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The Theory of Open Quantum Systems$
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Heinz-Peter Breuer and Francesco Petruccione

Print publication date: 2007

Print ISBN-13: 9780199213900

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199213900.001.0001

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The Stochastic Simulation Method

The Stochastic Simulation Method

The Theory of Open Quantum Systems

Heinz-Peter Breuer

Francesco Petruccione

Oxford University Press

The formulation of the dynamics of open quantum systems by means of stochastic processes in Hilbert space leads to efficient Monte–Carlo simulation techniques which are introduced and examined in this chapter. Depending on whether the stochastic process is a piecewise deterministic process or a diffusion process, the corresponding individual realizations consist of intervals of deterministic evolution periods interrupted by instantaneous quantum jumps, or of continuous, nowhere differentiable paths. Various Monte Carlo algorithms for both types of processes are described in detail, and their convergence behaviour and their numerical performance is investigated for a number of applications, such as the damped harmonic oscillator, the driven two-level system, and the damped driven Morse oscillator.

Keywords:   piecewise deterministic processes, stochastic Schrödinger equations, stochastic differential equations, Monte-Carlo simulations, Euler algorithm, Heun algorithm, Runge–Kutta algorithm, Platen–Kloeden algorithm, harmonic oscillator, driven Morse oscillator

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