## 2 Materials and Methods  
For the calculations of two-photon absorption rates and time dependent population states, a three-level model described by rate equations was assumed. A similar solution was proposed in [13], where a constant intensity for the incident X-rays was assumed and, furthermore, a stationary case was considered. This allowed for an analytical solution of the rates equations. Herein, in contrast, we focus on an exact time-dependent solution of these rate equations for a given initially assumed time-profile of the incident X-ray pulse. Consequently, this approach allows the study of the time evolution of both the state population and the X-ray emission for a given pulse shape, i.e., a full description of the dynamics of the OPA and TPA processes. The schematic representation of atomic states included in the three-level rate equations model is shown in Figure 1.  
This model is based on the conclusions drawn from the experimental report [13], where the two-photon absorption mechanism leading to sequential ionization through an intermediate virtual atomic state was observed. In this 3-level model (see Figure 1), N\({}_{1}\) corresponds to the population of the ground state, N\({}_{2}\) to the population of the virtual intermediate state, and N\({}_{3}\) to the ionized atom through the sequential (\(1\to 2\to 3\)) two-photon absorption process. According to this picture, a subsequent radiative decay of the N\({}_{2}\) state carries information on the direct OPA mechanism and a  
Figure 1: A schematic representation of the three-level model, representing the possible electronic transitions in one- and two-photon absorption processes on atomic core states. N\({}_{1}\) corresponds to the population of the ground state, N\({}_{2}\) to the population of the virtual intermediate state, and N\({}_{3}\) to the ionized state. \(\sigma_{1,2}\), and \(\sigma_{1,2}\) are cross-sections for the first and second photon absorption. \(\tau_{2}\) and \(\tau_{3}\) corresponds to the lifetimes of the N\({}_{2}\) and N\({}_{3}\) states. radiative decay of the N\({}_{3}\) state on the TPA process. We assumed that the excitation of the N\({}_{3}\) state is possible only through the intermediate N\({}_{2}\) state, i.e., only a sequential two-photon absorption process is possible. Direct, simultaneous two-photon absorption is neglected here, because it is expected that the cross-sections for this process are three orders of magnitude lower than the sequential TPA mechanism [9]. Assuming a sequential model for the two-photon absorption mechanism, the following rate equations are considered:  
\[\frac{\mathrm{d}\mathrm{N}_{1}}{\mathrm{d}t}=-\mathrm{I}(t)\times(\mathrm{N}_ {1}(t)\times\sigma_{1,2}-\mathrm{N}_{2}(t)\times\sigma_{1,2})+\frac{\mathrm{N} _{2}(t)}{\tau_{2}}+\frac{\mathrm{N}_{3}(t)}{\tau_{3}}, \tag{1}\]  
\[\frac{\mathrm{d}\mathrm{N}_{2}}{\mathrm{d}t}=\mathrm{I}(t)\times(\mathrm{N}_ {1}(t)\times\sigma_{1,2}-\mathrm{N}_{2}(t)\times\sigma_{2,3}-\mathrm{N}_{2}(t) \times\sigma_{1,2})-\frac{\mathrm{N}_{2}(t)}{\tau_{2}}, \tag{2}\]  
\[\frac{\mathrm{d}\mathrm{N}_{3}}{\mathrm{d}t}=\mathrm{I}(t)\times\mathrm{N}_{ 2}(t)\times\sigma_{2,3}-\frac{\mathrm{N}_{3}(t)}{\tau_{3}}, \tag{3}\]  
\[\mathrm{N}_{1}(t)+\mathrm{N}_{2}(t)+\mathrm{N}_{3}(t)=\mathrm{N}, \tag{4}\]  
where I(t) describes the time structure of the incident X-ray pulse and \(\mathrm{d}\mathrm{N}_{1}/\mathrm{d}t\), \(\mathrm{d}\mathrm{N}_{2}/\mathrm{d}t\), \(\mathrm{d}\mathrm{N}_{3}/\mathrm{d}t\) are the decay rates of the initial, intermediate, and ionized states of an atom at time t. The cross-sections \(\sigma_{1,2}\) and \(\sigma_{2,3}\) represent transition probabilities between the N\({}_{1}\) and N\({}_{2}\), and the N\({}_{2}\) and N\({}_{3}\) states, respectively. In Equation (1), the first term describes the transitions between states N\({}_{1}\) and N\({}_{2}\) due to photon absorption and stimulated emission processes, respectively.