Phase space localization and approach to thermal equilibrium for a class of open systems

Abstract

We analyse the evolution of a quantum oscillator in a finite temperature environment using the quantum state diffusion (QSD) picture. Following a treatment similar to that of reference [7] we identify stationary solutions of the corresponding Itô equation. We prove their global stability and compute typical time scales characterizing the localization process. The recovery of the density matrix in approximately diagonal form enables us to verify the approach to thermal equilibrium in the long time limit and we comment on the connection between QSD and the decoherent histories approach.

Continuous stochastic Schrödinger equations and localization

“…When only one Wiener process N = 1 is present, the unitary invariance condition (12) implies α j1 = 0 for all j. As a consequence there is no invariant unraveling with only one Wiener process.…”

“…For the QSD unraveling c = 0, the two equations (19) and (20) are known to describe the localization of the state |ψ towards a coherent state in the case of a harmonic oscillator. Furthermore, this localization is known to be globally stable [12]. Taking the ensemble mean over equation (19) removes the noise terms but introduces statistical correlations since the statistical mean of a product is in general different from the product of the statistical means.…”

“…which correspond to the complex noise used in the QSD unraveling when the phases φ j are set to zero. The simplest case which can satisfy the invariance condition (12) is given by the QSD unraveling. The phases φ j and other choices of n introduce only irrelevant phase factors which can be neglected.…”

“…In the fundamental theory of quantum measurement, stochastic equations have been used to describe the general dynamical process of the state collapse into an eigenstate of the measured observable, i.e. localization [4,7,8,9,10,11,12,13]. In quantum optics, stochastic Schrödinger equations have been used to describe the system state conditioned by measurement outcomes.…”

“…Several well known results are recovered for a hermitian operator [4,32]. In the case of a non-hermitian operator the minimal rate of localization introduced for the QSD unraveling [9,11,12] is extended to the complete set showing that localization is a general feature shared by all the continuous unravelings, and a new unraveling is introduced. Some theoretical arguments supported by numerical simulations suggest that this new unraveling possesses the highest localization rate.…”