Coarse grained and fine dynamics in trapped ion Raman schemes

A novel result concerning Raman coupling schemes implemented using trapped ions is obtained. By means of an operator perturbative approach, it is shown that the complete time evolution of these systems can be expressed, with a high degree of accuracy, as the product of two unitary evolutions. The first one describes the time evolution related to an effective coarse grained dynamic. The second is a suitable correction restoring the fine dynamics suppressed by the coarse graining performed to adiabatically eliminate the nonresonantly coupled atomic level. The case where a decoherence source is present is also studied