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

Abstract

An open quantum system that is put in contact with an infinite bath is pushed towards equilibrium, while the state of the bath remains unchanged. If the bath is finite, the open system still relaxes to equilibrium but it induces a dynamical evolution of the bath state. In this work, we extend the weak-coupling master equation approach of open quantum systems interacting with finite baths to include imprecise measurements of the bath energy. Those imprecise measurements are not only always the case in practice, but they also unify the theoretical description. We investigate the circumstances under which our equation reduces to the more standard Born-Markov-secular master equation. As a result, we obtain a hierarchy of master equations that improve their accuracy by including more dynamical information about the bath. We discuss this formalism in detail for a particular non-interacting environment where the Boltzmann temperature and the Kubo-Martin-Schwinger relation naturally arise. Finally, we apply our hierarchy of master equations to study the central spin model.

Apparent heating due to imperfect calorimetric measurements

“…We are now in a position to derive a qubit-energy hybrid master equation for the partially reduced density ρ(E, t) by generalizing the usual steps of a weak coupling Born-Markov treatment [11]. An alternative route in terms of the Nakajima-Zwanzig projection operator technique [33,34] is presented in appendix A, see also [26,27,30,31] for similar applications and the recent work [35]. We will not perform the Davies weak coupling limit prescription here explicitly, as it is the equivalent to the Born-Markov treatment we present below.…”