Topological phase dynamics described by overtone-synthesized classical and quantum Adler equations

TL;DR

This paper extends the Adler equation to include overtone-synthesized coupling, revealing topological features and quantum effects like winding-number breakdown, with implications for optomechanical oscillator synchronization.

Contribution

It introduces a novel extension of the Adler equation incorporating overtone-synthesized coupling and explores its topological and quantum properties.

Findings

Winding-number quantization is observed in the classical regime.

Quantum regime shows breakdown of winding-number quantization.

Hysteretic dynamics reappear in non-adiabatic quantum calculations.

Abstract

The Adler equation is a well-known one-dimensional model describing phase locking and synchronization. Motivated by recent experiments using optomechanical oscillators, we extend the model to include overtone-synthesized sinusoidal coupling with adiabatic temporal modulation. This extension gives rise to unique topological features such as winding-number quantization, discontinuous phase-slip transitions, and hysteretic and non-reciprocal phase dynamics. We further extend the analysis to the quantum regime, where we find a counterintuitive result: the breakdown of winding-number quantization. This arises from the superposition of different winding-number states in a closed-space Thouless pump. Moreover, hysteretic dynamics, once eliminated in quantum adiabatic approximation, is recovered in non-adiabatic calculations, as the superposition of two Floquet states with different PT eigenvalues becomes the quantum counterpart of phase trajectory.