Quantum limit cycles and the Rayleigh and van der Pol oscillators
Self-oscillating systems, described in classical dynamics as limit cycles, are emerging as canonical models for driven dissipative nonequilibrium open quantum systems and as key elements in quantum technology. We consider a family of models that interpolates between the classical textbook examples of the Rayleigh and the van der Pol oscillators and follow their transition from the classical to the quantum domain, while properly formulating their corresponding quantum descriptions. We derive an exact analytical solution for the steady-state quantum dynamics of the simplest of these models, applicable to any bosonic system—whether mechanical, optical, or otherwise—that is coupled to its environment via single-boson and double-boson emission and absorption. Our solution is a generalization to arbitrary temperature of existing solutions for very-low, or zero, temperature, often misattributed to the quantum van der Pol oscillator. We closely explore the classical to quantum transition of the bifurcation to self-oscillations of this oscillator, while noting changes in the dynamics and identifying features that are uniquely quantum.
© 2021 Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Received 6 November 2020; accepted 19 January 2021; published 10 February 2021. The authors thank Moshe Goldstein and Haim Suchowski for their insightful comments on an earlier version of this paper and are very grateful to Dariel Mok and Andy Chia for alerting us to the recent publication of their work, which very nicely complements ours. R.L. thanks Nadav Steiner for his contribution  during early stages of this work and Mark Dykman for fruitful and inspiring discussions. This research was supported by the U.S.-Israel Binational Science Foundation (BSF) under Grant No. 2012121.
Published - PhysRevResearch.3.013130.pdf
Submitted - 2011.02706.pdf