Stochastic Schroedinger equation from optimal observable evolution

Wilfried Grecksch, Hannelore Lisei

Nicolas Gisin

Our aim is to unify the Schroedinger dynamics and the projection postulate. We first prove that the Schroedinger evolution is the only quantum evolution that is deterministic and compatible with relativity. Next we present a non deterministic generalization of the Schroedinger equation compatible with relativity. This time continuous pure state valued stochastic process covers the whole class of quantum dynamical semigroups, as exemplified by the damped harmonic oscillator and spin relaxation. Finally a symmetry preserving generalization of the Ghirardi-Rimini-Weber model is presented. (orig.).

Crispin W. Gardiner, P. Zoller

Angelo Bassi, Detlef Duerr, Martin Kolb

We discuss the time evolution of the wave function which is solution of a stochastic Schroedinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. Observing that there exist three time regimes, namely the collapse regime, after which the wave function is localized in space; the classical regime, during which the collapsed wave function moves along a classical path and the diffusive regime, in which diffusion overlaps significantly the deterministic motion, we study the long time behavior of the wave function. We assert that the general solution converges a.s. to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this.

Alberto Barchielli, Matteo Gregoratti

Denis Lacroix A

The quantum-jump approach, where pairs of state vectors follow the stochastic Schroedinger equation (SSE) in order to treat the exact quantum dynamics of two interacting systems, is described. In this work the nonuniqueness of such stochastic Schroedinger equations is investigated to propose strategies to optimize the stochastic paths and reduce statistical fluctuations. In the proposed method, called the ``adaptative noise method,'' a specific SSE is obtained for which the noise depends explicitly on both the initial state and on the properties of the interaction Hamiltonian. It is also shown that this method can be further improved by introduction of a mean-field dynamics. The different optimization procedures are illustrated quantitatively in the case of interacting spins. A significant reduction of the statistical fluctuations is obtained. Consequently a much smaller number of trajectories is needed to accurately reproduce the exact dynamics as compared to the SSE without optimization.