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Physical Review E

Studying open quantum systems by means of a deterministic approach to approximate functional integration

V.D. Rushay, Yu. Yu. Lobanov

Published Jun 30, 2005• 0 citations
Publisher

Abstract

Representation of propagator for open quantum systems in the form of a double functional integral with respect to conditional Wiener measure is proposed. It allows one to apply the approximate formulas exact for functional polynomials of a certain power to calculation of such integrals. Within this deterministic approach the problem is reduced to evaluation of usual (Riemann) integrals of low multiplicity. The formulas are in fact the basis of a numerical method of studying time evolution of the systems. The features of the method are discussed and some examples of calculations are given.

DOI
10.1103/physreve.71.066708

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Representation of propagator for open quantum systems in the form of a double functional integral with respect to conditional Wiener measure is proposed. It allows one to apply the approximate formulas exact for functional polynomials of a certain power to calculation of such integrals. Within this deterministic approach the problem is reduced to evaluation of usual (Riemann) integrals of low multiplicity. The formulas are in fact the basis of a numerical method of studying time evolution of the systems. The features of the method are discussed and some examples of calculations are given.
Abstract The application of the numerical functional integration method to the solution of differential equations in quantum physics is discussed. We have developed a method of numerical evaluation of functional integrals in abstract complete separable metric spaces, which proves to have important advantages over the conventional Monte Carlo method of path integration. One of the considered applications is the investigation of open quantum systems (OQS), i.e., systems interacting with their environment. The density operator of OQS satisfies the known Lindblad differential equation. We have obtained the expression for matrix elements of this operator in the form of the double conditional Wiener integral and considered its application to some problems of nuclear physics. Another application is the solution of the Scr¨odinger equation with imaginary time and anticommuting variables for studying many-fermion systems. We have developed a numerical method based on functional integration over ordered subspaces. The binding energies of some nuclei are computed using this method. Comparison of the results with those obtained by other authors and with experimental values is presented.
We describe an efficient path integral scheme for calculating the propagator of an arbitrary quantum system, as well as that of a stochastic system in special cases where the Fokker–Planck equation obeys strict detailed balance. The basic idea is to split the respective Hamiltonian into two exactly solvable parts and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The resulting single step propagator allows rather large time steps in a path integral and leads to convergence with fewer time slices. Because it involves no system-specific reference system, the algorithm is amenable to all known numerical schemes available for evaluating quantum path integrals. In this way one obtains a highly accurate method, which is simultaneously fast, stable, and computationally simple. Numerical applications to the real time quantum dynamics in a double well and to the stochastic dynamics of a bistable system coupled to a harmonic mode show our method to be superior over the approach developed by the Makri group in their quasiadiabatic propagator representation, to say nothing about the propagation scheme based on the standard Trotter splitting.
The coherent-state representation of quantum-mechanical propagators as well-defined phase-space path integrals involving Wiener measure on continuous phase-space paths in the limit that the diffusion constant diverges is formulated and proved. This construction covers a wide class of self-adjoint Hamiltonians, including all those which are polynomials in the Heisenberg operators; in fact, this method also applies to maximal symmetric Hamiltonians that do not possess a self-adjoint extension. This construction also leads to a natural covariance of the path integral under canonical transformations. An entirely parallel discussion for spin variables leads to the representation of the propagator for an arbitrary spin-operator Hamiltonian as well-defined path integrals involving Wiener measure on the unit sphere, again in the limit that the diffusion constant diverges.
A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.
We propose the differential equation based path integral method to simulate the real-time evolution of open quantum systems. In this method, a system of partial differential equations is derived based on the continuation of a classical numerical method called the iterative quasi-adiabatic propagator path integral (i-QuAPI). While the resulting system has infinite equations, we introduce a reasonable closure to obtain a series of finite systems. New numerical schemes can be derived by discretizing these differential equations. It is numerically verified that in certain cases, by selecting appropriate systems and applying suitable numerical schemes, the memory cost required in the i-QuAPI method can be significantly reduced.
A number of problems in statistical physics can be reformulated in terms of a two-state system evolving in a random field. The corresponding evolution operator can be written in the form of time-ordered operator exponential. Functional formalism allows us to rewrite the latter as a product of usual matrix exponentials using a nonlinear change of functional integration variables. In this review I present this formalism applied to two physical systems the quantum Heisenberg magnet and one-dimensional quantum mechanics in a spatially random potential. First, I derive a representation of the partition function of a quantum Heisenberg ferromagnet as a functional integral over number valued fields (a real one and a complex one) free of constraints. The fields of integration as functions of time obey initial conditions instead of the usual periodic boundary conditions. This is a manifestation of the finite-dimensionality of the space of spin states. In the subsequent sections I study the one-dimensional localization problem. The change of functional integration variables gives simultaneously explicit expressions for the averaging weight and for the Green function of the stationary Schrödinger equation. It allows to compute density correlators of arbitrary orders. The generalization to the case of different energy correlators (Berezinskii-Gor’kov equations) is considered too. In the present review such technical points as regularizations of functional integrals and transformations are discussed in more details than in the original papers.