Mixed classical-quantal representation for open quantum systems

For all but the simplest open quantum systems, quantum trajectory Monte Carlo methods, including quantum jump and quantum state diffusion (QSD) methods, have besides their intuitive insight into the measurement process a numerical advantage over direct solutions for the density matrix, especially where many degrees of freedom are involved. For QSD the trajectories are continuous, and often localize to small near-minimum uncertainty wave packets which follow approximately classical paths in phase space. The mixed representation discussed here takes advantage of this localization to reduce computing space and time by a further significant factor, using a quantum oscillator representation that follows a classical path. The classical part of this representation describes the time evolution of the expectation values of position and momentum in classical phase space, while the quantal part determines the degree of localization of the quantum mechanical state around this phase space point. The method can be applied whether or not the localization is produced by a measuring apparatus.