In this paper, we use the su(1,1)-algebraic treatment to explore the effects of the intrinsic decoherence in a twomode
cavity containing a two-level atom. Each field is resonant with the qubit through a four-photon process.
The role of the intrinsic decoherence and the superposition of the initial generalized Barut-Girardello coherent
state on the quantum effects is investigated via different quantifiers as: atomic inversion, linear entropy and
negativity. It is found that the non-linearity of the four-photon process leads to generate non-classical effects
with a high oscillatory behavior. The superposition of the Barut-Girardello coherent state controls the dynamics
of the purity loss and the entanglement. It is found that the nonclassical effects are very sensitive to the nonlinear
interactions between the qubit and the two-mode cavity.