Fundamental Limits of the Dispersion of the Two-Photon Absorption Cross-Section

We rigorously apply the sum rules to the sum-over-states expression to calculate the fundamental limits of the dispersion of the two-photon absorption cross-section. A comparison of the theory with the data suggests that the truncated sum rules in the three-level model give a reasonable fundamental limit. Our ansatz that the two photon absorption cross-section near the limit must have only three dominant statesis is supported by a rigorous analytical calculation that the resonant term gets smaller as more states are added. We also find that the contributions of the non-explicitly resonant terms can not be neglected when analyzing real molecules with many excited states, even near resonance. However, puzzling as it may be, extrapolating an off-resonant result to resonance using only the resonant term of the three-level model is shown to be consistent with the exact result. In addition, the off-resonant approximation is shown to scale logarithmically when compared with the full three-level model. This scaling can be used to simplify the analysis of measurements. We find that existing molecules are still far from the fundamental limit; so, there is room for improvement. But, reaching the fundamental limit would require precise control of the energy-level spacing, independently of the transition dipole moments -- a task that does not appear possible using today's synthetic approaches. So, we present alternative methods that can still lead to substantial improvements which only require the control of the transition moment to the first excited state. While it is best to normalize measured two photon absorption cross-sections to the fundamental limits when comparing molecules, we show that simply dividing by the square of the number of electrons per molecule yields a good metric for comparison.