I classify multiphoton absorption into separable, linked, and simultaneous processes. The first and second types can be distinguished when the rate-equation approximation is valid whereas the third type refers to the case when the full description of multiphoton absorption is essential. For this purpose, rate equations are solved analytically without decay processes which shows that even if many photons are absorbed the interaction with the light field is linear and one has the case of separable multiphoton absorption. Next a short-pulse approximation is investigated in which I first solve the rate equations without decay processes and then solve only rate equations for the ensuing decay. Finally, the full rate equations are examined and a successive approximation of the underlying Volterra integral equation of the second kind is derived leading to linked multiphoton absorption by the involved decay widths. The three methods are applied to a nitrogen atom in intense and ultrafast x rays from free-electron lasers (FELs). The linearity theorem barely approximates the results in the presence of decay processes which is also not satisfactorily corrected for by the short-pulse approximation. The successive approximation gives excellent agreement with the numerically-exact solution of the rate equations.
In analyzing atomic systems undergoing interactions with external fields that are nearly resonant with two, and only two of the atom's energy levels, one commonly employs a two-level or resonance approximation in which all states of the atom are neglected except for the two in resonance with the field. In this paper, we examine the assumptions necessary to make the transition from a differential equation for the exact polarization of an atomic system to the corresponding equation in the two-level approximation. Only when one includes contributions from states outside the two-level subspace does he find the exact equation reduces to the two-level equation within correctional factors of an intuitive nature.