Is Quantum Mechanics a Limit Cycle Theory?

Stanisław D. Głazek

Craig M. Savage

We investigate quantum-mechanical second-harmonic generation for parameters such that classical electrodynamics predicts oscillations. Specifically we calculate the Q distribution function in a Gaussian approximation about the classical limit cycle. In the classical limit initial rapid collapse of the Q distribution into the neighborhood of the limit cycle is followed by diffusion around the limit cycle. The experimental significance of this quantum diffusion is discussed.

Thomas Boyer‐Kassem

Nana Siddhartha

Quantum decoherence does not permit the operation of (new) miniaturised devices or protocols that can otherwise run unassisted in classical settings, without explicit user agent intervention in the quantum realm. Classical driving signals are inadequate to model a truly quantum device built entirely in the quantum realm. Synchronisation endures miniaturisation without sacrificing quantum coherence, unlike thermalisation. We quantise the Kuramoto model and the Van der Pol oscillator and couple it with qubit(s) to show a robust limit cycle even in typical quantum thermodynamical settings. Novel effects such as Entanglement State Revivals, Zeno locking and thermal current interference are observed. We propose toy model devices that can be implemented harnessing the Synchronisation of a qubit (or an arbitrary collection of qubits) with a nonlinear phase oscillator. The framework theory used here, Spectral Mechanics, is a fresh look at the foundations of conventional quantum mechanics and is extensible to completely new forms of interactions and models in experimental settings.

Stefano Scopa, Gabriel Teixeira Landi, Dragi Karevski

The operation of autonomous finite-time quantum heat engines relies 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 the state of the system is 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. 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.

D. Sen, S. Sengupta