  In this paper we study the behavior of (laser--cooled) m--atoms trapped in a
cavity interacting with a photon $...$ Cavity QED $...$ and attempt to solve
the Schr{\"o}dinger equation of this model in the strong coupling regime. In
the case of m = 2 we construct Bell--Schr{\" o}dinger cat states (in our
terminology) and obtain with such bases some unitary transformations by making
use of the rotating wave approximation under new resonance conditionscontaining
the Bessel functions, which will be applied to construct important quantum
logic gates in Quantum Computation. Moreover we propose in the case of m = 3 a
crucial problem to solve on Quantum Computation.
