## 3 Case Study Results and Discussion  
As mentioned in the introduction, our hot electron model can be used to understand and simulate the results acquired through pump-probe experiments, where the pump's role is to excite a substantial concentration of photocarriers, which eventually form a population of hot electrons and holes. The latter then gradually dissipate their energy into the lattice via multiple decay pathways, as depicted in Figure 1. Due to their exceedingly high temperature, resulting in a very broad distribution over energies, the hot electrons can impede further transitions between the valence and conduction bands, with the absorption of probe beam photons, because of the Pauli blocking, both below and above the pump photon energy. Notably, because of the inverted electron population above \(\varepsilon=0\), the real part of the conductivity takes on negative values within a certain spectral range, implying an optical gain in this region [42]. The spectral width of this region is determined by the condition of population inversion, \(f(\hbar\omega/2)>f(-\hbar\omega/2)\). It extends up to 0.6 eV for \(t=1\) ps (see Figure 4, right panel). Thermal equilibrium is reestablished within a few tens of ps after the peak of the pump pulse (black dashed curve in the plot). Given our goal of modeling pump-probe experiments, we introduced the weight function, Equation (24), describing the spatial overlap between the pump and probe beams. Assuming that both of them are Gaussian and, when focused onto the graphene surface, produce spots of radius \(w_{0}\), whose centers may be shifted with respect to each other by some vector \(\vec{r}_{0}\). The inset in Figure 5 provides a schematic illustration of this situation. As a test of the proposed model, we applied it to the calculation of differential reflectivity signals measured by [PERSON] et al. [7]. The experimental data presented in this article, where a normalized differential reflectivity was measured, are shown by black dots in Figure 5. To eliminate artifacts arising from power fluctuations, this quantity has also been normalized with respect to the pump fluence. The normalized differential reflectivity is defined as follows:  
\[\frac{\Delta R_{a}(\delta t)}{R_{a}}=\frac{R_{a}^{\prime}(\delta t)-R_{a}}{R_{a }}\, \tag{44}\]  
where \(R_{a}^{\prime}(\delta t)\) is the reflectivity of the pumped system for the central frequency of the probe beam (\(\omega_{a}\)) at a time delay \(\delta t\) between the two pulses, while \(R_{a}\) corresponds to the reflectivity in the absence of pump. Both \(R_{a}\) and \(R_{a}^{\prime}\) depend on the frequency \(\omega_{a}\), but it is skipped as an argument in (44) and below, to be short. The derivation of \(R_{a}^{\prime}(\delta t)\) is provided in Section S5 of the Supplemental Material, and the differential reflectivity is given by  
\[\frac{\Delta R_{a}(\delta t)}{R_{a}}\approx-\frac{4\pi}{c}\frac{\cos^{2}\theta }{\cos\alpha}\frac{t^{(0)}}{r^{(0)}}\bigg{\{}\frac{\text{Re}\{\Sigma_{a}( \delta t)\}}{W_{a}}-\text{Re}\{\sigma_{\text{eq}}\}\bigg{\}}\, \tag{45}\]  
Figure 4: Temporal evolution of the hot electron distribution function (**left**) and out-of-equilibrium electron conductivity (**right**). The inset in the left panel shows the equilibrium Fermi–Dirac distribution. The black dashed line indicates the initial Fermi level of graphene. For both plots, the parameters are the same as the ones used for Figure 3. Notice the small dip at \(\sim\)2 eV in the conductivity plot for \(t=0.1\) ps, which is due to non-thermalized carriers whose contribution is small for the relatively lower pump power (peak intensity \(I_{p}=2.44\) GW/cm\({}^{2}\)) corresponding to the experiments of Ref. [7].