Methods in Theoretical Quantum Optics

This work presents the mathematical methods widely used by workers in the field of quantum optics. It deals with the physical assumptions which lead to the models and approximations employed, but the text is meant to give a firm grounding in those techniques needed to derive analytical solutions to problems. Based on teachings by the authors, most of the text has been tested on students.

1 Foundations

1

(13)

1.1 Introduction

1

(1)

1.2 Basic quantum theory

1

(8)

1.3 Atoms and fields

9

(5)

2 Coherent interactions

14

(20)

2.1 Introduction

14

(1)

2.2 Interaction pictures

14

(4)

2.3 Two-state dynamics

18

(5)

2.4 The Jaynes-Cummings model

23

(4)

2.5 The Bixon-Jortner quasi-continuum

27

(2)

2.6 Interactions between field modes

29

(5)

3 Operators and states

34

(53)

3.1 Introduction

34

(1)

3.2 Atom and field operators

34

(5)

3.3 Functions of operators and ordering theorems

39

(11)

3.4 Number states

50

(3)

3.5 Thermal states

53

(4)

3.6 Coherent states

57

(10)

3.7 Squeezed states

67

(13)

3.8 Angular momentum coherent states

80

(7)

4 Quantum statistics of fields

87

(43)

4.1 Introduction

87

(2)

4.2 Photon number statistics

89

(6)

4.3 Optical phase

95

(11)

4.4 Characteristic functions

106

(8)

4.5 Quasi-probability distributions

114

(11)

4.6 Statistics of multimode states

125

(5)

5 Dissipative processes

130

(52)

5.1 Introduction

130

(1)

5.2 Perturbation theory and Fermi's golden rule

130

(2)

5.3 Weisskopf-Wigner decay

132

(8)

5.4 Final value theorem and spectra

140

(5)

5.5 Damping in the Heisenberg picture

145

(16)

5.6 Master equations

161

(15)

5.7 Evolution of field statistics

176

(6)

6 Dressed states

182

(40)

6.1 Introduction

182

(1)

6.2 Two-state systems

182

(7)

6.3 Three-state systems

189

(7)

6.4 Dressing the Bixon-Jortner model

196

(4)

6.5 Dressing dissipative systems: Fano theory

200

(13)

6.6 Dressed operators

213

(9)

Appendix 1 Kronecker delta and the permutation symbol

222

(3)

Appendix 2 The Dirac delta function

225

(3)

Appendix 3 Special functions

228

(6)

Appendix 4 Quadrature eigenstates

234

(2)

Appendix 5 Operator ordering theorems

236

(4)

Appendix 6 The pole approximation

240

(3)

Appendix 7 Principal part integrals

243

(2)

Appendix 8 Contour integrals

245

(5)

Appendix 9 Laplace transforms and the final value theorem

250

(5)

Appendix 10 Operator ordering in the Heisenberg equations

255

(3)

Appendix 11 The method of characteristics for partial differential equations

258

(4)

Appendix 12 Transformation of master equations into partial differential equations

262

(4)

Appendix 13 Fokker-Planck equations

266

(10)

Appendix 14 Cubic equations

276

(1)

Selected bibliography

277

(2)

Index

279

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Methods in Theoretical Quantum Optics

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