Semi-Markov Processes in Open Quantum Systems. III. Large Deviations of First Passage Time Statistics

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.