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Stochastic Master Equations in Thermal Environment

Stéphane Attal, université Lyon, Clément Pellegrini

Published Sep 1, 2008• 0 citations
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Abstract

Abstract. We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant. 1.

DOI
10.1142/s123016121000024

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Abstract. We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant. 1.
We derive stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. It is well-known that at zero temperature one obtains stochastic differential equations of jump-diffusion type. We show that only pure diffusion type equations are relevant at strictly positive temperatures.
In this paper, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath has a clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
In this article, we derive the stochastic master equations corresponding to the sta-tistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
In the presence of quantum measurements with direct photon detection, the evolution of open quantum systems is usually described by stochastic master equations with jumps. Heuristically, diffusion models can be obtained from these equations as an approximation. A condition for a general diffusion approximation for jump master equations is presented. This approximation is rigorously proved by using techniques for Markov processes, which are based on the convergence of Markov generators and martingale problems. This result is illustrated by rigorously obtaining the diffusion approximation for homodyne and heterodyne detection.
A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutioness Master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non Markovian stochastic Master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed. 1
A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutionless master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non-Markovian stochastic master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed.