Spectral properties of spontaneous photon emission by a material two-level system in a parabolic cavity

The spectral properties of a photon spontaneously emitted by a material two-level system, modelling an atom or ion, in a parabolic cavity are investigated. In particular, we concentrate on the special case of a motionless two-level system positioned exactly at the focus of a parabolic cavity with a dipole moment oriented along the symmetry axis of this cavity. Treating the corresponding atom-field coupling in the dipole- and rotating wave approximation, it is demonstrated that inside the parabolic cavity the position and frequency dependence of the spectrum of the spontaneously emitted photon exhibits interesting interference patterns. These patterns are explored in detail with the help of a photon path representation of the first-order electric field correlation function. In the radiation, zone the spatial behavior of the spectrum reveals strong interference in particular at distances from the two-level system of the order of the focal length of the parabola. With increasing distances, these interference patterns decay except for an undepleted component surrounding the symmetry axis at an almost constant radius. Furthermore, the maximum of the frequency dependence of the spectrum exhibits a position-dependent frequency shift with respect to the atomic resonance frequency.