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arXiv:quant-ph/9906001

QED in the Presence of Arbitrary Kramers-Kronig Dielectric Media

Stefan Scheel, D.‐G. Welsch

Published Jun 1, 1999• 0 citations
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Abstract

The phenomenological Maxwell field is quantized for arbitrarily space- and frequency-dependent complex permittivity. The formalism takes account of the Kramers--Kronig relation and the dissipation-fluctuation theorem and yields the fundamental equal-time commutation relations of QED. Applications to the quantum-state transformation at absorbing and amplifying four-port devices and to the spontaneous decay of an excited atom in the presence of absorbing dielectric bodies are discussed.

DOI
10.48550/arxiv.quant-ph/9906001

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The phenomenological Maxwell field is quantized for arbitrarily space- and frequency-dependent complex permittivity. The formalism takes account of the Kramers--Kronig relation and the dissipation-fluctuation theorem and yields the fundamental equal-time commutation relations of QED. Applications to the quantum-state transformation at absorbing and amplifying four-port devices and to the spontaneous decay of an excited atom in the presence of absorbing dielectric bodies are discussed.
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A quantization scheme for the phenomenological Maxwell theory of the full electromagnetic field in an inhomogeneous three-dimensional, dispersive, and absorbing dielectric medium is developed. The classical Maxwell equations with spatially varying and Kramers-Kronig consistent permittivity are regarded as operator-valued field equations, introducing additional current- and charge-density operator fields in order to take into account the noise associated with the dissipation in the medium. It is shown that the equal-time commutation relations between the fundamental electromagnetic fields $\mathbf{E}^$ and $\mathbf{B}^$ and the potentials $\mathbf{A}^$ and $\stackrel{^}{\ensuremath{\varphi}}$ in the Coulomb gauge can be expressed in terms of the Green tensor of the classical problem. From the Green tensors for bulk material and an inhomogeneous medium consisting of two bulk dielectrics with a common planar interface it is explicitly proven that the well-known equal-time commutation relations of QED are preserved.
A general theory of the interaction of the quantized electromagnetic field with atoms, in the presence of dispersing and absorbing dielectric bodies of given Kramers-Kronig consistent permittivities, is developed. It is based on a source-quantity representation of the electromagnetic field, in which the electromagnetic-field operators are expressed in terms of a continuous set of fundamental bosonic fields via the Green tensor of the classical problem. Introducing scalar and vector potentials, the formalism is extended to include in the theory the interaction of the quantized electromagnetic field with additional atoms. Both the minimal-coupling scheme and the multipolar-coupling scheme are considered. The theory replaces the standard concept of mode decomposition, which fails for complex permittivities. It enables us to treat the effects of dispersion and absorption in a consistent way and to give a unified approach to the atom-field interaction, without any restriction to a particular interaction regime in a particular frequency range. All relevant information about the dielectric bodies, such as form and intrinsic dispersion and absorption, is contained in the Green tensor. The application of the theory to the spontaneous decay of an excited atom in the presence of dispersing and absorbing bodies is addressed.