  We compare the derivation of the dynamic Stark shift of hydrogenic energy
levels in a classical framework with an adiabatically damped laser-atom
interaction, which is equivalent to the Gell-Mann-Low-Sucher formula, and a
treatment based on time-independent perturbation theory, with a
second-quantized laser-atom dipole interaction Hamiltonian. Our analysis
applies to a laser that excites a two-photon transition in atomic hydrogen or
in a hydrogenlike ion with low nuclear charge number. Our comparisons serve to
demonstrate why the dynamic Stark shift may be interpreted as a stimulated
radiative correction and illustrates connections between the two derivations.
The simplest of the derivations is the fully quantized approach. The classical
and the second-quantized treatment are shown to be equivalent in the limit of
large photon numbers.
