Recursive approach for non-Markovian time-convolutionless master equations

We consider a general open system dynamics and we provide a recursive method to derive the associated non-Markovian master equation in a perturbative series. The approach relies on a momenta expansion of the open system evolution. Unlike previous perturbative approaches of this kind, the method presented in this paper provides a recursive definition of each perturbative term. Furthermore, we give an intuitive diagrammatic description of each term of the series, which provides a useful analytical tool to build them and to derive their structure in terms of commutators and anticommutators. We eventually apply our formalism to the evolution of the observables of the reduced system, by showing how the method can be applied to the adjoint master equation, and by developing a diagrammatic description of the associated series.