  A single trapped ion interacting with laser light in a radiofrequency trap is
considered by diagonalization of full Hamiltonian of the system in a suitable
basis. The energies, eigenvectors, probabilities of finding the atom in the
ground state, density matrixes and its entanglement are computed. The study
repeated for the Hamiltonian under the rotated wave approximation in the same
basis. The results compared in general and in the Lamb-Dick regime for
ultra-cold atoms to reveal essential changes in the results.
