## 2 Materials and Methods
### Electron Relaxation Model
#### 2.2.1 Non-Thermalized Carriers  
We start by writing down equations that describe the populations of non-thermalized electrons, \(n_{\text{nt}}\), and holes, \(p_{\text{nt}}\), in optically excited graphene. Given the symmetry of their spectra (within the Dirac cone approximation) and the unique scattering time, \(\tau_{ee}\), both electrons and holes evolve in the same way, so the non-thermalized populations are equal:  
\[p_{\text{nt}}(\vec{r},t)=n_{\text{nt}}(\vec{r},t)\;. \tag{2}\]  
Owing to the carrier-carrier scattering, two populations of thermalized (i.e., having reached equilibrium among them but not with the environment) carriers are created. We denote the 2D concentration of the thermalized (hot) electrons by \(n_{\text{t}}(\vec{r},t)\) and the thermalized holes by \(p_{\text{t}}(\vec{r},t)\). Therefore, the total concentrations of electrons and holes are expressed as  
\[n(\vec{r},t)=n_{\text{nt}}(\vec{r},t)+n_{\text{t}}(\vec{r},t)\;; \tag{3}\]  
\[p(\vec{r},t)=p_{\text{nt}}(\vec{r},t)+p_{\text{t}}(\vec{r},t)\;. \tag{4}\]  
The balance equations governing the depletion of carriers from the non-thermalized to the thermalized population are  
\[\dot{p}_{\text{nt}}(\vec{r},t)=\dot{n}_{\text{nt}}(\vec{r},t)=\frac{A(\vec{r}, t)}{\hbar\omega_{b}}I_{b}(\vec{r},t)-\frac{n_{\text{nt}}(\vec{r},t)}{\tau_{ee}}\;, \tag{5}\]  
where  
\[I_{b}(\vec{r},t)=I_{b}^{\text{max}}\exp\left(-\frac{2t^{2}}{\tau_{b}^{2}} \right)\exp\left(-\frac{\text{ln}2}{w_{0}^{2}}r^{2}\right) \tag{6}\]  
is the pump intensity, \(I_{b}^{\text{max}}\) is the peak intensity value, \(w_{0}\) is the effective radius of the pump beam, and  
\[A(\vec{r},t)=A_{0}(\alpha,\theta)\left\{1-g_{h}\left(-\frac{\hbar\omega_{b}}{ 2},\vec{r},t\right)-g_{e}\left(\frac{\hbar\omega_{b}}{2},\vec{r},t\right)\right\} \tag{7}\]  
is the time-dependent absorbance of graphene (see Section S1 of the Supplementary Material for details). Here, \(g_{e}\) and \(g_{h}\) are the non-thermalized distribution functions for electrons and holes, respectively, and  
\[A_{0}(\alpha,\theta)=\frac{4\pi\alpha_{F}}{\left(n\cos\alpha+\cos\theta \right)^{2}}\cos\alpha\cos^{2}\theta \tag{8}\]  
is the absorbance amplitude, with \(\alpha\) and \(\theta\) denoting the incident and refraction angles of the pump beam (with respect to the normal to the surface) and \(n\) denoting the refractive index of the substrate. Equation (8) holds for transverse magnetic (TM) polarization. Notethat the first term in Equation (5) corresponds to the generation of non-thermalized carriers and the second one to their thermalization. According to Equation (2), the non-thermalized distributions \(g_{\epsilon}\) and \(g_{h}\) in Equation (7) should be equal:  
\[g_{h}(-\epsilon,\overline{r},t)=g_{\epsilon}(\epsilon,\overline{r},t)=g( \epsilon,\overline{r},t). \tag{9}\]  
Since the pump pulse is described by a Gaussian wavepacket in the frequency domain, \(g(\epsilon,\overline{r},t)\) is modeled as  
\[g(\epsilon,\overline{r},t)=\tilde{g}(\overline{r},t)\frac{\Delta_{\epsilon}( -\infty)}{\Delta_{\epsilon}(t)}\exp\left(-\frac{(\epsilon-\hbar\omega_{b}/2)^ {2}}{2\Delta_{\epsilon}^{2}(t)}\right)\,, \tag{10}\]  
where \(\tilde{g}(\overline{r},t)\) can be expressed in terms of the non-thermalized population of charge carriers:  
\[\tilde{g}(\overline{r},t)=\frac{n_{\text{nt}}(\overline{r},t)}{2n_{\text{nt} }^{\text{max}}}. \tag{11}\]  
Here, \(n_{\text{nt}}^{\text{max}}=\sqrt{\frac{\pi}{2}}\Delta_{\epsilon}\rho\left( \frac{\hbar\omega_{b}}{2}\right)\) is the maximum possible concentration of non-thermalized electrons (holes) (the factor of \(1/2\) in Equation (11) appears because the absorption vanishes for \(g(\epsilon,\overline{r},t)=\frac{1}{2}\); see Equations (7) and (9)). These expressions are derived in Section S2 of the Supplementary Material. In order to proceed, we make the following simplifying assumptions:  
1.