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

We solved time-dependent Hamiltonian of a three-level ion interacting with two-laser beams in Λ scheme using a unitary transformation method which transform the Λ scheme to a cascade Ξ scheme for the vibrational phonon transitions. Our analytical results for the probability amplitudes become precise after a certain period of time when the Lamb-Dicke regime is reached. We analyzed the entanglement created in the system.