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Physical Review A

Green-function approach to the radiation-field quantization for homogeneous and inhomogeneous Kramers-Kronig dielectrics

Toralf Gruner, D.‐G. Welsch

Published Mar 1, 1996• 388 citations
Publisher

Abstract

A quantization scheme for the radiation field in dispersive and absorptive linear dielectrics is developed, which applies to both bulk material and multilayer dielectric structures. Starting from the phenomenological Maxwell equations, where the properties of the dielectric are described by a permittivity consistent with the Kramers-Kronig relations, an expansion of the field operators is performed that is based on the Green function of the classical Maxwell equations and preserves the equal-time canonical field commutation relations. In particular, in frequency intervals with approximately vanishing absorption the concept of quantization through mode expansion for dispersive dielectrics is recognized. The theory further reveals that weak absorption gives rise to space-dependent mode operators that spatially evolve according to quantum Langevin equations in the space domain. To illustrate the applicability of the theory to inhomogeneous structures, the quantization of the radiation field in a dispersive and absorptive one-interface dielectric is performed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.1818

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The use of instruments in optical experiments needs careful examination with regard to their action on the quantum statistics of radiation. Instruments that respond linearly to radiation may be regarded as spatially structured dielectrics whose presence can be taken into consideration by quantizing the phenomenological Maxwell theory. Since radiation in dielectric matter undergoes dispersion and absorption, the losses must necessarily be included in a quantization scheme. This central problem can be solved by using a novel approach that is based on a Green’s function expansion of the radiation field in place of a mode expansion, which fails when the losses are taken into account. The theory is applied to calculate quantum-theoretically consistent inputoutput relations for dispersive and absorptive multi-slab dielectric plates. Assuming that the polarization field and the electric field are locally related to each other, the phenomenological Maxwell equations governing the propagation of radiation in linear dielectrics, without external sources, can be quantized by supplementing them by an appropriately chosen noise source associated with the losses in the dielectric matter. For the sake of transparency, let us consider a multi-slab dielectric plate with permittivity ε(x, ω) and restrict attention to radiation that propagates along the x axis and is polarized in z direction.
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.
Quantization of the phenomenological Maxwell theory of the electromagnetic field in dispersive and absorbing dielectrics has been of increasing interest over the last years since it allows for a consistent formulation of quantum optics in the presence of dielectric bodies. We show, using a Dyson-type integral representation of the Green tensor, that the quantization scheme is consistent with QED for both isotropic and anisotropic media and therefore applies to arbitrary linear Kramers-Kronig media including amplifying media.
Using the Green-function approach to the problem of quantization of the phenomenological Maxwell theory, the propagation of quantized radiation through dispersive and absorptive multilayer dielectric plates is studied. Input-output relations are derived, with special emphasis on the determination of the quantum noise generators associated with the absorption of radiation inside the dielectric matter. The input-output relations are used to express arbitrary correlation functions of the outgoing field in terms of correlation functions of the incoming field and those of the noise generators. To illustrate the theory, photons at dielectric tunneling barriers are considered. It is shown that inclusion in the calculations of losses in the photonic band gaps may substantially change the barrier traversal times. \textcopyright{} 1996 The American Physical Society.