D.F.Walls, G.J.MilburnQUANTUM OPTICS

ContentsThis text book originated out of a graduate course of lectures in Quantum Optics

given at the University of Waikato and the University of Auckland. A broad range ofmaterial is covered in this book ranging from introductory concepts to current researchtopics. A pedagogic description of the techniques of quantum optics and theirapplications to physical systems is presented. Particular emphasis is given to systemswhere the theoretical predictions have been confirmed by experimental observation.

The material presented in this text could be covered in a two semester course.Alternatively the introductory material in Chaps. 1-6 and selected topics from the laterchapters would be suitable for a one semester course. For example, for materialinvolving the interaction of light with atoms Chaps. 10-13 would be appropriate,whereas for material on squeezed light Chaps. 7 and 8 are required. Chaps. 14-16describe the interrelation of fundamental topics in quantum mechanics with quantumoptics. The final chapter on atomic optics gives an introduction to this new and rapidlydeveloping field.

Contents1. Introduction 12. Quantisation of the Electromagnetic Field 7

2.1 Field Quantisation 72.2 Fock or Number States 102.3 Coherent States 122.4 Squeezed States 152.5 Two-Photon Coherent States 182.6 Variance in the Electric Field 202.7 Multimode Squeezed States 222.8 Phase Properties of the Field 23Exercises 26

3. Coherence Properties of the Electromagnetic Field 293.1 Field-Correlation Functions 293.2 Properties of the Correlation Functions 313.3 Correlation Functions and Optical Coherence 323.4 First-Order Optical Coherence 343.5 Coherent Field 383.6 Photon Correlation Measurements 393.7 Quantum Mechanical Fields 41

3.7.1 Squeezed States 423.7.2 Squeezed Vacuum 44

3.8 Phase-Dependent Correlation Functions 443.9 Photon Counting Measurements 46

3.9.1 Classical Theory 46

3.9.2 Constant Intensity 483.9.3 Fluctuating Intensity - Short-Time Limit 48

3.10 Quantum Mechanical Photon Count Distribution 513.10.1 Coherent Light 523.10.2 Chaotic Light 523.10.3 Photo-Electron Current Fluctuations 53

Exercises 554. Representations of the Electromagnetic Field 57

4.1 Expansion in Number States 574.2 Expansion in Coherent States 58

4.2.1 P Representation 58a) Correlation Functions 60b) Covariance Matrix 61c) Characteristic Function 62

4.2.2 Wigner's Phase-Space Density 63a) Coherent State 64b) Squeezed State 64c) Number State 64

4.2.3 Q Function 654.2.4 P Representation 664.2.5 Generalized P Representations 68

a) Number State 71b) Squeezed State 71

4.2.6 Positive P Representation 71Exercises 72

5. Quantum Phenomena in Simple Systems in Nonlinear Optics 735.1 Single-Mode Quantum Statistics 73

5.1.1 Degenerate Parametric Amplifier 735.1.2 Photon Statistics 755.1.3 Wigner Function 75

5.2 Two-Mode Quantum Correlations 775.2.1 Non-degenerate Parametric Amplifier 775.2.2 Squeezing 805.2.3 Quadrature Correlations and the Einstein- Podolsky-Rosen Paradox 815.2.4 Wigner Function 825.2.5 Reduced Density Operator 83

5.3 Quantum Limits to Amplification 855.4 Amplitude Squeezed State with Poisson Photon Number Statistics 86Problems 89

6. Stochastic Methods 916.1 Master Equation 916.2 Equivalent c-Number Equations 98

6.2.1 Photon Number Representation 986.2.2 P Representation 996.2.3 Properties of Fokker-Planck Equations 1016.2.4 Steady State Solutions - Potential Conditions 1016.2.5 Time Dependent Solution 1036.2.6 Q Representation 1046.2.7 Wigner Function 1066.2.8 Generalized P Represention 108

a) Complex P Representation 109b) Positive P Representation 110

6.3 Stochastic Differential Equations 1116.3.1 Use of the Positive P Representation 115

6.4 Linear Processes with Constant Diffusion 1156.5 Two Time Correlation Functions in Quantum Markov Processes 117

6.5.1 Quantum Regression Theorem 1186.6 Application to Systems with a P Representation 118Exercises 119

7. Input-Output Formulation of Optical Cavities 1217.1 Cavity Modes 1217.2 Linear Systems 1247.3 Two-Sided Cavity 1267.4 Two Time Correlation Functions 1277.5 Spectrum of Squeezing 1297.6 Parametric Oscillator 1297.7 Squeezing in the Total Field 1327.8 Fokker-Planck Equation 132Exercises 135

