Fluctuations, Dissipation, and Noise

General concepts in the theory of fluctuations and dissipation are reviewed and applied to examples in quantum optics. Brownian motion, Fokker-Planck and Langevin equations, and the Wiener-Khintchine theorem are reviewed, followed by a derivation and discussion of the fluctuation-dissipation theorem. The general problem of an oscillator coupled to a heat bath is revisited, as is the nonrelativistic theory of radiation reaction. The general ideas about fluctuations and dissipation developed in the first part of the chapter are then applied to the theory of the fundamental laser linewidth, the photon statistics of linear amplifiers and attenuators, the noise figure, amplified spontaneous emission, and the quantum theory of the beam slitter and homodyne detection.

Interaction Hamiltonian and Spontaneous Emission

The atom-field interaction is formulated within the fully quantized-field theory, starting from a detailed analysis of the transformation from the fundamental minimal coupling interaction Hamiltonian to the electric dipole Hamiltonian used extensively in quantum optics. Spontaneous emission, radiative level shifts, and the natural radiative lineshape are treated in both the Schrodinger and Heisenberg pictures, with emphasis on the roles of vacuum field fluctuations, radiation reaction, and the fluctuation-dissipation relation between them. The shortcomings of semiclassical radiation theories are discussed.

Elements of Classical Electrodynamics

This chapter reviews some topics in classical electrodynamics that are fundamental for modern quantum optics and that appear throughout the remaining chapters, includingelectric dipole radiation, electromagnetic energy, Abraham and Minkowski momenta in dielectric media, photon momentum, and Rayleigh scattering. Other foundational topics treatedare Earnshaw’s theorem, gauges and Lorentz transformations of fields, radiation reaction, the Ewald-Oseen extinction theorem, different forms of stress tensors in dielectric media, and the optical theorem.

Dipole Interactions and Fluctuation-Induced Forces

Concepts considered in earlier chapters, especially vacuum field fluctuations and zero-point energy, are applied to van der Waals, Casimir, and dipole-dipole resonance interactions, and to field quantization in dissipative dielectric media. Detailed but physically motivated calculations are presented regarding the Lifshitz theory, Casimir and Casimir-Polder forces, and the many-body theory of van der Waals and Casimir interactions based on dyadic Green functions. Expressions for quantized fields in dispersive and dissipative media are derived straightforwardly from Langevin noise theory, and it is shown how this approach is related to the more complicated Fano diagonalization method. Zero-point field energy in dissipative media and its role in Casimir and other effects is discussed in relation to other physical interpretations. Other topics discussed are (Forster) fluorescence resonance energy transfer and the modification of spontaneous emission rates by reflectors and host dielectric media.

Atoms and Light: Quantum Theory

Some of the most basic aspects of the interaction of atoms with light are considered, with emphasis on distinctly quantum-electrodynamical effects. Absorption and stimulated emission are associated with interference between incident and scattered fields. The JaynesCummings model, collapses and revivals, and dressed states are discussed along with related experimental studies in cavity quantum electrodynamics. Entangled states are associated with the interference of probability amplitudes for indistinguishable processes. The no-cloning theorem is reviewed. Von Neumann’s proof concerning hidden variable theories is examined and used to introduce Bell’s theorem and its proof. Resonance fluorescence spectra and photon anti-bunching correlations are calculated and compared with experiment. Photon polarization correlations in atomic cascades are calculated from the perspectives of both source fields and entanglement, and experimental studies of these correlations and Bell inequalities are reviewed.

Atoms in Light: Semiclassical Theory

The atom-field interaction is treated in semiclassical radiation theory, starting from the transformation from the minimal coupling Hamiltonian to the electric dipole form used extensively in quantum optics. The Heisenberg and density matrix approaches are developed and applied to two-state atoms, Bloch equations, Rabi oscillations, Maxwell-Bloch equations, and transition rates for absorption and emission. Einstein’s theory of blackbody radiation based on momentum fluctuations and dissipation is reviewed. The Einstein fluctuation formula is derived and used to introduce wave-particle duality, Hanbury Brown-Twiss correlations, and photon bunching.

Quantum Theory of the Electromagnetic Field

The electromagnetic field is quantized for free space and for dispersive dielectric media. Different quantum states of the field (thermal, laser, squeezed) are discussed and their photon-statistical properties derived. The density matrix operator for the field is introduced along with a detailed discussion of the diagonal representation of the field density operator and its properties. Field commutation relations, correlation functions, and uncertainty relations are derived and discussed. Also discussed are the historically important Bohr-Rosenfeld analysis of field commutation and uncertainty relations; energy-time uncertainty relations; simultaneous (Arthurs-Kelly) measurement of non-commuting observables; and complementarity from the perspective of probability amplitudes and probabilities for distinguishable and indistinguishable processes.