The open quantum Brownian motions

Abstract

Using quantum parallelism on random walks as the original seed, we introduce new quantum stochastic processes, the open quantum Brownian motions. They describe the behaviors of quantum walkers—with internal degrees of freedom which serve as random gyroscopes—interacting with a series of probes which serve as quantum coins. These processes may also be viewed as the scaling limit of open quantum random walks and we develop this approach along three different lines: the quantum trajectory, the quantum dynamical map and the quantum stochastic differential equation. We also present a study of the simplest case, with a two level system as an internal gyroscope, illustrating the interplay between the ballistic and diffusive behaviors at work in these processes.Notation: orbital (walker) Hilbert space, in the discrete, in the continuum: internal spin (or gyroscope) Hilbert space: system Hilbert space: probe (or quantum coin) Hilbert space, : density matrix for the total system (walker + internal spin + quantum coins): reduced density matrix on : : system density matrix in a quantum trajectory:.If diagonal and localized in position: ρt: internal density matrix in a simple quantum trajectoryXt: walker position in a simple quantum trajectoryBt: normalized Brownian motionξt, : quantum noises