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We study the dynamics of heat flux in the thermalization process of a pair ofidentical quantum system that interact dissipatively with a reservoir in a {\itcascaded} fashion. Despite the open dynamics of the bipartite system S isglobally Lindbladian, one of the subsystems "sees" the reservoir in a statemodified by the interaction with the other subsystem and hence it undergoes anon-Markovian dynamics. As a consequence, the heat flow exhibits anon-exponential time behaviour which can greatly deviate from the case whereeach party is independently coupled to the reservoir. We investigate boththermal and correlated initial states of $S$ and show that the presence ofcorrelations at the beginning can considerably affect the heat flux rate. Wecarry out our study in two paradigmatic cases -- a pair of harmonic oscillatorswith a reservoir of bosonic modes and two qubits with a reservoir of fermionicmodes -- and compare the corresponding behaviours. In the case of qubits andfor initial thermal states, we find that the trace distance discord is at anytime interpretable as the correlated contribution to the total heat flux.

Heat flux dynamics in dissipative cascaded systems