The transition from non-Markovian to Markovian dynamics for generic environments

Using random matrices, we study the reduced dynamics of a two level system interacting with a generic environment. In the weak coupling limit, the result can be obtained directly from known results for purity decay, and result in Markovian dynamics. We then focus on the case of strong coupling, when the dynamics is known to be non-Markovian. In this regime, the coupling dominates over the local parts of the Hamiltonian, and thus we treat the latter as a perturbation of the former. With the help of the linear response approximation, this allows us to obtain an analytical description of the reduced dynamics. Finally, we find a transition from non-Markovian to Markovian dynamics at a point where the coupling and the local Hamiltonian are comparable in size.