Algebraic solution for a two-level atom in radiation fields and the Freeman resonances

Using techniques of complex analysis in an algebraic approach, we solve the wave equation for a two-level atom interacting with a monochromatic light field exactly. A closed-form expression for the quasienergies is obtained, which shows that the Bloch-Siegert shift is always finite, regardless of whether the original or the shifted level spacing is an integral multiple of the driving frequency ω. We also find that the wave functions, though finite when the original level spacing is an integral multiple of ω, become divergent when the intensity-dependent shifted energy spacing is an integral multiple of the photon energy. This result provides an ab initio theoretical explanation for the occurrence of the Freeman resonances observed in above-threshold ionization experiments.