## 4 Photoinduced Dynamics
### Cooperative Dynamics of Charge and Lattice  
Motivated by recent experiments, we study the electronic excitation driven by an ultrashort laser pulse. We take the following form for the vector potential:  
\[\mathbf{A}(t)=\mathbf{\epsilon}A_{0}\cos(\omega_{\mathrm{pump}}t)\exp\left(-\frac{t^ {2}}{2T_{\mathrm{pump}}^{2}}\right). \tag{15}\]  
Here, the real parameter \(A_{0}\) denotes the peak amplitude of the vector potential, and \(\mathbf{\epsilon}\) is the polarization vector and assumed to be \(\mathbf{\epsilon}=(1,0)\) in this paper. The results are not sensitive to the direction of \(\mathbf{\epsilon}\). Following the experiment [20], we set the central frequency \(\omega_{\mathrm{pump}}\) and the pulse width \(T_{\mathrm{pump}}\) so that \(\hbar\omega_{\mathrm{pump}}=1.55\,\mathrm{eV}\) and \(2T_{\mathrm{pump}}=35\,\mathrm{fs}\). The profile \(A_{x}(t)\) is shown in Figure 5a. We note that the peak amplitude of the electric field is approximately given by \(E_{0}=\omega_{\mathrm{pump}}A_{0}\) since \(T_{\mathrm{pump}}\gg 1/\omega_{\mathrm{pump}}=2.4\,\mathrm{fs}\). In the following, we shall use \(E_{0}\) rather than \(A_{0}\), making it easier to compare our results with experimental results. The laser electric field induces charge dynamics on a short time scale of order \(T_{\mathrm{pump}}\). In Figure 5, we show the numerical results of local charge densities at 13 sites in a hexagram unit for a typical peak amplitude of the electric field \(E_{0}=0.92\,\mathrm{MV/cm}\), which is comparable with the experimental condition [20] (see Figure 1 for the labeling of sites). Since we are not interested in the spin density, the charge density averaged between the two hexagrams is plotted. At our initial time of calculation \(t=t_{\mathrm{init}}=-52.5\,\mathrm{fs}\), the charge-density distribution is the one shown in Figure 3a. As noted before, the 6-fold rotational symmetry is slightly broken down to the 2-fold symmetry. As time advances, the electric field envelope gradually increases, and the charge density starts to oscillate in time at each site. To analyze the charge-density oscillations in detail, we pair up the six \"B\" sites as \((2,5)\), \((3,6)\), and \((4,7)\), each of which locates symmetrically about the central A site. We do the similar pairing for the \"C\" sites as well. We look into the average and the difference of the charge densities for each site pair, which are even and odd, respectively, under the 2-fold rotation. We note that the charge density at site 1 (A) itself is even under this operation. These even and odd quantities show different kinds of dynamics. The even quantities shown in panels (a-c) oscillate with frequency \(\sim\)2\(\omega_{\mathrm{pump}}\), whereas the odd ones (d,e) do with frequency \(\sim\omega_{\mathrm{pump}}\). This is because the vector potential \(\mathbf{A}(t)\) has an odd parity with respect to the 2-fold rotation, and these two quantities are coupled to \(\mathbf{A}(t)\) in the second and the first orders, respectively. In addition to those oscillations, the charge dynamics changes its spatial distribution and the CDW order decreases on the short time scale of order \(T_{\mathrm{pump}}\). The charge-density imbalance in panels (a-c) decreases during the laser irradiation, and the local charge density slightly approaches the uniform one. This CDW order decrease is caused not just by the decrease in the PLD. Figure 5f shows that the lattice distortion \(x(t)\) remains almost unchanged until \(t\sim 0\), while the significant changes have already appeared in the charge density. In addition, at \(0<t<50\,\mathrm{fs}\), \(|x(t)|\) gradually decreases, while the charge dynamics rather slows down. Therefore, the short-time dynamics is dominated by the electrons driven by the laser field, and the CDW is rapidly suppressed. On a longer time scale, the lattice distortion \(x(t)\) plays the main role in dynamics.