
Nd:YAG LASER

Test
Mass

AOM

PBS

Reference
CavityBSFI

Nd:YAG
MOPA LASER

PM2Interferometer

BS

PM6

HPD1

PD8
HPD2

PD11

LengthSensing
Detector

Figure 6-1: A schematic of the quantum-enhanced prototype gravitational wave detector that consists of the following parts: (1) a pre-stabilized laser (PSL) system that is the light source of theinterferometer and squeezed vacuum generator that includes a Nd:YAG Master Oscillator Power
Amplifier (MOPA) laser with a throughput of 5 W at 1064 nm and all the optical and electrooptical devices to stabilize the frequency, amplitude, and mode of the laser, (2) a triangular opticalcavity – or mode cleaner (MC) – which consists of three free hanging mirrors with a linewidth of
4 kHz, (3) a test interferometer in a SRMI configuration that comprises a 50/50 beamsplitter, twohigh-reflective end mirrors, and a SRM, all suspended as single loop pendulums, (4) a squeezed
vacuum generator – or squeezer – that consists of a second-harmonic generator (SHG), an optical parametric oscillator (OPO), a monitor homodyne detector, subcarrier optics to lock the OPO
cavity, and an optical circulator to inject the generated squeezed vacuum field to the antisymmetric port of the interferometer, and (5) a length sensing detector that consists of a high quantum
efficiency photodetector to sense differential motion of the interferometer mirrors. BS: 50/50 beamsplitter, DBS: dichroic beamsplitter, PD1-PD11: photodetectors, HPD1 and HPD2: homodyne
photodetectors, PO: pickoff mirror, λ/2: half-wave plate, λ/4: quarter-wave plate, PBS: polarizing
beamsplitter, PMC: pre-mode cleaner, AOM: acousto-optic modulator, PM1-6: electro-optic modulators as phase modulators, PZT1-6: piezo-electric transducers, FI: Faraday isolator, FR: Faraday
rotator, FIC: fiber input coupler, FOC: fiber output coupler, FM: flipper mirror. The reference
cavity, pre-mode cleaner, mode cleaner, signal-recycling cavity, SHG, and OPO are controlled using
the Pound-Drever-Hall locking technique.

161

6.3.2Pre-Stabilized Laser

The PSL system is the light source of the interferometer and squeezed vacuum generator that includes
a Lightwave MOPA laser (Lightwave Electronics Inc.) with a throughput of 5 W at 1064 nm and all
the optical and electro-optical devices to stabilize the frequency, amplitude, and mode of the laser[166]. It was designed and developed by Lightwave Electronics Inc. in collaboration with the LIGO
group. The MOPA laser is based on a diode-pumped, narrow-linewidth, single-frequency laser as themaster oscillator and a double-passed power amplifier. The PSL consists of three major components:
the frequency stabilization servo system (FSS), the intensity servo system (ISS), and the pre-modecleaner (PMC). The laser has three frequency actuators: a PZT of the master oscillator (fast), an
electro-optic modulator (medium), and a Peltier element that controls the temperature of the master
oscillator (slow). The electro-optic modulator (EOM) can also be used as an amplitude stabilization
actuator.
In the FSS, the laser is frequency-locked to one of the resonance frequencies of a monolithic
optical cavity suspended in a vacuum chamber as a frequency reference, using the Pound-DreverHall (PDH) locking technique. An error signal for controlling the laser frequency is fed back to
the frequency actuators. In this way the frequency noise is reduced by the FSS and limited by thestability of the reference cavity. To match the frequency of the laser with the resonance frequencyof the cavity, an acousto-optic modulator (AOM) shifts the laser frequency by 3.3 MHz and allows
additional servos (mode cleaner and the arm common mode) to further change the laser frequency,
while keeping the laser resonant in the reference cavity. The length, free spectral range (FSR),
finesse, and FWHM bandwidth of the reference cavity are 203.3 mm, 736.5 MHz, 9518, and 77.4
kHz, respectively. The EOM modulation frequency for the PDH technique is 21.5 MHz [211].
In the ISS, a small portion of the laser is measured by a photodetector after the PMC and thesignal is sent to feedback control electronics, yielding an error signal [212]. It is fed back to thediode current actuator and EOM. A relative intensity noise of 10−8 Hz−1/2 between 100 Hz and 1
kHz and 5 × 10−6 Hz−1/2 between 1 kHz and 100 kHz is achieved.
The PMC is a triangular cavity that filters out higher-order Gaussian modes of the input beam.
The laser after the PMC is mainly a Gaussian TEM00 mode, and this helps decrease the intensity
noise of the laser. The PDH technique is used to generate an error signal, which is fed back to the
PZT on the concave mirror located at the edge of the PMC. The length, FSR, finesse (high), andFWHM bandwidth (high) of the PMC are 210 mm, 713.8 MHz, 4100, and 174 kHz, respectively.
The EOM modulation frequency for the PDH technique is 35.5 MHz [211].
162

6.3.3Mode Cleaner

Before the PSL-enhanced laser is injected into the interferometer, the triangular optical cavity or
the MC with a linewidth of 4 kHz serves to further suppress the laser intensity and frequency
fluctuations, and to filter out higher-order spatial modes of the input beam. It also plays a rolein giving a low beam position and pointing jitter. The MC consists of three free hanging mirrors
with a half-length of 13.5 m. The length is carefully chosen such that two sets of RF sidebands (33MHz and 166 MHz) that are used to lock the Michelson interferometer and signal-recycling cavity(SRC) respectively can transmit through the MC. The suspended optics are placed on passive seismic
isolation stacks within a single vacuum volume with a pressure of 10−6 torr. The MC is locked to
the laser using the PDH technique.

6.3.4

Interferometer

Components
The interferometer consists of three free hanging mirrors and a balanced beamsplitter, all suspended
as single loop pendulums that attenuate seismic noise at frequencies above the pendulum resonant
frequency (∼ 1 Hz), and therefore, the optics behave as free masses above the pendulum resonant
frequency. A feedback control system suppresses the seismically driven motion of the suspended
mirrors. Four pairs of magnets affixed to each optic, along with current-carrying coils, act as
actuators to keep the SRC locked on resonance to the laser frequency. The SRM and Michelsoninterferometer form the SRC to enhance the detector sensitivity. The suspended optics are placed on
passive vibration isolation stacks within the single vacuum volume with the same pressure of 10−6
torr as the MC. More details about the interferometer components can be found in Refs. [95, 165].
Configuration
The SRMI configuration is chosen since it is an important feature of the optical configuration envisioned for Advanced LIGO [7, 95]. The Michelson interferometer is locked on a dark fringe witha small offset such that a small amount of the carrier light comes out of the SRC, while the SRC
is locked on resonance to the carrier. The carrier light at the antisymmetric port acts as a local
oscillator field (LO) that a GW-induced signal beats against – this is called the DC readout scheme[7].
Two sets of RF sidebands at 33 MHz and 166 MHz are generated by the phase modulators, PM3
and PM4, respectively. The photodetector at the symmetric port (PD6) is used to lock the SRC andthe photodetector in one of the Michelson arms (PD7) is used to lock the Michelson interferometer.
Once the Michelson offset is introduced, the photodetector used for locking the offseted Michelson isswitched from the PD7 to the length sensing photodetector (PD11) for stabilizing the optical power163

at the PD11. The Michelson interferometer and SRC locking schemes will be described in Sec. 6.3.7.
Displacement Sensitivity
The displacement sensitivity of the interferometer is shown in Fig. 6-5. The comparison between themeasured sensitivity without squeezing and the theoretically predicted shot-noise-limited displacement sensitivity based on the measured optical power of 100 µW indicates that the interferometer isshot-noise-limited at frequencies above 42 kHz. At frequencies below 42 kHz, the noise is dominatedby laser intensity noise and interferometer length noise that appear in the differential motion of thearms. The peaks at frequencies above 42kHz are also due to the interferometer length noise. In√
addition, the optical power at the antisymmetric port is changed to verify the P scaling of theshot noise as shown in Fig. 6-2. The figure shows that as the optical power doubles, the noise floor
increases by 3 dB at frequencies above 42 kHz and by 6 dB at frequencies below 42 kHz, indicatingthat the detector is shot-noise-limited above 42 kHz. Below 42 kHz, the detector is dominated by
the laser noise. Quantum radiation pressure noise is buried under the laser noise as well as othertechnical noise, and only the shot noise is accessible for showing quantum-enhancement1 .

−120

Noise Level [10log(V2rms/Hz)]

−125
−130

200 µW−135−140
100 µW
−145
−150

50 µW

−155
−160
−165 410

Photodetector Dark Noise5

Frequency [Hz]10

Figure 6-2: The noise floor of the SRMI detector for different optical powers at the length sensing
photodetector. As the optical power doubles, the noise floor increases by 3 dB at frequencies above
42 kHz and by 6 dB at frequencies below 42 kHz, indicating that the detector is shot-noise-limited
above 42 kHz. Below 42 kHz, the detector is dominated by the laser noise. For the proper comparisonof the noise floors at different powers, the photodetector noise must be subtracted from each noise
floor.
1 As of today, no interferometer has ever seen quantum radiation pressure noise, not just the experiments with
injected squeezing.

164

The ultimate displacement sensitivity of the SRMI detector is given by [96]

∆L̃SRMI (Ω) = .

1|GSR |

:

h̄cλ
,
πηPin

(6.1)

where Pin is the input power to the beamsplitter, λ is the laser wavelength, GSR is the signal-recycling
gain given by GSR = [ts /(1 − rs rm e−2iωd LSRC /c )]2 (assuming Ω + ωd ), and η is the composite
detection efficiency (or 1 − η is the optical loss) from the SRM to the length sensing photodetector(including the quantum efficiency of the photodetector). Here rs and ts are the amplitude reflectivity
and transmissivity of the SRM, rm is the reflectivity of the Michelson interferometer seen from theantisymmetric port, ωd is the SRC detuning, and LSRC is the length of the SRC. In the shot-noise√
limited band, the displacement sensitivity of the detector is 7.0 × 10−17 m/ Hz. η, rs , and rm are√√measured to be 0.83, 0.925, and 0.995 respectively. Pin and ωd LSRC /c are found to be 57.0 mW
and zero or ωd LSRC /c + 1 in the frequency of interest.

High Voltage
AmplifierOscillator
Steering
M irror 2
Steering
M irror 1

Oven

Beam
Sampler

Dichroic
M irror

PPKTP

M ode-M atching
Telescope
λ/2
Faraday
Rotator

PZT5

M onitor
Photodetector

Input
Coupler

OutputCoupler

Focusing
Lens

Flipper
λ/2Polarizing
Beamsplitter

PZT6
Oscilloscope

Steering
M irror 3
Length Sensing
Photodetector

Figure 6-3: A detailed schematic of the squeezer-interferometer interface. The squeezed vacuum
field is injected into the interferometer by use of the polarizing beamsplitter (PBS), Faraday rotator,and half-wave plate (λ/2). When the input squeezed vacuum is in the S polarization to the PBS,
the interferometer output field enters the PBS in the P polarization due to the effect of the Faraday
isolation.

165

6.3.5

Squeezed Vacuum Generator

Overview
The squeezed vacuum generator or squeezer is essentially identical to the squeezer discussed in Chap.
3 except for the photodetector. The squeezer consists of a SHG, an OPO, a monitor homodyne
detector, subcarrier optics to lock the OPO cavity, and an optical circulator to inject the generated
squeezed vacuum field to the antisymmetric port of the interferometer. The squeezer is powered by
1.5 W of the Nd:YAG MOPA laser that also powers the SRMI detector. In this section, the squeezed
vacuum injection optics are mainly described while the SHG, OPO, subcarrier optics, and monitor
homodyne detector have been described in Chap. 3.

SHG, OPO, and Subcarrier Optics
The SHG, OPO, and subcarrier optics are essentially identical to the squeezer components in Chap.
3, and therefore, they will not be discussed here. For details, Chap. 3 should be reviewed.

