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Quantum phase synchronisation enhanced via Coulomb interaction in an optomechanical system

In this work, we investigate the dynamics of quantum synchronization in a four-mode optomechanical system, focusing on the influence of the Coulomb interaction between two mechanical resonators. We analyze the effect of the Coulomb coupling on three distinct synchronization regimes, i.e., complete quantum synchronization, $ϕ$-synchronization, and quantum phase synchronization. Our results show that while the Coulomb interaction plays a pivotal role in significantly enhancing quantum phase synchronization by facilitating energy exchange and phase coherence, it has little impact on complete and $ϕ$-synchronization. This indicates that amplitude and frequency locking are primarily determined by the optical driving, whereas phase alignment depends critically on inter-resonator coupling. We also demonstrate that the oscillations of the two optical cavities, which are indirectly coupled via the mechanical resonators, can become aligned over time, resulting in classical synchronization. These findings provide a robust mechanism for controlling collective quantum dynamics and offer a foundation for applications in quantum communication, precision sensing, and the development of synchronized quantum networks.