Motional rotating-wave approximation for harmonically trapped particles

We present a family of generalized unitary transformations that simplifies the Hamiltonian for a harmonically trapped two-level atom (or ion) interacting with a plane wave laser field. Near-resonant single as well as double vibrational phonon dynamical regimes are found. The validity condition of the often used motional rotating-wave approximation (MRWA) is examined both numerically and analytically. Large errors are found within typical regimes of the MRWA with respect to the motional degrees of freedom. The effects of the MRWA in trapped ion systems are shown to be opposite to that of the rotating-wave approximation in the usual Jaynes-Cummings model. Our study points to a more restrictive condition on the particle localization (Lamb-Dicke) parameter for the validity of the MRWA in the single phonon dynamical regime. It also sheds light on quantum information storage and processing with trapped atoms