Tunable Optical Response and Fast (Slow) Light in Optomechanical System with Phonon Pump

We theoretically investigate the tunable optical response of a weak probe field in a single phonon mechanical driven cavity optomechanical system where an oscillating membrane acts as a mechanical mode. This membrane is also coupled to the optical cavity mode through both the linear optomechanical coupling (LOC) and the quadratic optomechanical coupling (QOC) simultaneously. An external single phonon mechanical driving field present in our scheme gives an additional coherence effect as well as its relative phase and intensity can control significantly the absorption (amplification) profile of the output probe field. We have also shown that the phase dispersion and group delay of the probe field can be controlled flexibly through both types of optomechanical coupling strengths as well as the mean number of thermal phonons. Our results provide a flexible and generalised route to control the light propagation in such kinds of complex cavity optomechanical systems with coherent mechanical excitation.