  We study Stokes and anti-Stokes processes in cavity optomechanics in the
regime of strong coupling. The Stokes and anti-Stokes signals exhibit
prominently the normal-mode splitting. We report gain for the Stokes signal. We
also report lifetime splitting when the pump power is less than the critical
power for normal-mode splitting. The nonlinear Stokes processes provide a
useful method for studying strong coupling regime of cavity optomechanics. We
also investigate the correlations between the Stokes and anti-Stokes photons
produced spontaneously by the optomechanical system. At zero temperature, our
nanomechanical system leads to the correlations between the spontaneously
generated photons exhibiting photon antibunching and those violating the
Cauchy-Schwartz inequality.
