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Physical Review Letters

Spontaneous Emission of an Atom in a Cavity with Nonorthogonal Eigenmodes

Yuh‐Jen Cheng

Published Aug 30, 2006• 11 citations
Publisher

Abstract

The spontaneous emission of an excited atom in a lossy cavity with nonorthogonal eigenmodes is analyzed. The quantum Langevin formalism is used to describe the dynamics of the spontaneous decay. The analysis shows that the spontaneous decay is modified by the Q value and the effective mode volume factor of each cavity eigenmode. The effective mode volume is generalized for cavities with nonorthogonal modes, which can be a very significant modification in the microcavity regime. It is shown that the spontaneous decay is not enhanced by the excess noise factor as claimed by other analyses.

DOI
10.1103/physrevlett.97.093601

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The spontaneous emission of an excited atom in a lossy cavity with nonorthogonal eigenmodes is analyzed. The quantum Langevin formalism is used to describe the dynamics of the spontaneous decay. The analysis shows that the spontaneous decay is modified by the Q value and the effective mode volume factor of each cavity eigenmode. The effective mode volume is generalized for cavities with nonorthogonal modes, which can be a very significant modification in the microcavity regime. It is shown that the spontaneous decay is not enhanced by the excess noise factor as claimed by other analyses.
The spatial dependence of the cavity modified spontaneous emission of an atom in a cavity with finite output coupling loss is analyzed. The analysis is based on the quantized nonorthogonal eigenmodes defined by the open cavity boundary condition. The spatial dependence of the Rabi oscillation frequency in the strong coupling regime is also analyzed. For spontaneous decay, the spatial dependence is a sinusoidal modulation on top of a dc value, where the interaction with all cavity modes is considered. For Rabi oscillations, where the atom interacts mainly with only one cavity mode, the oscillation frequency also has a sinusoidal spatial dependence.
For pt.IV see J. Phys. Soc. Japan, vol.47, p.217-224 (1982). Spontaneous emission in a lossy one-dimensional cavity is analyzed under the perturbation approximation and the Wigner-Weisskopf approximation. The cavity loss due to output coupling is treated quantum mechanically by the use of a set of continuously distributed radiation modes. The enhancement and reduction factors in the spontaneous emission probability due to the presence of the cavity structure are derived. Interesting transients in spontaneous emission during a time of the order of the cavity decay time are predicted, which leads to an interpretation of the enhancement or the reduction in spontaneous emission in terms of feedback and self-interference effects. The detection of the emitted photon by a detector atom outside the cavity is also discussed.>
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.
In the effective Hamiltonian representation, we have obtained a quantum stochastic differential equation of a generalized Langevin type for the evolution operator of an atomic ensemble in a microcavity in an external broadband quantized field and in a nonresonant field of the microcavity. We show that, depending on the number of particles in the atomic ensemble, its dynamics demonstrates both the Langevin and the generalized Langevin types of the two-photon spontaneous decay. In this case, one photon is emitted into the cavity mode, whereas the other photon is emitted into the external broadband electromagnetic field. The Langevin type is determined by a considerable Stark interaction of the atomic ensemble with the broadband photon-free quantized field. We show that, here, the Stark interaction is represented by a quantized Poisson process and, depending on its magnitude (at certain numbers of atoms in the ensemble), the two-photon collective spontaneous emission of microcavity atoms can be completely suppressed. In this case, the two-photon spontaneous emission of the singly excited atomic ensemble is described by the two-level model, while the atom-photon cluster of the microcavity under the described conditions is an artificial two-level quantum particle with a strong Stark interaction.
We show that the spontaneous emission of a moving atom inside a cavity is modified by two different mechanisms. The first one is a Doppler-induced detuning of the cavity line that leads to observable changes in the spontaneous emission rate for atomic velocities in excess of a characteristic velocity vcav{\mathit{v}}_{\mathrm{cav}}vcav​=c/Q, Q being the quality factor of the cavity. In addition, spatial mode effects can lead to a temporal modulation of the exponential decay rate. This modulation is observable for velocities that do not exceed another characteristic velocity vmod{\mathit{v}}_{\mathrm{mod}}vmod​ which is typically several orders of magnitude smaller than vcav{\mathit{v}}_{\mathrm{cav}}vcav​.
The results of a quantum-electrodynamical theory of excess spontaneous emission noise in lossy resonators will be presented. The " Petermann K factor" does not enter into the spontaneous emission rate of a single atom in the cavity. The QED theory allows different interpretations of the K factor and we use this fact to justify semiclassical analyses and to provide in one example a simple derivation of K in terms of the amplification of the quantum vacuum field entering the resonator through its mirrors.