In this paper, we provide an analytical description of the intrinsic noise model of the two-mode cavity, containing
a single two-level atom through su(1,1)-algebraic treatment. Each field interacts with the qubit through a fourphoton
process and is assumed initially in Barut-Girardello coherent state. The Atomic Quantum Fisher information
(AQFI), atomic entropy and the correlation function are analyzed under the effects of the intrinsic
damping and the superposition of the initial generalized Barut-Girardello Coherent States (B-GCS). Under the
nonlinear interactions, the AQFI has irregular oscillatory behavior that depends on the superposition of the BGCS.
Due to the high nonlinear interactions, the damping work with a very small value and leads to the
degradation of the AQFI. This degradation can be enhanced in the case of the even B-GCS. The generated
entanglement and mixedness of the atomic entropy are explored, they are very sensitive to the physical parameters
of the intrinsic damping, the detuning and the initial cavity state. It is found that the second-order
correlation depends on the mean photon number, the superposition of the B-GCS, and the damping. For the
small value of the mean photon number and with the damping, the sub-Poissonian effects increase with the BGCS,
while the super-Poissonian behavior increases with its even states. For the off-resonant case, the frequency,
the regularity and stability of the NSOC function, the atomic entropy and the AQFI are enhanced