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

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.This paper discusses several approaches for describing the dynamics of open quantum systems. Open quantum systems play an important role in modelling physical systems coupled to their environment and, in particular, for the emerging field of quantum feedback control theory (see [12]). Thus, in studying coherent quantum feedback, models consisting of a quantum system related to quantum control system, so that each of these systems turns out to be open, are considered.Generally the master equation is presented in the theoretical physics literature as the central description of an open quantum system. In practice, however, it is these solutions that are important for potential applications, rather than the master equations themselves. Our goal is to solve the exact master equations, which describe the reduced dynamics of subsystems of certain large systems generated by the dynamics of these large systems: these master equation arise from a number of different approaches which we will consider.In fact, we examine four approaches for describing subsystems dynamics, and in each case we exploit Feynman type formulas (see [1,2]). Moreover, we assume that the quantum systems under consideration are obtained by the Schrödinger quantization [3] of classical Hamiltonian systems.1. Our first approach is based on a representation of mixed states as random pure ones, where the dynamics of a subsystem of the isolated quantum

Wigner Measures and Coherent Quantum Control

“…If K = E = Q × P and F K (q, p) = e i q,p , then F K is called the Hamiltonian Feynman measure; it can be used to introduce the Fourier transform that acts on functions defined on infinite-dimensional spaces and maps them into measures. Here the structure of the Hilbert space matters little, and, like the Gaussian measure, the Feynman measure can be defined on any LCS; in particular, the Hamiltonian Feynman measure can be defined on any symplectic LCS (see [3,12,15] for more information).…”

Invariant states of linear quantum stochastic systems under Weyl perturbations of the Hamiltonian and coupling operators

“…The QCF and its Fourier transform -the Wigner quasi-probability density function (QPDF) -constitute the basis of the Wigner-Moyal phase-space approach [16,38] which avoids the "burden of the Hilbert space" by representing the quantum state dynamics in the more conventional form of PDEs and IDEs involving only real or complex variables and functions thereof. Although the Moyal equations [38] for the QCF and QPDF dynamics were obtained originally for isolated quantum systems, there also are extensions of the phase-space approach to different classes of open quantum systems [14,15,29,33,62].…”

A phase-space formulation of the Belavkin-Kushner-Stratonovich filtering equation for nonlinear quantum stochastic systems

“…The phase-space approach allows the quantum dynamics to be represented without the "burden of the Hilbert space" and leads to partial differential and integro-differential equations for the QPDFs and QCFs, which involve only real or complex variables and encode the moments of the system operators. Although the Moyal equations [33] for the QPDF dynamics were originally obtained for closed systems, the phase-space approach has also extensions to different classes of open quantum systems; see, for example, [16], [17], [28], [31], [45].…”