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 optomechanical system and this two phonon process makes the mean amplitude, which plays the role of atomic coherence in traditional 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.