Stochastic unraveling of positive quantum dynamics

Stochastic unravelings represent a useful tool to describe the dynamics of open quantum systems, and standard methods, such as quantum state diffusion (QSD), call for the complete positivity of the open-system dynamics. Here, we present a generalization of QSD, which also applies to positive, but not completely positive evolutions. The rate and the action of the diffusive processes involved in the unraveling are obtained by applying a proper transformation to the operators which define the master equation. The unraveling is first defined for semigroup dynamics and then extended to a definite class of time-dependent generators. We test our approach on a prototypical model for the description of exciton transfer, keeping track of relevant phenomena, which are instead disregarded within the standard, completely positive framework.

Figure 1

Trajectories for the evolution of the population of (a) site one and (b) site two; each trajectory corresponds to a different realization of the solution of the SDE in Eq. (2), with $A_{{\psi }_t}$ as in Eq. (3) and $B_{{\psi }_t}$ as in Eq. (21), and derived by diagonalizing the GTRO at each point of the computational time domain. The deterministic initial state is $\left|\psi \left(0\right)\right.〉=\left|2\right.〉$, while the state at time $t$ is $\left|\psi \left(t\right)\right.〉=\alpha \left(t\right)\left|1\right.〉+\beta \left(t\right)\left|2\right.〉+\gamma \left(t\right)\left|3\right.〉$. Evolution of the population of (c) site one and (d) site two given by the ensemble average of 1000 trajectories of our unraveling (blue/dark), and the solution of the Lindblad equation after full SA (yellow/light). In the inset, the ensemble average (blue/dark) and the solution of the P ME (green dotted) are shown to agree within the standard deviation of the mean (vertical bars) of the trajectories. The initial state is set as $\rho \left(0\right)=\left|2\right.〉\left.〈2\right|$.