Abstract
We develop a general theoretical framework of semiclassical phase reduction for analyzing synchronization of quantum limit-cycle oscillators. The dynamics of quantum dissipative systems exhibiting limit-cycle oscillations are reduced to a simple, one-dimensional classical stochastic differential equation approximately describing the phase dynamics of the system under the semiclassical approximation. The density matrix and power spectrum of the original quantum system can be approximately reconstructed from the reduced phase equation. The developed framework enables us to analyze synchronization dynamics of quantum limit-cycle oscillators using the standard methods for classical limit-cycle oscillators in a quantitative way. As an example, we analyze synchronization of a quantum van der Pol oscillator under harmonic driving and squeezing, including the case that the squeezing is strong and the oscillation is asymmetric. The developed framework provides insights into the relation between quantum and classical synchronization and will facilitate systematic analysis and control of quantum nonlinear oscillators.
Highlights
Spontaneous rhythmic oscillations and synchronization arise in various science and technology fields, such as laser oscillations, electronic oscillators, and spiking neurons [1,2,3,4,5,6]
The reduced phase equation is essentially the same as that for the classical limit-cycle oscillator driven by noise, and synchronization dynamics of the weakly perturbed quantum nonlinear oscillator in the semiclassical regime can be analyzed on the basis of the reduced phase equation by using the standard methods for the classical limit-cycle oscillator
We have developed a general framework of the phase reduction theory for quantum limit-cycle oscillators under the semiclassical approximation and confirmed its validity by analyzing synchronization dynamics of the quantum van der Pol (vdP) model
Summary
Spontaneous rhythmic oscillations and synchronization arise in various science and technology fields, such as laser oscillations, electronic oscillators, and spiking neurons [1,2,3,4,5,6]. A standard theoretical framework for analyzing limit-cycle oscillators in classical dissipative systems is the phase reduction theory [1,2,3,7,8,9]. By using this framework, we can systematically reduce multidimensional nonlinear dynamical equations describing weakly perturbed limit-cycle oscillators to a onedimensional phase equation that approximately describes the oscillator dynamics.
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