## 4 Conclusions  
In this work, we extended the ab initio pump-probe simulation of ATAS for 2D materials in order to provide a platform to understand the microscopic origin of the features observed in ATAS experiments. As an example, we investigated ultrafast modification of optical properties of monolayer \(h\)-BN under intense IR pulses. First, we performed the pump-probe simulations, setting the polarization of the pump and probe pulses along the \(x\)-direction in Figure 1a, which is perpendicular to a B-N bond. As a result of the simulation, we obtained the transient absorption spectrum during the laser irradiation. The resulting  
Figure 6: (**a**) real part of the transient conductivity of monolayer \(h\)-BN under the IR pump pulse for the \(y\)-direction in Figure 1a; (**b**) the time-profile of the applied pump electric field. The result is computed by TDDFT with ALDA. spectrum presented an oscillatory behavior with a V-shaped energy dispersion: the _fishbone_ structure. The frequency of the oscillatory behavior was found to be twice the frequency of the pump pulse. In previous studies, similar structures have been observed in bulk materials [10; 13; 39] and understood in terms of the dynamical Franz-Keldysh effect [40; 41]. To clarify the microscopic mechanism of the observed transient absorption spectrum of \(h\)-BN, we studied the electronic structure of the material with the DFT and found relevant transitions at the \(M\) and \(\Gamma\) points. To elucidate the contribution from these transitions, we further analyzed the transient absorption spectra with a parabolic two-band model. As a result, we identified that the fishbone structure in Figure 2 consists of two major contributions. One is the dynamical Franz-Keldysh effect around the \(\Gamma\) point. The other is also the dynamical Franz-Keldysh effect but with an anomalous band dispersion around the \(M\) point. Therefore, we clearly demonstrated that the IR-induced intraband transitions are significant for the transient absorption even in 2D materials. Furthermore, the discovered mechanism based on intraband motion with the anomalous band dispersion indicates the possibility of introducing exotic dynamical properties of the 2D material since the classical analog of the intraband motion in the anomalous dispersion would be a classical motion of a negative mass particle. Next, we performed the TDDFT pump-probe simulation, setting the polarization of the pump and probe pulses along the \(y\)-direction. For this direction, the inversion symmetry of the material is broken, and the system displays even-order nonlinear responses. Indeed, reflecting the nature of the even-order nonlinearity, the obtained transient absorption spectrum shows an oscillation with the frequency of the pump pulse, \(\omega_{pump}\). Therefore, it was demonstrated that the system presents a qualitatively different behavior for different polarization directions. However, despite the significant differences for the different polarizations in Figure 2 and 6, the characteristic features emerge around the probe photon energy of 15 eV. This fact indicates that the \(\omega_{pump}\)-oscillation structure in the transient absorption spectrum with the \(y\)-direction polarization may also be related to the anomalous band dispersion at the \(M\) point. In order to understand the microscopic origin of the \(\omega\)-oscillation structure, an extension of the parabolic two-band model is required. The demonstrated ab initio ATAS simulation for 2D materials is a very general approach that is applicable not only to single-layered materials but also to multi-layered systems including heterostructures. Here, the interplay of multi-layer and interface effects with geometrical tilting, such as moire patterns, could strongly affect the electron dynamics and be observed in the time domain. **Author Contributions:** S.A.S. performed the calculations. All of the authors participated in the interpretation of the results and wrote the paper.