  The analytic derivation of the dynamic Stark shift of hydrogenic energy
levels in the presence of the circularly polarized laser light is presented. We
use the classical framework with considering an adiabatically damped laser+atom
interaction and an approach relies on time-independent perturbation theory with
a second-quantized laser+atom dipole interaction Hamiltonian. We thus
investigate the excitation process in atomic hydrogen or in a hydrogen like ion
with low nuclear charge number in the presence of the circularly polarized
laser light. The derived formula for the dynamic AC Stark shift of hydrogenic
energy levels using these two methods are shown to be same in the limit of
large photon numbers.
