Quantum φ synchronization in a coupled optomechanical system with periodic modulation

Based on the concepts of quantum synchronization and quantum phase synchronization proposed by A. Mari et al. [Phys. Rev. Lett. 111, 103605 (2013)], we introduce and characterize the measure of a more generalized quantum synchronization called quantum phi synchronization under which pairs of variables have the same amplitude and possess the same phi phase shift. Naturally, quantum synchronization and quantum antisynchronization become special cases of quantum phi synchronization. Their relations and differences are also discussed. To illustrate these theories, we investigate the quantum phi synchronization and quantum phase synchronization phenomena of two coupled optomechanical systems with periodic modulation and show that quantum phi synchronization is more general as a measure of synchronization. We also show the phenomenon of quantum antisynchronization when phi = pi.