  We study the non-degenerate parametric amplifier problem from an algebraic
approach of the $SU(1,1)$ group. We write the Hamiltonian of this problem in
terms of the boson generators of the $SU(1,1)$ group and the difference
operator. We apply the tilting transformation to our results to exactly solve
this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by
assuming that our Hamiltonian is an explicit function of time we calculate its
Berry phase. Finally we obtain the Mandel $Q-$parameter of the photon numbers
$n_a$ and $n_b$.
