Doubly resonant two-photon absorption cross-sections: It does not get any bigger than this

Ever since researchers first probed materials with light, a natural question they have asked pertains to the strength of this interaction. Can materials be made to more strongly absorb light at certain wavelengths, thereby yielding vibrant colors? Can the nonlinear interaction between two beams be used to make an optical switching device? How efficiently can we convert the frequency of light through a nonlinear interaction? All of these questions, in effect, are the same. Equivalently, the question can be phrased as, "how large can we make the linear or the nonlinear susceptibility?" In this paper, we apply sum rules to calculate the maximum limit of the doubly resonant two-photon absorption cross-section. We find that even for a single electron system, the cross-section can exceed delta = 10(5) GM.