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Journal of Optics B Quantum and Semiclassical Optics

Microscopic calculation of noise current operators for electromagnetic field quantization in absorbing material systems

Omar Di Stefano, Salvatore Savasta, R. Girlanda

Published Aug 1, 2001• 16 citations
Publisher

Abstract

We present a microscopic quantization scheme for the electromagnetic field in dispersive and lossy dielectrics of arbitrary geometry. This method also describes anisotropic media and media driven by light field via a spatially nonlocal permittivity. The method removes the need for complicated diagonalization of material, reservoir and field variables and allows us to include the effect of all the material excitations. Dissipation inside the medium is described by considering the coupling of the polarization quanta of the system with the reservoir oscillators in the usual Langevin approach.

DOI
10.1088/1464-4266/3/4/314

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We present a microscopic quantization scheme for the electromagnetic field in dispersive and lossy dielectrics of arbitrary geometry. This method also describes anisotropic media and media driven by light field via a spatially nonlocal permittivity. The method removes the need for complicated diagonalization of material, reservoir and field variables and allows us to include the effect of all the material excitations. Dissipation inside the medium is described by considering the coupling of the polarization quanta of the system with the reservoir oscillators in the usual Langevin approach.
We quantize the electromagnetic field in an absorbing resonant medium with translation symmetry broken along one direction. In this case the general polarization of the material system is driven by the field via a nonlocal complex susceptibility. We extend recently developed schemes for field quantization in absorbing dielectric media with local susceptibilities to dielectric systems described by a nonlocal susceptibility. Dissipation inside the medium is described by considering the coupling of polarization quanta of the system with the reservoir oscillators in the usual Langevin approach. The method is applied to the case of a semiconductor quantum well embedded in infinite barriers. Expressions for the electromagnetic-field operators are obtained in the different spatial regions and the spatial and frequency dependencies of the vacuum-field fluctuations are calculated.
The electromagnetic field is quantized in dielectric media that show both loss and dispersion. The complex dielectric function of the medium is assumed to be a known function and the loss is modeled by Langevin forces in the forms of noise current operators. The noise current correlation function is related to the assumed dielectric function by the fluctuation-dissipation theorem. Field quantization is carried out for the infinite homogeneous dielectric, the semi-infinite dielectric, and the dielectric slab, where the fields in the second and third cases are restricted to propagation perpendicular to the dielectric surfaces. The forms of the vector potential operator are obtained in the different spatial regions for all three geometries, and in each case the required canonical commutation relation for the vector potential and its conjugate generalized momentum operator is verified. The spatial dependence of the vacuum field fluctuations is calculated for the two dielectric geometries that have surfaces.
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.
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 present a one-dimensional quantization scheme for the electromagnetic field in arbitrary planar absorbing and dispersive dielectrics, taking into account explicitly the finite extent of media. The complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the light modes of the dielectric structure, we obtain a natural generalization to absorbing media of the familiar method of mode expansion which strictly applies to non-absorbing media. As an application of our mode expansion, we obtain general quantum-optical input-output relations relating the output photon operators in vacuum to the input photon operators and to the reservoir noise operators. The input-output relations obtained are uniquely determined by the knowledge of the classical light modes of the planar system. We finally show that our quantization scheme for arbitrary planar dielectrics satisfies the prescribed equal-time commutation relations of QED.
There are both loss and dispersion characteristics for most dielectric media. In quantum theory the loss in medium is generally described by Langevin force in the Langevin noise (LN) scheme by which the quantization of the radiation field in various homogeneous absorbing dielectrics can be successfully actualized. However, it is invalid for the anisotropic dispersion medium. This paper extends the LN theory to an anisotropic dispersion medium and presented the quantization of the radiation field as well as the transformation relation between the homogeneous and anisotropic dispersion media.