8. Generation and Applications of Squeezed Light 1378.1 Parametric Oscillation and Second Harmonic Generation 137

8.1.1 Semi-classical Steady States and Stability Analysis 1398.1.2 Parametric Oscillation 1398.1.3 Second Harmonic Generation 1408.1.4 Squeezing Spectrum 1418.1.5 Parametric Oscillation 1428.1.6 Experiments 143

8.2 Twin Beam Generation and Intensity Correlations 1468.2.1 Second Harmonic Generation 1508.2.2 Experiments 1528.2.3 Dispersive Optical Bistability 153

8.3 Applications of Squeezed Light 1588.3.1 Interferometric Detection of Gravitational Radiation 1588.3.2 Sub-Shot-Noise Phase Measurements 172

Exercises 1749. Nonlinear Quantum Dissipative Systems 177

9.1 Optical Parametric Oscillator: Complex P Function 1779.2 Optical Parametric Oscillator: Positive P Function 1829.3 Quantum Tunnelling Time 1869.4 Dispersive Optical Bistability 1919.5 Comment on the Use of the Q and Wigner Representations 193Exercises 1939.A Appendix 194

9.A.1 Evaluation of Moments for the Complex P function for ParametricOscillation (9.17)

194

9.A.2 Evaluation of the Moments for the Complex P Function for OpticalBistability (9.48)

195

10. Interaction of Radiation with Atoms 19710.1 Quantization of the Electron Wave Field 19710.2 Interaction Between the Radiation Field and the Electron Wave Field 19910.3 Interaction of a Two-Level Atom with a Single Mode Field 20410.4 Quantum Collapses and Revivals 20510.5 Spontaneous Decay of a Two-Level Atom 20610.6 Decay of a Two-Level Atom in a Squeezed Vacuum 20810.7 Phase Decay in a Two-Level System 210Exercises 211

11. Resonance Fluorescence 21311.1 Master Equation 21311.2 Spectrum of the Fluorescent Light 21711.3 Photon Correlations 22111.4 Squeezing Spectrum 225Exercises 228

12. Quantum Theory of the Laser 22912.1 Master Equation 22912.2 Photon Statistics 232

12.2.1 Spectrum of Intensity Fluctuations 23312.3 Laser Linewidth 23512.4 Regularly Pumped Laser 23612.A Appendix: Derivation of the Single-Atom Increment 240Exercises 244

13. Intracavity Atomic Systems 24513.1 Optical Bistability 24513.2 Nondegenerate Four Wave Mixing 25213.3 Experimental Results 258Exercises 259

14. Bells Inequalities in- Quantum Optics 261

14.1 The Einstein-Podolsky-Rosen (EPR) Argument 26114.2 Bell Inequalities and the Aspect Experiment 26214.3 Violations of Bell's Inequalities Using a Parametric Amplifier Source 26814.4 One-Photon Interference 273Exercises 279

15. Quantum Nondemolition Measurements 28115.1 Concept of a QND measurement 28215.2 Back Action Evasion 28415.3 Criteria for a QND Measurement 28415.4 The Beam Splitter 28715.5 Ideal Quadrature QND Measurements 29015.6 Experimental Realisation 29115.7 A Photon Number QND Scheme 294Exercises 296

16. Quantum Coherence and Measurement Theory 29716.1 Quantum Coherence 29716.2 The Effect of Fluctuations 30316.3 Quantum Measurement Theory 30616.4 Examples of Pointer Observables 31016.5 Model of a Measurement 311Exercises 313

17. Atomic Optics 31517.1 Young's Interference with Path Detectors 316

17.1.1 The Feynman Light Microscope 31917.2 Atomic Diffraction by a Standing Light Wave 32117.3 Optical Stern-Gerlach Effect 32517.4 Quantum Non-Demolition Measurement of the Photon Number by Atomic

Beam Deflection330

17.5 Measurement of Atomic Position 33417.5.1 Atomic Focussing and Contractive States 337

Exercises 33917.A Appendix 339

References 341Subject Index 347

Subject Indexadiabatic elimination 179, 249, 254antibunching, photon 2

laser 3resonance fluorescence 221second-order correlation function 42sub-Poissonian statistics 42

atomic decay rate, transverse 246atomic operators, collective 247atomic optics 5back action evasion 284Bargmann state 99beam splitter 287

Bell inequality 4, 265CHSH form 266parametric down conversion 268

bistability, dispersive (linear) 153squeezing spectrum 156

bistability, dispersive (nonlinear) 191Fokker-Planck equation 191

bistability, optical 245atomic model 245noise correlation 249

Brownian motion 112bunching, photon 2

second-order correlation function 40,42

sub-Poissonian statistics 42Cauchy-Schwarz inequality 32, 78cavity, boundary condition 121cavity, laser 229cavity, two-sided 126chaotic field, see thermal fieldcharacteristic function 62

optical bistability 247P representation 62Q representation 65quantum coherence 303Wigner representation 63, 106

