  A novel result concerning Raman coupling schemes in the context of trapped
ions is obtained. By means of an operator perturbative approach, it is shown
that the complete time evolution of these systems (in the interaction picture)
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 dynamics. The second is a suitable correction restoring the {\em
fine} dynamics suppressed by the coarse graining performed to adiabatically
eliminate the nonresonantly coupled atomic level.
