Limit-cycle existence and convergence for repeated non-quasi-static protocols

Determine whether repeated application of the finite-time non-quasi-static quantum Otto protocol for a time-dependent harmonic oscillator converges to a periodic steady state (limit cycle), and characterize that state when convergence occurs.

Background

In the instantaneous energy representation, finite-rate driving leaves the working medium in a state that generally differs from its initial state after the Hamiltonian returns to its starting value. Consequently, subsequent cycles begin with a different population distribution, so the long-time thermodynamic behavior cannot be inferred from the first cycle alone.

The paper notes that repeated-cycle dynamics in finite-time quantum thermal machines may approach a periodic steady state, or limit cycle, but that such behavior is not assured for arbitrary unitary protocols. Establishing existence and convergence would be necessary for evaluating the asymptotic thermodynamic performance of the non-quasi-static cycle rather than only its first-cycle energetic penalty.

References

In some finite-time quantum thermal machines, the repeated cycle dynamics may converge to a periodic steady state, commonly referred to as a limit cycle, whose thermodynamic performance can then be evaluated in the long-time regime. However, the existence and convergence to such a limit cycle are not guaranteed for arbitrary unitary protocols.

— Adiabatic Otto-like quantum thermodynamical cycle in the non-quasi-static regime  (2608.26690 - Robles-Pérez et al., 27 Aug 2026) in Section IV, subsection “Thermodynamic performance in the non-quasi-static regime”