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arXiv:0907.3996 · Journal of Mathematical Physics

Diffusion approximation of stochastic master equations with jumps

Clément Pellegrini, F. Petruccione

Published Dec 1, 2009• 8 citations
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Abstract

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.

DOI
10.1063/1.3263941

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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.
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 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 upon the convergence of Markov generators and martingale problems. This result is illustrated by rigorously obtaining the diffusion approximation for homodyne and heterodyne detection.
"Quantum trajectories" are solutions of stochastic differential equations also called Belavkin or Stochastic Schrödinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually considered, one is driven by a one-dimensional Brownian motion and the other is driven by a counting process. In this article, we present a way to obtain more advanced models which use jump-diffusion stochastic differential equations. Such models come from solutions of martingale problems for infinitesimal generators. These generators are obtained from the limit of generators of classical Markov chains which describe discrete models of quantum trajectories. Furthermore, stochastic models of jump-diffusion equations are physically justified by proving that their solutions can be obtained as the limit of the discrete trajectories.
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.
Les trajectoires quantiques sont des solutions d’équations différentielles stochastiques décrivant des phénomènes aléatoires associés aux prinicpes de mesure (quantique) des systèmes quantiques ouverts. Ces équations, également appelées équations de Belavkin ou équations maîtresses stochastiques sont habituellement de deux types: soit diffusif soit de type saut. Dans cet article, nous considérons des modèles plus avancés où des équations de type saut-diffusion apparaissent. Ces équations sont obtenues comme solutions de problèmes de martingales. Ces problèmes de martingales sont obtenus comme limites continus (en temps) à partir de chaînes de Markov classiques décrivant des trajectoires quantiques pour des modèles à temps discret. Les résultats de cet article généralisent ceux de [Ann. Probab. 36 (2008) 2332–2353] et [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. Ici, les techniques probabilistes utilisés sont complétement différentes afin de pouvoir mixer les deux types d’évolutions: diffusives et poissoniennes.