Non-Markovian thermalisation of a two-level atom

We consider thermalisation and spontaneous decay of a two-level atom beyond the Markovian approximation. While the standard elimination of the continuum of radiation modes results in exponential decay represented by a Lindblad equation of motion, we use a simple toy model that takes into account the finite relaxation rate of the environment and present an exact non-Markovian master equation of the Nakajima-Zwanzig form. Because the exact derivation of non-Markovian equations has proved very difficult for all more realistic (and hence much more complicated) models, we analyze the master equation obtained and also discuss difficulties that are likely to arise with non-Markovian evolution operators.