CHSH inequality 266Clauser-Horne inequality 269coherence, first order optical 34

Young's interference experiment 35coherence function 298coherence, optical 29coherence, quantum 303

visibility 303coherent field 38coherent state 12, 108

completeness 14displacement operator 12number state expansion 12Poissonian statistics 13, 58second-order correlation function 42,

58

coherent state, two photon 18coincidence probability 275collisional broadening limit 246complementarity 5, 261, 316contractive state 282, 337correlated state 261correlated state, polarization 263correlation function 31

general inequalities 31phase information 44

correlation function, first order 30correlation function, higher order 30correlation function, phase dependent

44correlation function, photon number 222correlation function, QND 285correlation function, second-order 39

antibunching 42bunching 40coherent state 42number state 42resonance fluorescence 222thermal field 58

correlation function, two-time 53Brownian motion 113input-output formalism 127master equation 91Ornstein-Uhlenbeck process 116parametric oscillator 133phase diffusion 236photon counting 53regularly pumped laser 237resonance fluorescence 217, 225two-level atom 207

correlation, intensity 39Hanbury-Brown Twiss experiment 1,

30, 39parametric amplifier 75

covariance matrix 61covariance matrix, stationary 116density operator, reduced 92detailed balance 98

harmonic oscillator 98laser 232

duality, wave-particle 315efficiency, quantum 52Einstein-Podolsky-Rosen paradox, see

EPR paradoxelectric dipole approximation 200,

245electromagnetic field 7

commutation relations 10Hamiltonian 10quantisation 7vacuum state 10

entangled state 275,318EPR paradox 4, 261

parametric amplifier 81error ellipse 16

coherent state 16Wigner function 63

Fock state, see number stateFokker-Planck equation 101

degenerate parametric oscillator 178dispersive bistability 155, 191Green's function 103potential condition 101stochastic differential equation 111

four-wave mixing 3, 245, 252degenerate operation 257noise correlation 254phase matching 253QND 294side-band modes 253squeezing spectrum 256

gain, QND 290, 295gravitational radiation 281Hanbury-Brown Twiss experiment 1,

30, 37harmonic oscillator 11

electromagnetic field 11harmonic oscillator, damped 115harmonic oscillator, master equation 93

Fokker-Planck equation 100, 110

transition probabilities 98heat bath, see reservoirhidden variable theory 264homodyne detection 45

four-wave mixing 255parametric oscillator 132

input-output formalism 121boundary condition 124laser fluctuations 233parametric oscillator 129photon counting distribution 51squeezing spectrum 129two-time correlation function 127

intensity fluctuations, laser 234, 239interaction picture 201interferometer, gravity wave 158

semiclassical behaviour 162signal variance 166total noise 167

interferometer, Mach-Zehnder 172interferometer, polarization 174interferometry, matter 5, 315, 327interferometry, optical 1

squeezed light 3Jaynes Cummings model 204Kapitza-Dirac regime 323Kerr effect 87, 191KTP crystal 291Langevin equation 112

Brownian motion 112constant diffusion process 115damped harmonic oscillator 113four-wave mixing 254input-output formalism 124

laser 2, 229diode 229gain 232phase diffusion 235Scully-Lamb form 231sub-Poissonian statistics 3threshold 232

laser cooling 5, 315

laser, fluctuations 229intensity 234photon number 233, 238pump 236sub-shot noise 229

laser, quantum theory 229laser, regularly pumped 239laser, semiconductor 236, 240lens, atomic 337linewidth, laser 236Liouvillian operator 117Markov approximation, first 95, 123master equation 91

gravitational wave interferometer161, 165

optical bistability 155, 246parametric oscillator 138resonance fluorescence 213Scully-Lamb laser 231two-level atom 206two-time correlation function 117

Maxwell's equations 8measurement limit, strong 313measurement theory, quantum 306micro-cavity 210microscope, Feynman light 315minimum uncertainty state 16

parametric oscillator 145squeezed state 15

mixture, classical 297noise spectrum 116normal ordering 51, 54

P representation 118number state 10, 57observable, QND 282optical Bloch equations 215

oscillation threshold 216optical tap 284, 294Ornstein-Uhlenbeek process 112

Brownian motion 112constant diffusion process 116damped harmonic oscillator 115

P representation 58, 61Bell inequality 266chaotic state 59coherent state 59harmonic oscillator 99normal ordering 118quadrature variance 61

P representation, complex 68coherent state 69dispersive bistability 192, 195harmonic oscillator 109number state 69parametric oscillator 132, 180, 194squeezed state 70

P representation, generalised 68, 138bistability 155harmonic oscillator 108parametric oscillator 138, 178

