Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving

We investigate theoretically the response of the output field from an optomechanical system consisting of $N$ nearly degenerate mechanical resonators each coupled to a common cavity mode. When the cavity is driven simultaneously by a strong control field and a weak probe field and each mechanical resonator is driven by a coherent mechanical pump, we obtain the analytical expression for the probe transmission. We show that the probe transmission spectrum can exhibit multiple optomechanically induced absorption (OMIA) with at most $N$ narrow absorption dips, which can be tuned by the phase and amplitude of the mechanical driving field as well as the control field. Moreover, it is shown that the peak probe transmission can be enhanced or suppressed by increasing the amplitude of the mechanical pump, which depends on the phase difference. This phase-dependent effect plays an important role in controlling the propagation of the probe field between OMIA and parametric amplification.