Monitor Homodyne Detector
A balanced homodyne detector is used to monitor the generated squeezed vacuum field before
injecting it to the interferometer. The monitor homodyne detector is essentially identical to thehomodyne detector discussed in Chap. 3, and therefore, it will not be discussed here. For details,
Chap. 3 should be reviewed.
As shown in Fig. 6-1, a flipper mirror is installed in the path of the squeezed vacuum field. Whenthe flipper mirror is up, the squeezed vacuum is directed toward the monitor homodyne detector tomonitor the level of squeezing before the injection. When the flipper mirror is down, the squeezed
vacuum is injected into the interferometer.

Squeezed Vacuum Injection Optics
A detailed schematic of the interface of the squeezer to the interferometer is shown in Fig. 6-3.
The squeezed vacuum field is injected into the interferometer via a polarizing beamsplitter (PBS),
Faraday rotator, and half-wave plate. When the input squeezed vacuum is in the S polarization to
the PBS, the squeezing-enhanced interferometer output field enters the other port of the PBS in the
P polarization due to the effect of the Faraday isolation.
Another important set of optical components is steering mirrors and mode-matching optics.
However, to align and mode-match the input squeezed vacuum field to the interferometer mode
is not an easy task since the squeezed vacuum field and interferometer output field are counterpropagating. This is done by optimizing the transmission of the interferometer output field through166the OPO cavity in a TEM00 mode using the steering mirrors and mode-matching telescope. The
alignment and mode-matching procedure is as follows:
1. Block the seed and subcarrier and scan the OPO cavity using the PZT on the input coupler (PZT5). Set up a photodetector in the seed path to monitor the transmission of theinterferometer field through the OPO cavity.
2. Flip the flipper half-wave plate in between the Faraday rotator and PBS. The orientation of
the half-wave plate is such that the interferometer field is directed toward the OPO cavity.
When the flipper is not in place, the orientation of the half-wave plate between the Faraday
rotator and mode-matching telescope is such that the field transmits through the PBS toward
the length sensing photodetector.
3. Align and mode-match the interferometer output field through the OPO cavity using steering
mirrors 1 and 2 and the mode-matching telescope. When the OPO cavity is scanned, the mode
structure measured by the monitor photodetector indicates the coupling of the interferometermode to the OPO cavity mode or equivalently the optical loss due to the mode-mismatch
between the interferometer and squeezed vacuum field. Once it is optimized, the procedureensures the squeezed vacuum field to be aligned and mode-matched to the interferometer.
4. Now that the squeezed vacuum injection path has been optimized, the interferometer field path
has also been determined. Use steering mirror 3 to align the beam onto the length sensing
photodetector.
The squeeze angle is locked to the amplitude quadrature of the interferometer field by the noiselocking technique using the PZT-actuated mirror (PZT6) in Fig. 6-1. The squeeze angle readout
and control scheme will be described in Sec. 6.3.7.

6.3.6

Length Sensing Photodetector

To measure the quantum-enhancement in the differential motion in the arms of the prototype GW
detector, a high quantum efficiency photodiode in a transimpedance amplifier is used. The photodiode is an ETX500T photodiode (JDS Uniphase) with a quantum efficiency of 93%. The operational amplifier used for the transimpedance amplifier is OPA657 (Texas Instrument) and the√transimpedance is 10 kΩ. Its input voltage noise and input current noise are 4.8 nV/ Hz and 1.3√
fA/ Hz at frequencies above 100 Hz respectively. This operational amplifier is chosen because its
low input current noise allows the photodetector to be Johnson-noise-limited in the frequency band
between 100 Hz and 10 MHz. This ensures about 8 dB of separation between the electric noise of
the photodetector and the shot noise of 100 µW. The measured noise floor of the photodetector is167√shown in Fig. 6-2. A broadband noise floor of about 7 nV/ Hz is achieved with this photodetector
circuit.
(b)

Low-Pass Filter
1kHz Cutoff

Variable Gain
AmplifierDifferential
Amplifier
OffsetInput(a)

OpticalCirculator
Interferometer
Output Field

Michelson
OffsetPhotodetector
50Ω Source
Impedance

Buffer
1MΩ Load Impedance
600 Ω Source Impedance

RF Low-Pass Filter
5MHz Cutoff
50Ω System

PZT6

(d)

Oscilloscope
1MΩ Load Impedance(e)Oscillator

Adder

Active High-Pass Filter
100Hz Cutoff
1MΩ Load Impedance
600 Ω Source Impedance

RF Band-Pass Filter
2MHz-100MHz
Frequency Band(c)
Low-Pass Filter
1kHz Cutoff

Mixer

Envelope Detector
50Ω Load Impedance

AmplifierSR785 Spectrum Analyzer
40dB Gain
100M Ω Load Impedance
1MΩ Load Impedance
600 Ω Source Impedance

RF Amplifier
20dB Gain
50Ω Load Impedance
50Ω Source Impedance

RF Power Splitter50Ω System

HP8591E Spectrum Analyzer
50Ω Load Impedance

Figure 6-4: A detailed schematic of the readout and control schemes to (a) detect the squeezingenhanced interferometer output field, (b) control the Michelson offset, (c) monitor the level ofsqueezing, (d) control the squeeze angle using the noise-locking technique and the PZT-actuatedmirror (PZT6), and (e) measure the broadband quantum-enhancement.

6.3.7

Readout and Control Schemes

The readout and control schemes to (a) detect the squeezing-enhanced interferometer output field,(b) control the Michelson offset, (c) monitor the level of squeezing, (d) control the squeeze angle
using the noise-locking technique, and (e) measure the broadband quantum-enhancement are shownin Fig. 6-4. The length sensing photodetector measures the squeezing-enhanced interferometer
output field. The output of the photodetector is low-pass filtered with a cutoff frequency of 5 MHz
to remove the beats between the carrier and 33 MHz sidebands and between the carrier and the 166
MHz sidebands. The output of the low-pass filter is high-pass filtered with a cutoff frequency of
100 Hz, amplified by an amplification factor of 100, and measured by the SR785 spectrum analyzer(Stanford Research Systems) to see the noise spectrum of the interferometer. The 5 MHz low-passfilter output is also measured via the RF amplifier by the HP8591E spectrum analyzer (HewlettPackard) to monitor the level of squeezing at MHz frequencies. The output of the RF amplifier isalso used to lock the squeeze angle using the quantum noise locking technique. The phase modulator
168

used in this experiment is the PZT (PZT6) in Fig. 6-1. The output of the 5 MHz low-pass filter isalso measured by the oscilloscope to monitor the DC power fluctuations of the interferometer output
field. It is also compared with a Michelson offset setting value by the differential amplifier and its
output is fed back to the coil drivers that actuate on the Michelson mirror positions.
Some preparation is required to calibrate the shot noise and observe quantum-enhancement in theinterferometer. The procedure for controlling the Michelson offset and squeeze angle and measuring
the effect of the broadband quantum-enhancement in the interferometer is the following:
1. Block the squeezed vacuum not to inject it into the interferometer, or equivalently let random
vacuum noise enter the interferometer. This is necessary to calibrate the shot noise level inthe interferometer before observing the effect of squeezing.
2. Lock the interferometer using the PD6, PD7, and RFPD which is located at the interferometer
output port between the optical circulator and PD11 via a flipper mirror (not shown in the
figure). Also, make sure that the carrier-to-sideband power ratio is sufficiently high so thatthe contribution of the sideband shot noise to the overall shot noise level is small2 . This canbe verified by using an optical spectrum analyzer (scanning Fabry-Perot cavity).
3. Flip down the flipper mirror after switching to the PD6 and PD7 from the RFPD (so that theinterferometer lock does not drop) and detect the interferometer output field with the PD11.
Filter out the 33 MHz and 166 MHz beats with the carrier. Make sure on the oscilloscope thatthe DC power level is constant.
4. Turn on the Michelson offset feedback control system and set the offset to an optimum value
so that the power of the interferometer output field at the PD11 is 100 µW. Increase thecontrol gain to stabilize the shot noise so that its fluctuation level is within the possible levelof squeezing.
5. Measure the broadband noise power of the interferometer field with the SR785 spectrum analyzer. Make sure that the shot noise level is sufficiently higher than the electronic noise of themeasurement system. Compare the measured shot noise level with the theoretically predicted
shot noise level.
6. Change the laser power at the PD11 to see how the noise power changes with the increasingor decreasing laser power. If it is shot noise, it goes up by 3 dB as the laser power doubles. If
it is laser intensity or frequency noise, it goes up by 6 dB as it doubles. This gives an estimateof the cutoff frequency of squeezing.
2 Another way of minimizing the effect of the sidebands is to place an OMC between the output port of the interferometer beamsplitter and the photodetector and reject the sidebands outside the linewidth of the OMC. However,this might degrade the level of squeezing by introducing an extra optical loss due to the possible mode-mismatch
between the OMC and the squeezed vacuum field.

169

7. Inject the squeezed vacuum into the interferometer. Start with a low squeezing level (byattenuating the OPO pump power) not to disturb the lock of the interferometer (although, inprinciple, it should not disturb it). Make sure that the optical crosstalk between the OPO andinterferometer is sufficiently attenuated by tweaking the Faraday isolator.
8. Scan the squeeze angle using the PZT6 to verify the effect of squeezing with the HP8591E
spectrum analyzer. This gives an estimate of how much the injection of squeezing reduces theshot noise.
9. Turn on the noise-locking feedback control system to lock the squeeze angle to the squeezed
quadrature. Optimize the control gain and phase and modulation frequency.
10. Measure the broadband noise power of the interferometer with the SR785 spectrum analyzer
to verify the broadband quantum-enhancement. Monitor the HP8591E spectrum analyzer andthe oscilloscope to check if the interferometer stays locked during the measurement.
11. Excite the interferometer beamsplitter to simulate a GW signal at a frequency where the shotnoise is flat and verify an increase in SNR by the injection of the squeezed vacuum. If theeffect of squeezing is proper, the simulated GW signal strength does not change regardless ofthe presence of squeezing.

6.4

Experimental Results

6.4.1

Overview

With all the necessary components ready, the squeezing injection has been tested. In the following
sections, experimental results of broadband squeezing-enhancement in the prototype GW detector
in the quantum noise limited frequency band and an increase in SNR with a simulated GW signalby the squeezing-enhancement are presented.

6.4.2

Broadband Squeezing-Enhancement in the Interferometer

The effect of quantum-enhancement in the SRMI detector is represented by squeeze factor r in thedisplacement sensitivity of the detector in Eq. 6.1 such that

∆L̃SRMI (Ω) = .

1|GSR |or in terms of the familiar quadrature variance,/:h̄cλ −r
e
πηPin

(6.2)

Ṽ (Ω) = e−r . This is an approximate form of themost rigorous equation given in Refs. [96, 82]. The squeeze factor r is a function of η, rm , rs , and ωdas discussed in Sec. 5.2. Since the frequency of interest Ω is much smaller than the linewidth of the
170

−15

Displacement [m/Hz

1/2

]

10

Shot Noise−16

10

Theoretically Predicted Shot Noise
Squeezed Shot Noise
−17

10

4

10

5

Frequency [Hz]10

Figure 6-5: The noise floor of the SRMI detector with and without the injection of a squeezed vacuum
field. The theoretically predicted shot noise level based on the measured optical power is comparedwith the measured noise floor of the SRMI detector. The interferometer is shot-noise-limited at
frequencies above 42 kHz. With the squeezing injection, the shot noise is reduced broadband in
the shot-noise-limited frequency band by a squeeze factor of r = 0.34. The peak at 18 kHz in thesqueezing spectrum is due to the PZT modulation for the noise locking technique. The other peaks
are due to the interferometer input PZT noise coupled to the interferometer length noise.

