Decoherence and classical predictability of phase-space histories.

Abstract

We consider the decoherence of phase-space histories in quantum Brownian motion models, consisting of a particle moving under a potential $V(x)$ in contact with a heat bath of temperature $T$ and dissipation constant $\ensuremath{\gamma}$ in the Markovian regime. The phase-space histories are described by quasiprojectors consisting of Gaussian density matrices smeared over large phase-space cells and are characterized by the size [$\ensuremath{\Gamma}$] of the phase-space cell together with the size [$M$] of the margin (the region at the boundary of $\ensuremath{\Gamma}$ in which the Weyl symbol of the projector goes from 1 to 0). We generalize earlier results of Hagedorn to show that an initial Gaussian density matrix remains approximately Gaussian under nonunitary time evolution deriving a bound giving the validity of this approximation. Following the earlier work of Omnes [J. Stat. Phys. 51, 351 (1989)] we use this result to compute the time evolution of the phase-space projectors under the master equation and show that histories of phase-space samplings approximately decohere, and that the probabilities for these histories are peaked about classical dissipative evolution, but with an element of unpredictability which is reflected in the increase of the size of the margin.