Spontaneous phase locking of mechanical multimodes in anti-parity-time optomechanics

We propose a system for observing the spontaneous phase locking of two frequency separate mechanical modes in an anti-parity-time symmetric optomechanical system. In our approach, a common optical cavity mode mediates the coupling between two phonon modes, leading to the phase locking of the coupled mechanical modes to a common frequency in the symmetry unbroken regime. We furthermore observe the change of quantum correlation near the exceptional point. Our results are also directly relevant to numerous other physical platforms, such as atomic ensembles in cavity quantum electrodynamics (QED) systems and spin interaction mediated by collective motional mode in trapped ions.