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

TL;DR

This paper develops a semi-Markov process approach to analyze large deviations in first passage time statistics of open quantum systems, deriving core formulas and applying them to a two-level system to explore quantum violations of classical uncertainty relations.

Contribution

It introduces a method to calculate large deviations of first passage times in open quantum systems using pole equations, with analytical solutions for specific models.

Findings

Derived the core pole equation for large deviations analysis.

Obtained analytical solutions for first passage time statistics in a two-level system.

Explored quantum violations of classical uncertainty relations using these statistics.

Abstract

In a specific class of open quantum systems with finite and fixed numbers of collapsed quantum states, the semi-Markov process method is used to calculate the large deviations of the first passage time statistics. The core formula is an equation of poles, which is also applied in determining the scaled generating functions (SCGFs) of the counting statistics. For simple counting variables, the SCGFs of the first passage time statistics are derived by finding the largest modulus of the roots of this equation with respect to the $z$-transform parameter and then calculating its logarithm. The procedure is analogous to that of solving for the SCGFs of the counting statistics. However, for current-like variables, the method generally fails unless the equation of pole is simplified to a quadratic form. The fundamental reason for this lies in the nonuniqueness between the roots and the region of convergence for the joint transform. We illustrate these results via a resonantly driven two-level quantum system, where for several counting variables the solutions to the SCGFs of the first passage time are analytically obtained. Furthermore, we apply these functions to investigate quantum violations of the classical kinetic and thermodynamic uncertainty relations.