Thermodynamic Uncertainty of Work in Time-Dependently Driven Open Quantum Systems

Abstract

We derive a thermodynamic uncertainty relation for work in an open quantum system driven by a time-dependent protocol and subject to repeated projective energy measurements. The protocol is represented by a sequence of protocol quenches separated by finite-time Lindblad evolution, so that work is accumulated at the quenches while dissipation occurs between them. The resulting work statistics obey the Gallavotti-Cohen symmetry to give rise to the Crooks and Jarzynski fluctuation relations. By perturbing the dissipative dynamics and combining the Cramér--Rao inequality with the quantum Fisher information, we obtain $\mathrm{Var}W/[\tau {\partial }_{\tau }⟨W⟩{]}^2\ge 1/F\ge \max (1/A,2/E)$, where $⟨W⟩$ is the work expectation value, $\mathrm{Var}W$ is the variance of work, $F$ is the Fisher information, $A$ is the dynamical activity, $E$ is the dimensionless entropy production, and $\tau$ is the switching time interval between quenches. The two thermodynamic bounds follow from two independent perturbations that generate the same response of the work statistics. We illustrate the relation for a dissipative two-level system under square-wave and sinusoidal driving and show that the tighter thermodynamic bound can depend on the driving protocol and switching time scale.

5 pages, 3 figures, 1 supplementary material