  We study an optomechanical transistor, where an input field can be
transferred and amplified unidirectionally in a cyclic three-mode
optomechanical system. In this system, the mechanical resonator is coupled
simultaneously to two cavity modes. We show that it only requires a finite
mechanical gain to achieve the nonreciprocal amplification. Here the
nonreciprocity is caused by the phase difference between the linearized
optomechanical couplings that breaks the time-reversal symmetry of this system.
The amplification arises from the mechanical gain, which provides an effective
phonon bath that pumps the mechanical mode coherently. This effect is analogous
to the stimulated emission of atoms, where the probe field can be amplified
when its frequency is in resonance with that of the anti-Stokes transition. We
show that by choosing optimal parameters, this optomechanical transistor can
reach perfect unidirectionality accompanied with strong amplification. In
addition, the presence of the mechanical gain can result in ultra-long delay in
the phase of the probe field, which provides an alternative to controlling
light transport in optomechanical systems.
