We study the interaction of a three two-level atoms (3-2LA) with a one-mode
op?tical coherent field in coherent state in the presence of non-linear Kerr
medium. The three atoms are initially prepared in upper and entangled states
while the field mode is in a coherent state. The constants of motion, 3-2LA
and field density matrix are obtained. The analytic results are employed to
perform some investigations of the temporal evolution of the von Neumann
entropy as measure of the degree of entanglement between the 3-2LA and
optical coherent field. The effect of the detuning and the initial atomic
states on the evolution of geometric phase and en?tanglement is analyzed.
Also, we demonstrate the link between the geometric phase and non-classical
properties during the evolution time. Additionally the effect of detuning
and initial conditions on the Mandel parameter is studied. The obtained
results are emphasize the impact of the detuning and the initial atomic
states of the feature of the entanglement, geometric phase and photon
statistics of theoptical coherent field.