P representation, positive 71, 115Fokker-Planck equation 72harmonic oscillator 110optical bistability 247parametric oscillator 182twin beams 146

parametric amplifier, QND 291parametric amplifier, degenerate 73parametric amplifier, nondegenerate 77

selected state 84thermal reduced state 83two-mode squeezing 80

parametric down conversion 3, 73Bell inequality 268

parametric oscillator (linear) 3, 129, 137critical fluctuations 131Fokker-Planck equation 132, 134linearistation 129, 132, 139minimum uncertainty state 145output correlation matrix 133squeezing spectrum 129, 142threshold 129, 131

parametric oscillator (nonlinear) 177quadrature variance 183

semiclassical solution 180threshold distribution 184

Paul trap 225Pauli spin operators 202phase decay, collisionally induced 210,

246phase diffusion, laser 235

linewidth 236two-time correlation function 236

phase instability 147phase matching 253phase operator, Pegg-Barnett 25phase operator, Susskind-Glogower 23photo-detection 29, 39, 233photon counting 299photon counting, classical 46

depletion effects 50ergodic hypothesis 49generating function 48

photon counting, conditional 84parametric amplifier 84

photon counting distribution 52photon counting, nonclassical 70

number state 71squeezed state 71

photon counting, quantum mechanical51

photon number, QND 294Planck distribution 58pointer basis 308Poissonian pumping 230polarization rotator 291projection postulate, von Neumann 84pump depletion 180Q function 65

coherent state 66harmonic oscillator 104number state 66squeezed state 66

QND measurement 4,315,329,330pointer basis 310sub-Poissonian statistics 295

two photon transition 294QND measurement, ideal 290Q parameter 239quadrature phase measurement 44quadrature phase operator 16quantisation, electromagnetic field 7quantisation, electron field 197quantum nondemolition measurement,

see QND measurementquantum recurrance phenomena 205

Kerr effect 89quantum regression theorem 118

resonance fluorescence 221R representation 66Rabi frequency 204radiation pressure 159Raman-Nath regime 322, 334recoil energy 322refractive index, nonlinear 153relative states 309reservoir 91

collisional process 210spectrum 96spontaneous emission 214two-level atom 206

reservoir, squeezed 94, 208correlation function 208two-level atom 208

reservoir, thermal 96resonance fluorescence 2, 213

elastic scattering 219inelastic scattering 219quantum features 221spectrum 217rotating wave approximation 121,

202, 245Rydberg atom 205, 327Schrodinger cat 297second harmonic generation 137

critical behaviour 140, 150self pulsing 141squeezing spectrum 151

steady-state solution 140shot-noise limit 54

laser 235, 239signal-to-noise ratio 291, 295spectroscopy, photon correlation 40spontaneous emission 206

master equation 206Wigner-Weisskopf theory 207

squeezed state 15photon number distribution 21super-Poissonian statistics 43

squeezed state, multimode 22squeezed state, two-mode 22

parametric amplifier 80squeezing generator 23

squeezing 3, 137four-wave mixing 255input-output formalism 129nonlinear susceptibility 154parametric oscillator 129

squeezing, generator 16parametric amplifier 74, 80

squeezing spectrum 129dispersive bistability 156parametric oscillator 129, 142resonance fluorescence 225second harmonic generation 151twin beams 148

stability analysis 139standard quantum limit 169

gravity wave 281state preparation 285, 329statistics, photon 41

laser 232micromaser cavity 327regularly pumped laser 238

statistics, photon counting 53statistics, Poissonian 2, 13

coherent state 13, 58Kerr effect 86laser 232

statistics, sub-Poissonian 3

antibunching 42bunching 42laser 3, 236QND 295resonance fluorescence 225

statistics, super-Poissonian 42squeezed state 43

Stern-Gerlach effect, optical 32, 315stochastic differential equation 111

Fokker-Planck equation 111gravity wave interferometer 161, 165Ito equation 111Langevin equation 111optical bistability 248parametric oscillator 138

superposition, coherent 297superposition, macroscopic 4, 297superposition principle 297superposition state 177, 186susceptibility, nonlinear 154switching time 179thermal field 58

laser 232second-order correlation function 58

transition probability 98tunnelling time 177, 186twin beams 146

critical behaviour 146pulsed 150semiclassical steady-state 146squeezing spectrum 148

two-level atom 202master equation 206

uncertainty principle 16, 288, 313gravity waves 281

vector potential 8, 199visibility 32, 34, 318, 320

Bell inequality 271one-photon interference 276quantum coherence 303

wavelength, de Broglie 315Wigner function 63

coherent state 64covariance matrix 63harmonic oscillator 106number state 64parametric amplifier 75, 82squeezed state 64

Wigner-Weisskopf theory 207resonance fluorescence 213, 219

Young's interference experiment 1, 32coherent superposition 297correlation function 34experiments 37quantum explanation of 37visibility 34with atoms 315