  We study an optomechanical system in which the collective density excitations
(Bogoliubov modes) of a Bose Einstein condensate (BEC) is coupled to a cavity
field. We show that the optical force changes the frequency and the damping
constant of the collective density excitations of the BEC. We further analyze
the occurrence of normal mode splitting (NMS) due to mixing of the fluctuations
of the cavity field and the fluctuations of the condensate with finite atomic
two-body interaction. The NMS is found to vanish for small values of the
two-body interaction. We further show that the density excitations of the
condensate can be used to squeeze the output quantum fluctuations of the light
beam. This system may serve as an optomechanical control of quantum
fluctuations using a Bose Einstein condensate.
