Memory kernel and divisibility of Gaussian collisional models

Memory effects in the dynamics of open systems have been the subject of significant interest in the last decades. The methods involved in quantifying this effect, however, are often difficult to compute and may lack analytical insight. With this in mind, we study collisional models where non-Markovianity is introduced by means of additional interactions between neighboring environmental units. We show that the dynamics can be cast in terms of a Markovian embedding of the covariance matrix, which yields closed-form expressions for the memory kernel that governs the dynamics, a quantity that can seldom be computed analytically. The same is also possible for a divisibility monotone, based on the complete positivity of intermediate maps. By focusing on continuous-variable Gaussian dynamics, we are able to analytically study models of arbitrary size. We analyze in detail two types of interactions, a beam splitter implementing a partial SWAP and a two-mode squeezing, which entangles the ancillas and, at the same time, feeds excitations into the system. By analyzing the memory kernel and divisibility for these two representative scenarios, our results help to shed light on the intricate mechanisms behind memory effects in the quantum domain.