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Physical Review A

Relation between Huttner-Barnett QED in dielectrics and classical electrodynamics: Determining the dielectric permittivity

S. M. Dutra, K. Furuya

Published Apr 1, 1998• 13 citations
Publisher

Abstract

In the Huttner-Barnett theory of QED in linear dispersive dielectrics with absorption [Europhys. Lett. 18, 487 (1992); Phys Rev. A 46, 4306 (1992)], a certain function of frequency $\ensuremath{\varepsilon}(\ensuremath{\omega})$ is interpreted as the dielectric permittivity of the medium. As the concept of permittivity is independent of the quantum nature of radiation, to fully justify their interpretation it should be demonstrated that the Huttner-Barnett $\ensuremath{\varepsilon}(\ensuremath{\omega})$ coincides with the result of a classical calculation of the permittivity for the microscopic model adopted. We give such a demonstration here. In addition, we show that the expression for $\ensuremath{\varepsilon}(\ensuremath{\omega})$ given by Huttner and Barnett can be considerably simplified. The simplified expression makes evident what classes of dielectric permittivity can be described within the Huttner-Barnett theory.

DOI
10.1103/physreva.57.3050

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In the Huttner-Barnett theory of QED in linear dispersive dielectrics with absorption [Europhys. Lett. 18, 487 (1992); Phys Rev. A 46, 4306 (1992)], a certain function of frequency \ensuremathε(\ensuremathω)\ensuremath{\varepsilon}(\ensuremath{\omega})\ensuremathε(\ensuremathω) is interpreted as the dielectric permittivity of the medium. As the concept of permittivity is independent of the quantum nature of radiation, to fully justify their interpretation it should be demonstrated that the Huttner-Barnett \ensuremathε(\ensuremathω)\ensuremath{\varepsilon}(\ensuremath{\omega})\ensuremathε(\ensuremathω) coincides with the result of a classical calculation of the permittivity for the microscopic model adopted. We give such a demonstration here. In addition, we show that the expression for \ensuremathε(\ensuremathω)\ensuremath{\varepsilon}(\ensuremath{\omega})\ensuremathε(\ensuremathω) given by Huttner and Barnett can be considerably simplified. The simplified expression makes evident what classes of dielectric permittivity can be described within the Huttner-Barnett theory.
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.
In this work the electromagnetic field in a dielectric medium is quantized in a simple and straightforward way based on the classical expression for the energy density for the case of weak absorption. The quantized field leads to expressions for spontaneous and stimulated emission rates, and gain and absorption coefficients, for the general case of a dispersive dielectric host medium. Local field corrections, when they apply, arise in the same manner as in classical electromagnetic theory. The theory also leads to a radiative level shift dependent on the dispersion of the dielectric, and this shift may be interpreted as a van der Waals interaction between the guest atom and the atoms of the host medium.>
The electromagnetic field in a dispersive dielectric medium is quantized in a straightforward manner based on the classical expression for the field energy away from absorption resonances. Results for spontaneous and stimulated radiative rates, and absorption and gain coefficients, are obtained and compared with other expressions in the literature. Local field corrections, when they exist, are included. A level shift for an atom in a host dielectric medium is obtained and interpreted as the van der Waals interaction between that atom and its neighbours in the host medium.
The correlation of the quantum-mechanical ground-state fluctuations of the electric-field strength in a dispersive and lossy linear dielectric are studied in terms of the symmetrized autocorrelation function, with special emphasis on the optical frequency domain. Starting from the canonical quantization scheme developed by Huttner and Barnett [Phys. Rev. A 46, 4306 (1992)], the analysis is based on a quantization of the phenomenological Maxwell theory, the effect of the medium being described by a frequency-dependent complex permittivity. In this way, the spectrally resolved ground-state autocorrelation function of the electric-field strength can be expressed in terms of the real and imaginary parts of the refractive index. Both analytical and numerical results are presented and the effects of dispersion and absorption including their dependence on the frequency interval chosen are discussed. A comparison with the vacuum fluctuations in free space is given.
Classical dispersion theory Many aspects of the interaction between radiation and matter can be described quite accurately by a classical theory in which the medium is represented by model atoms consisting of positive and negative parts bound by an attraction which depends linearly on their separation. Although quantum theory is necessary to calculate from first principles the magnitude of the parameters involved, in this chapter we shall show that many optical effects can be interpreted physically in terms of this model by the use of classical mechanics. In §13.5 we shall relax the restriction of linearity. Some of the quantum mechanical foundations will be discussed briefly in Chapter 14, but most are outside the scope of this book (see Yariv, 1989; Loudon, 1983). The term dispersion means the dependence of dielectric response (dielectric constant and refractive index) on frequency of the wave field. This will be the topic of the present section. Afterwards we shall see some of the applications of dielectric response to spatial effects. The classical atom Our classical picture of an atom consists of a massive positive nucleus surrounded by a light spherically-symmetrical cloud of electrons with an equal negative charge. We imagine the two as bound together by springs as in Fig. 13.1, so that in equilibrium the centres of mass and charge of the core and electron charge coincide. As a result the static atom has zero dipole moment.
We present a fully canonical quantization scheme for the electromagnetic field in dispersive and lossy linear dielectrics. This scheme is based on a microscopic model, in which the medium is represented by a collection of interacting matter fields. We calculate the exact eigenoperators for the coupled system and express the electromagnetic field operators in terms of them. The dielectric constant of the medium is explicitly derived and is shown to satisfy the Kramers-Kronig relations. We apply these results to treat the propagation of light in dielectrics and obtain simple expressions for the electromagnetic field in the medium in terms of space-dependent creation and annihilation operators. These operators satisfy a set of equal-space commutation relations and obey spatial Langevin equations of evolution. This justifies the use of such operators in phenomenological models in quantum optics. We also obtain two interesting relationships between the group and the phase velocity in dielectrics.