A master equation approach for the interaction of an atom with a dielectric semi-infinite medium

We use the master equation approach to calculate the energy level shifts of an atom in the presence of a general dielectric semi-infinite medium characterized by a dielectric constant $\epsilon(\omega)$. Particularly, we analyze the case of a non-dispersive medium for which we obtain a general expression for the interaction as well as the asymptotic behaviors for $k_0 z \ll 1$ (non-retarded regime) and $k_0 z \gg 1$ (retarded regime), where $\omega_0 = k_0 c$ is the main transition frequency of the atom. The limiting cases $\epsilon \simeq 1$ and $\epsilon \gg 1$ are discussed for both retarded and non-retarded limits. For the retarded limit, we compute the non-additivity contribution of van der Waals forces.