# Synchronization dynamics of two nanomechanical membranes within a Fabry-Perot cavity

## TL;DR

This paper investigates the quantum synchronization of two nanomechanical membranes in a cavity driven by a blue-detuned laser, revealing robustness against noise and the relationship between phase variance and quantum correlations.

## Contribution

It introduces a quantum phase difference operator and analyzes synchronization dynamics using covariance matrix formalism, highlighting quantum discord behavior.

## Key findings

- Quantum synchronization persists at high driving strengths.
- Membranes are not entangled during synchronization.
- Quantum discord correlates with phase difference variance.

## Abstract

Spontaneous synchronization is a significant collective behavior of weakly coupled systems. Due to their inherent nonlinear nature, optomechanical systems can exhibit self-sustained oscillations which can be exploited for synchronizing different mechanical resonators. In this paper, we explore the synchronization dynamics of two membranes coupled to a common optical field within a cavity, and pumped with a strong blue-detuned laser drive. We focus on the system quantum dynamics in the parameter regime corresponding to synchronization of the classical motion of the two membranes. With an appropriate definition of the phase difference operator for the resonators, we study synchronization in the quantum case through the covariance matrix formalism. We find that for sufficiently large driving, quantum synchronization is robust with respect to quantum fluctuations and to thermal noise up to not too large temperatures. Under synchronization, the two membranes are never entangled, while quantum discord behaves similarly to quantum synchronization, that is, it is larger when the variance of the phase difference is smaller.