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arXiv:quant-ph/9511041

Quantum optical input--output relations for dispersive and lossy multi-slab dielectric plates

Toralf Gruner, D.‐G. Welsch

Published Nov 29, 1995• 0 citations
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Abstract

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, the effect of a single-slab plate on the mean photon-number densities in the frequency domain is discussed in more detail.

DOI
10.48550/arxiv.quant-ph/9511041

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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.
By using the Green-function concept of quantization of the electromagnetic field in dispersing and absorbing media, the quantized field in the presence of a dispersing and absorbing dielectric multilayer plate is studied. Three-dimensional input-output relations are derived for both amplitude operators in the k\mathbf{k}k space and the field operators in the coordinate space. The conditions are discussed, under which the input-output relations can be expressed in terms of bosonic operators. The theory applies to both (effectively) free fields and fields, created by active atomic sources inside and/or outside the plate, including also evanescent-field components.
The theory developed by Gruner and Welsch [Phys. Rev. A 54, 1661 (1996)] for calculating the 1D input-output relations of the quantized electromagnetic field at dispersing and absorbing dielectric (multilayer) plates is generalized to the three-dimensional case. First a general recipe for the derivation of the reflection and transmission coefficients at an arbitrary body that separates two half spaces from each other is presented. The general theory is then applied to the case of planar multilayer structures, for which the Green tensor is well-known.
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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.