  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.
