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COHERENT EFFECTS IN OPTICAL AND ATOMIC-MOLECULAR PROCESSES Change of Quantum Statistical Properties of Resonance Radiation during Its Scattering by Excited Systems

B. A. Veklenko

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Quantum electrodynamics is a very cumbersome theory, and one tries to avoid it in applied problems. Experience shows that the semiclassical theory of radiation, which operates with a nonquantized elec� tromagnetic field, provides, as a rule, required accu� racy of calculations. The authority of the semiclassical theory of radiation has grown greatly, because it describes adequately the operation of quantum devices such as lasers. In this work we present an example that demon� strates the inadequacy of the semiclassical theory of radiation. It is shown that, upon resonance scattering of an electromagnetic field by excited systems, quanti� zation of the electromagnetic field is necessary and the consequences of this operation manifest themselves at the macroscopic level. Suppose that resonance radiation is scattered by a certain system, the initial state of which is described in the interaction representation by the wave function . The total wave function of the electromagnetic field and scattering system after the scattering is denoted by . Expansion of the function in terms of the basis of the wave functions of the scatterer has the form

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Quantum electrodynamics is a very cumbersome theory, and one tries to avoid it in applied problems. Experience shows that the semiclassical theory of radiation, which operates with a nonquantized elec� tromagnetic field, provides, as a rule, required accu� racy of calculations. The authority of the semiclassical theory of radiation has grown greatly, because it describes adequately the operation of quantum devices such as lasers. In this work we present an example that demon� strates the inadequacy of the semiclassical theory of radiation. It is shown that, upon resonance scattering of an electromagnetic field by excited systems, quanti� zation of the electromagnetic field is necessary and the consequences of this operation manifest themselves at the macroscopic level. Suppose that resonance radiation is scattered by a certain system, the initial state of which is described in the interaction representation by the wave function . The total wave function of the electromagnetic field and scattering system after the scattering is denoted by . Expansion of the function in terms of the basis of the wave functions of the scatterer has the form
Scattering of a resonance electromagnetic field from excited atoms cannot be described by the semiclassical theory of radiation operating with nonquantized electromagnetic fields. Field quantization effects are manifested in this case on the macroscopic level and lead to evolution of statistical properties of radiation in the course of scattering. It is found that a combined process coupling elastic scattering from an atom and induced emission from the same atom, which cannot be studied by the methods of the standard perturbation theory, plays a significant role in this effect. The process of combined scattering in extended media exhibits coherent properties that cannot be described by the standard refractive index.
According to quantum electrodynamics, the cross section for resonant scattering of radiation on an aggregate of excited atoms can be written as a sum of positive definite terms. This type of structure is not consistent with the Fresnel formulas for the reflection coefficient of radiation from thermally excited media. The difference shows up on a macroscopic level and indicates that semiclassical radiation theory cannot be used. A study of the correlation between elastic scattering and stimulated emission processes clarifies the reason for the discrepancies. The resulting singularities require summing of Feynman diagrams which appear beginning in the sixth order of perturbation theory. A lower bound estimate for the reflection coefficient from a plane layer is given, including processes which violate the statistics of radiation. The contribution of stimulated emission processes caused by the initially scattered photon are examined specifically. An experiment is proposed which would settle the choice of theories.
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.
Abstract The present chapter is devoted to a description of the procedure by which quantum mechanics is applied to the electromagnetic field. We obtain expressions for the operators that represent the field observables and we examine some of their properties. The quantization introduces characteristic quantum-mechanical effects into the properties of the radiation field, for example field operators that do not commute with each other and thus represent incompatible measured quantities. The Hamiltonian of the coupled radiation-atomic system is expressed wholly in terms of quantum-mechanical operators. We show that the full quantum theory of the interaction between the radiation field and atoms produces the same transition rates for absorption and stimulated emission as does the semiclassical theory. The rate of spontaneous emission confirms that obtained from the phenomenological Einstein theory. The treatment of absorption of radiation by atoms also identifies the quantum-mechanical operator that represents a measurement of the energy or intensity of a light beam, and this has important consequences for the descriptions of the experiments that can be performed in practice.
The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c)2{)}^{2})2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/c2{\mathit{c}}^{2}c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/c3{\mathit{c}}^{3}c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2e2{\mathit{e}}^{2}e2/3c3{\mathit{c}}^{3}c3)(d〈v〉/dt)2{)}^{2})2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.