Quantum entanglement and quantum dynamics of a three level trapped ion under a time-dependent interaction with two laser beams

It is calculated time-dependent Hamiltonion of a three-level ion interacting with two laser beams in shema using aunitary transformation method. The unitariy transformation method transform a sheme for vibrational phonon transitions. The analytical results fort he probability amplutudes become precise after a certain period of time when the Lamb-Dicke regime is reached. It is analyzed the qantum entanglement created in the ion-phonon system. It is studed time dependent ion-phonon system via previus time-independent works. With 12 probability amplitudes, we analytically analyzed quantum dynamics and quantum entanglement in a time-dependent interaction of ion-phonon system via concurrence, negativity and quantum entropy Keywords: Entanglement, Trapped Ion, Concurrence, Negativity, Entropy, amount of entanglement.