Generation of long-living entanglement using cold trapped ions with pair cat states

With the reliance in the processing of quantum information on a cold trapped ion, we analyze the entanglement entropy in the ion-field interaction with pair cat states. We investigate a long-living entanglement allowing the instantaneous position of the center-of-mass motion of the ion to be explicitly time dependent. An analytic solution for the system operators is obtained. We show that different nonclassical effects arise in the dynamics of the population inversion, depending on the initial states of the vibrational motion. We study in detail the entanglement degree and demonstrate how the input pair cat state is required for initiating the long living entanglement. This long living entanglement is damp out with an increase in the number difference $q$. Owing to the properties of entanglement measures, the results are checked using another entanglement measure (high order linear entropy).