## 3 Case Study Results and Discussion
where \(\sigma_{\rm eq}\) is the equilibrium conductivity, given by Equation (38) but with equilibrium Fermi energy and temperature, and \(t^{(0)}\) and \(r^{(0)}\) are the transmission and reflection coefficients, respectively, in the absence of graphene:  
\[t^{(0)}=\frac{2\cos\alpha}{n\cos\alpha+\cos\theta}\quad;\quad r^{(0)}=-\frac{n \cos\alpha-\cos\theta}{n\cos\alpha+\cos\theta}\,. \tag{46}\]  
The term \(\mathrm{Re}\{\Sigma_{a}(\delta t)\}/W_{a}\), describing the graphene absorbance due to the convolution of two beams separated by a time interval \(\delta t\), is given by  
\[\frac{\mathrm{Re}\{\Sigma_{a}(\delta t)\}}{W_{a}}=\frac{1}{\tau_{a}}\sqrt{ \frac{2}{\tau}}\int_{-\infty}^{\infty}dt\exp\left[-\frac{2t^{2}}{\tau_{2}^{2}} \right]\mathrm{Re}\{\sigma_{\rm ne}(\omega_{a},t+\delta t)\}\,, \tag{47}\]  
where \(\sigma_{\rm ne}\) is the out-of-equilibrium conductivity provided by our hot electron model and \(\tau_{a}\) is the duration of the probe beam, which is assumed to have a temporal Gaussian shape. Our model contains three time parameters, \(\tau_{ee}\), \(\tau_{e}\), and \(\tau_{e}\) (Section 2.1), and their exact values are not known (and depend on the graphene sample), but the inequality (1) tells us that \(\tau_{ee}\) is the most important one and can be treated as a fitting parameter whose order of magnitude is known (few tens of fs). Parameters of the experiment by [PERSON] et al. are listed in Table 1.  
In Figure 5, we present the normalized differential reflectance calculated using our model (Equation (45)) juxtaposed with the experimental data obtained from Ref. [7]. Two theoretical curves are plotted in this figure: (i) the blue one represents the outcome \begin{table}
\begin{tabular}{c c c c} \hline \(\lambda_{a},\lambda_{b}\) & 615 nm & \(\tau_{a}\), \(\tau_{b}\) & 85 fs \\ \hline \(\alpha_{b}\) & 40\({}^{\circ}\) & \(\alpha_{a}\) & 20\({}^{\circ}\) \\ \hline \(\phi_{b}\) & 0.26 mJ/cm\({}^{2}\) & \(\phi_{a}\) & 0.0028 mJ/cm\({}^{2}\) \\ \hline \(w_{0}\) & 300 \(\mu\)m & \(T_{0}\) & 300 K \\ \hline \(n\) & 1.46 & \(E_{F}\) & 0.3 meV \\ \hline \end{tabular}
\end{table} 
Table 1: Values from the quantities that are listed in the article. Figure 5: Differential reflectivity (normalized to the pump fluence) for a pump–probe experiment with two Gaussian pulses. The black line with dots represents the experimental results taken from Ref. [7]. The blue dashed curve is obtained by applying our hot electron model with \(\tau_{\rm ee}=10\) fs, \(\tau_{e}=0.50\) ps, and \(\tau_{e}=10\) ps, while the remaining parameters are taken from Table 1. The red solid curve is obtained for the same set of parameters of the blue curve but with the intrinsic time shift \(\Delta t=0.102\) ps. The inset provides a schematic illustration of both pulses (blue and green circles), explicitly depicting the shift vector \(\vec{r}_{0}\) (the shift between the spots is exaggerated, we used \(r_{0}=0.62w_{0}\) in the calculation). of our model with the parameters listed in Table 1 and characteristic times given in the figure caption, and (ii) the red curve that nicely fits the experimental data is obtained by using an \"intrinsic\" time shift \(\Delta t=0.102\) ps. Both curves account for a shift of \(r_{0}\approx 0.62w_{0}\) between the pump and probe spots. Among the time parameters, \(\tau_{ex}\) is the critical one, and \(\tau_{e}\) influences the signal decay for large \(\delta t\), while \(\tau_{r}\) has a very little effect on the differential reflectance curve. The least-squares fitting method was applied in order to obtain the curve (ii). The adjustment of \(\Delta t\) can be explained by taking into account the following argument.