  A semi-Markov process method is used to calculate large deviations of first
passage time statistics of counting variables in open quantum systems. The core
formula is an equation of poles. Although it also calculates large deviations
of counting statistics of the same variables, the degrees of the equation are
distinct with respect to the two statistics. Because the former is usually
lower than the latter in the quantum case, analytical solutions for the first
passage time statistics are possible. We illustrate these results via a driven
two-level quantum system and apply them to explore quantum violations of the
classical kinetic and thermodynamic uncertainty relations.