SRC, Eq. (6.2) can be approximated as a frequency-independent equation. When the displacement
sensitivity is normalized to the shot noise level to see the effect of squeezing, Eq. (6.2) can be divided
by Eq. (6.1) to obtain
E
E∆L̃SRMI E
E Squeezed = e−r .
E∆L̃SRMI E(6.3)

Unsqueezed

The result of the broadband squeezing-enhancement in the interferometer is shown in Fig. 6-5.
The comparison between the two spectra shows that the quantum noise floor of the interferometeris reduced by the injection of the squeezed vacuum field by a squeeze factor of r = 0.34 in the shotnoise-limited frequency band. This is equivalent to a factor of 2 increase in laser power (withoutsqueezing) or equivalently a factor of 2 decrease in laser power for the same shot noise level when theinterferometer is squeezed. The injected squeezing level is 9.3 dB (inferred based on the measured
level of squeezing and optical losses) or 7.4 ± 0.1 dB (measured by the monitor homodyne detector).
The measurable squeezing effect is limited to frequencies above 42 kHz in this experiment since thequantum noise is masked by the classical noise at lower frequencies. However, squeezing-enhancement
171is expected to be effective on the quantum noise at lower frequencies as well. This is the first
implementation of quantum-enhancement in a laser-interferometric GW detector prototype withsuspended optics and readout and control schemes compatible with Advanced LIGO.
−15

10

Displacement [m/Hz

1/2]

Simulated GW Signal

Shot Noise

−16

10

Squeezed Shot Noise
−17

10

40

45

50

55
60
Frequency [kHz]65

70

Figure 6-6: The noise floor of the SRMI detector with a simulated GW signal at 50 kHz with andwithoutof squeezed
vacuum. The shot noise floor is reduced broadband from 7.0×10−17√ the injection√−17
m/ Hz to 5.0 × 10
m/ Hz by the injection of squeezing while the strength of the simulated GW
signal is retained. This corresponds to a 40% increase in SNR or detector sensitivity, which isequivalent to a factor of 1.43 = 2.7 increase in GW detection rate for isotropically distributed GW
sources that are confined to the frequency band in which squeezing is effective.

6.4.3

Increase in Signal-to-Noise Ratio by Squeezing-Enhancement

Fig. 6-6 shows the noise floor with a simulated GW signal at 50 kHz, with and without the injectionof the squeezed vacuum field. The GW signal is simulated by exciting the beamspliter at 50 kHz
to cause differential motion of the arms. The quantum noise floor is reduced broadband3 from√√
7.0 × 10−17 m/ Hz to 5.0 × 10−17 m/ Hz, while the strength of the simulated GW signal isretained. This corresponds to a 40% increase in SNR or detector sensitivity (by the same squeeze
factor), which is equivalent to a factor of 1.43 = 2.7 increase in GW detection rate for isotropically
distributed GW sources that are confined to the frequency band in which squeezing is effective [15].
Other peaks in the squeezing spectrum are due to optical crosstalk between the interferometer and
OPO, arising from inadequate isolation of the OPO from the interferometer output.
3 Systematic uncertainty in the displacement calibration is 10%, but does not affect the relative improvement
achieved by the squeezing injection that was observed.

172

Chapter 7

The Future
7.1

Overview

In Chap. 6, quantum-enhancement in the Caltech 40m LIGO Interferometer Prototype using thetechniques described in Chaps. 2, 3, 4, and 5 has been experimentally demonstrated. Although theproof-of-principle of quantum-enhanced gravitational wave (GW) detectors has been demonstrated,several possible improvements can be incorporated into the techniques used in Chap. 6. In thischapter, some possible future improvements and investigations are proposed and studied.

7.2

Possible Future Improvements

7.2.1

Bow-Tie Optical Parametric Oscillators

It has been shown in Sec. 4.2 that any scattered light (which is not shot noise limited) from ahomodyne detector or an interferometer couples into an optical parametric oscillator (OPO) asa seed and degrades squeezing at low frequencies. In Advanced LIGO, the optical power of theinterferometer output field in the DC readout scheme [7, 95] is about 100 mW. Although backscattered light can be substantially avoided by placing a multiple number of Faraday isolators inthe squeezing injection path, this solution is not realistic since Faraday isolators are typically lossy(about 10%) and using even a few of them leads to a significant loss in the level of squeezing.
One way to go around this problem is to use a cavity configuration that geometrically isolatesinterferometer back-scattered light from the input squeezed vacuum. A good example is a bow-tie
cavity configuration as shown in Fig. 7-1. Even if back-scattered light couples into the OPO cavity, itcannot couple out of the OPO cavity, and therefore, do not co-propagate with the squeezed vacuum
field. A bow-tie OPO has been demonstrated by Grosse et al. [115] although it has not been tested
with an interferometer yet. It is expected to improve the performance of squeezing in it, especially173

at low frequencies.

Squeezed
VacuumVacuumSeed

Nonlinear
CrystalFigure 7-1: A schematic of a bow-tie OPO cavity that geometrically isolates back-scattered lightfrom the squeezed vacuum field. Even if back-scattered couples into the OPO cavity, it cannot
couple out of the OPO cavity, and therefore, does not co-propagate with the squeezed vacuum field.

7.2.2

Doubly-Resonant Optical Parametric Oscillators

Another improvement is a doubly-resonant OPO. It has three advantages: (1) to increase the pump
power, (2) to mode-match the pump to the seed perfectly, and (3) to lock the OPO cavity withthe pump field. Since a high-power laser with a throughput of several watts at 1064 nm is costlyor unavailable in many situations such as LIGO, it is difficult to obtain about a few watts at 532
nm from a second-harmonic generator (SHG). However, the OPO cavity can be made resonant at
both 1064 nm and 532 nm so that the nonlinear interaction between the two fields can be enhanced.
The doubly-resonant cavity also makes it possible for both beams to be mode-matched perfectly,
indicating that no extra pump that does not interact with the seed resonates in the cavity. Since
nonlinear crystals are typically absorptive at 532 nm, the doubly-resonant OPO helps stabilize thetemperature of the crystal. The other advantage is that the OPO cavity can be locked by use of thesecond-harmonic pump, due to which, a frequency-shifted subcarrier used to lock the OPO cavity
at 1064 nm is not necessary. A doubly-resonant OPO has also been demonstrated by Grosse et al.
[115].

7.3

Possible Future Investigations

7.3.1

Generation of Squeezed States in Optical Waveguides

In the past, various efforts have been made to obtain squeezed states without optical cavities so thatsqueezing occurs broadband, not limited by the linewidth of the cavities. Shelby et al. demonstrated
the generation of 0.58 dB of squeezing from four-wave mixing in a 114 m long optical fiber which
was cooled down to 4.2 K to suppress Brillouin scattering [213]. It has also been shown recently that174

optical confinement using waveguides helps increase the generation rate of correlated photon pairs.
Periodically poled waveguides, for example, produce large numbers of correlated photon pairs with
only a few µW of pump [214, 215, 216]. Fiorentino et al. demonstrated the generation of 2.9 × 106
correlated photon pairs/s per mW of pump from a periodically poled KTiOPO4 (PPKTP) waveguidein a 1-nm band, which was more than 50 times higher than the bulk PPKTP crystal generation
rate [217]. If the optical loss in the PPKTP waveguide is decreased, the PPKTP waveguide may be
an interesting approach to the cavity-less generation of squeezed states since it is compact without
the need for an optical cavity. The cavity-less generation of squeezed states is strongly desired since
instability in the performance of squeezing mostly comes from cavities used in the squeezer such as
the OPO and SHG.

7.3.2

Generation of Squeezed States in Microcavities

Recently, Kerr-nonlinearity induced OPO in a toroidal microcavity that is a silica-based whisperinggallery-type resonator [218, 219] has been shown by Kippenberg et al. [220]. The OPO threshold
was measured to be only 174 µW, which is more than two orders of magnitude lower than for opticalfiber-based OPO. The micro-OPO showed a high conversion efficiency of 36% and a signal-to-idler
ratio of 97%. A phase-sensitive amplification process that can exclude competing Brillouin or Ramanprocesses with a highly idieal coupling junction can provide an excellent candidate system for thegeneration of squeezed states of light in a microcavity.

7.3.3

Metamaterial-Enhanced Optical Parametric Oscillation

The optics community has been excited by recent demonstrations of negative-index metamaterials.
The unique properties of metamaterials were verified in Caloz et al. in 2001 [221]. Eleftheriades
et al. and Caloz et al. invented a method to realize left-handed metamaterials using artificial
lumped-element loaded transmission lines in microstrip technology [222]. The first superlens with anegative refractive index provided resolution three times better than the diffraction limit and wasdemonstrated at microwave frequencies by Grbic and Eleftheriades [223]. Subsequently, the first
optical superlens (an optical lens that exceeds the diffraction limit) was invented and demonstrated
in 2005 by Fang et al. [224].
Negative-index metamaterials can also be used in an OPO and SHG to enhance their efficiencies.
A theoretical model of the OPO and SHG enhanced by negative-index metamaterials was studied
by Popov and Shalaev [225, 226] and the metamaterial-enhanced SHG was demonstrated in 2006
by Klein et al. [227, 228]. OPO enables transparency, amplification, and oscillation with no cavity
in strongly absorbing negative index-metamaterials. The opposite directions of the wave vector and
the Poynting vector in such materials result in extraordinary optical properties, including backward
phase-matching and the generation of entangled pairs of left- and right-handed counter-propagating
175

photons. Metamaterial-enhanced OPO may be an interesting approach to increasing the level ofsqueezing dramatically.

7.3.4

Quantum-Enhanced Laser Interferometers in Resonant Sideband
Extraction

The resonant sideband extraction (RSE) scheme is the interferometer configuration envisioned for
Advanced LIGO. However, squeezing-enhancement has not been tested with the RSE yet. Although in principle, squeezing-enhancement should also be effective in the RSE configuration justas squeezing-enhancement in the signal-recycled Michelson interferometer configuration has been
demonstrated, an interesting feature of the RSE is that the detuned signal-recycling cavity (SRC)
rotates the squeeze angle in the interferometer. Since a squeezed vacuum field from an OPO has aconstant squeeze angle, the injection of the squeezed vacuum into the antisymmetric port reduces
quantum noise at some frequencies, but increases it at other frequencies, especially near the arm
cavity resonance and optical spring frequencies. Therefore, to reduce the interferometer quantum
noise broadband, squeezing with a frequency-dependent squeeze angle is required. As discussed inSec. 4.5, filter cavities such as kilometer-scale or high-finesse empty cavities, squeeze amplitudefilter cavities, and cavities with electromagnetically induced transparency (EIT) media can be used
to overcome this problem. When radiation pressure noise becomes significant in the Advanced LIGO
interferometers, frequency-dependent squeezing in the GW detection band needs to be investigatedbefore implementing squeezing in the detectors.

7.3.5

Quantum-Enhanced Laser Interferometers with an Output Mode
Cleaner

An output mode cleaner (OMC) is planned to be installed in Advanced LIGO [7, 95]. The purposeof the OMC is to clean the mode of the interferometer output field and to remove all RF sidebands
so that oscillator phase noise does not couple in and the perfect overlap of the LO and GW signals
at the length sensing photodetector is ensured. Therefore, the OMC will play an important role in
Advanced LIGO. However, squeezing with the OMC has not been tested yet.
Presumably there will be some mode-mismatch between the OMC and interferometer output
field, which leads to some optical loss which is expected to be about 20%, indicating that about
20% of squeezing is lost. Although this test is fairly simple, squeezing with the OMC should be
investigated to make sure that nothing bad happens.
176

7.3.6

Coherent Control of Squeezing with an Output Mode Cleaner

As discussed in the previous section and Sec. 4.4.4, one of the purposes of the OMC is to remove
RF sidebands that travel along with the interferometer carrier field. This is because the carrier
field that possible GW signals beat with would be contaminated by RF sidebands with oscillator
phase noise that is not filtered by the interferometer arms. For the same reason, signal and idler
fields along with an input squeezed vacuum field should not transmit through the OMC since theywould otherwise contaminate the interferometer carrier field. Hence, an error signal for the coherent
control of squeezing needs to be obtained from the beat between the signal/idler and interferometerRF sidebands (not the interferometer carrier) at the reflection/rejection port of the OMC. One could
also suggest putting a pickoff mirror in between the optical circulator and OMC to pick off some tiny
amount of the beat between the signal/idler and interferometer field, but this requires the pickoff
mirror to be suspended since the OMC is planned to be suspended as well. Either way, the coherent
control of squeezing with the OMC needs to be investigated.

