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The optomechanical cavity system has been a paradigm in the manifestation of continuous variable quantum information over the past decade. This paper investigates how quantum phase synchronization is associated with bipartite Gaussian entanglement in coupled gain-loss mechanical oscillators, where the gain and loss rates can be controlled by driving the cavities with blue and red detuned lasers, respectively. We examine the role of the exceptional point (EP) in a deterministic way of producing self-sustained oscillations that generate robust quantum correlations among quadrature fluctuations of the mechanical oscillators. Particularly, steady phase synchronization dynamics along with the entanglement phenomena are observed in the effective weak-coupling regime above a critical driving power. These phenomena are further verified by observing the mechanical squeezing and phase-space rotations of the Wigner distributions. Moreover, the fidelity fluctuation shows how the quantum correlation dynamics are related to the EP of the system. We also discuss the impact of the mechanical oscillator's frequency mismatch and decoherence due to thermal phonons on system dynamics. These findings hold promise for applications in phonon-based quantum communication and information processing on the macroscopic scale.
Ghosh et al. (Mon,) studied this question.