Quantum Optics

by D. F. Walls and G. J. Milburn

Table of Contents

Figures (98 in all)
Preface
1. Introduction
2. Quantisation of the Electromagnetic Field

2.1 Field Quantisation
2.2 Fock or Number States
2.3 Coherent States
2.4 Squeezed States
2.5 Two-photon coherent states
2.6 Variance in the Electric Field
2.7 Multimode Squeezed States
2.8 Phase Properties of the Field
Exercises

3. Coherence Properties of the Electromagnetic Field

3.1 Field-Correlation Functions
3.2 Properties of the Correlation Functions
3.3 Corelation Functions and Optical Coherence
3.4 First-Order Optical Coherence
3.5 Coherent Field
3.6 Photon Correlation Measurements
3.7 Quantum Mechanical Fields

3.7.1 Squeezed States
3.7.2 Squeezed Vacuum

3.8 Phase-Dependent Correlation Functions
3.9 Photon Counting Measurements

3.9.1 Classical Theory
3.9.2 Constant Intensity
3.9.3 Fluctuating Intensity - Short-Time Limit

3.10 Quantum Mechanical Photon Count Distribution

3.10.1 Coherent Light
3.10.2 Chaotic Light
3.10.3 Photo-Electron Current Fluctuations

Exercises

4. Representations of the Electromagnetic Field

4.1 Expansion in Number States
4.2 Expansion in Coherent States

4.2.1 P Representation

(a) Correlation Functions
(b) Covariance Matrix
(c) Characteristic Function

4.2.2 Wigner's Phase-Space Density

(a) Coherent State
(b) Squeezed State
(c) Number State

4.2.3 Q Function
4.2.4 R Representation
4.2.5 Generalized P Representations

(a) Number State
(b) Squeezed State

4.2.6 Positive P Representation

Exercises

5. Quantum Phenomena in Simple Systems in Nonlinear Optics

5.1 Single-Mode Quantum Statistics

5.1.1 Degenerate Parametric Amplifier
5.1.2 Photon Statistics
5.1.3 Wigner Function

5.2 Two Mode Quantum Correlations

5.2.1 Non-degenerate Parametric Amplifier
5.2.2 Squeezing
5.2.3 Quadrature Correlations and the Einstein-Podolsky-Rosen Paradox
5.2.4 Wigner Function
5.2.5 Reduced Density Operator

5.3 Quantum Limits to Amplification
5.4 Amplitude Squeezed State with Poisson Photon Number Statistics
Exercises

6. Stochastic Methods

6.1 Master Equation
6.2 Equivalent c-number Equations

6.2.1 Photon Number Representation
6.2.2 P Representation
6.2.3 Properties of Fokker-Planck Equations
6.2.4 Steady State Solutions - Potential Conditions
6.2.5 Time Dependent Solution
6.2.6 Q Representation
6.2.7 Wigner Function
6.2.8 Generalized P Representation

(a) Complex P Representation
(b) Positive P Representation

6.3 Stochastic Differential Equations

6.3.1 Use of the Positive P Representation

6.4 Linear Processes with Constant Diffusion
6.5 Two Time Correlation Functions in Quantum Markov Processes

6.5.1 Quantum Regression Theorem

6.6 Application to Systems with a P Representation
6.7 Stochastic Unravellings
Exercises

7. Input-Output Formulation of Optical Cavities

7.1 Cavity Modes
7.2 Linear Systems
7.3 Two-Sided Cavity
7.4 Two Time Correlation Functions
7.5 Spectrum of Squeezing
7.6 Parametric Oscillator
7.7 Squeezing in the Total Field
7.8 Fokker-Planck Equation
Exercises

8. Generation and Applications of Squeezed Light

8.1 Parametric Oscillation and Second Harmonic Generation

8.1.1 Semi-Classical Steady States and Stability Analysis
8.1.2 Parametric Oscillation
8.1.3 Second Harmonic Generation
8.1.4 Squeezing Spectrum
8.1.5 Parametric Oscillation
8.1.6 Experiments