177

178

Conclusion
Theoretical analysis and experimental demonstration of techniques necessary for quantum-enhanced
gravitational-wave (GW) detectors have been performed and presented in this thesis. Chapter 1
has described the physics of GWs, possible sources of GWs, the basic theory of laser-interferometric
GW detectors, and quantum noise in the GW detectors (photon shot noise and quantum radiation pressure noise), and has briefly addressed the proposed scheme of quantum-enhancement in
the GW detectors. The goal of the thesis has also been addressed. Chapter 2 has described thetheory of squeezed states including the generation, degradation, detection, and control of squeezed
states using sub-threshold optical parametric oscillators (OPOs) and homodyne detectors in the
two-photon formalism. Chapter 3 has described the experimental characterization of the techniques
discussed in Chapter 2 and presented experimental results. Chapter 4 has discussed key requirements for quantum-enhanced GW detectors such as the frequency band, level, long-term stability,and frequency-dependence of squeezing. Chapter 5 has described the theory of quantum-enhanced
GW detectors, especially in a quantum-enhanced signal-recycled Michelson interferometer (SRMI)
configuration, using a two-photon mathematical framework of quadrature field propagation. Chapter 6 has described the experimental demonstration of quantum-enhancement in a prototype GW
detector using the techniques that have been developed throughout the period of my Ph.D. work.
Finally, Chapter 7 has discussed possible future improvements and investigations toward the actual
implementation of the quantum-enhancement in long baseline laser-interferometric GW detectors
such as Advanced LIGO.
In the demonstration of quantum-enhancement in the prototype GW detector, by injecting asqueezed vacuum field of 9.3 dB (inferred) or 7.4 ± 0.1 dB (measured) at frequencies above 3 kHz
and a cutoff frequency for squeezing at 700 Hz into the antisymmetric port of the prototype GW
detector in the SRMI configuration, the shot noise floor of the detector was reduced broadband from√√
7.0×10−17 m/ Hz to 5.0×10−17 m/ Hz while the strength of a simulated GW signal was retained,
resulting in a 40% increase in signal-to-noise ratio or detector sensitivity, which is equivalent to afactor of 1.43 = 2.7 increase in GW detection rate for isotropically distributed GW sources that are
confined to the frequency band in which squeezing was effective. This is the first implementation ofquantum-enhancement in a prototype GW detector with suspended optics and readout and control
179schemes similar to those used in LIGO and Advanced LIGO. It is, therefore, a critical step toward
implementation of quantum-enhancement in long baseline GW detectors.

180

Appendix A

Tables of Constants
A.1

Physical Constants
Parameter
Speed of Light in Vacuum
Planck ConstantBoltzmann ConstantElementary Charge
Electron Rest Mass
Proton Rest Mass
Permeability of vacuum
Permittivity of vacuum
Gravitational ConstantFine Structure ConstantStefan-Boltzmann ConstantAvogadro ConstantAtomic Mass Unit

Symbol
c
h̄
kB
e
me
mp
µ0
+0
G
α
σB
NA
u

Value
299792458
1.05457266 × 10−34
1.380658 × 10−23
1.60217733 × 10−19
9.1093897 × 10−31
1.6726231 × 10−27
4π × 10−71/(µ0 c2 )6.67300 × 10−11
1/137.0359895
5.67 × 10−8
6.022 × 1023
1.66054 × 10−27

Units
m/s
Js
J/K
C
kg
kg
H/m
C2 /J/m
m3 /kg/s2
W/m2 /K4
mol−1
kg

Table A.1: Values of the constants from the National Institute of Standards and Technology (NIST)[229].

181

A.2

Astrophysical Constants

Parameter
Astronomical Unit
Tropical Year (1994)
Sidereal Year (1994)
JanskyParsecLight Year
Schwarzschild Radius of the Sun
Solar Mass
Solar Luminosity
Solar Equatorial Radius
Earth Mass
Earth Equatorial Radiusv# around Center of Galaxy
Solar Distance from Galactic Center
Hubble Expansion Rate
Age of the Universe

Symbol
AU
yr
yr
Jy
pc
ly
2GM# /c2
M#
L#
R#
M⊕
R⊕
Θ◦
R◦
H0
t0

Value
1.495978706×1011
31556925.2
31558149.8
10−26
3.085677580 × 1016
0.9461 × 1016
2.95325008
1.98892 × 1030
(3.846 ± 0.008) × 1026
6.96 × 108
5.97370 × 1024
6.378140 × 106
220(20)8.0(5)
100h0 (0.6 < h0 < 0.8)
11.5 + 1 ± 1.5

Units
m
s
s
W/m2 /Hz
m
m
km
kg
W
m
kg
m
km/s
kpc
km/s/Mpc
Gyr

Table A.2: Values of the constants from Lawrence Berkeley National Laboratory (LBNL) [230] and
CERN [231].

182

Appendix B

Tables of Acronyms
B.1

LIGO Related Acronyms
Acronym
BH
BS
DRFPMI
DRMI
FPMI
GEO
GW
IMC
LHO
LIGO
LLO
LMXB
MC
MOPA
NS
OMC
PMC
PRC
PRM
PSL
PSR
RSE
SQL
SRC
SRM
SRMI
TAMA
VIRGO

Full Name
black hole
50/50 beamsplitter
dual-recycled Fabry-Perot Michelson interferometer
dual-recycled Michelson interferometer
Fabry-Perot Michelson interferometer
English-German Collaboration/Gravitational-Wave Detector
gravitational wave
input mode cleaner
LIGO Hanford Observatory
Laser Interferometer Gravitational-Wave Observatory
LIGO Livingston Observatory
low-mass X-ray binarymode cleaner
master oscillator power amplifier
neutron star
output mode cleaner
pre-mode cleaner
power-recycling cavity
power-recycling mirror
pre-stabilized laser
pulsarresonant sideband extraction
standard quantum limit
signal-recycling cavity
signal-recycling mirror
signal-recycled Michelson interferometerJapanese Laser-Interferometric Gravitational-Wave Detector
French-Italian Collaboration/Gravitational-Wave Detector

Table B.1: LIGO related acronyms and their full names in alphabetical order.
183

B.2

Squeezing Related Acronyms
Acronym
AM
AOM
AR
BPM
BS
DBS
DOPO
EIA
EIT
EOM
FI
FM
FR
FSR
FWHM
GRIIRA
HPD
HR
HWHM
KTP
LO
NDOPO
OPO
PBS
PD
PDH
PM
PO
PPKTP
PPLN
PZT
QPM
RF
SHG
TEM

Full Nameamplitude modulation
acousto-optic modulator
anti-reflection
birefringent phase-matching
50/50 beamsplitter
dichroic beamsplitter
degenerate optical parametric oscillator/oscillationelectromagnetically induced absorption
electromagnetically induced transparency
electro-optic modulator
Faraday isolator
flipper mirror / frequency modulation
Faraday rotator
free spectral range
full width at half maximum
green-induced infrared absorption
homodyne photodetector
high-reflection
half width at half maximum
potassium titanyl phosphate
local oscillator
nondegenerate optical parametric oscillator/oscillation
optical parametric oscillator/oscillation
polarizing beamsplitter
photodetector
Pound-Drever-Hallphase modulator
pickoff mirror
periodically poled potassium titanyl phosphate
periodically poled lithium niobate
piezo-electric transducer
quasi-phase-matchingradio frequency
second-harmonic generator/generationtransverse electromagnetic

Table B.2: Squeezing related acronyms and their full names in alphabetical order.

184

Appendix C

Tables of Crystal Properties
C.1

Potassium Titanyl PhosphateParameter
Chemical Formula
Crystal Structure
Lattice ConstantDensity at 293 K
Melting Temperature
Thermal Expansion at 373 K

Hardness
Transmittance Range
Average Refractive Index
Nonlinear Coefficients at 1064 nm / 532 nm

Optical Damage Threshold

Value
KTiOPO4
Orthorhombic
a = 12.814×10−10 m
b = 6.404×10−10 m
c = 10.616×10−10 m
3.03 g/cm31450 Kα1 = 8.7 × 10−6 K−1
α2 = 10.5 × 10−6 K−1
α3 = −0.2 × 10−6 K−1
5 Mohs0.35 - 4.5 µm
1.8d31 = 6.5 pm/V
d32 = 5.0 pm/V
d33 = 13.7 pm/V
d34 = 6.1 pm/V
10 - 40 MW/cm2

Table C.1: Properties of Potassium Titanyl Phosphate (KTiOPO4 ). The values are from Almaz
Optics Inc. [232].

185

C.2

Lithium Niobate
Parameter
Chemical Formula
Crystal Structure
Lattice ConstantDensity at 293 K
Melting Temperature
Curie Temperature
Dielectric Constant at 100 kHz
Thermal Conductivity at 300 K
Thermal Expansion at 300 K
Hardness
Transmittance Range
Bandgap
Electro-Optic Coefficients at 633 nm at high frequencies

Average Refractive Index
Nonlinear Coefficients at 1064 nm / 532 nm

Optical Damage Threshold at 1064 nm, t = 10 ns

Value
LiNbO3
Trigonal, 3m
a = 5.148×10−10 m
b = 13.863×10−10 m
4.644 g/cm3
1530 K1415
ea = 85ec = 29
5.6 W/m/K
αa = 15 × 10−6 K−1
αc = 5 × 10−6 K−1
5 Mohs0.35 - 5.5 µm
4.0 eVr33 = 31 pm/V
r31 = 9 pm/V
r22 = 3.4 pm/V
1.8d31 = −4.5 pm/V
d33 = −0.27 pm/V
d22 = 2.1 pm/V
250 MW/cm2

Table C.2: Properties of Lithium Niobate (LiNbO3 ). The values are from Almaz Optics Inc. [232].

186

Appendix D

Expression of Noise in Decibels
In squeezing experiments, the decibel (dB) is often used to characterize the level of noise power[12, 11, 173, 16, 18, 112, 155, 160]. It is defined by a dimensionless logarithmic unit to show theratio of the magnitude of a physical quantity relative to a specified reference level [233]. Because of
the logarithmic property, very large or very small ratios can be expressed by convenient numbers.
To assume a specific reference level, dBm is often used in RF electronics where the reference level is
1 mW [234].
When measuring P (such as electric power, optical power, and noise power), it is conventional
to use the base-10 logarithm of a ratio of the measured quantity P to the specified reference levelP0 so that x dB is given byx dB = 10log10

'

P
P0

(

.

(D.1)

This expression of decibels is often used in seismology, acoustics, electrical or optical power measurements (especially, in RF electronics), and squeezing experiments in quantum optics.
When measuring amplitude V (such as electrical signals), it is conventional to use the base-10
logarithm of a ratio of the square of the measured quantity V to the square of the specified reference
level V0 . This is due to the fact that in most cases, power P is proportional to the square of amplitude
V , so that P = CV 2 , where C is a constant. Thus, x dB is given byx dB = 10log10

'

V2
V02

(

= 20log10

'

V
V0

(

.

