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arXiv:1403.7172 · Doklady Mathematics

Feynman, Wigner, and Hamiltonian structures describing the dynamics of open quantum systems

John Edward Gough, Tudor Stefan Ratiu, Oleg Georgievich Smolyanov

Published Jan 1, 2014• 12 citations
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Abstract

This paper discusses several methods for describing the dynamics of open quantum systems, where the environment of the open system is infinite-dimensional. These are purifications, phase space forms, master equation and liouville equation forms. The main contribution is in using Feynman-Kac formalisms to describe the infinite-demsional components.

DOI
10.1134/s1064562414010190

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This paper discusses several methods for describing the dynamics of open quantum systems, where the environment of the open system is infinite-dimensional. These are purifications, phase space forms, master equation and liouville equation forms. The main contribution is in using Feynman-Kac formalisms to describe the infinite-demsional components.
This paper describes the states of quantum versions of finite dimensional Hamilton–Dirac systems in terms of (analogues of) the Wigner function (various definitions of Wigner functions for classical Hamiltonian systems can be found in [8]). We give a system of equations, which we call the Moyal–Dirac system, describing the evolution of this Wigner function. The passage from the Moyal equation describing the evolution of the quantum version of a classical Hamiltonian system to the Moyal–Dirac system describing the evolution of the quantum version of the corresponding Hamilton–Dirac system is similar to the passage from the Liouville equation describing the evolution of a classical Hamiltonian system to the Liouville–Dirac system of equations describing the evolution of the corresponding Hamilton–Dirac system. Thus, our Moyal–Dirac system is a generalization of both the Liouville–Dirac system and the Moyal equation
A theoretical framework for the nonequilibrium statistical mechanics of open systems is constructed. This is concerned with a formulation of a generalized master equation governing the evolution of an arbitrary system S in interaction with a ``large'' reservoir R. The dynamics of S are analyzed on the basis of a precise quantum-mechanical treatment of the microscopic equations of motion for the combined system S + R. On proceeding to the thermodynamical limit for R we obtain a generalized master equation for S, subject to specified conditions on the many-particle structure of R, its initial state, and its coupling to S. This master equation corresponds to a self-contained law of motion for S, in which the R variables appear only in the forms of certain thermal averages, taken over the initial state. This dynamical law is a generalization of the quantum-mechanical Liouville equation to a form appropriate to open systems.
This review gives a summary of the many techniques that are used in the analysis of open quantum systems. Emphasis is on those cases where it is unsuitable to use a memoryless or Markovian point of view, generally because there is no large separation of time scales between system and environment dynamics. The approaches reviewed include master equations, Heisenberg equations of motion, chain mapping representations, and various stochastic methods such as path integral Monte Carlo and stochastic equations. Guidance is given on how to evaluate the suitability of each of these methods for application in different physical problems.
Using the age-structure formalism, we definitely establish connections between semi-Markov processes and the dynamics of open quantum systems that satisfy the Markov quantum master equations. A generalized Feynman-Kac formula of the semi-Markov processes is also proposed. In addition to inheriting all statistical properties possessed by the piecewise deterministic processes of wave functions, the semi-Markov processes show their unique advantages in quantum counting statistics. Compared with the conventional method of the tilted quantum master equation, they can be applied to more general counting quantities. In particular, the terms involved in the method have precise probability meanings. We use a driven two-level quantum system to exemplify these results.