## 2 Materials and Methods
### Electron Relaxation Model
#### 2.2.4 Non-Equilibrium Conductivity due to Hot Carriers  
Given the quasi-Fermi levels and the electronic temperature as functions of \(t\), we can introduce the transient optical conductivity. We assume that, to the first order, the interband conductivity of graphene can be described by the usual formula [49]:  
\[\sigma_{\rm inter}(\omega,t) =\frac{4\sigma_{0}}{\pi}h(\gamma_{\rm inter}-i\omega)\int_{0}^{+ \infty}d\varepsilon\frac{1-f^{(e)}(\varepsilon,t)-f^{(h)}(\varepsilon,t)}{[h \gamma_{\rm inter}+i(2\varepsilon-\hbar\omega)][h\gamma_{\rm inter}-i(2 \varepsilon+\hbar\omega)]} \tag{38}\]  
Figure 3: Time evolution of the quasi-Fermi levels for electrons (solid lines) and holes (dashed lines) (**left** panel) and the electronic temperature (**right** panel) for different radii of the pump beam (\(w_{0}\)). For both plots, the parameters are as follows: equilibrium conditions, \(T_{0}=300\) K and \(E_{F}=300\) meV; characteristic times, \(\tau_{ee}=10\) fs, \(\tau_{e}=0.5\) ps, and \(\tau_{r}=10\) ps; pump pulse fluence \(\varrho_{b}=0.26\) mJ/cm\({}^{2}\) and \(\tau_{b}=\)85 fs; peak intensity \(2.44\) GW/cm\({}^{2}\); diffusion coefficient \(D=1.1\times 10^{4}\) cm\({}^{2}\)/s; thermal diffusivity \(D_{\varepsilon}=D\). The shift parameter in the weight function (24) \(r_{0}=0\). with \(\gamma_{\rm inter}\) meaning the interband scattering rate. Here, \(f^{(e)}\) and \(f^{(h)}\) are the full distribution functions for the electrons and holes with contributions from thermalized and non-thermalized carriers:  
\[f^{(e,h)}(\varepsilon,t)=g_{e,h}(\varepsilon,t)+f_{e,h}(\varepsilon,F_{e,h}(t)) \tag{39}\]  
with \(g_{e,h}\) given in Equation (9). The extension of the usual formula for the interband conductivity to the time-dependent version (38), similar to the slowly varying amplitude approximation for optical pulses, is valid as long as the electric field oscillates (with the frequency \(\omega\)) several times before the physical conditions change. In other words, this approach works if the two timescales, \(1/\omega\) and the time it takes for chemical potentials and temperature to change, are sufficiently different, and  
\[\omega\tau_{\rm ee}\gg 1\,. \tag{40}\]  
Here, we take \(\tau_{ee}\) as the shortest characteristic time in the system. Since it is expected that \(\tau_{ee}\) lies in the range of 10-30 fs [4], this condition is satisfied for optical frequencies. As we see below, to the first order, the differential reflectivity depends only on the real part of the transient conductivity. Also, it is expected that \(\hbar\gamma_{\rm inter}\ll k_{B}T\)[37], so the real part of the expression (38) can be written as  
\[\mathrm{Re}\sigma_{\rm inter}(\omega,t)=\mathrm{Re}\sigma_{\rm nt}(\omega,t)+ \mathrm{Re}\sigma_{\rm t}(\omega,t) \tag{41}\]  
with \(\sigma_{\rm t}(\omega,t)\) coming from the thermalized (hot) carriers,  
\[\mathrm{Re}\sigma_{\rm nt}(\omega,t)=\sigma_{0}\frac{\sinh\biggl{[}\frac{ \hbar\omega-(F_{e}(t)-F_{h}(t))}{2k_{B}T}\biggr{]}}{\cosh\biggl{[}\frac{F_{h}( t)+F_{e}(t)}{2k_{B}T}\biggr{]}+\cosh\biggl{[}\frac{\hbar\omega-(F_{e}(t)-F_{h}(t))}{2 k_{B}T}\biggr{]}}\,\,, \tag{42}\]  
and \(\sigma_{\rm nt}(\omega,t)\) from the non-thermalized ones [50],  
\[\mathrm{Re}\sigma_{\rm nt}(\omega,t)=-\sigma_{0}g\biggl{(}\frac{\hbar\omega}{ 2},t\biggr{)}\,. \tag{43}\]  
Figure 4 presents the time evolution of both the hot-electron distribution function and the real part of the optical conductivity calculated using Equations (39) and (41), respectively. The distribution function begins in a state of equilibrium; however, after the carriers undergo thermalization, two Fermi-Dirac-like distributions emerge for electrons and holes, characterized by distinct quasi-Fermi levels and unique electronic temperatures. Both distribution functions exhibit a small peak corresponding to the non-thermalized carrier population, which can be seen for some time after the excitation.