  We theoretically study photon transmission and mechanical ground state
cooling in a two-dimensional optomechanical system that is formed of a
suspended graphene sheet on an one-dimensional optomechanical crystal. When the
frequencies of graphene resonator and nanobeam resonator(phononic mode of
optomechanical crystal) are approximately the same, the $\Lambda$-type
degenerate four-level system of two-dimensional optomechanics shows two-color
optomechanically-induced transparency , and the transparency window could be
switched among probe signal's absorption, transparency, and amplification.
According to our calculations, the graphene resonator could also effectively
assist the ground state cooling of large damping nanobeam resonator in
two-dimensional optomechanics.
