Quantum Trajectories in Random Environment: the Statistical Model for a Heat Bath

In this article, 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 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. Introduction The theory of Quantum Trajectories consists in studying the evolution of the state of an open quantum system undergoing continuous indirect measurement. The most basic physical setting consists of a small system, which is the open system, in contact with an environment. Usually, in quantum optics and quantum communication, the measurement is indirectly performed on the environment [7, 8, 13, 10, 28, 39, 40]. In this framework, the reduced time evolution of the small system, obtained by tracing over the degrees of freedom of the environment, is described by stochastic differential equations called stochastic Schrödinger equations or stochastic Master equations. The solutions of these equations are called Continuous Quantum Trajectories. In the literature, two generic types of equations are usually considered 1. Diffusive equations dρt = L(ρt)dt + (