Stochastic dynamics of quantum jumps

Abstract

The dynamics of an open quantum system coupled to an external reservoir is studied on the basis of a recently proposed formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. The previous result is generalized to include interaction Hamiltonians of the form ${\mathit{tsum}}_{\mathit{i}}$${\mathit{A}}_{\mathit{i}}$\ensuremath{\bigotimes}${\mathit{B}}_{\mathit{i}}$, where ${\mathit{A}}_{\mathit{i}}$ and ${\mathit{B}}_{\mathit{i}}$ are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schr\"odinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.