Quantum Trajectories for Brownian Motion

We present the stochastic Schr\"odinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it to the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our powerful stochastic approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger as $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$.