Atomic decay and re-absorption in a one-dimensional cavity: Analytical results

I use Laplace transforms to ¯nd a complete analytical solution to the problem of a single two-level atom interacting with the quantized modes of a one-dimensional multimode optical cavity. The techniques were used by Stey and Gibberd [Physica, 60, 1-26 (1972)] on several model Hamiltonians, and this paper is essentially a translation of one of their model Hamiltonians into the language of quantum optics. The results of this paper are a complement to the numerical work presented in Ligare and Becker [Am. J. Phys., 63, 788-796 (1995)] and Ligare and Oliveri [Am. J. Phys., 70 58-66 (2002)]. 1 Simple fully quantized models of spontaneous emission and re-absorption have been investigated previously by the author [1,2]. In this paper I show how Laplace transforms can be used to ¯nd analytical solutions to the previously considered problems using the techniques introduced by Stey and Gibberd [3]. In this note I develop a Hamiltonian for a quantum optical system that is identical to a Hamiltonian considered in their paper. Additional mathematical details are presented in reference [3]. The system considered consists of a single two-level atom located at the position xa in an optical cavity of length L. The atom is coupled to all of the quantized radiation modes of the cavity. The angular frequencies of the modes are given by!j = (j0 + j)c=L; (1) where j0 is the label for the even-numbered mode closest to the atomic resonance,!eg. The modes are evenly spaced with angular frequency separation ¢c