  The propagation of a probe laser field in a cavity optomechanical system with
a Bose-Einstein condensate is studied. The transmission properties of the
system are investigated and it is shown that the group velocity of the probe
pulse field can be controlled by Rabi frequency of the pump laser field. The
effect of the decay rate of the cavity photons on the group velocity is studied
and it is demonstrated that for small values of the decay rates, the slope of
phase dispersion switches from positive to negative just by changing the Rabi
frequency of the pump field. Then, the gain-assisted superluminal light
propagation due to the cross-Kerr nonlinearity is established in cavity
optomechanical system with a Bose-Einstein condensate. Such behavior can not
appear in the pump-probe two-level atomic systems in the normal phase. We also
find that the amplification is achieved without inversion in the population of
the quantum energy levels.
