Topological phase dynamics described by overtone-synthesized classical and quantum Adler equations
Phys. Rev. A 113, 052220 – Published 19 May, 2026
DOI: https://doi.org/10.1103/vcqz-m2r1
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 this model to include overtone-synthesized sinusoidal coupling with adiabatic temporal modulation. This extension gives rise to unique topological features such as an integer-valued winding number, discontinuous phase-slip transitions, and hysteretic and nonreciprocal phase dynamics. We further extend the analysis to the quantum regime, where we obtain a counterintuitive result: a continuously valued winding number. This behavior arises from averaging over different winding-number components in a closed-system Thouless pump. Moreover, hysteretic dynamics—suppressed within the quantum adiabatic approximation—is recovered in nonadiabatic calculations, where Landau-Zener-Stückelberg oscillations induce the nonreciprocal phase evolution observed in classical systems.
