Under the condition of resonance,the entanglement of field-atom is investigated using the quantum reduced entropy.The effects of the initial field intensity,the interaction intensity between the optical field and atoms and that among BoseEinstein condensate(BEC) atoms on the evolution of the quantum entanglement are discussed.The results indicate that when the strength of the light field is strong enough,the system of the Schr¨odinger cat state optical field interacting with BEC atoms is always at the entan...