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.
Highlights
Self-oscillating systems are ubiquitous—from humanmade clocks and transistors, through heart cells and neurons in the living body, to flashing fireflies and circadian rhythms— and 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 Rayleigh [1] and the van der Pol [2] oscillators, which are probably the most common textbook examples of limit cycles in classical nonlinear dynamics
The frequency of the oscillation is set by the physical parameters of the oscillator, while the magnitude of the oscillation is set by the ratio of the linear to the nonlinear damping rates
Summary
Self-oscillating systems are ubiquitous—from humanmade clocks and transistors, through heart cells and neurons in the living body, to flashing fireflies and circadian rhythms— and 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 Rayleigh [1] and the van der Pol (vdP) [2] oscillators, which are probably the most common textbook examples of limit cycles in classical nonlinear dynamics. These models consist of a simple harmonic oscillator, driven by a timeindependent energy pump in the form of “negative damping.”.
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