Multiparty Spohn's theorem for a combination of local Markovian and non-Markovian quantum dynamics

We obtain a Gorini-Kossakowski-Sudarshan-Lindblad-like master equation for two or more quantum systems connected locally to a combination of Markovian and non-Markovian heat baths. The master equation was originally formulated for multiparty systems with either exclusively Markovian or non-Markovian environments. We extend it to encompass the case of multiple quantum systems connected to a mixture of Markovian and non-Markovian heat baths. The coexistence of both non-Markovian and Markovian environments is a plausible scenario, particularly when studying hybrid physical systems such as atom-photon arrangements. We analyze the thermodynamic quantities for such a set of local environments, and derive a modified form of the Spohn's theorem for the setup. The modification of the theorem naturally leads to a witness as well as an easily computable quantifier of non-Markovianity. Expectedly, we find that for multiparty situations, where a combination of Markovian and non-Markovian heat baths are active, the response in thermodynamic system characteristics due to non-Markovian baths is prominent at times close to the initial time of evolution, whereas the long-time behavior is predominantly controlled by the Markovian ones.