Approximation scheme and non-Hermitian renormalization for the description of atom-field-system evolution

Interactions between a source of light and atoms are ubiquitous in nature. The study of them is interesting on the fundamental level as well as for applications. They are at the core of quantum information processing tasks and quantum thermodynamics protocols. However, even for a two-level atom interacting with a field in rotating wave approximation there exists no exact solution. This touches upon a basic problem in quantum field theory, where we can only calculate the transitions in the time asymptotic limits (i.e., minus and plus infinity), while we are not able to trace the evolution. In this paper we want to get more insight into the time evolution of a total system of a two-level atom and a continuous-mode quantum field. We propose an approximation, which we are able to apply systematically to each order of Dyson expansion, resulting in a greatly simplified formula for the evolution of the combined system at any time. Our tools include a proposed non-Hermitian renormalization method. As a sanity check, by applying our framework, we derive the known optical Bloch equations.