(D.2)

This expression of decibels is often used in analysis of amplifiers, attenuators, and noise in electronics,
and gravitational wave (GW) physics in which the strain sensitivity of a laser-interferometric GW
detector is expressed in linear scale. It is important to note that Eq. (D.2) is essentially equivalent
to Eq. (D.1) under the condition P = CV 2 as in most applications. Eqs. (D.1) and (D.2) can
187

also be expressed in terms of squeeze factor r. Since the squeeze factor is defined by V /V0 = e−r ,substituting this into Eq. (D.2) yields
x dB = 20log10 e−r = −20
r % −8.69r.
ln10

(D.3)

For example, a squeeze factor of r = 0.5 corresponds to 4.34 dB of squeezing.
Historically, the spectrum of squeezed shot noise is measured in units of dBm and compared tothe spectrum of shot noise which is also measured in units of dBm, yielding the level of squeezing√in units of dB. In the GW community, the amplitude spectral density in units of V/ Hz is often
used to analyze the strain sensitivity of an interferometer. Thus, it is instructive to state that,
following the definitions in Eqs. (D.1) and (D.2), 10 dB of squeezing means 10 dB of decrease innoise power, which is equivalent to a factor of 3.16 decrease in noise voltage. For example, in alaser-interferometric GW detector, 10 dB of squeezing (with a frequency-dependent squeeze angle
perfectly matched to the ponderomotive squeeze angle of the interferometer) corresponds to a factorof 3.16 increase in strain sensitivity at frequencies where the detector is quantum-noise-limited.

188

Appendix E

Publications
The majority of this thesis has been published, submitted to, or accepted for publication in peerreviewed journals. Some selected articles resulting from the work performed during my Ph.D. are
as follows:
1. Quantum Enhancement in a Prototype Gravitational Wave Detector, K. Goda,
O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A.
Weinstein, and N. Mavalvala, Submitted (2007).
2. Generation of a Stable Low-Frequency Squeezed Vacuum Field with PeriodicallyPoled KTiOPO4 at 1064 nm, K. Goda, E. E. Mikhailov, O. Miyakawa, S. Saraf, S. Vass,
A. Weinstein, and N. Mavalvala, Submitted, arXiv:quant-ph/0703001v1 (2007).
3. Utility Investigation of Artificial Time-Delay in Displacement-Noise-Free Interferometers, K. Somiya, K. Goda, Y. Chen, and E. E. Mikhailov, Phys. Rev. D 76, 022002(2007).
4. Frequency-Dependent Squeeze Amplitude Attenuation and Squeeze Angle Rotation by Electromagnetically Induced Transparency for Gravitational Wave Interferometers, E. E. Mikhailov, K. Goda, T. Corbitt, and N. Mavalvala, Phys. Rev. A 73,
053810 (2006).
5. Interferometers for Displacement-Noise-Free Gravitational Wave Detection, Y.
Chen, A. Pai, K. Somiya, S. Kawamura, S. Sato, K. Kokeyama, R. L. Ward, K. Goda, and E.
E. Mikhailov, Phys. Rev. Lett. 97, 151103 (2006).
6. Optical Measurement of Cell Membrane Tension, G. Popescu, T. Ikeda, K. Goda, C.
A. Best, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. Feld, Phys. Rev.
Lett. 97, 218101 (2006).
189

7. Noninvasive Measurements of Cavity Parameters by Use of Squeezed Vacuum, E.
E. Mikhailov, K. Goda, and N. Mavalvala, Phys. Rev. A 74, 033817 (2006).
8. Photothermal Fluctuations as a Fundamental Limit to Low-Frequency Squeezing
in a Degenerate Optical Parametric Amplifier, K. Goda, K. McKenzie, E. E. Mikhailov,
P. K. Lam, D. E. McClelland, and N. Mavalvala, Phys. Rev. A 72, 043819 (2005).
9. Quantum Noise Locking, K. McKenzie, E. E. Mikhailov, K. Goda, P. K. Lam, N. Grosse,
M. B. Gray, N. Mavalvala, and D. E. McClelland, J. Opt. B: Quantum Semiclass. Opt. 7,
S421 (2005).
10. Frequency Resolving Spatiotemporal Wavefront Sensor, K. Goda, D. Ottaway, B.
Connelly, R. Adhikari, N. Mavalvala, and A. Gretarsson, Opt. Lett. 29, 1452 (2004).

190

Appendix F

Other Work
F.1

Overview

During my Ph.D., other than the work on the quantum-enhancement in gravitational wave (GW)
detectors, I have worked on several different projects, some of which are related to squeezing andothers are not. All the projects have been published in peer-reviewed journals. In this chapter, theseprojects are briefly presented. For the details of each project, the published papers in Appendix Eshould be consulted.

F.2

Noninvasive Measurements of Cavity Parameters by Useof Squeezed Vacuum

F.2.1

Introduction

High Q cavities such as whispering gallery mode (WGM) cavities have recently demonstrated quality
factors (Q) as high as 2 × 1010 and have shown the potential to reach even higher Q values [235, 219,
236]. However, there are difficulties in measurement of the linewidth and Q of such high Q cavities.
While in theory, the Q factor could be as high as 1012 and is limited only by Rayleigh scattering [237],
in practice, it is limited by other losses in the cavity. They include absorption and scattering losses
due to impurities in the cavity material, and light-induced losses due to nonlinear processes. Due tothe extremely small mode volume and high Q-factor of the cavity, the cavity build-up intensity isextremely high, even in the case of an input with small power (as small as several mW). Such a highresonator intensity leads to very efficient nonlinear processes inside WGM cavities, such as Raman
scattering, second-harmonic generation, and four-wave mixing [238]. Whereas this is beneficial inmany applications, it causes additional losses in the cavity and thus makes the Q factor measurement
191

unreliable (at least, making it power-dependent).
In the following sections, an alternative method of measuring Q factors by use of a squeezed
vacuum field is proposed and demonstrated. This technique is advantageous over traditional opticalmethods in that it utilizes the injection of squeezed vacuum into a test cavity not to excite any nonlinear processes in the cavity. When the input field is detuned from the cavity resonance frequency,
it transmits only the upper or lower quantum sidebands within the cavity linewidth while reflecting
the counterparts (associated upper or lower sidebands) and all the other sidebands. The linewidth ofthe cavity can then be measured by observing the destruction of the correlation between the upperand lower quantum sidebands with respect to the carrier frequency. The linewidth and Q factor
of a test cavity using the method agree with those measured by traditional optical methods. Moredetails about the technique can be found in Ref. [155].

F.2.2

TheoryOverview
A theory of the noninvasive method of measuring cavity parameters by use of squeezed vacuum fields
is identical to the theory of quantum-enhanced GW detectors in Chap. 5, except that the Michelsoninterferometer is replaced by a test cavity whose parameters are of interest and no interferometerfield is required. The test cavity is also identical to the squeeze amplitude filter cavity in Sec. 4.5.4
except that the center frequency of the probe squeezed vacuum field is detuned from the cavity
resonance of the test cavity.

Destruction of Quantum Sideband Correlation as a Probe for Cavity Parameter Measurements
When a squeezed vacuum field [ã = (ã1 , ã2 )t ] is injected into a test cavity with a resonance frequencyof ωc as shown in Fig. 4-9, assuming that the input squeezed vacuum is perfectly mode-matched tothe test cavity1 , the reflection is given byb̃ = Ma ã + Mv ṽ,(F.1)

where ṽ is a vacuum field that couples in associated with the transmission and intra-cavity loss of
the input field and Ma and Mv are the matrices that represent quadrature field propagation through1 The case of imperfect mode-matching is discussed in Ref. [155]192the cavity, given by

Ma

Mv



cos α+− sin α+sin α+

V+

iV−

−iV−

V+

cos α+
= eiα− 


= 




A+

iA−

−iA−

A+



,

(F.2)

,

(F.3)

where Ma comprises overall phase shift α− and quadrature rotation by angle α+ . Ma and Mv also
comprise field attenuation by factors A+ and V+ . Here we have defined
α± ≡

6
15
arg(ras ) ± arg(rai ) ,
2

A± ≡

1
10 s
|ra | ± |rai | ,
2

V± ≡

1
10 s
|lv | ± |lvi | ,
2

(F.4)

where ras and rai are the frequency-dependent reflection coefficients of the test cavity and lvs and lvi
are the frequency-dependent cavity transmission coefficients. They are respectively given byras (ω + Ω)=

lvs (ω + Ω)=

t21 r2 r3 e−2i(ωd +Ω)Lc /c,rai (ω − Ω) = ras (ω − Ω),r1 −1 − r1 r2 r3 e−2i(ωd +Ω)Lc /c.
1 − |ras (ω + Ω)|2 ,lvi (ω − Ω) = lvs (ω − Ω),

(F.5)
(F.6)

where ωd is the detuning from the cavity resonance given by ωd = ω − ωc and Lc is the half-lengthof the test cavity.
When the reflection field is measured by a homodyne detector, the amplitude and phase quadrature variances of the field are found in terms of the amplitude and phase quadrature variances of
the input fields a and v to be



Ṽb1
Ṽb2





=

cos2 α+sin2 α+sin2 α+cos2 α+




A2+
A2−

A2−
A2+




Ṽa1
Ṽa2





+

V+2
V−2

V−2
V+2




1
1



.

(F.7)

Note that if the input field is a vacuum field or in the coherent state such that Ṽa1 = Ṽa2 = 1, thenṼb1 = Ṽb2 = 1, as expected, and no cavity information is contained in the output state.
If the carrier frequency is detuned downward from the cavity resonance frequency, the cavity
transmits only the upper sidebands within the cavity linewidth and replaces them by vacuum at
those frequencies while reflecting the associated lower sidebands and all the other sidebands. Hence,the cavity-coupled reflected field is composed of the uncorrelated sidebands within the linewidth
and the reflected correlated sidebands outside of it. The consequence is the destruction of thecorrelation within the linewidth between the upper and lower quantum sidebands. This is analogous
to the destruction of the correlation between electro-optically modulated coherent sidebands in pairs,
in which the beat between the carrier and the upper or lower sideband can be measured only wheneither sideband is absorbed into the cavity, reflecting the carrier and other sideband. The beat193

could not be observed if all the fields were reflected. Similar measurements could be done with thetransmission of the squeezed vacuum field through the cavity. However, the signal-to-noise ratio
would not be as good as in the reflection method because the background of the transmission signal
is shot noise.
It is convenient to define the test cavity linewidth γ, the quality factor Q, and the finesse F, as
γ

=

Q =
F

=

,
2
1 − r1 r2 r3−1 1 − r1 r2 r3
ωFSR sin% √ωFSR ,√
π2 r1 r2 r3
π r1 r2 r3ω,γ√
π r1 r2 r3
ωFSR%
1 − r1 r2 r3
γ

(F.8)
(F.9)
(F.10)

respectively. The approximations made in Eqs. (F.8) and (F.10) are valid for high Q cavities. r1 ,r1 r2 r3 , ωd , and ωFSR will be treated as free fitting parameters. We also assume the input mirror islossless.

Squeezed Vacuum vs. Classically Noisy Light
Since we are interested in having as little light (at the carrier frequency) as possible in the test
cavity, it is instructive to calculate the average photon number in the field we use. The average
photon number in squeezed light with squeeze factor r and squeeze angle θ is given by Eq. (1.88).
As the number of coherent photons becomes zero (α → 0), resulting in squeezed vacuum, Eq. (1.88)
becomes8 9-n. = a† a = sinh2 r.

