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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

A derivation of optical field quantization from absorber theory

Alan V. Durrant, George William Series

Published Mar 12, 1980• 6 citations
Publisher

Abstract

Abstract A quantum absorber theory of optical field quantization is developed, in which only matter is quantized by postulate. The amplitudes of the optical field in an absorbing universe are expressed, by using Maxwell’s source equations, in terms of the atomic source operators of the surrounding absorber atoms treated as a single quantum system. Stationary twolevel absorbing atoms are assumed and only electric dipole fields are considered. The resulting optical field is found to be quantized with a mathematical description identical to that of the conventionally quantized optical field, but the physical interpretation is different.

DOI
10.1098/rspa.1980.0021

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Abstract A quantum absorber theory of optical field quantization is developed, in which only matter is quantized by postulate. The amplitudes of the optical field in an absorbing universe are expressed, by using Maxwell’s source equations, in terms of the atomic source operators of the surrounding absorber atoms treated as a single quantum system. Stationary twolevel absorbing atoms are assumed and only electric dipole fields are considered. The resulting optical field is found to be quantized with a mathematical description identical to that of the conventionally quantized optical field, but the physical interpretation is different.
An approach to the quantum theory of light is described using absorber theory and source field theory. It is assumed that all optical fields have atomic sources, and that the so-called free field is a source field of the surrounding absorbing material of the universe. The field operators are defined to be functions of the atomic operators and act on the states of the atomic sources. The source field of a completely enclosing perfect absorber is found to be mathematically equivalent to the quantised free field of conventional theory, but the physical basis of the theory is different.
Chapter 17 begins by approaching the general problem of a quantum system subject to a time-dependent perturbation. We then apply perturbation theory to the interaction of atomic matter with the radiation field and derive the most important radiative corrections. Atomic transitions – both induced and spontaneous – are discussed in detail, with some important applications. The formal introduction of field quantization allows us to make contact with quantum electrodynamics and quantum optics and discuss the Jaynes–Cummings model. Stochastic electrodynamics, in which the matter–field interaction is not a perturbation but is present ab initio, is used to explain atomic stability and its breakdown leading to radiative transitions. In the same spirit, the quantization of the field is physically justified by deriving the conventionally postulated field operators and their basic commutators. The chapter ends with a brief introduction to second quantization, an elegant and powerful formalism for describing and analyzing quantum many-body systems.
Abstract 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.
After giving an outline of the quantization scheme based on the microscopic Hopfield model of a dielectric bulk material, we show how the classical phenomenological Maxwell equations of the electromagnetic field in the presence of dielectric matter of given space- and frequency-dependent complex permittivity can be transferred to quantum theory. Including in the theory the interaction of the medium-assisted field with atomic systems, we present both the minimal-coupling Hamiltonian and the multipolar-coupling Hamiltonian in the Coulomb gauge. To illustrate the concept, we discuss the input--output relations of radiation and the transformation of radiation-field quantum states at absorbing four-port devices, and the spontaneous decay of an excited atom near the surface of an absorbing body and in a spherical micro-cavity with intrinsic material losses. Finally, we give an extension of the quantization scheme to other media such as amplifying media, magnetic media, and nonlinear media.
Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically. Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983). Quantum theory of particles and fields There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.
Are we to accept quantization as a fundamental property of nature, the origin of which does not require or admit further investigation? To get an insight into this question we consider atomic systems as open systems, since they are by necessity in contact with the electromagnetic radiation field. This includes not only photonic radiation, but, more importantly for our purposes, the random zero-point or nonthermal radiation that pervades the Universe. The Heisenberg inequalities, atomic stability and the existence of discrete solutions are explained as a result of the permanent action of this field upon matter and the balance between mean absorbed and emitted powers in the equilibrium regime. A detailed study carried out along the years has led to the usual quantum-mechanical formalism as a powerful and revealing statistical description of the behavior of matter in the radiationless approximation, as well as to the radiative corrections of nonrelativistic QED. The theory presented gives thus a response to the question posed above, within a local, realist and objective framework: quantization appears as an emergent phenomenon due to the matter-field interaction.