Equivalence of two Bochkov-Kuzovlev equalities in quantum two-level systems

We present two kinds of Bochkov-Kuzovlev work equalities in a two-level system that is described by a quantum Markovian master equation. One is based on multiple time correlation functions and the other is based on the quantum trajectory viewpoint. We show that these two equalities are indeed equivalent. Importantly, this equivalence provides us a way to calculate the probability density function of the quantum work by solving the evolution equation for its characteristic function. We use a numerical model to verify these results.

Figure 1

The ratio of the second and first moments of the quantum work (in units $\hslash \omega$) with respect to different perturbation strength ${\lambda }_0$ (in units $\hslash \omega$) for the TLS model in Ref. [30], where $T\omega /2\pi =10$, $\beta \hslash \omega =2.0$. The crosses (${\gamma }_{\downarrow }=0.02\omega$) and stars (${\gamma }_{\downarrow }=0.01\omega$) are the data of the quantum jump simulation [30], while the open squares and circles are the numerical results of Eqs. (19) and (23). The inset shows the pdf of the quantum work. The dashed bars are from the simulation of Ref. [30], and the solid black bars are obtained by our characteristic function method, where $\beta \hslash \omega =1.0$, ${\gamma }_{\downarrow }=0.05\omega$, ${\lambda }_0=0.05\hslash \omega$.