Failure of the two-level and sum over states methods in nonlinear optics, demonstrated by ab initio methods

When calculating nonlinear susceptibilities, a widely used two-level approximation in a sum-over states formulation is the exclusion of all but the ground state and one single excited state. With the goal of efficient optical frequency conversion, the basis of the two-level model is an assumption that just one excited energy level dominates, when determining the response of a nonlinear optical material. Naturally, any system that can be justifiably modelled as comprising just two energy levels affords numerous advantages, most notably calculational simplicity. However, caution is required; the two-level model can deliver potentially misleading results if it is applied without regard to the criteria for its validity. In a series of recent works, analytical results regarding the unsuitability of the two-level approximation have been proven. Ab initio computations of the hyperpolarizability for a class of merocyanine dyes have further demonstrated a drastic inaccuracy from not including higher energy levels in the calculations. In this paper, we report the results of our recent work testing the general validity of two-level calculations in nonlinear optics, constructed with a precise quantum electrodynamical framework as a basis for the theory. These new results show that, for the first-order dynamic polarizability, successive terms contribute progressively less to the final value of the tensorial components, guaranteeing convergence. In contrast, the values of second harmonic optical susceptibility components, similarly calculated, reveal that contributions from successive energy levels, often assumed to be diminishing, in fact fail to deliver the assumed convergence.