Non-Markovian Quantum Trajectories: An Exact Result

We analyze the non-Markovian stochastic Schrödinger equation describing a particle subject to spontaneous collapses in space (in the language of collapse models), or subject to a continuous measurement of its position (in the language of continuous quantum measurement). For the first time, we give the explicit general solution for the free particle case ($H=p^2/2m$) and discuss the main properties. We analyze the case of an exponential correlation function for the noise, giving a quantitative description of the dynamics and of its dependence on the correlation time.

Figure 1

Time evolution, under the assumption of an exponential correlation function, of the spread in position $\sigma (t)$ of a Gaussian wave function. $\sigma (0)$ has been set $=1\mathrm{m}$. The value $\gamma =\infty$ corresponds to the Markovian case. The other parameters have been chosen as follows: $m=1\mathrm{Kg}$, ${\lambda }_0={10}^{-2}{\mathrm{m}}^{-2}{\sec }^{-1}$. Time is measured is sec, distances in m.