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Theoretical and Mathematical Physics

Memory Effects and Nonequilibrium Correlations in the Dynamics of Open Quantum Systems

V. G. Morozov

Published Jan 1, 2018• 9 citations
Publisher

Abstract

We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.

DOI
10.1134/s0040577918010075

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We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.
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.
We study the dynamical correlations in open quantum systems on an example of an exactly solvable dephasing model. The system of non-Markovian kinetic equations for the generalized coherence and quasi-temperature is derived up to the 2-nd order in the coupling constant using the generalized quantum master equation [20]. Numerical and analytical solutions of the kinetic equations are obtained. The analytical result shows a redistribution of the real and imaginary parts of the coherence as compared to the exact one. We give a physical interpretation to the quasi-temperature, relating it to the nonequilibrium kinetic energy of the environment.
Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level quantum system with finite number N of quantum eigenstates interacting with arbitrary external classical fields and dissipative environment simultaneously. It was shown that the structure of these equations can be simplified significantly if the free Hamiltonian driven dynamics of an arbitrary quantum multi-level system under the influence of the external driving fields as well as its Markovian and non-Markovian evolution, stipulated by the interaction with the environment, are described in terms of the SU(N) algebra representation. As a consequence, efficient numerical methods can be developed and employed to analyze these master equations for real problems in various fields of theoretical and applied physics. It was also shown that literally the same master equations hold not only for the reduced density operator but also for arbitrary nonequilibrium multi-time correlation functions as well under the only assumption that the system and the environment are uncorrelated at some initial moment of time. A calculational scheme was proposed to account for these lost correlations in a regular perturbative way, thus providing additional computable terms to the correspondent master equations for the correlation functions.
A systematic approach to the non-Markovian quantum dynamics of open systems is given by the projection operator techniques of nonequilibrium statistical mechanics. Combining these methods with concepts from quantum information theory and from the theory of positive maps, we derive a class of correlated projection superoperators that take into account in an efficient way statistical correlations between the open system and its environment. The result is used to develop a generalization of the Lindblad theory to the regime of highly non-Markovian quantum processes in structured environments.
The nonlinear quantum kinetic theory for many-body systems either near or far from equilibrium that a nonequilibrium ensemble formalism provides is revisited. In this communication we consider an important limit of such transport equations, consisting of the memoryless approximation, which leads to the so-called Markovian kinetic equations. They are derived in Zubarev's approach to the method, and next applied to a particular model of a spin system in interaction with a thermal bath of lattice vibrations. The limitations of the approach, as well as some criticism it has received, are discussed.
Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level open quantum system with finite number NNN of quantum eigenstates interacting with arbitrary external deterministic fields and dissipative environment simultaneously. Unlike conventional master equations derived by the projection operator method for the quantum systems of this kind so far, these equations are homogeneous in the reduced statistical operator while, at the same time, taking into account initial correlations between the multi-level system and the environment. Consistent representation of the multi-level system evolution in terms of the SU(N)SU(N)SU(N) operator algebra facilitates efficient numerical analysis of these equations derived for various physically realistic quantum models.