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arXiv:quant-ph/9811067 · Physical Review A

Quantum local-field corrections and spontaneous decay

Stefan Scheel, L. Knöll, Dirk‐Gunnar Welsch, Stephen Mark Barnett

Published Aug 1, 1999• 85 citations
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Abstract

A recently developed scheme [S. Scheel, L. Kn\"oll, and D.-G. Welsch, Phys. Rev. A 58, 700 (1998)] for quantizing the macroscopic electromagnetic field in linear dispersive and absorbing dielectrics satisfying the Kramers-Kronig relations is used to derive the quantum local-field correction for the standard virtual-sphere-cavity model. The electric and magnetic local-field operators are shown to become approximately consistent with QED only if the polarization noise is fully taken into account. It is shown that the polarization fluctuations in the local field can dramatically change the spontaneous decay rate, compared with the familiar result obtained from the classical local-field correction. In particular, the spontaneous emission rate strongly depends on the radius of the local-field virtual cavity.

DOI
10.1103/physreva.60.1590

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A recently developed scheme [S. Scheel, L. Kn"oll, and D.-G. Welsch, Phys. Rev. A 58, 700 (1998)] for quantizing the macroscopic electromagnetic field in linear dispersive and absorbing dielectrics satisfying the Kramers-Kronig relations is used to derive the quantum local-field correction for the standard virtual-sphere-cavity model. The electric and magnetic local-field operators are shown to become approximately consistent with QED only if the polarization noise is fully taken into account. It is shown that the polarization fluctuations in the local field can dramatically change the spontaneous decay rate, compared with the familiar result obtained from the classical local-field correction. In particular, the spontaneous emission rate strongly depends on the radius of the local-field virtual cavity.
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 have computed the spontaneous decay rate of an atom placed inside an absorbing dielectric adopting the real cavity model. The calculations are performed applying a quantization scheme which is causal and fully consistent with the dissipation-fluctuation theorem. We have started with a full electric field operator and found that the decay rate contains only contributions from the transverse field and sensitively depends on the cavity radius - results that contradict earlier results derived on the basis of heuristic introduction of local-field correction factors.
We consider the spontaneous emission in an absorbing medium by calculating the quantum flow of energy from an emitting atom. For this purpose, we extend the analysis by Power and Thirunamachandran of quantized Maxwell fields in the vicinity of an atom in the free space to a situation where the emitter is in a dielectric medium. We represent the quantum flow in terms of the non-equal time commutators involving electric and magnetic fields, as well as the local displacement field. The commutators are calculated applying a previously developed microscopic theory of quantization of radiation in dielectrics. Consequently, we derive from first principles the rate of spontaneous emission in an absorbing medium containing local field factors and other refractive contributions. The results support the previous introduction of the (virtual cavity) local field correction by Barnett et al on a phenomenological basis, as well as our previous approach in which the spontaneous emission is considered as a resonance energy transfer to the far-zone atoms. A distinctive feature of the present formalism is that it deals with the spontaneous emission directly by calculating the quantum flow from the emitting atom. Therefore, the method is free from divergences appearing in other approaches analysing the spontaneous decay in absorbing media.
The decay rate and the classical radiation power of an excited molecule (atom) in the center of a dispersive and absorbing dielectric sphere, taken as a simple model of a cavity, are calculated adopting the Onsager model for the local field. The local-field correction factor to the external (radiation and absorption) power loss of the molecule is found to be $|3\ensuremath{\varepsilon}(\ensuremath{\omega})/[3\ensuremath{\varepsilon}(\ensuremath{\omega})+1]{|}^{2},$ with $\ensuremath{\varepsilon}(\ensuremath{\omega})$ being the dielectric function of the sphere. However, local-field corrections to the total decay rate (power loss) of the molecule are found to be much more complex including those to the decay rate in the infinite cavity medium, as derived recently by Scheel et al. [Phys. Rev. A 60,60,60, 4094 (1999)], and similiar corrections to the cavity-induced decay rate. The results obtained can be cast into general forms that do not rely on the specific cavity shape and molecule position considered. This suggests the general results for the local-field corrections to the decay rate and to the external power loss of a molecule in an absorbing cavity that are valid for molecule positions away from the cavity walls.
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.