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An open quantum system in contact with an infinite bath approachesequilibrium, while the state of the bath remains unchanged. If the bath isfinite, the open system still relaxes to equilibrium, but it induces adynamical evolution of the bath state. In this work, we study the dynamics ofopen quantum systems in contact with finite baths. We obtain a hierarchy ofmaster equations that improve their accuracy by including more dynamicalinformation of the bath. For instance, as the least accurate but simplestdescription in the hierarchy we obtain the conventional Born-Markov-secularmaster equation. Remarkably, our framework works even if the measurements ofthe bath energy are imperfect, which, not only is more realistic, but alsounifies the theoretical description. Also, we discuss this formalism in detailfor a particular non-interacting environment where the Boltzmann temperatureand the Kubo-Martin-Schwinger relation naturally arise. Finally, we apply ourhierarchy of master equations to study the central spin model.

Open quantum systems coupled to finite baths: A hierarchy of master equations