Work fluctuations due to partial thermalizations in two-level systems

TL;DR

This paper investigates work fluctuations in two-level quantum systems undergoing partial thermalizations, deriving analytic results for average work, variance bounds, and efficiency in modified Carnot cycles.

Contribution

It introduces a stochastic model for work extraction with partial thermalizations, providing analytic expressions and bounds for work fluctuations and efficiency in finite-time quantum processes.

Findings

Work fluctuations cannot be eliminated in general.

An upper bound for work variance can be derived using Jarzynski's relation.

Efficiency at maximum power is characterized for modified Carnot cycles with partial thermalizations.

Abstract

We study work extraction processes mediated by finite-time interactions with an ambient bath -- \emph{partial thermalizations} -- as continuous time Markov processes for two-level systems. Such a stochastic process results in fluctuations in the amount of work that can be extracted and is characterized by the rate at which the system parameters are driven in addition to the rate of thermalization with the bath. We analyze the distribution of work for the case where the energy gap of a two-level system is driven at a constant rate. We derive analytic expressions for average work and lower bound for the variance of work showing that such processes cannot be fluctuation-free in general. We also observe that an upper bound for the Monte Carlo estimate of the variance of work can be obtained using Jarzynski's fluctuation-dissipation relation for systems initially in equilibrium. Finally, we analyse work extraction cycles by modifying the Carnot cycle, incorporating processes involving partial thermalizations and obtain efficiency at maximum power for such finite-time work extraction cycles under different sets of constraints.