  We investigate a dual membrane active-passive cavity where each mechanical
membrane individually quadratically coupled to passive and active cavities via
two-phonon process. Due to the fact that in the quadratically coupled
optomechanical system mean-field approximation fails, hence to analyze the
system completely, we switch to a more generalized out of equilibrium approach,
namely Keldysh Green's functional approach. We calculate transmission rate
using predetermined full retarded Green's function, and then numerically
examine the effect of the various parameters on the transmission coefficient
and discuss the features and physics behind them in detail. On the basis of the
optical responsivity we further extend our study of fast and slow light
phenomenon. The results show that our proposed system can not only realize
ultra-fast light/ultra-slow light under proper choice of cavity parameters, but
realization of the conversion between fast and slow light and vice versa.
