Quantum Statistics of Linear and Nonlinear Optical Phenomena

1. Introduction.- 2. Quantum Theory of the Electromagnetic Field.- 2.1 Quantum description of the field.- 2.2 Statistical states.- 2.3 Multimode description.- 2.4 Calculation of commutators of the field operators.- 2.5 Time development of quantum states.- 3. Optical Correlation Phenomena.- 3.1 Definition of quantum correlation functions.- 3.2 Properties of quantum correlation functions.- 3.2.1 Analytic properties.- 3.2.2 Spectral properties.- 3.2.3 Wave equations in vacuo.- 3.2.4 Symmetries and inequalities.- 3.2.5 Examples of the second-order degrees of coherence.- 3.3 Quantum coherence.- 3.3.1 Second-order phenomena.- 3.3.2 Higher-order phenomena.- 3.4 Measurements corresponding to antinormally ordered products of field operators — quantum counters.- 3.5 Quantum characteristic functionals.- 3.6 Measurements of mixed-order correlation functions.- 3.7 Photocount distribution and photocount statistics.- 3.8 Determination of the integrated intensity probability distribution from the photocount distribution.- 3.9 Short-time measurements.- 3.10 Bunching and antibunching of photons.- 3.11 Hanbury Brown — Twiss effect — correlation interferometry and correlation spectroscopy.- 4. Coherent-State Description of the Electromagnetic Field.- 4.1 Coherent states of a harmonic oscillator and of the electromagnetic field.- 4.1.1 Definitions.- 4.1.2 Expansions in terms of coherent states.- 4.1.3 Minimum-uncertainty wave packets.- 4.1.4 Properties of the displacement operator $$\hat{D}(\alpha)$$.- 4.1.5 Expectation values of operators in coherent states.- 4.1.6 Generalized coherent states.- 4.1.7 Multimode description.- 4.1.8 Time development of the coherent states.- 4.1.9 Even and odd coherent states.- 4.2 Glauber — Sudarshan representation of the density matrix.- 4.3 The existence of the Glauber — Sudarshan representation.- 4.4 The phase operators.- 4.5 Multimode description.- 4.6 Relation between the quantum and classical descriptions.- 4.6.1 Quantum and classical correlation functions.- 4.6.2 Photon-number and photocount distributions.- 4.7 Stationary conditions for the field.- 4.7.1 Time invariance properties of the correlation functions.- 4.7.2 Stationary conditions in phase space.- 4.8 Ordering of field operators.- 4.8.1 ?- and s-ordering and general decompositions.- 4.8.2 Connecting relations.- 4.8.3 Multimode description.- 4.9 Interference of independent light beams.- 4.10 Two-photon coherent states, atomic coherent states and coherent states for general potentials.- 4.10.1 Two-photon coherent states.- 4.10.2 Atomic coherent states.- 5. Special States of the Electromagnetic Field.- 5.1 Chaotic (Gaussian) light.- 5.1.1 Distributions and characteristic functions.- 5.1.2 The second-order correlation function for blackbody radiation.- 5.1.3 Photocount statistics.- 5.2 Laser radiation.- 5.2.1 Ideal laser model.- 5.2.2 Real laser model.- 5.3 Superposition of coherent and chaotic fields.- 5.3.1 One-mode field.- 5.3.2 Multimode field-characteristic generating function.- 5.3.3 Integrated intensity probability distribution.- 5.3.4 The photocount distribution.- 5.3.5 Factorial moments.- 5.3.6 Factorial cumulants.- 5.3.7 Accuracy of approximate M-mode formulae.- 6. Review of Nonlinear Optical Phenomena.- 6.1 General classical description.- 6.2 The second-order phenomena.- 6.3 The third- and higher-order phenomena.- 6.4 Transient coherent optical effects.- 6.4.1 Self-induced transparency.- 6.4.2 Photon echo.- 6.4.3 Superradiance.- 7. Heisenberg — Langevin and Master Equations Approaches to the Statistical Properties of Radiation Interacting With Matter.- 7.1 The Heisenberg — Langevin approach.- 7.2 The master equation and generalized Fokker — Planck equation approaches.- 7.3 The interaction of radiation with the atomic system of a nonlinear medium.- 8. Quantum Statistics of Radiation in Random Media.- 8.1 Phenomenological description of propagation of radiation through turbulent atmosphere and Gaussian media.- 8.2 The hamiltonian for radiation interacting with a random medium.- 8.3 Heisenberg—Langevin equations and the generalized Fokker — Planck equation.- 8.4 Solutions of the generalized Fokker — Planck equation and the Heisenberg — Langevin equations.- 8.5 Photocount statistics.- 8.6 Diament—Teich and Tatarskii descriptions.- 8.7 Comparison of the quantum and phenomenological descriptions.- 8.8 Speckle phenomenon.- 9. Quantum Statistics of Radiation in Nonlinear Media.- 9.1 Optical parametric processes with classical pumping.- 9.1.1 Degenerate case.- 9.1.2 Non-degenerate case.- 9.2 Interaction of three one-mode boson quantum fields.- 9.3 Second and higher harmonic and subharmonic generation.- 9.4 Raman, Brillouin and hyper-Raman scattering.- 9.4.1 Reservoir phonon system.- 9.4.2 Dynamics of photon and phonon modes.- 9.4.3 Completely quantum description.- 9.4.4 Hyper-Raman scattering.- 9.5 Multiphoton absorption.- 9.6 Multiphoton emission.- 9.7 Resonance fluorescence.- 9.8 Other interesting nonlinear phenomena.- 9.8.1 Coherent ?-emission by stimulated annihilation of electron — positron pairs.- 9.8.2 A solvable model for light scattering.- 9.9 Phase-transition analogies.- 10. Conclusions.- References.