Quantum Trajectories for Quantum Optical Systems

Abstract

A quantum trajectory refers to the conditional state of a quantum system, conditioned on a time sequence of measurement records [1–3]. This state is thus dependent on a measurement history. The measurement record is a stochastic process. The source of the randomness is two-fold; it is due to the irreducible quantum fluctuations in the measured system, the source of the information, and it is due to noise added by the measurement apparatus itself. As the stochastic measurement record is accumulated, the conditional state of the measured system will undergo a stochastic evolution. If the measurement record is complete, and if the initial state is pure, the stochastic evolution of the conditional state can be represented by a pure quantum state, otherwise the conditional state is represented by a mixed state. A quantum trajectory thus connects a classical stochastic process, the measurement record, to a quantum stochastic process. We will primarily be concerned with continuous measurement, for which the measured system is always coupled to an environment from which information can be extracted. From this point of view there is nothing particularly new about quantum trajectories. The conditional evolution of a measured system has been considered by many authors [4–8]. Stochastic Schroedinger equations have also been postulated as a means to describe all open quantum systems, not just measured systems [9–11]. In this paper we will primarily be concerned with particular applications of the theory of continuous measurement for quantum optical systems.