Dynamic Stark shift of Hydrogen-like atoms in the presence of circularly polarized laser light

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.