## 1 Introduction
They successfully detected signals up to the seventh order in single-layer graphene that were driven by electric fields reaching tens of kilovolts per centimeter, with very high conversion efficiencies. The field intensities were well beyond the range of validity of the perturbation theory used to describe the third-order nonlinearity owing to intraband electronic transitions [24]. The generation of such terahertz high-harmonics was probably made possible by the creation of hot Dirac fermions that oscillate at these new frequencies. Therefore, the explanation, in this case, does not rely solely on the conventional nonlinear theory but rather is a combination of coherent nonlinear effects with the generation and dynamics of hot electrons, resulting in a strongly nonlinear intraband THz conductivity. Furthermore, it is possible that the broad dispersion of experimental data for the third-order susceptibility (\(\chi^{(3)}\)) in graphene, measured through the Kerr effect [23], is partially due to hot electrons generated in those experiments. It is well known that monolayer graphene possesses inversion symmetry, which means that, under normal incidence, even-order nonlinear effects vanish. However, this symmetry can be broken by using oblique incidence. In a work published by [PERSON] et al. [7; 29], two femtosecond pulses in a pump-probe scheme with different incident angles were used to excite graphene surface plasmons (GSPs). The differential reflectance of the probe beam was then measured, yielding two types of signals: one non-resonant, present for any combination of incident angles, and the other that resonantly depended on the angles (for a given frequency difference between the pump and probe). In the latter case, the two oblique pulses, through phase-matching, were able to excite GSPs. One interesting observation from the results of Refs. [7; 29] is that the predicted order of magnitude using the conventional theory for \(\chi^{(2)}\) is three orders of magnitude smaller than what is needed to explain the experimental results [30; 31]. This suggests, once again, that the usual perturbative nonlinear approach is not suitable for their description or, at least, that some further effects, such as strong deviations of the electronic system from thermal equilibrium, are to be taken into account. \"Hot carriers\" is the term used to describe such out-of-equilibrium situations that typically arise in semiconductors subjected to high-intensity steady or optical fields, and the charge carriers rapidly acquire more energy than they would have in equilibrium with the crystal lattice. If the energy gain rate is higher than the rate at which they lose energy to the lattice, the carriers enter a quasi-equilibrium state with each other, with a much higher temperature than that of the lattice. Because of this high temperature, these carriers are commonly referred to as \"hot\" [32; 33]. Their importance has been demonstrated for semiconductors [32; 33] and metals [34; 35]. In graphene, owing to its particular linear band structure and, consequently, low density of states around the Dirac points, this phenomenon is extremely relevant and fundamental to understanding the ultrafast dynamics of photoexcited carriers. Because of its short carrier-carrier scattering characteristic time, graphene rapidly creates a population of hot electrons in the conduction band and holes in the valence band that can be described by two Fermi-Dirac distributions with different Fermi levels but a common temperature [36; 37]. Unlike assumed in the perturbative approach, where the Fermi level and temperature remain fixed at their equilibrium values, for hot electrons, these parameters are replaced by quasi-Fermi levels and an effective temperature, which are dynamic and depending on the intensity, the duration of the optical pulses, and other influencing factors. This situation is typical of pump-probe spectroscopy experiments.