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3 ON THE PROBLEM OF ELECTROMAGNETIC-FIELD QUANTIZATION

Christian Krattenthaler, Sergey Kryuchkov, Alex Mahalov, Sergei, K. Suslov

Published Aug 17, 2016• 12 citations
OpenAlex

Abstract

We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg's equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.

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We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg's equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.
The electromagnetic field is quantized in nonlinear media whose dielectric functions and nonlinear susceptibilities may vary with space. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg equations we prove that Maxwell's equations are satisfied for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure developed in Ref.[1] for linear inhomogeneous media. The general formalism is illustrated by applications to quantization of the second-order and third-order nonlinear quasi-steady-state processes.
A simple approach is proposed for the quantization of the electromagnetic field in nonlinear and inhomogeneous media. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg's equations we derive Maxwell's equations for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure for linear inhomogeneous media [Phys. Rev. A 43, 467 (1991)]. The explicit Hamiltonians for the second-order and third-order nonlinear quasi-steady-state processes are obtained based on this quantization procedure.
The macroscopic electromagnetic field is quantized in a linear isotropic permeable-dielectric medium by associating a proper damped quantum-mechanical harmonic oscillator with each mode of the radiation field. We introduce a particular form of the Langevin equation with two memory functions and two random impulses to describe the motion of a damped oscillator. A fully canonical approach is used to quantize the damped oscillator, the conjugate momentum is defined and the quantum-mechanical Hamiltonian is introduced. This brings forth a direct method for the quantization of the radiation field and at the same time introduces an effective Hamiltonian operator. Special attention is paid to the vacuum field fluctuation in permeable dielectrics.
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.
We discuss the quantum fluctuations of the fields associated with a broad class of optical scattering and transmission problems by developing the quantum electrodynamics of an idealized linear, but nonuniform, dielectric medium. We present and compare two quantization schemes for this purpose. The first is based on the expansion of the field in terms of a set of single-frequency solutions of the Maxwell equations. The second involves expanding the field in the set of plane-wave solutions of the Maxwell equations in the vacuum. The relation between the two quantization schemes is discussed in the framework of the scattering theory that connects them. The methods presented are used to show that various field components within a dielectric medium may be either superfluctuant or subfluctuant relative to their fundamental uncertainties in the vacuum. These alterations of the fluctuation properties of the fields are shown to lead to changes in the spontaneous emission rates for both electric and magnetic dipole transitions of excited atoms within or near dielectric media. We also analyze the quantum properties of the transition radiation emitted by a fast charged particle in passing from one dielectric medium to another.