## 4 Discussion  
The presented calculations regarding the time evolution of the state populations and X-ray emission rates for the OPA and TPA processes can be discussed in different aspects. First, we would like to stress that the presented 3-level model is very simple in terms of number of atomic states, and hence the possible excitations and transitions, that are considered. We are aware that the XFEL beam interaction may lead, in specific cases, to more complicated distributions of atomic states that very often may be difficult or even impossible to treat with such a simple model. Nonetheless, as shown in the present report, the three-level model rate equations can be employed as a general tool for the description of nonlinear phenomena relevant for two-photon absorption via X-ray interaction with core atomic states. The presented model describes quite well the reported experimental dependences of the X-ray emission rates on the incident X-ray flux for both one- and two-photon absorption processes. The linear and quadratic dependences observed for the OPA and TPA processes were able to be well reproduced by the 3-level model calculations, and the extracted total-cross section values are in agreement with the experiment. The observed underestimation of the TPA cross-section by our calculations will be discussed below, together with the narrowing effect found for the population of the continuum state through the TPA mechanism. Figure 4: Calculated width (FWHM-full width at half maximum) for the N\({}_{2}\) (blue) and N\({}_{3}\) (red) states population as a function of the incident X-ray flux. The assumed width of the X-ray pulse was 30 fs and is drawn for comparison as a horizontal dotted line. As shown in Figures 3 and 4, the population N\({}_{3}\) of continuum states is about 30% narrower than the population N\({}_{2}\) of the intermediate state and thus also narrower than the applied incident X-ray flux. While the incident X-ray flux width is 30 fs, the estimated width for the population of the continuum state is about 21.4 fs. We found that this effect can be explained by an effective square-dependence of the excitation of these atomic states on the incident photon flux due to the two-photon absorption process. In fact, this dependence manifests, in the occurrence of the nonlinear X-ray interaction with matter. Indeed, the two-photon absorption process scales with the square of the applied X-ray intensity. Therefore, the employed Gaussian-like incident X-ray pulse will effectively act as the squared Gaussian profile. The Gauss-square function, in comparison with the nominal Gauss-shape function, will exhibit a width that is narrower by a factor of \(2^{-1/2}\) (i.e., \(\approx 1.41\)). Indeed, this observation is in perfect agreement with the observed narrowing, i.e., the ratio of 30 fs to 21.4 fs is equal to 1.4. This is also confirmed in Figure 3c, where the Gaussian-square function (dotted line) is compared with the calculated population of the continuum state, and both profiles are in perfect agreement. Finally, we would like to note that the discussed narrowing effect should be taken into consideration, in general, for the cross-section determination of nonlinear processes. As reported in [13], the experimental data were normalized while assuming that the incident X-ray pulse has a Gaussian shape. While this normalization is valid for linear processes, it seems that in nonlinear interaction cases the normalization factor resulting from the time distribution of the incident X-ray flux should be, in addition, divided by a factor if \(2^{-1/2}\) for the two-photon processes. It is expected that, for this reason, the cross-sections calculated here for TPA mechanism should again be lower, by a factor of about 1.4, than the discussed experimental results. Finally, we would like to comment on the observed increase of widths of population profiles for excited states, here both N\({}_{2}\) and N\({}_{3}\), at higher incidence X-ray flux.