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We theoretically study photon transmission and mechanical ground statecooling in a two-dimensional optomechanical system that is formed of asuspended graphene sheet on an one-dimensional optomechanical crystal. When thefrequencies of graphene resonator and nanobeam resonator(phononic mode ofoptomechanical crystal) are approximately the same, the $\Lambda$-typedegenerate four-level system of two-dimensional optomechanics shows two-coloroptomechanically-induced transparency , and the transparency window could beswitched among probe signal's absorption, transparency, and amplification.According to our calculations, the graphene resonator could also effectivelyassist the ground state cooling of large damping nanobeam resonator intwo-dimensional optomechanics.

Two-dimensional optomechanics formed by the graphene sheet and photonic crystal cavity