Quantum control in nearly non-degenerate qubits interacting with light

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.