The time evolution of the atomic Wehrl entropy and long-lived entanglement generation using a single trapped ion interacting with a laser field are analyzed. Starting from the Heisenberg equation of motion, an exact solution of the system is obtained by indicating that there are some interesting features when a time-dependent modulating function is considered. We demonstrate that the long-living quantum entanglement can be obtained using the time-independent interaction when the field is initially in a pair cat states.