Class of exact memory-kernel master equations

A well-known situation in which a non-Markovian dynamics of an open quantum system $S$ arises is when this is coherently coupled to an auxiliary system $M$ in contact with a Markovian bath. In such cases, while the joint dynamics of $S\text{−}M$ is Markovian and obeys a standard (bipartite) Lindblad-type master equation (ME), this is in general not true for the reduced dynamics of $S$. Furthermore, there are several instances (e.g., the dissipative Jaynes-Cummings model) in which a closed ME for the $S$'s state cannot even be worked out. Here, we find a class of bipartite Lindblad-type MEs such that the reduced ME of $S$ can be derived exactly and in a closed form for any initial product state of $S\text{−}M$. We provide a detailed microscopic derivation of our result in terms of a mapping between two collision models.

Figure 1

(a) CM with internal memory: interancillary collisions are interspersed with system-ancilla collisions. (b) Equivalent Markovian collision model for $S\text{−}M$: only the auxiliary system $M$ undergoes successive interactions with the bath ancillas, which are interspersed with unitary $S\text{−}M$ collisions. In the latter model no interancillary collisions occur.