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arXiv:quant-ph/0604034

A master equation approach for the interaction of an atom with a dielectric semi-infinite medium

T. N. C. Mendes, Carlos Farina

Published Apr 5, 2006• 0 citations
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Abstract

We use the master equation approach to calculate the energy level shifts of an atom in the presence of a general dielectric semi-infinite medium characterized by a dielectric constant $ε(ω)$. Particularly, we analyze the case of a non-dispersive medium for which we obtain a general expression for the interaction as well as the asymptotic behaviors for $k_0 z \ll 1$ (non-retarded regime) and $k_0 z \gg 1$ (retarded regime), where $ω_0 = k_0 c$ is the main transition frequency of the atom. The limiting cases $ε\simeq 1$ and $ε\gg 1$ are discussed for both retarded and non-retarded limits. For the retarded limit, we compute the non-additivity contribution of van der Waals forces.

DOI
10.48550/arxiv.quant-ph/0604034

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We use the master equation approach to calculate the energy level shifts of an atom in the presence of a general dielectric semi-infinite medium characterized by a dielectric constant ε(ω)ε(ω)ε(ω). Particularly, we analyze the case of a non-dispersive medium for which we obtain a general expression for the interaction as well as the asymptotic behaviors for k0z≪1k_0 z \ll 1k0​z≪1 (non-retarded regime) and k0z≫1k_0 z \gg 1k0​z≫1 (retarded regime), where ω0=k0cω_0 = k_0 cω0​=k0​c is the main transition frequency of the atom. The limiting cases ε≃1ε\simeq 1ε≃1 and ε≫1ε\gg 1ε≫1 are discussed for both retarded and non-retarded limits. For the retarded limit, we compute the non-additivity contribution of van der Waals forces.
The energy–level shifts and the change in the rate of spontaneous emission are calculated for an atom located at a distanceZ from a dielectric half–space. The dielectric is a non–dispersive and non–absorbing medium characterized by a constant real refractive indexn. The explicit analytic formulae derived are applicable for arbitrary values ofn. All results are analysed in the non-retarded and retarded limits, which apply to the atom being close to or far from the interface, respectively, i.e. toZ being small or large on the scale of a wavelength of a typical atomic transition. For ground-state atoms, the energy–level shift varies as 1/Z3 in the non–retarded regime and as 1/Z4 in the retarded regime, which agrees with the Casimir–Polder result for the limitn→α. For excited–state atoms, the energy–level shifts receive additional contributions that oscillate with the distanceZ from the interface.
We calculate the energy-level shifts of an atom near a dielectric microsphere. With the quantum electrodynamic linear-response formalism and the exact field susceptibility satisfying the boundary conditions, the energy shifts of a ground-state atom are calculated numerically. From the numerical results, in particular, we are able to identify the short-range instantaneous van der Waals interaction energy and the long-range Casimir-Polder retarded interaction energy between two polarizable objects (atom and cavity).
The Casimir-Polder potential for interaction between an excited atom and a ground-state one in the retarded case is obtained with the help of perturbation technique. The potential drops as ${R}^{\ensuremath{-}2}$ with the distance between the atoms [E. A. Power and T. Thirunamachandran, Phys. Rev. A 47, 2539 (1993)]. It results in divergent integrals for interaction between an excited atom and a dilute gas medium. We investigate interaction between two atoms embedded in a dielectric medium with the help of a method that enables us to make calculations beyond the perturbation technique. We take into account absorption of photons in the medium. This approach solves the problem of divergence. We consider interaction between an excited atom and a planar dielectric gas medium of ground-state atoms. We show that the retarded interaction between an excited atom and a gas of ground-state atoms is not oscillating but follows a simple power law. We show that to obtain a conventional nonretarded expression for the van der Waals force between an excited atom and a dilute gas, the distance between the atom and the interface should be much smaller than the free mean path of a photon in the medium. Interaction between an excited atom and a hemisphere of ground-state atoms is considered.
Within the frame of macroscopic quantum electrodynamics in causal media, a rigorous derivation of the van der Waals energy of an atomic system in the presence of an arbitrary arrangement of dispersing and absorbing dielectric bodies is given. As an application, the van der Waals potential of an atom in the presence of a dispersing and absorbing sphere is derived. Closed expressions for the long-distance (retardation) and short-distance (non-retardation) limits are given, and the effect of material absorption is discussed.
The energy–level shifts of an atom located at a distance Z outside a dielectric half–space are calculated at finite temperatures. The refractive index of the medium is real and frequency independent; the limiting case of perfect reflectors is also considered. The analytic expressions for the energy–level shifts are analysed in various regimes of temperature and of the distance of the atom from the surface. The significance of thermal corrections in comparison with zero–temperature effects is analysed in each regime and discussed for recent experiments.
In this work the electromagnetic field in a dielectric medium is quantized in a simple and straightforward way based on the classical expression for the energy density for the case of weak absorption. The quantized field leads to expressions for spontaneous and stimulated emission rates, and gain and absorption coefficients, for the general case of a dispersive dielectric host medium. Local field corrections, when they apply, arise in the same manner as in classical electromagnetic theory. The theory also leads to a radiative level shift dependent on the dispersion of the dielectric, and this shift may be interpreted as a van der Waals interaction between the guest atom and the atoms of the host medium.>