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Physical Review Letters

Reduced System Dynamics from theN-Body Schrödinger Equation

A. Muriel

Published Sep 18, 1995• 11 citations
Publisher

Abstract

We derive the time-dependent wave function for one particle in the company of $N\ensuremath{-}1$ other particles using the method of iterated projection. Using this approach, we confirm the general philosophy of describing a quantum system by a flow component and another component which may be given various descriptions as stochastic, heat bath, or quantum jump processes. The possible equivalence of the Breuer-Petruccione reduced system dynamics [Phys. Rev. Lett. 74, 3788 (1995)] and ours is explored.

DOI
10.1103/physrevlett.75.2255

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Comment on “Reduced System Dynamics from theN-Body Schrödinger Equation”

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We argue that the "reduced wave function", proposed recently [Phys.Rev.Lett. {\bf 75}, 2255 (1995)], contains conditional and restricted information on the reduced system. The concept of "reduced wave function" can thus not represent a relevant alternative to the common reduced dynamics methods.

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We derive the time-dependent wave function for one particle in the company of N\ensuremath−1N\ensuremath{-}1N\ensuremath−1 other particles using the method of iterated projection. Using this approach, we confirm the general philosophy of describing a quantum system by a flow component and another component which may be given various descriptions as stochastic, heat bath, or quantum jump processes. The possible equivalence of the Breuer-Petruccione reduced system dynamics [Phys. Rev. Lett. 74, 3788 (1995)] and ours is explored.
Employing a formulation of quantum statistical ensembles in terms of classical probability distributions on projective Hilbert space, it is shown that the wave function of an open quantum system represents a stochastic process. The stochastic dynamics of the state vector is obtained from a microscopic system-plus-reservoir model by deriving within the Markov approximation the differential Chapman-Kolmogorov equation for the classical distribution of the reduced system. The realizations of the stochastic process are found to be similar to those of the Monte Carlo wave function simulation method proposed, in general form, by Zoller et al. [Phys. Rev. A 46, 4363 (1992)].
The dynamics of an open quantum system coupled to an external reservoir is studied on the basis of a recently proposed formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. The previous result is generalized to include interaction Hamiltonians of the form tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​Ai{\mathit{A}}_{\mathit{i}}Ai​\ensuremath{\bigotimes}Bi{\mathit{B}}_{\mathit{i}}Bi​, where Ai{\mathit{A}}_{\mathit{i}}Ai​ and Bi{\mathit{B}}_{\mathit{i}}Bi​ are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schr\"odinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.
We discuss in detail how non-Markovian open system dynamics can be described in terms of quantum jumps [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. Our results demonstrate that it is possible to have a jump description contained in the physical Hilbert space of the reduced system. The developed non-Markovian quantum jump approach is a generalization of the Markovian Monte Carlo wave function (MCWF) method into the non-Markovian regime. The method conserves both the probabilities in the density matrix and the norms of the state vectors exactly and sheds new light on non-Markovian dynamics. The dynamics of the pure state ensemble illustrates how local-in-time master equation can describe memory effects and how the current state of the system carries information on its earlier state. Our approach solves the problem of negative jump probabilities of the Markovian MCWF method in the non-Markovian regime by defining the corresponding jump process with positive probability. The results demonstrate that in the theoretical description of non-Markovian open systems, there occurs quantum jumps which recreate seemingly lost superpositions due to the memory.
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 report a study of a stochastic Schrödinger equation corresponding to the Redfield master equation with slipped initial conditions, which describes the dynamics of a slow subsystem weakly coupled to a fast thermal bath. Using the projection-operator method of Feshbach, we derive a non-Markovian stochastic Schrödinger equation of the generalized Langevin type, which simulates the time evolution of the quantum wave functions of the subsystem driven by the fluctuating bath. For δ-correlated baths, the non-Markovian stochastic Schrödinger equation reduces to the previously derived Markovian one. Numerical methods are proposed to simulate the time evolution under our non-Markovian stochastic Schrödinger equation. These methods are illustrated with the spin-boson model.