Computation of the asymptotic states of modulated open quantum systems with a numerically exact realization of the quantum trajectory method

Abstract: Quantum systems out of equilibrium are presently a subject of active research, both in theoretical and experimental domains. In this work we consider time-periodically modulated quantum systems which are in contact with a stationary environment. Within the framework of a quantum master equation, the asymptotic states of such systems are described by time-periodic density operators. Resolution of these operators constitutes a non-trivial computational task. To go beyond the current size limits, we use the quantum trajectory method which unravels master equation for the density operator into a set of stochastic processes for wave functions. The asymptotic density matrix is calculated by performing a statistical sampling over the ensemble of quantum trajectories, preceded by a long transient propagation. We follow the ideology of event-driven programming and construct a new algorithmic realization of the method. The algorithm is computationally efficient, allowing for long 'leaps' forward in time, and is numerically exact in the sense that, being given the list of uniformly distributed (on the unit interval) random numbers, ${\eta_1, \eta_2,...,\eta_n}$, one could propagate a quantum trajectory (with $\eta_i$'s as norm thresholds) in a numerically exact way. %Since the quantum trajectory method falls into the class of standard sampling problems, performance of the algorithm %can be substantially improved by implementing it on a computer cluster. By using a scalable $N$-particle quantum model, we demonstrate that the algorithm allows us to resolve the asymptotic density operator of the model system with $N = 2000$ states on a regular-size computer cluster, thus reaching the scale on which numerical studies of modulated Hamiltonian systems are currently performed.