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Physical Review A

Longitudinal fields in interactions between atoms and absorptive dielectrics

A. Tip

Published Oct 23, 2007• 0 citations
Publisher

Abstract

The properties of atoms close to an absorptive dielectric are studied using the phenomenological Maxwell's equations. Although radiative decay has been considered by many authors, the coupling of atoms with longitudinal modes does not seem to have been treated in detail. Here we show that there are two main effects. The first is a change in the atomic interaction potential from the Coulomb one to a static potential, i.e., one that satisfies a Poisson equation featuring the static dielectric function ${\ensuremath{\epsilon}}_{\mathrm{stat}}=\ensuremath{\epsilon}(\ensuremath{\omega}){\ensuremath{\mid}}_{\ensuremath{\omega}=0}$. The second is the decay of excited atomic states through longitudinal field interactions. We find that the corresponding decay constant is nonzero only for atom-dielectric distances in the order of an atomic diameter and that it decreases exponentially fast on an atomic scale with increasing distance. We also show that the Hamiltonian used by the Jena group [Dung et al., Phys. Rev. A 65, 043813 (2002)], featuring the Coulomb potential, is unitarily equivalent to one containing the static potential.

DOI
10.1103/physreva.76.043835

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The properties of atoms close to an absorptive dielectric are studied using the phenomenological Maxwell's equations. Although radiative decay has been considered by many authors, the coupling of atoms with longitudinal modes does not seem to have been treated in detail. Here we show that there are two main effects. The first is a change in the atomic interaction potential from the Coulomb one to a static potential, i.e., one that satisfies a Poisson equation featuring the static dielectric function \ensuremathϵstat=\ensuremathϵ(\ensuremathω)\ensuremath∣\ensuremathω=0{\ensuremath{\epsilon}}_{\mathrm{stat}}=\ensuremath{\epsilon}(\ensuremath{\omega}){\ensuremath{\mid}}_{\ensuremath{\omega}=0}\ensuremathϵstat​=\ensuremathϵ(\ensuremathω)\ensuremath∣\ensuremathω=0​. The second is the decay of excited atomic states through longitudinal field interactions. We find that the corresponding decay constant is nonzero only for atom-dielectric distances in the order of an atomic diameter and that it decreases exponentially fast on an atomic scale with increasing distance. We also show that the Hamiltonian used by the Jena group [Dung et al., Phys. Rev. A 65, 043813 (2002)], featuring the Coulomb potential, is unitarily equivalent to one containing the static potential.
We use the auxiliary field method to obtain a quantized Hamilton formalism for charged particles interacting with absorptive dielectrics. We give various equivalent representations. For absorptive dielectrics extending over all space, such as absorptive photonic crystals, our formalism is equivalent to that of Dung et al. [Phys. Rev. A 65, 043813 (2002)]. We also show that, as for nonabsorptive photonic crystals, to leading order the excited states of embedded two-level atoms do not decay radiatively if their transition frequencies are in a band gap.
We present calculations of the rates of decay of an excited atom embedded in an absorbing dielectric. Decay can occur by spontaneous emission into transverse radiative modes of the electromagnetic field and by Joule heating via longitudinal coupling of the atom to the dielectric. The spontaneous emission (transverse) decay rate is modified in a dielectric, being the free-space rate multiplied by the real part of the refractive index at the transition frequency of the atom. There is a further modification due to the difference between the macroscopic dielectric field and the local field at the position of the atom. In addition there is a longitudinal decay rate which is proportional to the imaginary part of the dielectric constant and therefore vanishes in non-absorbing media. We derive expressions for each of these rates of decay and discuss the physical mechanisms leading to them.
Starting from the quantized version of Maxwell's equations for the electromagnetic field in an arbitrary linear Kramers-Kronig dielectric, the Heisenberg equations of motion for a two-level atom resonantly coupled to the radiation field in the presence of dispersive and absorbing dielectric bodies are derived. The theory is applied to the problem of spontaneous decay of excited atoms in absorbing media. The decay rate is calculated for the (Glauber-Lewenstein) real-cavity model, and a comparison with the recently studied (Clausius-Mosotti) virtual-cavity model [S. Scheel, L. Kn\"oll, D.-G. Welsch, and S. M. Barnett, Phys. Rev. A 60, 1590 (1999)] is given. It is shown that owing to nonradiative decay associated with absorption, the rate of spontaneous decay sensitively depends on the cavity radius, particularly when the atomic transition frequency approaches an absorption band of the medium. Only when the effect of absorption is fully disregarded, is the familiar local-field correction factor recovered.
The change in the decay rate of an excited atom that is brought about by extinction and thermal-radiation effects in a nearby dielectric medium is determined from a quantum-mechanical model. The medium is a collection of randomly distributed thermally excited spherical scatterers with absorptive properties. The modification of the decay rate is described by a set of correction functions for which analytical expressions are obtained as sums over contributions from the multipole moments of the scatterers. The results for the modified decay rate as a function of the distance between the excited atom and the dielectric medium show the influence of absorption, scattering and thermal-radiation processes. Some of these processes are found to be mutually counteractive. The changes in the decay rate are compared to those following from an effective-medium theory in which the discrete scatterers are replaced by a continuum.
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.
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.