Electromagnetically induced transparency in a quadratically coupled optomechanical system with an atomic medium

We investigate a quadratically coupled optomechanical cavity system filled with a two-level atomic medium. The output of the cavity field exhibits analogous electromagnetically induced transparency when the optomechanical system interacts with the coupling and probe fields, respectively. We show that the introduction of the atomic medium can enhance the fluctuation of the displacement of the membrane as well as its energy. With the increasing of the atomic number, the dip of the absorption becomes deep.