Non-Markovian Poissonian Spontaneous Collapse Models

Spontaneous collapse models provide a possible solution to the measurement problem by modifying standard quantum dynamics. The modification consists of adding non-linear and stochastic terms inducing wavefunction collapse in space. Non-Markovian versions of these models are motivated by physical reasons, phenomenological consistency, and potential for relativistic extensions. Here, we investigate a non-Markovian version of the Poissonian Spontaneous Localization (PSL) model, i.e., a model characterized by instantaneous and localized collapse events. We assume that our model is characterized by a typical time scale $\tau_C$ so that, given an initial state $\rho_0$ of standard quantum matter at time $t=0$, we derive an effective long-time ($t\gg \tau_C$) statistical dynamics in terms of a CPTP map $\Phi_t$. We then show how $\Phi_t$ can be made equal to that obtained by non-Markovian CSL models. Moreover, given $\Phi_t$, we obtain the associated time-convolutionless master equation by means of a supercumulant expansion. Finally, we characterize the collapse events process (for events with $t \gg \tau_C$) given an initial quantum state $\rho_0$ at $t=0$.