  In this paper, we investigate theoretically a system consisting of a one
dimensional Bose-Einstein condensate trapped inside the optical lattice of an
optical cavity. In the weak-interaction regime and under the Bogoliubov
approximation, the wave function of the Bose-Einstein condensate can be
described by a classical field (condensate mode) having some quantum
fluctuations (the Bogoliubov mode) about the mean value. Such a system behaves
as a so-called atomic parametric amplifier, similar to an optical parametric
amplifier, where the condensate and the Bogoliubov modes play respectively, the
roles of the pump field and the signal mode in the degenerate parametric
amplifier and the s-wave scattering frequency of atom-atom interaction plays
the role of the nonlinear gain parameter. We show that using the nonlinear
effect of atomic collisions, how one can manipulate and control the state of
the Bogoliubov mode and produce squeezed states.
