  We theoretically investigate the dispersive interaction of a Bose-Einstein
condensate (BEC) trapped inside an optomechanical cavity with a moving end
mirror in the presence of the laser phase noise (LPN) as well as the atomic
collisions. We assume that the effective frequency of the optical mode is much
greater than those of the mechanical and the Bogoliubov modes of the movable
mirror and the BEC . In the adiabatic approximation where the damping rate of
the cavity is faster than those of the other modes, the system behaves as an
effective two-mode model in which the atomic and mechanical modes are coupled
to each other through the mediation of the optical field by an effective
coupling parameter. We show that in the effective two-mode model, the LPN
appears as a classical stochastic pump term which drives the amplitude
quadratures of the mechanical and the Bogoliubov modes. It is also shown that a
strong stationary mirror-atom entanglement can be established just in the
dispersive and Doppler regimes where the two modes come into resonance with
each other and the effect of the LPN gets very low.
