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

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