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The Scientific World JOURNAL

Simulation of Quantum Dynamics Based on the Quantum Stochastic Differential Equation

Ming Li

Published Jan 1, 2013• 4 citations
Publisher

Abstract

The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.

DOI
10.1155/2013/424137

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The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.
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The simulation methods based on stochastic realizations of state vector evolutions derived from the time convolutionless master equation are investigated. The quantum jumps for non-Markovian systems and their waiting time distribution is mainly focused on. With the help of an exact analytical waiting time distribution, the non-Markovian quantum trajectory for a stochastic process of a driven two-level system is generated. The effectiveness and superiority of the algorithm is verified by the performance comparing to our previous work in the sense of Markovian prosperities.
The quantum-stochastic-differential-equation formulation of driven quantum-optical systems is carried out in the interaction picture, and quantum stochastic differential equations for wave functions are derived on the basis of physical principles. The Ito form is shown to be the most practical, since it already contains all the radiation reaction terms. The connection between this formulation and the master equation is shown to be very straightforward. In particular, a direct connection is made to the theory of continuous measurements, which leads directly to the method of quantum-jump simulations of solutions of the master equation. It is also shown that all conceivable spectral and correlation-function information in output fields is accessible by means of an augmentation of the simulation process. Finally, the question of the reality of the jumps used in the simulations is posed.
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.
A model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation, derived from a density operator equation such as the Bloch equation, and consistent with it. The advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described. The method is applied to the nonlinear absorber, cascades of quantum transitions, second-harmonic generation and a measurement reduction process. The model provides graphic illustrations of these processes, with statistical fluctuations that mimic those of experiments. The stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics. The model is compared with the quantum-jump model of Dalibard (1992), Carmichael and others, which originated among experimenters looking for intuitive pictures and rules of computation.
Stochastic differential equation (SDE) and Fokker-Planck equation (FPE) are two general approaches to describe the stochastic drift-diffusion processes. Solving SDEs relies on the Monte Carlo samplings of individual system trajectory, whereas FPEs describe the time evolution of overall distributions via path integral alike methods. The large state space and required small step size are the major challenges of computational efficiency in solving FPE numerically. In this paper, a generic continuous-time quantum walk formulation is developed to simulate stochastic diffusion processes. Stochastic diffusion in one-dimensional state space is modeled as the dynamics of an imaginary-time quantum system. The proposed quantum computational approach also drastically accelerates the path integrals with large step sizes. The new approach is compared with the traditional path integral method and the Monte Carlo trajectory sampling.