(F.11)

This is the average photon number in squeezed vacuum generated by squeezing. Note that if the field
is unsqueezed (r = 0), -n. = 0. For a squeeze factor of 1.5 corresponding to the squeezed or antisqueezed level of −13 dB which is the current experimental limit [142, 138], -n. = 4.53. Therefore,
it is fair to say that the optical influence of ideal squeezed vacuum on cavities is negligible.
Similarly, it is instructive to compare this technique to using a classical state. For simplicity,
assuming that the quadrature variance in both quadratures is frequency-independent, we consider
the case in which the lower sideband is fully transmitted through an impedance-matched cavity andthe upper sideband is fully reflected at the input mirror such that rai = 0 and ras = 1 at Ω = ωd ,respectively, which gives A+ = A− = 1/2 from Eq. (F.4). Thus, the amplitude and phase quadrature
variances of the reflected field are found to beṼb1 (ωd ) = Ṽb2 (ωd ) =< 1
1;Ṽa1 + Ṽa2 + .
4
2

194

(F.12)

In the absence of coherent light, the signal contrast can be defined as the quadrature varianceat detuning frequency ωd compared to the cavity-uncoupled quadrature variance at off-resonancefrequencies (|Ω − ωd | ) γ), in which case Ṽb1 = Ṽa1 and Ṽb2 = Ṽa2 , and the signal contrasts at thetwo orthogonal quadratures are respectively given by
S1 (ωd )=1(Ṽa1 + Ṽa2 ) + 12
Ṽb1 (ωd )
= 4
,
Ṽa1
Ṽa1

(F.13)

S2 (ωd )

=

Ṽa2
Ṽa2
= 1
.
1
Ṽb2 (ωd )4 (Ṽa1 + Ṽa2 ) + 2(F.14)In the limiting case of Ṽa2 ) Ṽa1 and Ṽa2 ) 1, we obtain
Ṽa2
,
4Ṽa1
S2 (ωd ) % 4.
S1 (ωd ) %(F.15)(F.16)We see that S2 has about the same limiting level as in the classical case, while S1 grows if Ṽa1gets smaller. Classically, Ṽa1 ≥ 1 (the shot noise limit), but using squeezed vacuum we can obtainṼa1 < 1, or improved signal contrast for a measurement in the squeezed quadrature. The cavitycoupled responses of the classical and anti-squeezed quadrature variances behave almost identicallyin the case of the impedance-matched cavity, whereas squeezing improves the signal contrast of themeasurement.

Fundamental Limit on Measurement Uncertainty
It is important to note that even in the absence of technical noise, quadrature variance measurements
are intrinsically contaminated by quantum noise itself. The standard deviation of the quadrature
variances is given by [160]∆Ṽb1 =√

2Ṽb1 ,

∆Ṽb2 =√
2Ṽb2 .

(F.17)

Thus, the noise of the measurement is proportional to the measured value itself, and many averages
can be performed to achieve smaller uncertainty levels.
This is different from the classical case where the parameters of a cavity are measured by measuring the transmission of a probe optical field incident on the cavity as a function of cavity detuning.
In this case, the measurements are fundamentally limited by shot noise: the number of measured√
photons (n) has uncertainty proportional to n. Therefore, the signal-to-noise ratio grows as thenumber of the transmitted photons increases.
195F.2.3

Experimental Demonstration

To demonstrate the proposed technique, a squeezed vacuum field is prepared by a squeezed vacuum
generator (different from the squeezer in Chap. 3) that consists of an optical parametric oscillator(OPO) with a monolithic 5%MgO:LiNbO3 crystal and a second-harmonic generator (SHG). The
SHG pumped by the Nd:YAG laser generates 250 mW at 532 nm, which then pumps the OPO
below threshold with a vacuum seed. The resultant field generated by the OPO is a squeezed
vacuum field with a squeezing bandwidth of 66.2 MHz defined by the OPO cavity linewidth. A
subcarrier field, frequency-shifted by an acousto-optic modulator (AOM) to a frequency that iscoincident with the cavity TEM01 mode, is injected into the other end of the OPO cavity. The
cavity is thus locked to the TEM01 mode, offset by 220 MHz from the carrier frequency, using thePound-Drever-Hall (PDH) locking technique [162]. The frequency-shift is necessary to ensure thatno cavity transmitted light at the fundamental frequency is injected into the OPO cavity since it acts
as a seed and degrades broadband squeezing due to the imperfect isolation of the Faraday isolator[17, 170]. This is especially important for high Q cavities with linewidths as narrow as kHz because

Quadrature Variance Relative to Shot Noise (dB)low-frequency squeezing is difficult to achieve.

7
6
5
4
3
2
1
0
−1
−2
−3
6

7

8

9

10
11
12
Frequency (MHz)

13

14

15

16

Figure F-1: Measured squeezed and anti-squeezed shot noise spectra relative to shot noise (dots) andfits to the data points (curves). The resolution bandwidth of the spectrum analyzer is 100 kHz. The
data are exponentially averaged 100 times. The apparent peak at 13.3 MHz is due to the couplingof the phase modulation at the frequency for the Pound-Drever-Hall (PDH) locking technique. The
overall decrease in the squeezing and anti-squeezing levels with frequency is due to the OPO cavity
linewidth. With the optically measured FSR, the linewidth is found from the fits to be γ = 844 ± 40
kHz.
196

The squeezed vacuum is injected into a triangular test cavity with a FSR of 713 MHz and
FWHM of γ = 856 ± 34 kHz, both measured by traditional methods using light. The frequency
shift of the subcarrier is 231 ± 0.1 MHz so that the carrier frequency is detuned from the TEM00
mode by 11.0 ± 0.1 MHz. As a result of this frequency shift, only the upper sidebands are within
the cavity linewidth, destroying the correlation between the upper and lower sidebands and, therefore, destroying the squeezing or anti-squeezing. This cavity-coupled squeezed vacuum reflection ismeasured by a balanced homodyne detector. The difference of the two photodetector signals is sent
to an HP4195A spectrum analyzer (Hewlett-Packard) to measure the quadrature variance of thesqueezed or anti-squeezed quadrature. The results are shown in Fig. F-1. The experimental data
are exponentially averaged 100 times. The resolution bandwidth of the spectrum analyzer is 100
kHz. Since the squeezed vacuum does not carry any coherent amplitude, the noise-locking technique[160] is employed to lock the homodyne angle to either the squeezed or anti-squeezed quadrature at
2 MHz.
We have assumed that the input mirror is lossless. This assumption is valid since it is a singlepass loss and does not influence the linewidth of the cavity. We then fit Eq. (F.7) to the measured
data points with free parameters r1 , r2 , r3 , and ωd . Both the data and the fits are shown in Fig.
√
F-1. The resulting fitting values are r1 r2 r3 = 0.99628 ± 0.00016, r1 = 0.99783 ± 0.00005, andωd /(2π) = 11.098 ± 0.017 MHz. Therefore, the FWHM linewidth of the cavity is found to beγ = 844 ± 40 kHz, which agrees with the classically measured linewidth of the cavity within theuncertainty (γ = 856 ± 34 kHz). We note that ωFSR can be determined from the fit, but here wehave used the optically measured value to estimate the linewidth.

F.3

Frequency-Resolving Spatiotemporal Wavefront Detection

F.3.1

Introduction

A variety of wavefront sensing techniques for spatial profiling of laser beams exist [239]. ShackHartmann detectors [240], for example, provide high spatial resolution, and heterodyne techniques[241, 242, 243] afford high sensitivity to lower-order spatial modes of a laser field with specificsymmetries. None of these techniques, however, addresses the need for spatial profiling of a singlefrequency component of a laser field, such as the sidebands of a phase-modulated laser field.
In the following sections, a technique that constructs the spatial wavefront of a RF sideband
of a phase-modulated laser field is theoretically introduced and experimentally demonstrated. The
development of this technique was motivated by the need to separately measure the spatial modesof the carrier and phase-modulated sidebands exiting the output port of a laser-interferometric GW
197

detector [91], where different frequency components of the phase-modulated laser field that are
incident upon the interferometer experience different spatial filtering effects. More details of thetechnique can be found in Ref. [244].
Many laser-interferometric GW detectors such as LIGO comprise a Michelson interferometerwith kilometer-long Fabry-Perot cavities in each arm and a several-meter long power-recycling cavity (PRC) at the input [91, 2]. Interferometric signals are used to hold all three cavities on resonanceand the Michelson interferometer on a dark fringe. Discriminants for these interferometer lengths,
as well as for mirror misalignments, are derived by injecting a phase-modulated laser field into theinterferometer. Laser fields induced by passing GWs at the output port are measured by heterodyne
detection in which the RF phase-modulated sidebands of the carrier field beat with the carrier field[245]. The carrier is resonant in the arm cavities, which have g parameters of ∼ 0.33 [152] and are
effective spatial filters. The RF sidebands, however, resonate only in the PRC, which is nearly degenerate with a g parameter of ∼ 1, and do not experience any significant spatial filtering. Moreover,the RF sidebands are significantly more sensitive to misalignments or other spatial distortions of the
PRC than the carrier field [246]. Consequently, the spatial modes of the carrier and RF sidebands
exiting the interferometer may be quite different. Maximum signal sensitivity requires perfect spatial
overlap between the transverse modes of the carrier and RF sidebands. It is, therefore, desirable tomeasure the spatial mode of the RF sideband field independent of the carrier field. Furthermore,
before the heterodyne detector is fully optimized, the RF sidebands at frequencies above and below
the carrier frequency, referred as the upper and lower sidebands respectively, can experience different
spatial distortions, and knowledge of the spatial profiles of the upper and lower sidebands circulatingin various parts of the interferometer is expected to be a valuable tool in optimizing the sensitivityof the GW detectors2 .

F.3.2

Description of Frequency-Resolving Wavefront Sensing

The basic principle behind the wavefront sensing technique is to measure the beat note between
a test laser and a reference laser that spatially overlaps it. The wavefront sensor has two distinct
properties: (1) high spatial resolution and (2) frequency discrimination. The high spatial resolutionis achieved by used of a reference field with high modal purity for interference with the test laser, as
well as a high spatial resolution scan. A relatively fast (up to 10 Hz) scan rate is achieved, which isnecessary to measure profiles faster than the dominant angular pointing fluctuations of the beams
in the LIGO interferometers. Frequency discrimination is realized by heterodyne detection, which
is used to measure the beat note between the reference laser and the frequency component of thetest laser that is of interest. The frequency discrimination implemented by heterodyne detection is
2 The wavefront camera is currently used in the LIGO interferometers for aligning the interferometers and studying
an output mode cleaner (OMC) [247].

198

distinct from previous heterodyne spatial profiling techniques [242, 243]. Both these techniques arelimited in the spatial resolution that can be obtained, and they cannot measure the spatial propertiesof the upper and lower sidebands independently, which is a key feature of this method.

Test
Laser

Galvo Y

Band-Pass
FilterPinholeBS
PD1

Reference
Laser

Galvo X
Mixer

Low-Pass
Filter

PD2

Mixer
VI

Low-Pass
Filter

Band-Pass
Filter

Band-Pass
FilterOscillator

VQ

π/2

Phase
Shifter

Mixer

Low-Pass
FilterFigure F-2: A schematic of the wavefront sensor that consists of a reference laser which is frequencyshifted from an input test laser, two photodetectors (PD1 and PD2), a beamsplitter (BS), a pairof galvanometers (Galvos X and Y), a pinhole in front of the PD1, a demodulator that generates
the I-phase and Q-phase of the combined field, and a feedback control loop that phase-locks thereference laser to the test laser.

A schematic of the wavefront sensor is shown in Fig. F-2. The wavefront sensor consists of areference laser, two photodetectors (PD1 and PD2), a beamsplitter (BS), a pair of galvanometers, apinhole, a demodulator, and a phase-locking feedback control loop. The reference laser is expanded
by a pair of lenses before the BS to give a fairly flat phase front at the photodetection plane of
the PD1 when it interferes with the test laser. The beat between the two lasers measured by thePD2 is demodulated by a mixer at a fixed frequency of the oscillator after band-pass filtering thebeat between the reference laser and the sideband of the test laser of interest, and the demodulator
output is fed back to the reference laser to phase-lock it to the test laser so that the frequency of thereference laser is stable during the measurement time interval. The beat between the two lasers is
also measured by the PD1 and then demodulated in two orthogonal quadratures at the frequency ofthe beat between the reference laser and the sideband to obtain the I-phase and Q-phase, which areused to obtain the amplitude and phase of the sideband. The PD1 measures the spatial content of
the amplitude and phase of the combined field. The galvanometers (Galvos X and Y) make a spiralscan over the PD1 with the pinhole to measure each pixel of the beat. The spiral pattern is chosen
to circumvent a scan rate limit resulting from the inertia of the galvanometers. The spatial profilesof the amplitude and phase of the combined field are reconstructed from the measured I-phase and199

Q-phase and given by/

VI2 (x, y) + VQ2 (x, y),,
VQ (x, y),φ(x, y) = tan−1
VI (x, y)|E(x, y)| =(F.18)(F.19)

where VI (x, y) and VQ (x, y) are the demodulated voltages in the I-phase and Q-phase respectively,
and the x and y are the transverse coordinates of the combined beam at the detection plane of the
PD1.

