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-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-M. We provide a detailed microscopic derivation of our result in terms of a mapping between two collision models.