We propose two methods for computing the large deviations of the first-passage-time statistics in general open quantum systems. The first method determines the region of convergence of the joint Laplace transform and the $z$ transform of the first-passage time distribution by solving an equation of poles with respect to the $z$-transform parameter. The scaled cumulant generating function, which is the logarithm of the boundary values within this region, is subsequently obtained. The theoretical basis is that the dynamics of open quantum systems can be unraveled into a piecewise deterministic process and that a tilted Liouville master equation exists in Hilbert space. The second method uses a simulation-based approach that is built on the wave-function cloning algorithm. To validate both methods, we derive analytical expressions for the scaled cumulant generating functions in field-driven two-level and three-level systems. In addition, we present numerical results and cloning simulations for a field-driven system comprising two interacting two-level atoms.

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• Large deviation & rare event statistics