F.3.3

Experimental Demonstration

To perform an experimental demonstration of the wavefront sensor, a bench-top experimental setup
was constructed to measure the spatial amplitude and phase variation of a RF sideband of a phasemodulated input test field. An important feature of the setup was the need to generate a test
field similar to one that is expected from a LIGO interferometer, but with a controllable spatialmode. The test field was generated by transmitting a RF sideband of the test laser in a higher-order
Hermite-Gaussian mode of a high-finesse cavity, while transmitting the carrier as well, such thatthe cavity transmission is composed of the carrier and the single sideband. A low-power, monolithic
Nd:YAG laser (Lightwave Model 120) was used as a laser source. In the experiment, the test field
was phase-modulated by a phase-modulator at 81.9 MHz. The frequency of one of the 81.9-MHz
sidebands was matched to the resonant frequency of a TEM21 mode of the cavity. A RF sideband
at twice the 81.9-MHz frequency also passed through the cavity in a TEM42 mode.
Reference
Laser

103.4 MHz21.5 MHz

TEM00

Test
Laser

81.9 MHz

TEM21

TEM00

Figure F-3: The frequency spectrum that was incident upon the wavefront sensor. The frequency of
the reference laser was offset from one of the 81.9-MHz sidebands of the test laser by 21.5 MHz. One
can choose this frequency by sweeping the reference laser frequency to coincide with any frequency
component of the test laser to be probed.
The frequency spectrum that was incident upon the wavefront sensor is shown in Fig. F-3. In thisdemonstration, the frequency of the reference laser was offset from one of the 81.9-MHz sidebands ofthe test laser by 21.5 MHz. One can choose this frequency by sweeping the reference laser frequency
to coincide with any frequency component of the test laser to be probed. The 21.5-MHz beat was200

detected on the PD2 and demodulated with the mixer. The resulting discriminant was used forlocking the reference laser frequency to that of the test laser. All the band-pass filters have a center
frequency of 21.5 MHz to transmit only the beat between the reference laser frequency and the81.9-MHz sideband. A scan frequency of 5 Hz with a sampling rate of 1000 points per scan withno dwell time at each point were achieved. The limitation here was the sampling rate of the dataacquisition system used (National Instruments Model PCI 6052E).

Figure F-4: Top maps: The measured amplitude and phase of the first-order sideband in the TEM21
mode. Left: The amplitude profile, color coded according to the bars at the right to show spatial
variation in the relative amplitude. Right: The phase profile, showing the sudden phase transitions
that appear alternately as the amplitude changes polarity. Bottom maps: Theoretical predictions of
the amplitude and phase of the first-order sideband in the TEM21 mode, with color coding identicalto that of the top maps and with no free parameters except for an overall phase shift.
Measured maps of the spatial variation in the amplitude and phase of the first-order TEM21
sideband are shown at the top of Fig. F-4. The amplitude plot clearly shows the six lobes that characterize a TEM21 mode, and the sudden phase transitions that appear alternately as the amplitude
switches polarity are evident in the phase plot. These sharp phase transitions are superimposed on
the gradual phase variation in the radial direction because of the spherical curvature of the TEM21
mode phase front. The optical power of the sideband was 40 times less than that of the carrier during the experiment. Both maps show good qualitative agreement with the theoretical predictions
illustrated at the bottom of Fig. F-4.
The theoretical maps were generated by use of the exact optical parameters of the test and reference lasers. The Hermite-Gaussian mode of the cavity through which the test laser was transmitted201

overlapped the flat wavefront of the reference field on the photodetection plane of the PD1. The
blurring of the sharp phase transitions in the center of the image is due to the limited isolationthat the cavity provides for transmission of a TEM12 mode. The slight astigmatism of the cavity
prevented complete degeneracy of the TEM21 and TEM12 modes. The best isolation that could beachieved was a reduction of 100 in optical power. Theoretical calculations clearly show this blurringof the phase transitions when a TEM12 of one-tenth the amplitude is added to a TEM21 mode.
Although they are not shown, clear images of a TEM42 mode with 300 times less power than the
carrier power were also obtained.

F.4

Displacement-Noise-Free Interferometers and Utility ofTime-Delay Devices

F.4.1

Introduction

Displacement-Noise-Free Interferometry
GW detection does not require freely falling test masses because non-geodesic test-mass motion
affects travel times of pulses only when they arrive and leave the test masses, whereas the effectof GWs is distributed [248]. This idea was further explored in Ref. [249], which showed thatonce the number N of test masses is large enough, the number of available light-pulse-travel-timemeasurement channels [O(N 2 )] exceeds the total number of clock- and displacement-noise channels[O(N )], and there must exist clock- and displacement-noise-free channels. Ref. [249] also showedthat interferometers can be combined to realize displacement- and laser-noise free GW detection. As
argued there, when lasers are used as clocks, motions of laser devices cause Doppler-shifts to laser
frequencies, and are indistinguishable from laser noise. Therefore, displacement-noise-free detection,
strictly speaking, requires the cancellation of laser noise. Henceforth, the term displacement-noisefree interferometry (DFI) is used in the following sections to describe displacement- and laser-noise
free interferometer configurations.
Refs. [248] and [249] study DFI by calculating pulse time delays between emitters and receivers,
which are fixed on point test masses. This approach, although mathematically simpler and inprinciple applicable to laser interferometry, does not provide practical interferometer configurations.
Configurations that have been adopted and constructed so far require composite mirrors, namely
mirrors with multiple reflective surfaces. Apart from being experimentally challenging, the useof composite mirrors gives rise to the fundamental difficulty that thermal fluctuations of relative
positions between the multiple reflective surfaces are not canceled. In addition, so far only 2dimensional (2-D) configurations have been explored, for which it can be proved that shot-noiselimited sensitivity to GWs can be no better than ∼ f 3 at low frequencies.
202

tB

2

out B1

A1

ou

2

inA

in

A

D2 z
C2

C1
D1

y

B

ou

tA

in B2

B1

out
A1

in

x

2

Figure F-5: The 3-D octahedron configuration that consists of four Mach-Zehnder interferometers;
A1 : At C1 Br − Ar D1 Bt , B1 : Bt C1 Ar − Br D1 At (shown as solid lines) and A2 : At D2 Br −Ar C2 Bt , B2 : Bt D2 Ar − Br C2 At (shown as dashed lines). The subscripts r, t indicate reflection andtransmission.
Quantum-Enhancement in Displacement-Noise-Free Interferometers
In general, DFI detectors in any configuration are free of quantum radiation pressure noise sincedisplacement noise is canceled out, which means that the sensitivity of the DFI detectors is completely limited by photon shot noise, assuming that the displacement noise cancellation is perfect.
Therefore, unlike conventional GW detectors in Michelson interferometer configurations that require frequency-dependent squeezing, DFI does not require frequency-dependent squeezing when asqueezed state is injected into a DFI detector to reduce its quantum noise (which is only shot noisein DFI).

Displacement-Noise-Free Interferometry with Time-Delay Devices
Although DFI detectors are attractive, they require very long arms because of its f 2 response to GWs
at low frequencies. To circumvent this issue, time-delay devices that delay the propagation of light orstore light for some period of time, such as optical fibers, multiple cavities, and electromagneticallyinduced transparency (EIT) media, can be used in a short-arm DFI detector to achieve sensitivity
similar to that of a DFI detector with long arms. It has been found that a DFI detector with atime-delay device at each mirror (C1 , C2 , D1 , and D2 ) can increase the sensitivity of a DFI detector
by a factor of f at frequencies between the cutoff frequency fcutoff and 1/τ , where τ is the lightdelay/storage time.
203

Overview
In the following sections, a simple 3-D configuration that implements DFI without composite mirrors is discussed. First of all, Mach-Zehnder (MZ) interferometry is used so that laser noise can becanceled right in the beginning. Moreover, each mirror participates in at least two MZ interferometers, and thus has its location sensed redundantly. Finally, by combining the MZ output signals,
the redundant displacement information can be canceled out, leaving non-vanishing response to
GWs. In particular, it is shown that the 3-D configuration has ∼ f 2 sensitivity at low frequencies,
which is the best one can achieve with DFI. A 3-D DFI configuration with time-delay devices is also
discussed. The details about the DFI and the DFI with time-delay devices can be found in Refs.
[25, 64] respectively.

F.4.2

Detector Description

The 3-D DFI detector configuration is shown in Fig. F-5. The mirrors are located on the eight
vertices of a regular octahedron with an edge length of 2L. All laser beams in the detector propagate
along the edges of the octahedron. A Cartesian coordinate system is attached to the octahedron,with the origin coinciding with its center, the z axis coinciding with its B-A axis, the x axis in parallelwith the C1 -D1 (D2 -C2 ) direction, and the y axis in parallel with the C2 -D1 (D2 -C1 ) direction. ξand η are defined as shown in the figure. A balanced beamsplitter is located at each of vertices A
and B with normal directions in parallel with the x axis. The four perfectly reflective mirrors at
C1,2 and D1,2 are oriented such that the laser beams from A are reflected directly toward B. We
assume all the perfect mirrors to have an amplitude reflectivity of r = 1, and both beamsplitters to√have an amplitude reflectivity of −1/ 2 for the laser beams that travel from the +x side (traveling√toward the −x direction), and an amplitude reflectivity of +1/ 2 for the laser beams that travelfrom the −x side. The edge lengths are assumed to be an integer multiple of the laser wavelengthin absence of laser noise, mirror motion, and GWs.
Likewise, four MZ interferometers can be constructed in a configuration as shown in Fig. F-5.
At the zero point, the output ports ( outA1 ,A2 ,B1 ,B2 ) are kept dark, while each input port is also the(t)bright port for another interferometer. During operations, for each MZ, I = A1 , A2 , B1 , B2 , if φI(r)and φI

represent the additional phase shifts gained by the beams transmitted and reflected fromits first beamsplitter, respectively, then the output optical amplitude is proportional to
(t)

(r)

(t)

(r)

eiφI − eiφI ∝ φI − φI ≡ δφI .

(F.20)For interferometers A1 and A2 , the first beamsplitter means A, while for B1 and B2 , it means B. In(r)

Eq. (F.20), we have a minus sign in front of eiφI , because light always incident from the +x side√of the beamsplitters with the first reflection always encounters a −1/ 2 reflectivity.
204

z
eZ
eY

θ

eX
y
ϕ
x
Figure F-6: The orthonormal system (eX , eY , eZ ) used to describe a generic plane GW. eZ is the
propagation direction.

These additional phase shifts can arise from laser noise, displacement noise, and GWs. Because
we only consider linear order in GWs and the noises, we can first include only effects of laser anddisplacement noises, construct a combination from the outputs of the four MZ interferometers thatis free from these noises, and then calculate its response to GWs. For dark-port detection, each MZ
is already free from laser noise, and therefore, we only need to consider displacement noise. For amirror with normal direction n and incident wavevector k, the phase shift gained by the reflected
light when the mirror moves spatially by δx is 2(n · k)(n · δx). For A1 and B1 interferometers, wehave
φA1 (t)(t)=

φA1 (t)(r)=

φB1 (t)

(t)

=

(r)

=

φB1 (t)√2ω
[ηC1 (t − 2L/c) + xB (t)] ,c√2ω
[ξD1 (t − 2L/c) − xA (t − 4L/c)] ,√c
2ω
[ηC1 (t − 2L/c) + xA (t)] ,c√2ω
[ξD1 (t − 2L/c) − xB (t − 4L/c)] ,c(F.21)(F.22)(F.23)(F.24)

where ω is the laser frequency, c is the speed of light in vacuum, and ηC1 is the motion of C1 alongthe η axis and so on. Thus, we haveδφA1 − δφB1

=
> =
>
(t)
(r)
(t)
(r)
φA 1 − φA 1 − φB1 − φB1√2ω
=[xB (t) − xA (t) − xB (t − 4L/c) + xA (t − 4L/c)] .
c=(F.25)

Note that motions of C1 and D1 are already canceled in this subtraction because the two MZs sense
their motions equally, due to the fact that |AD1 | = |BD1 | = |AC1 | = |BC1 | = 2L. Similarly, the205combination of the other two MZs is given by
δφA2 − δφB2 =√2ω[xB (t) − xA (t) − xB (t − 4L/c) + xA (t − 4L/c)] .
c(F.26)

Consequently, the total combination
φDFI ≡ [δφA1 − δφB1 ] − [δφA2 − δφB2 ](F.27)

is free from any displacement noise. This is also anticipated because A1 and A2 sense the beamsplitters in the same way, and so do B1 and B2 .

