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