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arXiv:quant-ph/0003138 · Physical Review A

Spontaneous decay in the presence of dispersing and absorbing bodies: General theory and application to a spherical cavity

Ho Trung Dung, L. Knöll, Dirk‐Gunnar Welsch

Published Oct 12, 2000• 269 citations
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Abstract

A formalism for studying spontaneous decay of an excited two-level atom in the presence of dispersing and absorbing dielectric bodies is developed. An integral equation, which is suitable for numerical solution, is derived for the atomic upper-state-probability amplitude. The emission pattern and the power spectrum of the emitted light are expressed in terms of the Green tensor of the dielectric-matter formation, including absorption and dispersion. The theory is applied to the spontaneous decay of an excited atom at the center of a three-layered spherical cavity, with the cavity wall being modeled by a band-gap dielectric of Lorentz type. Both weak and strong coupling are studied, the latter with a special emphasis on cases where the atomic transition is (i) in the normal-dispersion zone near the medium resonance, and (ii) in the anomalous-dispersion zone associated with the band gap. In a single-resonance approximation, conditions of the appearance of Rabi oscillations and closed solutions to the evolution of the atomic state population are derived, which are in good agreement with the exact numerical results.

DOI
10.1103/physreva.62.053804

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A formalism for studying spontaneous decay of an excited two-level atom in the presence of dispersing and absorbing dielectric bodies is developed. An integral equation, which is suitable for numerical solution, is derived for the atomic upper-state-probability amplitude. The emission pattern and the power spectrum of the emitted light are expressed in terms of the Green tensor of the dielectric-matter formation, including absorption and dispersion. The theory is applied to the spontaneous decay of an excited atom at the center of a three-layered spherical cavity, with the cavity wall being modeled by a band-gap dielectric of Lorentz type. Both weak and strong coupling are studied, the latter with a special emphasis on cases where the atomic transition is (i) in the normal-dispersion zone near the medium resonance, and (ii) in the anomalous-dispersion zone associated with the band gap. In a single-resonance approximation, conditions of the appearance of Rabi oscillations and closed solutions to the evolution of the atomic state population are derived, which are in good agreement with the exact numerical results.
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.
We present a formalism for studying the influence of dispersive and absorbing dielectric bodies on a radiating atom in the framework of quantization of the phenomenological Maxwell equations for given complex permittivities of the bodies. In Markov approximation, the rate of spontaneous decay and the line shift associated with it can then be related to the complex permittivities and geometries of the bodies via the dyadic Green function of the classical boundary value problem of electrodynamics -- a result which is in agreement with second-order calculations for microscopic model systems. The theory is applied to an atom near a planar interface as well as to an atom in a spherical cavity. The latter, also known as the real-cavity model for spontaneous decay of an excited atom embedded in a dielectric, is compared with the virtual-cavity model. Connections with other approaches are mentioned and the results are compared.
Spontaneous decay of an excited atom near a dispersing and absorbing microsphere of given complex permittivity that satisfies the Kramers-Kronig relations is studied, with special emphasis on a Drude-Lorentz permittivity. Both the whispering gallery field resonances below the band gap (for a dielectric sphere) and the surface-guided field resonances inside the gap (for a dielectric or a metallic sphere) are considered. Since the decay rate mimics the spectral density of the sphere-assisted ground-state fluctuation of the radiation field, the strengths and widths of the field resonances essentially determine the feasible enhancement of spontaneous decay. In particular, strong enhancement can be observed for transition frequencies within the interval in which the surface-guided field resonances strongly overlap. When material absorption becomes significant, then the highly structured emission pattern that can be observed when radiative losses dominate, reduces to that of a strongly absorbing mirror. Accordingly, nonradiative decay becomes dominant. In particular, if the distance between the atom and the surface of the microsphere is small enough, the decay becomes purely nonradiative.
The decay rate of an excited molecule (atom) embedded in a dispersive and absorbing planar cavity is derived by using a recently obtained compact form of the Green's function for a multilayer. As a by-product, a hint is provided for a straightforward extension of the results obtained for lossless cavities of other shapes to the corresponding absorbing cavities. The decay rate in an absorbing cavity consists of the spontaneous emission rate and of the nonradiative rates caused by the near-field interaction of the molecule with the cavity medium and, for nearby molecules, with the cavity mirrors. Only the spontaneous emission rate is satisfactorily described in the macroscopic approach adopted. The theory is applied to an analysis of the effects of the weak cavity absorption on the decay rate in a dielectric microcavity formed by two metallic mirrors. As expected, dissipation in the cavity medium spoils the conditions for controlled spontaneuos emission and strongly suppresses the intensity of spontaneous emission. However, its effect on the spontaneous emission rate is much less pronounced.
The spontaneous-emission rate of a radiating atom reaches its time-independent equilibrium value after an initial transient regime. In this paper, we consider the associated relaxation effects of the spontaneous-decay rate of atoms in dispersive and absorbing dielectric media for atomic-transition frequencies near material resonances. A quantum mechanical description of such media is furnished by a damped-polariton model in which absorption is taken into account through coupling to a bath. We show how all field and matter operators in this theory can be expressed in terms of the bath operators at an initial time. The consistency of these solutions for the field and matter operators are found to depend on the validity of certain velocity sum rules. The transient effects in the spontaneous-decay rate are studied with the help of several specific models for the dielectric constant, which are shown to follow from the general theory by adopting particular forms of the bath coupling constant.
After giving a summary of the basic-theoretical concept of quantization of the electromagnetic field in the presence of dispersing and absorbing (macroscopic) bodies, their effect on spontaneous decay of an excited atom is studied. Various configurations such as bulk material, planar half space media, spherical cavities, and microspheres are considered. In particular, the influence of material absorption on the local-field correction, the decay rate, the line shift, and the emission pattern are examined. Further, the interplay between radiative losses and losses due to material absorption is analyzed. Finally, the possibility of generating entangled states of two atoms coupled by a microsphere-assisted field is discussed.