The interaction between a Three-Level Atom and Four Systems of N-Two Level Atoms

We have four systems, each of them contains N-atoms, and each atom of them consists of two levels, and all of them interact with an atom consisting of three levels. Moreover, the atom is also related to the systems by coupling parameters that are time-dependent. By considering particular conditions, the exact solution for the wave function is realized. Then we survey the atomic population inversion in addition to the normalized correlation functions. We examine the influence of many parameters on the previous statistical aspects. We deduce from the conclusions that the behavior of the correlation function is affected by the existence of the time-dependent coupling parameters between spins. We prove that changing the values of the coupling parameters between spins λk, the parameters e, and the quantum numbers Nγ enables us to control the correlated behavior in addition to the atomic population. So, we can control this model through these parameters.