Entropy, non-classical property of the trapped ion system driven by running-wave lasers

We study the dynamical behaviour of the ionic inversion in a trapped ion system in running-wave lasers, and we discuss the properties of entanglement between the ion and phonons and the statistics of the vibrational phonon field. It is shown that the appearance of the super-revivals of the ion inversion is dependent on the initial parameters of the vibrational phonon field. The Schr?dinger cat and sub-Poissonian statistics of the phonon field can be also exhibited in this system for suitable initial values.