Measuring the Quantum State of Light

Quantum mechanics sets fundamental limits on the amount of information one can extract from a system with a single set of measurements. Recent results of new theoretical analyses and optical experiments have given rise to a more complete knowledge of the quantum properties of light. This book gives the first detailed description of this fascinating branch of quantum optics. This self-contained book begins with a description of some key results and tools from quantum optics. The author discusses the quantum mechanical description of simple optical instruments before giving a detailed treatment of quantum tomography. The book concludes with a chapter devoted to the problem of the simultaneous measurement of position and momentum. This book will appeal to graduate students and researchers in quantum optics, as well as anyone interested in the foundations of quantum mechanics or more general problems of quantum measurement.

1 Introduction

1

(16)

1.1 A note to the reader

1

(1)

1.2 Questions

2

(3)

1.3 Quantum states

5

(10)

1.3.1 Classical physics

6

(1)

1.3.2 Quantum mechanics

7

(1)

1.3.3 Axioms

8

(3)

1.3.4 General quantum states

11

(3)

1.3.5 Remarks

14

(1)

1.4 Further reading

15

(2)

2 Quantum theory of light

17

(20)

2.1 The electromagnetic oscillator

17

(3)

2.2 Single-mode states

20

(10)

2.2.1 Quadrature states

20

(1)

2.2.2 Fock states

21

(4)

2.2.3 Coherent states

25

(5)

2.3 Uncertainty and squeezing

30

(6)

2.4 Further reading

36

(1)

3 Quasiprobability distributions

37

(30)

3.1 Wigner representation

37

(17)

3.1.1 Wigner's formula

38

(3)

3.1.2 Basic properties

41

(4)

3.1.3 Examples

45

(9)

3.2 Other quasiprobability distributions

54

(5)

3.2.1 Q function

54

(2)

3.2.2 P function

56

(1)

3.2.3 s-parameterized quasiprobability distributions

57

(2)

3.3 Examples

59

(6)

3.4 Further reading

65

(2)

4 Simple optical instruments

67

(31)

4.1 Beam splitter

67

(16)

4.1.1 Heisenberg picture

67

(4)

4.1.2 Schrödinger picture

71

(6)

4.1.3 Fock representation and wave–particle dualism

77

(2)

4.1.4 Beam-splitter model of absorption

79

(4)

4.2 Homodyne detector

83

(14)

4.2.1 Photodetector

84

(1)

4.2.2 Balanced homodyne detection

85

(3)

4.2.3 Spatial–temporal modes

88

(6)

4.2.4 Inefficiencies in homodyne detection

94

(3)

4.3 Further reading

97

(1)

5 Quantum tomography

98

(46)

5.1 Phase-space tomography

98

(11)

5.1.1 Basics of tomography

101

(1)

5.1.2 Random phase and Abel transformation

102

(3)

5.1.3 Filtered back-projection algorithm

105

(2)

5.1.4 Quantum mechanics without probability amplitudes

107

(2)

5.2 Quantum-state sampling

109

(22)

5.2.1 Pattern functions and Hilbert transformation

110

(3)

5.2.2 Random phase and photon statistics

113

(4)

5.2.3 A theorem on the Schrödinger equation

117

(5)

5.2.4 Irregular wave functions

122

(4)

5.2.5 Numerical recipes

126

(2)

5.2.6 Quantum and classical tomography

128

(3)

5.3 How precisely can we measure quantum states?

131

(11)

5.3.1 Detection losses

132

(2)

5.3.2 Finite number of reference phases

134

(2)

5.3.3 Quadrature resolution

136

(2)

5.3.4 Statistical errors

138

(4)

5.4 Further reading

142

(2)

6 Simultaneous measurement of position and momentum

144

(28)

6.1 Prologue

144

(3)

6.1.1 An abstract Gedanken experiment

145

(2)

6.2 Quantum-optical scheme

147

(12)

6.2.1 Heisenberg picture

149

(3)

6.2.2 Phase-space density and squeezing

152

(3)

6.2.3 Optical interference and squared Wigner function

155

(4)

6.3 Quantum-optical phase

159

(11)

6.3.1 Canonical phase distribution

162

(5)

6.3.2 Measured phase distribution

167

(3)

6.4 Further reading

170

(2)

7 Summary

172

(3)

7.1 Acknowledgments

173

(2)

Appendix 1 Semiclassical approximation

175

(5)

Appendix 2 A theorem on the Schrödinger equation

180

(4)

Bibliography

184

(9)

Index

193

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Measuring the Quantum State of Light

Summary

Table of Contents

Supplemental Materials