8.2 Twin Beam Generation and Intensity Correlations

8.2.1 Second Harmonic Generation
8.2.2 Experiments
8.2.3 Dispersive Optical Bistability

8.3 Applications of Squeezed Light

8.3.1 Interferometric Detection of Gravitational Radiation
8.3.2 Sub-Shot-Noise Phase Measurements
8.3.3 Quantum Information

Exercises

9. Nonlinear Quantum Dissipative Systems

9.1 Optical Parametric Oscillator : Complex P Function
9.2 Optical Parametric Oscillator : Positive P Function
9.3 Quantum Tunnelling Time
9.4 Dispersive Optical Bistability
9.5 Comment on the Use of the Q and Wigner Representations
Exercises
9.A Appendix

9.A.1 Evaluation of Moments for the Complex P Function for Parametric Oscillation (9.17)
9.A.2 Evaluation of Moments for the Complex P Function for Optical Bistability (9.48)

10. Interaction of Radiation with Atoms

10.1 Quantization of the Electron Wave Field
10.2 Interaction of a Two-Level Atom with a Single Mode Field
10.3 Spontaneous Emission from a Two-Level Atom
10.4 Phase Decay in a Two-Level System
10.5 Resonance Fluorescence
Exercises

11. Cavity QED

11.1 Cavity QEDn

11.1.1 Vacuum Rabi Splitting
11.1.2 Single Photon Sources
11.1.2 Cavity QED with N atoms

11.2 Circuit QED
Exercises

12. Quantum Theory of the Laser

12.1 Master Equation
12.2 Photon Statistics

12.2.1 Spectrum of Intensity Fluctuations

12.3 Laser Linewidth
12.4 Regularly Pumped Laser
12.A Appendix: Derivation of the Single Atom Increment
Exercises

13. Bells Inequalities in Quantum Optics

13.1 The Einstein-Podolsky-Rosen (EPR) Argument
13.2 Bell Inequalities and the Aspect Experiment
13.3 Violations of Bell's Inequalities Using a Parametric Amplifier Source
13.4 One-Photon Interference
Exercises

14. Quantum Nondemolition Measurements

14.1 Concept of a QND Measurement
14.2 Back Action Evasion
14.3 Criteria for a QND Measurement
14.4 The Beam Splitter
14.5 Ideal Quadrature QND Measurements
14.6 Experimental Realisation
14.7 A Photon Number QND Scheme
Exercises

15. Quantum Coherence and Measurement Theory

15.1 Quantum Coherence
15.2 The Effect of Dissipation

15.2.1 Experimental Observation of Coherence Decay

15.3 Quantum Measurement Theory
15.4 Examples of Pointer Observables
15.5 Model of a Measurement
15.6 Conditional States and Quantum Trajectories
Exercises

16. Quantum Information

16.1 Introduction

16.1.1 The Qubit
16.1.2 Entanglement

16.2 Quantum Key Distribution
16.3 Quantum Teleportation
16.4 Quantum Computation

16.4.1 Linear Optical Quantum Gates
16.4.2 Single Photon Sources

Exercises
References
Subject Index

17. Ion Traps

17.1 Introduction
17.2 Trapping and Cooling
17.3 Novel Quantum States
17.4 Trapping Multiple Ions
17.5 Ion trap Quantum Information Processing
Exercises
References
Subject Index

18.Light Forces

18.1 Radiative Forces in the Semiclassical LImit
18.2 Mean Force for a Two-Level Atom
18.3 Frictional Force
18.4 Dressed State Description
18.5 Atomic Diffraction by a Standing Wave
18.6 Optical Stern-Gerlach
18.7 Quantum Chaos
18.7 Decoherehce due to Spontaneous Emission
Exercises
References
Subject Index

19. Bose-Einstein Condensation.

19.1 Hamiltonian and Binary Collision model
19.2 Mean Field Theory
19.3 Single Mode Approximation
19.4 Quantum State of the Condensate
19.5 Quantum Phase Diffusion
19.6 Interference of Two BECs
19.7 Quantum Tunneling
19.8 Coherence Properties
Exercises
References
Subject Index

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