Dephasing and dissipation in qubit thermodynamics

We analyze the stochastic evolution and dephasing of a qubit within the quantum jump (QJ) approach. It allows one to treat individual realizations of inelastic processes, and in this way it provides solutions, for instance, to problems in quantum thermodynamics and distributions in statistical mechanics. As a solvable example, we study a qubit in the weak dissipation limit, and demonstrate that dephasing and relaxation render the Jarzynski and Crooks fluctuation relations (FRs) of non-equilibrium thermodynamics intact. On the contrary, the standard two-measurement protocol, taking into account only the fluctuations of the internal energy $U$, leads to deviations in FRs under the same conditions. We relate the average $\langle e^{-\beta U} \rangle$ (where $\beta$ is the inverse temperature) with the qubit's relaxation and dephasing rates, and discuss this relationship for different mechanisms of decoherence.