Quantum limit-cycles and the Rayleigh and van der Pol oscillators

TL;DR

This paper explores the quantum behavior of limit-cycle oscillators, specifically the Rayleigh and van der Pol models, providing an exact solution for their steady-state dynamics and analyzing the classical-quantum transition.

Contribution

It derives a general analytical solution for the steady-state quantum dynamics of these oscillators, extending previous low-temperature results to arbitrary temperatures.

Findings

Exact steady-state quantum solution applicable to various bosonic systems

Identification of quantum-specific features in the classical-quantum transition

Analysis of bifurcation changes from classical to quantum regimes

Abstract

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.