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Journal of Optics B Quantum and Semiclassical Optics

Atomic dynamics in microcavities: absorption spectra by Green function method

Ottavia Jedrkiewicz, R. Loudon

Published Nov 21, 2000• 20 citations
Publisher

Abstract

The method of thermodynamic Green functions is applied to the determination of the susceptibilities and absorption spectra of atoms in microcavities. The method is introduced by its application to an atom in free space, where the well known expression for the spectrum is very simply rederived. The calculations remain simple for more complicated systems, and results are obtained for atoms in single-mode cavities with lossy mirrors and with the additional damping by emission into free-space modes that occurs in open-sided cavities. The expressions for some limiting cases of the general theory are compared with the results of previous work and contact is made with the dressed-state theory. Special consideration is given to an atom placed in a planar microcavity; it is shown that non-Lorentzian spectra may in principle occur but that the parameter values of practical cavities produce Lorentzian spectra with linewidths given by Fermi's golden rule.

DOI
10.1088/1464-4266/2/6/01

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The method of thermodynamic Green functions is applied to the determination of the susceptibilities and absorption spectra of atoms in microcavities. The method is introduced by its application to an atom in free space, where the well known expression for the spectrum is very simply rederived. The calculations remain simple for more complicated systems, and results are obtained for atoms in single-mode cavities with lossy mirrors and with the additional damping by emission into free-space modes that occurs in open-sided cavities. The expressions for some limiting cases of the general theory are compared with the results of previous work and contact is made with the dressed-state theory. Special consideration is given to an atom placed in a planar microcavity; it is shown that non-Lorentzian spectra may in principle occur but that the parameter values of practical cavities produce Lorentzian spectra with linewidths given by Fermi's golden rule.
A compact and transparent plane-wave expansion of the complete Green function for an absorbing multilayered system is derived. The plane-wave solutions involved match, in the appropriate limit, the space-mode functions recently employed for quantization of the radiation field in a planar cavity and make this Green function particularly convenient for consideration of optical processes in realistic cavities and multilayers in general. This is illustrated by considering classically light scattering on a molecule in a planar cavity. It is demonstrated that the cross sections derived, besides generally describing light scattering on molecules embedded in a multilayer, correctly reproduce the effects of field confinement on light scattering in a planar cavity as predicted recently by the quantum-mechanical theory of Cairo et al. [Phys. Rev. Lett. 70, 1413 (1993)]. It is also pointed out that scattering into waves guided by the cavity lossy (metallic) mirrors may become an effective process in low-order microcavities. As a by product, a classical approach to spontaneous emission in absorbing planar cavities emerges from these considerations.
This chapter gives an overview of the field as it stands at the time of writing in mid 1995. It shows how the picture radically changes when there is more than one atom in the cavity. The simplest light-atom interaction one can think of is the coupling between a single atom and a single mode of the electromagnetic field. This system provides considerable insight into the fundamental physics of the light-atom coupling but the reader is warned that the more general many mode-many atom interactions are much richer in their dynamics. An atom in free space couples not only to one, but to a continuum of modes. The mathematics of the Weisskopf-Wigner theory of spontaneous emission tends to obscure the relatively simple physics of the emission process. An alternative, shorter way of calculating the single atom spontaneous emission rate is to use Fermi’s Golden rule, which is a prescription for calculating general transition rates to first order in perturbation theory.
We study the spontaneous emission of atoms coupled to frequency-dependent photon-mode reservoirs, and find atomic emission spectra qualitatively different from those of free space. Frequency-dependent photon-mode reservoirs are found in a variety of physical situations (e.g., in waveguides, microcavities, dielectrics, systems with photonic spectral gaps, etc.), and are shown to lead to nonexponential decay and concomitant complicated non-Lorentzian emission line shapes. By studying Autler-Townes spectra in the presence of a frequency-dependent photon-mode density, we present simple examples of dynamical modifications of spontaneous decay properties.
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.
This chapter presents a tutorial review of spontaneous emission in optical resonators. Approximating correlation functions as delta functions means that the reservoir is assumed to lose its memory in a short time, compared to the time scale over which the dynamics of the small system occurs, the Markov approximation. In a generic cavity quantum electrodynamics experiment, an atom is coupled to one, or possibly a few, cavity mode(s). These modes are in turn coupled to the external world via mirror losses and diffraction. In addition, the atom might also be coupled to a continuum of vacuum modes, in the case where the cavity is open-sided. A simple physical interpretation of the spontaneous emission spectrum can be obtained if one adopts the point of view that the atom-cavity mode small system represents a single quantum system, and by analyzing its eigenenergy spectrum.
Starting from a quantum Maxwell equation for the vector potential operator, we study the spontaneous emission properties of a two-level atom interacting with the electromagnetic field in a lossless and inhomogeneous dielectric structure and express the spontaneous emission rate, the external quantum efficiency, and the spontaneous emission factor in terms of the classical Green function of the dielectric structure. Comparing these results with the corresponding classical results, we show that the above quantities can be calculated from the radiation field of a classical dipole. Based on this correspondence, a numerical algorithm is developed to calculate the modification of spontaneous emission in microcavities. The line shape of the emission spectrum of the light source and the decay of the cavity resonant modes are automatically taken into account in this algorithm. The approximate expressions for the spontaneous emission rate and the spontaneous emission factor are discussed.