In this work, we analyze the effects of quantum synchronization Sq\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{S}}_{\varvec{q}}$$\end{document} and ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{\phi }$$\end{document} synchronization Sqϕ\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{S}}_{\varvec{q}}^{\varvec{\phi }}$$\end{document} in a hybrid optomechanical system composing a representative coupled optomechanical component and a two-level atom. Although an atom may induce some perturbations to synchronization, good quantum synchronization and quantum ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{\phi }$$\end{document} synchronization can also be achieved in the hybrid optomechanical system. Moreover, we investigate the responses of the two synchronization measures on different periodic modulations in the presence and absence of an atom respectively. In addition, the effects of Kerr nonlinearity on quantum synchronization are also investigated. Our results can provide a theoretical foundation for quantum synchronization in hybrid optomechanical systems and realize some valuable applications in quantum information processing and quantum networks.
