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Annalen der Physik

On the Theory of Spontaneous Emission by a Single Atom Located in an Ideal Cavity

Henryk Paul

Published Jan 1, 1986• 1 citation
Publisher

Abstract

Abstract With the help of the well‐known eigenfunctions for a two‐level system coupled to a single mode of the radiation field, the correlation functions for both the atomic dipole and the electric field oscillation are calculated, in rotating‐wave approximation, for an atom emitting spontaneously into an ideal cavity. Using the Eberly‐Wódkiewicz formalism, the corresponding spectra are evaluated in the long‐time, narrow‐band detection limit. The two spectra exhibit the same Rabi splitting, but the ratio of the strengths of the two lines is significantly different, in the off‐resonance case, for the dipole and the field oscillation.

DOI
10.1002/andp.19864980614

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Journal of physics. A, Proceedings of the Physical Society. General•Nov 1, 1972•12 citations
Abstract With the help of the well‐known eigenfunctions for a two‐level system coupled to a single mode of the radiation field, the correlation functions for both the atomic dipole and the electric field oscillation are calculated, in rotating‐wave approximation, for an atom emitting spontaneously into an ideal cavity. Using the Eberly‐Wódkiewicz formalism, the corresponding spectra are evaluated in the long‐time, narrow‐band detection limit. The two spectra exhibit the same Rabi splitting, but the ratio of the strengths of the two lines is significantly different, in the off‐resonance case, for the dipole and the field oscillation.
The spectrum of spontaneously emitted photons from an atom interacting with a single cavity mode depends on the relative sizes of the mode coupling constant and the cavity loss rate. If the coupling constant is not too large, perturbation theory can he used to show that the atomic linewidth is either increased (enhanced spontaneous emission) or decreased (inhibited spontaneous emission) by the presence of the cavity. When the coupling constant is so large that perturbation theory cannot be used, the spontaneous- emission spectrum forms a doublet, with peaks located at the normal-mode frequencies for the coupled-dipole and cavity-mode oscillators (the so-called "vacuum" Rabi splitting).
The problem of a single two level atom in its excited state, placed in a cavity in which no photons are present initially, and interacting with N modes of the electromagnetic field is solved exactly in the electric dipole, rotating wave approximation. Expressions for the energy eigenvalues and eigenvectors are derived, and an exact representation for the time dependent wavefunction of the system is given. These quantities are evaluated analytically for N=1,2 and 3, and the time dependent properties of the system for these values of N are investigated and discussed.
A composite atom-cavity system exhibits complex behavior with respect to spontaneous emission, resonance fluorescence, and transmission spectra.1–3 An early investigation1 showed that for an atom in an ideal cavity, the spontaneous-emission spectra show a doublet splitting in the case of very weak excitation. This splitting, which arises from vacuum-field Rabi oscillation, is regarded as an inherently quantum phenomenon.1,2 However, recent experiments and theoretical analyses have demonstrated that the multiatom vacuum Rabi splitting may be regarded as a feature of linear dispersion theory.3 This approach is dramatically different from the one employed in QED descriptions of the problem.
We present a nonperturbative result for the spectrum of spontaneous photons emitted by an atom coupled on resonance to a standing-wave cavity mode. Motion of the atom through the standing-wave mode function is included in a classical way. We discuss the form of the spectrum as a function of the velocity of the atom in the limit of strong dipole coupling. For slow atoms the normal-mode doublet (``vacuum'' Rabi spectrum) survives with small changes. The motion of a fast atom through the standing wave averages the coupling constant to zero, and the spectrum reverts to a single peak. At intermediate velocities the spectrum has a complicated multipeaked structure that can be understood in terms of interfering probability amplitudes for different quantum-mechanical emission paths or as a frequency-modulation spectrum for radiating coupled oscillators.