F.4.3

Detector Response to Gravitational Waves

The response of φDFI to GWs is calculated in this section. For a particular case with a plane GW
coming directly along the z axis such that eZ = ez in Eq. (1.5) andhTT (t, x) = h(t − z/c) [eξ ⊗ eξ − eη ⊗ eη ] ,(F.28)

it is relatively simple to argue, based on the symmetry of the octahedron and the scheme of combining
the output signals from the interferometers, that the beams in all four branches or those involving
C1 , D1 , C2 and D2 respectively experience equal GW contributions. For the branch involving D1 ,the response to GWs is given by
1
φGW
DFI
=
ωL/c
2

G
√
√
dζ h[t + (2Lζ − 2L(1 − ζ))/c] + h[t + (2L(1 + ζ) + 2Lζ)/c]0H√√−h[t + (2Lζ + 2L(1 − ζ))/c] − h[t + (2L(1 + ζ) − 2Lζ)/c] .

) 1(F.29)In the frequency domain defined in Eq. (1.60), Eq. (F.29) becomes
√
>
>H
√
√
√ =
√ =
iω h̃e−i 2ΩL/c G
GWφ̃DFI (Ω) =(2 − 2) 1 − e(4+2 2)iΩL/c + (2 + 2) e4iΩL/c − e2 2iΩL/c , (F.30)

4Ω

GW
where φ̃GW
DFI and h̃ are the Fourier transforms of φDFI and h., and Ω is the GW frequency. Eq. (F.30)

shows a non-vanishing response to GWs.
For GWs with generic propagation directions and polarizations, the detector response to GWs
at low frequencies goes like ∼ f 2 with
=

>
φ̃GW
DFI ΩL
c

,1

=

√ '
(2 '
(
>
4 2 ΩLωL =h̃× (1 + cos2 θ) cos 2ϕ + 2h̃+ cos θ sin 2ϕ ,3cc(F.31)

where θ and ϕ are shown in Fig. F-6. For general frequencies and generic incoming GWs, the
206

1

10

0

|φGW
/(h ω L/c)|
DFI

10

−1

10

−2

10

−3

10

−4

10 −2
10

−1

10

0

10
2Ω L/(π c)

1

10

2

10

Figure F-7: The response of the DFI detector to GWs. It is peaked when 2ΩL/(πc) ∼ 1 and decaysas ∼ f 2 at low frequencies.
analytical formula for the transfer function is complicated, and therefore, not presented here.

F.4.4

Practicability of 3-D DFI Detectors

Without displacement noise, ideal DFI configurations are shot-noise-limited at all frequencies, andtherefore, the detector sensitivity can be improved by simply turning up the laser power. At frequencies f > fc ≡ c/L, the 3-D configuration has comparable shot-noise-limited sensitivity toconventional GW detector configurations with similar arm lengths and optical power, but without
optical cavities, at frequencies f < fc , the shot-noise-limited sensitivity degrades as (f /fc )2 .
In practice, the effectiveness of displacement noise cancellation is limited by experimental imperfections. For example, the following requirements must be satisfied to optimize the performanceof DFI: (1) all interferometer arms have equal lengths, (2) beamsplitters are exactly 50/50, (3) mirrors are lossless, (4) beams incident on each mirror must all hit the same point. If a DFI detector
does not perfectly satisfy these requirements, it limits the displacement-noise reduction factor (inamplitude) at max(δL/L, |RBS − TBS |, +), where δL is the length accuracy, RBS and TBS are thepower reflectivity and transmissivity of the beamsplitters, and + is the optical loss in the mirrors (inpower). It is reasonable to expect a 100 to 1000-fold reduction in displacement noise.
207F.4.5

Utility of Time-Delay Devices in DFI Detectors

Although the DFI detector can evade displacement noise, it requires an arm length to be comparable
to the wavelength of GWs for detection since the detector response decays as f 2 at low frequencies
up to the cutoff frequency
fcutoff =c.
2L(F.32)

The f 2 response comes from the unique nature of the DFI detector with folded interferometer arms.
To decrease the f 2 cutoff frequency, the DFI detector requires a very long arm, which is also arequirement for Michelson-type GW detectors, but to detect an appreciable GW signal against
background displacement noise.
To circumvent this problem in DFI detectors, time-delay devices that delay the propagation oflight or store light for some period of time, such as optical fibers, multiple cavities, and electromagnetically induced transparency (EIT) media, can be used in a short-arm DFI detector to achievesensitivity similar to that of a DFI detector with long arms. It has been found that a DFI detector
with a time-delay device at each mirror (C1 , C2 , D1 , and D2 ) can increase the sensitivity of a DFI
detector by a factor of f at frequencies between the cutoff frequency fcutoff and 1/τ , where τ is thelight delay/storage time. Much of the derivation can be found in Ref. [64].

F.5

Quantitative Phase Imaging by Use of Stabilized Optical
Interferometry

F.5.1

Introduction

Optical microscopy has been the most commonly used method of investigation in medicine andbiology. Various related technologies have been developed over the past years [250]. Numerous
biological samples, including live cells, are quite transparent at visible wavelengths and behave essentially as phase objects. Techniques such as phase contrast and Nomarski microscopy provide
contrast of nearly invisible samples by transforming the phase information into the intensity distribution, thus revealing morphometric features of biological systems without the need for exogenous
contract agents [251]. However, the information obtained with these techniques about the phaseshift is only qualitative. Retrieving quantitative phase information from transparent objects withhigh accuracy and low noise allows for novel applications in the biological investigation of structureand dynamics [252]. Both interferometric [253, 254, 255] and non-interferometric [256] techniques
have been proposed for quantitative phase imaging of biological samples.
Recently, a stabilized Hilbert phase microscope as a quantitative phase imager with sub-nanometer208

phase sensitivity and millisecond time resolution has been developed [255]. Its long-term stability
over extended periods of time allows for investigating the dynamics of biological systems on timescales from seconds to a cell lifetime. Its high time resolution also enables investigation of many
processes that take place at the cellular level at kHz frequencies, including cytoskeletal dynamics[257], cell membrane fluctuations [258], and neural activity [259]. In the next section, the stabilized
Hilbert phase microscope is introduced, and then, in the last section, measurement of cell membranetension using the microscope is carried out. The details of the microscope can be found in Ref. [255].

F.5.2

Stabilized Hilbert Phase Microscopy

A schematic of the Hilbert phase microscope is shown in Fig. F-8. The laser source is a He-Ne laser
at 632.8 nm. The laser is collimated and divided into the sample and reference arms to form a MachZehnder interferometer. The sample beam provides an illumination field for an inverted microscope
equipped with an immersion 100× objective with an NA of 1.2 that allows for a transverse resolutionof 0.26 µm. The tube lens is positioned such that the image of the sample is formed at the CCD
plane via the beamsplitter cube. The reference beam is collimated and expanded by a telescope thatconsists of another microscope objective and recombined with the sample beam at the beamsplitter.
At the CCD plane, the reference beam can be approximated by a plane wave, which interferes withthe sample beam that contains the information of the sample.
The reference beam is purposely misaligned with respect to the sample beam such that uniform
fringes are created at an angle of 45◦ with respect to the x and y axes in the CCD detection plane,such that for a given sample, the spatially varying irradiance at the image (CCD) plane across either
the x or y axis has the form.
I(x) = IR + IS (x) + 2 IR IS (x) cos[qx + φ(x)],(F.33)

where IR and IS (x) are the reference and sample irradiance distributions respectively, q is thespatial frequency of the fringes, and φ(x) is the spatially varying phase associated with the object(the quantity of interest in this microscope). The CCD (Hamamatsu Photonics C7770) has an
acquisition rate of 291 frames/s with 640×480 pixels per image and the fringes are sampled by.
4-6 CCD pixels per period. High-pass filtering Eq. (F.33) gives I(x) = 2 IR IS (x) cos[qx + φ(x)],.
which is then demodulated at the spatial frequency q to obtain I(x) = 2 IR IS (x) cos φ(x). Hilberttransforming the cosine function using the Kramers-Kronig relation
1P
z(x) = I(x) + i
2
2π

)

209

"

I(x ) "dx
x − x"(F.34)

FiberFiber
SAMPLE ARM

Computer

LASER
SampleBeamsplitter
Fiber

Objective

Lens
REFERENCE
ARM

Lens
Objective
CCD

PZT
PinholeBeamsplitter
Grating

LO
PD
HV Adder
Amp

LP

Mixer

Pre
Amp

HP

Figure F-8: A schematic of the stabilized Hilbert phase microscope in a Mach-Zehnder interferometer
configuration that consists of a laser source, two balanced beamsplitters, two objectives, a sample,
a tube lens, a pinhole, a photodetector (PD), a CCD camera, and an amplitude grating. Theinterference fringe spatially filtered by the grating is detected by the PD, high-pass filtered, amplified,and demodulated by a local oscillator (LO) that modulates the optical phase in the reference arm,
yielding an error signal that is fed back to the interferometer via a piezo-electric transducer (PZT).
retrieves the phase information of the sample from a single-exposure image, given by
φ(x) = tan−1

Im[z(x)].
Re[z(x)](F.35)In order to suppress fluctuations in relative phase between the two arms, a feedback control
system is used to lock the interferometer on an interference fringe. A small mirror in front of the
CCD deflects a portion of the interfering beams before they reach the CCD. At a plane conjugate to
the image (CCD) plane, an amplitude grating with the same period as the interferogram is placed.
Due to this spatial matching, each diffraction order n produced by the sample beam overlaps withthe order n − 1 of the reference beam. The interfering beams are spatially isolated by an apertureand detected by a photodetector. The interference fringe is then detected by the PD, high-passfiltered, amplified, and demodulated by a local oscillator (LO) that modulates the optical phase in
the reference arm, yielding an error signal that is fed back to the interferometer via a piezo-electrictransducer (PZT). The cutoff frequency of the high-pass filter is 1 kHz, the LO modulation frequency
is 15 kHz, and the cutoff frequency of the low-pass filter is 300 Hz.
The long-term stability of the microscope over extended periods of time due to the feedback
control system allows for investigating the dynamics of biological systems on time scales from seconds
210

to a cell lifetime. Its high time resolution due to the high sampling rate of the CCD also enables
investigation of many processes that take place at the cellular level at kHz frequencies.

F.5.3

Measurements of Cell Membrane Tension

The stabilized Hilbert phase microscope allows for transferring the information encoded in the phaseof the imaging field into the intensity distribution of the final image, and therefore, the optical phaseshift through a given sample can be regarded as a powerful endogenous contrast agent since it
contains information about both the thickness and refractive index of the sample. In this respect,mature erythrocytes or red blood cells (RBCs) represent a very particular type of structure in thatthey lack nuclei and major organelles, and hence, they can be modeled as optically homogeneous
objects and are useful for studying cell membranes, which have broad applications in both scienceand technology [260, 261]. The RBC has a composite membrane that consists of a lipid bilayer
coupled to a two-dimensional spectrin network, which grants the cell its characteristic properties ofboth softness and strong shear elasticity [262, 263].
The membrane tension of RBCs was measured by the stabilized Hilbert phase microscope, analyzed and compared with a theoretical model. The result is described in Ref. [255], and therefore,
will not be discussed any further here.

211

212

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