Non-Markovian Quantum Trajectories

Abstract

The technique of quantum trajectories (stochastic Schrödinger equations or Monte Carlo wave functions) for open systems is generalized to the non-Markovian regime. I consider a microscopic model of an open system consisting of a boson field coupled linearly (with an excitation preserving coupling) to a localized system. The model allows for a field with an arbitrary dispersion relation and an arbitrary mode-dependent coupling to the system. The trajectories are formulated as continuous measurements of the output field from the system. For a general dispersive field these measurements must be distributed in space for this formulation to be possible. The result of this formulation is a non-Markovian equation for the system conditioned on the measurements. A method of numerically simulating this equation has been determined and implemented in some test cases. Numerical simulation is possible if one can introduce a finite memory time for the evolution of the reduced system. As an illustration, the method is applied to the spectral detection of the emission from a driven two-level atom and also to an atom radiating into an electromagnetic field where the free space modes of the electromagnetic field are altered by the presence of a cavity. In both cases the non-Markovian behaviour arises from the uncertainty in the time of emission of a photon that is later detected (or reabsorbed), although, in the second case, the non-Markovian behaviour is intrinsic to the system environment coupling whereas, in the spectral detection case, it is a consequence of the choice of measurement process. The generalization of the techniques of quantum trajectories to the non-Markovian regime promises to make a range of open system problems where the Born-Markov approximation is invalid tractable to numerical simulation.