  We investigate the properties of the optical output fields from a cavity
optomechanical system, where the cavity is driven by a strong coupling and a
weak probe optical fields and the mechanical resonator is driven by a coherent
mechanical pump. When the frequency of the mechanical pump matches the
frequency difference between the coupling and probe optical fields, due to the
interference between the different optical components at the same frequency, we
demonstrate that the large positive or negative group delay of the output field
at the frequency of probe field can be achieved and tuned by adjusting the
phase and amplitude of the mechanical driving field. Moreover, the strength of
the output field at the frequency of optical four-wave-mixing (FWM) field also
can be controlled (enhanced and suppressed) by tuning the phase and amplitude
of the mechanical pump. We show that the power of the output field at the
frequency of the optical FWM field can be suppressed to zero or enhanced so
much that it can be comparable with and even larger than the power of the input
probe optical field.
