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Physica A Statistical Mechanics and its Applications

Quasi-equilibrium states of open systems via quantum dynamical master equations

W. E. Hagston, M. Roberts, I. D. Cox

Published Oct 1, 1982• 2 citations
Publisher
DOI
10.1016/0378-4371(82)90032-2

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The master equation for an open quantum system is derived in the weak-coupling approximation when the additional dynamical variable—the mean interaction energy—is included into the generic relevant statistical operator. This master equation is nonlocal in time and involves the “quasi-temperature”, which is a non- equilibrium state parameter conjugated thermodynamically to the mean interaction energy of the composite system. The evolution equation for the quasi-temperature is derived using the energy conservation law. Thus long-living dynamical correlations, which are associated with this conservation law and play an important role in transition to the Markovian regime and subsequent equilibration of the system, are properly taken into account.
Open quantum system dynamics is efficiently described by the quantum master equation formalism. Therein, quantum master equations in Lindblad form constitute an important subclass describing Markovian dynamics. When an open quantum system is in an out-of-equilibrium state, an exchange of particles between the open system and reservoirs takes place yielding to a non-zero average net current and associated current fluctuations, which can be characterised with the quantum jump formalism for quantum master equations expressed in Lindblad form. However, a large class of quantum master equations cannot be described by Lindblad dynamics. Here we assess the validity and the effectiveness of the quantum jump formalism when the dissipators in the quantum master equation describe a non-additive, open quantum system dynamics. We find that an additive unravelling of the non-additive quantum master equation does not generate a completely-positive dynamics in the scenario of perfect jump detection. Nevertheless, allowing for an imperfect jump detection scenario, we find that an additive unravelling is possible that reproduces the current and the fluctuations obtained via the Landauer-Büttiker formalism.
An open quantum system in contact with an infinite bath approaches 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 paper, we study the dynamics of open quantum systems in contact with finite baths. We obtain a hierarchy of master equations that improve their accuracy by including more dynamical information of the bath. For instance, as the least accurate but simplest description in the hierarchy, we obtain the conventional Born-Markov-secular master equation. Remarkably, our framework works even if the measurements of the bath energy are imperfect, which not only is more realistic but also unifies the theoretical description. Also, we discuss this formalism in detail for a particular noninteracting 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.