  We consider the time evolution of nearly degenerate two-state systems with an
external interaction. Conditions in which full population transfer is possible
when the two-states become degenerate are considered. A new variation of the
coupled differential equations for the probability amplitudes suggests a
possible singularity in the time evolution of the system. The singularity
occurs at the point in time where full control in the degenerate limit is
achieved. We solve this new variation to derive more information about
non-degenerate population control. The new results of numerical calculations
for occupational probabilities of non-degenerate systems are presented and
interpreted. These results are used to understand how population control breaks
down as the photon energy and the energy of the system are changed.
Applications are discussed, with an emphasis on interactions of twisted-vortex
and plane-wave photons with a variety of targets, including macroscopic gas
cells of atoms, of large single crystals, and of molecules.
