Lindblad-Floquet description of finite-time quantum heat engines

TL;DR

This paper introduces a Floquet-Liouvillian framework for analyzing finite-time quantum heat engines, providing clear criteria for limit cycle convergence and characterizing the steady state within the cycle.

Contribution

It develops a novel Floquet theory approach for Lindblad dynamics, enabling the analysis of limit cycles and steady states in quantum heat engines with periodic driving.

Findings

Spectrum of Floquet Liouvillian determines convergence to limit cycle.

Steady state within the cycle is the zero eigenstate of the Floquet Liouvillian.

Applied to a harmonic oscillator with periodic protocols, illustrating practical use.

Abstract

The operation of autonomous finite-time quantum heat engines rely on the existence of a stable limit cycle in which the dynamics becomes periodic. The two main questions that naturally arise are therefore whether such a limit cycle will eventually be reached and, once it has, what is the state of the system within the limit cycle. In this paper we show that the application of Floquet's theory to Lindblad dynamics offers clear answers to both questions. By moving to a generalized rotating frame, we show that it is possible to identify a single object, the Floquet Liouvillian, which encompasses all operating properties of the engine. First, its spectrum dictates the convergence to a limit cycle. And second, the state within the limit cycle is precisely its zero eigenstate, therefore reducing the problem to that of determining the steady-state of a time-independent master equation. To illustrate the usefulness of this theory, we apply it to a harmonic oscillator subject to a time-periodic work protocol and time-periodic dissipation, an open-system generalization of the Ermakov-Lewis theory. The use of this theory to implement a finite-time Carnot engine subject to continuous frequency modulations is also discussed.