Pump and probe nonlinear processes: new modified sum rules from a simple oscillator model

The nonlinear oscillator model is useful to basically understand the most important properties of nonlinear optical processes. It has been shown to give the correct asymptotic behaviour and to provide the general features of harmonic generation to all orders, in particular dispersion relations and sum rules. We investigate the properties of pump and probe processes using this model, and study those cases where general theorems based on the holomorphic character of the Kubo response functions cannot be applied. We show that it is possible to derive new sum rules and nem Kramers-Kronig relations for the two lowest moments of the real and of the imaginary part of the third order susceptibility and that new specific contributions become relevant as the intensity of the prole increases. Since the analytic properties of the susceptibility functions depend only upon the time causality of the system we are confident tl-lat these results are not model dependent and therefore have a general validity, provided one substitutes for the equilibrium values of the potential derivatives the density matrix expectation values of the corresponding operators.