Abstract
It is solved a time-dependent Hamiltonian using a unitary transformation method which Λ(t) type is used to engineer a cascade Ξ scheme interaction between the vibrational phonons and trapped three-level ion. Quantum entanglement is characterized by comparing concurrence and negativity of the time-dependent ionic-phononic system. In this quantum system, we obtain that the amount of concurrence can be tuned between 0 and 0.99 while the amount of negativity changes between 0 and 0.49.
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