Calculating work in weakly driven quantum master equations: backward and forward equations

TL;DR

This paper demonstrates the equivalence of two methods for calculating work distribution in weakly driven quantum master equations, connecting quantum jump trajectories and energy measurements, and reveals a new heat-related equality.

Contribution

It shows the equivalence of backward and forward approaches for work calculation in quantum systems and clarifies the microscopic basis of one method.

Findings

Two methods are mathematically equivalent.

A new heat-related equality is identified.

The microscopic basis of the quantum jump approach is clarified.

Abstract

I present a technical report indicating that the two methods used for calculating characteristic functions for the work distribution in weakly driven quantum master equations are equivalent. One involves applying the notion of quantum jump trajectory [Phys. Rev. E 89, 042122 (2014)], while the other is based on two energy measurements on the combined system and reservoir [Silaev, et al., Phys. Rev. E 90, 022103 (2014)]. These represent backward and forward methods, respectively, which adopt a very similar approach to that of the Kolmogorov backward and forward equations used in classical stochastic theory. The microscopic basis for the former method is also clarified. In addition, a previously unnoticed equality related to the heat is also revealed.