Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes

We describe how electromagnetically induced transparency can arise in quadratically coupled optomechanical systems. Due to quadratic coupling, the underlying optical process involves a two-phonon process in an optomechanical system, and this two-phonon process makes the mean displacement, which plays the role of atomic coherence in traditional electromagnetically induced transparency (EIT), zero. We show how the fluctuation in displacement can play a role similar to atomic coherence and can lead to EIT-like effects in quadratically coupled optomechanical systems. We show how such effects can be studied using the existing optomechanical systems.