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Physical Review A

Quantum analysis and the classical analysis of spontaneous emission in a microcavity

Yong Xu, Reginald K. Lee, Amnon Yariv

Published Feb 14, 2000• 118 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.61.033807

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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.
We present a general, semimicroscopic, self-consistent treatment of spontaneous emission for a two-level atom in a dielectric microcavity with arbitrary shape. Both lossless and absorbing media are considered. Our approach is based on linear-response theory and the coupled-dipole method. We illustrate our method on spherical and cubic cavities. We discuss the Purcell effect, the frequency shift, the local-field effect, and the dipole-dipole coupling experienced by a source placed in a microcavity.
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.
Dynamics of the spontaneous emission from a two-level atom embedded in a bisphere microcavity is analyzed theoretically. A bisphere supports a morphology-dependent resonance having such a high quality factor with local field enhancement that the strong coupling between the atom and the cavity can be realized. By taking full account of the photon degree of freedom, we derive theoretically the coupling constant between the atom and the cavity as well as the radiation damping constants, which are used in the conventional cavity QED approach. In addition, we show that the power spectrum of the spontaneous emission at a local observation point is strongly affected by the photon Green function.
A complete set of cavity modes in planar dielectric microcavities is presented which naturally includes guided modes. We show that most of these orthonormal fields can be derived from a coherent superposition of plane waves incoming on the stack from the air and from the substrate. Spontaneous emission of a dipole located inside the microcavity is analyzed, in terms of cavity modes. Derivation of the radiation pattern in the air and in the substrate is presented. The power emitted into the guided modes is also determined. Finally, a numerical analysis of the radiative properties of an erbium atom located in a Fabry-P\'erot multilayer dielectric microcavity is investigated. We show that a large amount of light is emitted into the guided modes of the structure, in spite of the Fabry-P\'erot resonance, which increases the spontaneous emission rate in a normal direction. \textcopyright{} 1996 The American Physical Society.
In this chapter, the author calculates the modification of the spontaneous emission pattems and the corresponding rate change using classical physics. To derive the spontaneous emission pattems or rates the readers must average the emission rate at each frequency over this distribution function. It is important to understand that this assumption is only valid if the emission line is homogeneously broadened (albeit not necessarily at its natural linewidth limit due to the radiative decay). One more thing needs to be pointed out before the readers move on to a more exact calculation of the spontaneous emission pattern and the emission rate of a dielectric cavity. In the ideal mirror the reflection takes place at a well-defined plane. Integration of the emission over all solid angles and all wavelengths gives us information about the total spontaneous emission rate, or, equivalently, about the spontaneous emission lifetime of an emitter in a dielectric cavity.
The rate of spontaneous emission of an atom in the presence of an interface between two dielectrics was calculated using the quantum theory of electromagnetic radiation. Spatial distribution of the field for quasi-continuous spectrum inside an infinite cavity with ideally conducting walls was determined for multiple values of real refractive indices. Spontaneous emission in the continuous spectrum of the field was calculated using the known spatial Green’s function for a one-dimensional dielectric interface. The rate of spontaneous emission of an atom may either be higher or much lower than that in the free space, depending on the refractive indices and the distance between the atom and the interface between dielectrics.