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We study the non-degenerate parametric amplifier problem from an algebraicapproach of the $SU(1,1)$ group. We write the Hamiltonian of this problem interms of the boson generators of the $SU(1,1)$ group and the differenceoperator. We apply the tilting transformation to our results to exactly solvethis Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, byassuming that our Hamiltonian is an explicit function of time we calculate itsBerry phase. Finally we obtain the Mandel $Q-$parameter of the photon numbers$n_a$ and $n_b$.

Berry phase and the Mandel parameter of the non-degenerate parametric amplifier