## 3 Results
### Extended Hubbard model
The energy difference between the peaks is around 0.4 eV, which is almost the same as the energy of the new oscillatory component, as specified by the horizontal arrows. The appearance of this component is quite natural because the coherence between the excited states is maintained in our calculation that assumes no dissipation. Regarding this new oscillation, we add one more remark. We easily notice a repeated pattern in the region of \(t\geq 30\) (\(h/\)eV) and \(E\geq 0.5\) eV of Figure 3b. Checking the period of this time structure, we judge that this is another manifestation of the new oscillation. As the last comment to Figure 3, the peak distribution  
Figure 3: Dynamical behaviors for \(\Delta_{eff}=0.28\) eV during and after the pulse excitation with \(h\omega_{c}=1.0\) eV and \(\tau_{d}=30\) (\(h/\)eV). (**a**) The site density at the odd-numbered site; (**b**) a spectrogram obtained from (**a**); and (**c**) the time integration of (**b**) (red line) and optical conductivity (blue line). The two green arrows in (**c**) show that the energy difference is the same at the two parts. of the optical conductivity is rather sparse. Since it is a continuum in an infinite system on the phase boundary, this sparse distribution should be interpreted as due to the finite size effect. We next argue the cases of which the initial states are far from the phase boundary. In Figure 4, we summarize the results calculated for \(\Delta_{eff}\) = 0.50 eV and \(\hbar\omega_{c}\) = 1.5 eV. The larger value for \(\Delta_{eff}\) lowers the site energy of the odd sites, leading to the stabilization of the neutral phase. Again, we show three figures that are analogous to those in Figure 3. In addition, in this case, we recognize an oscillational component that is absent in the optical conductivity, around 0.5 eV. We again attribute this to the energy difference between the first peak and the second energy band in the optical conductivity, as specified by the horizontal arrows. We emphasize that the peak positions in the spectrum are almost free from the size effect because the larger order parameter in this case makes the state rather localized [13]. We, therefore, assure that the energy of the new oscillatory component is well below the optical gap of about 1 eV, which suggests the feasibility of its detection. At this point, readers might have a question related to the covered energy region. Namely, the present duration time is 30 (\(\hbar/\)eV), of which the inverse is roughly 0.03 eV. Even if we multiply it by \(2\pi\), it becomes only 0.2 eV, which is much smaller than the concerned energy difference. We note that the present pulse form is a step function and that the covered energy region extends widely obeying a function like \(1/(E-\hbar\omega_{c})^{2}\). We confirm this region in Figure 4d, which shows energy distribution in the excited states immediately after the pulse, with the case of \(\Delta_{eff}\) = 0.50 eV as an example. We find that not only the directly excited region around \(\hbar\omega\) = 1.5 eV but also the state of the first peak is substantially excited due to this wide extension. We also see at least two oscillatory components at 0.25 eV and 0.2 eV. While detailed analyses are still required, we tentatively attribute the former to the interband coherence between the second energy band and the third energy band and the latter to the intraband coherence within the second energy band. Almost the same feature is also confirmed in a much more localized case with \(\Delta_{eff}\) = 0.7 eV. In Figure 5, we again show the calculated results, which are obtained with \(\hbar\omega_{c}\) = 1.2 eV. The extra peak appears around 0.7 eV, which coincides with the energy difference between the first and second peaks in the optical conductivity. From these three cases, we judge that the coincidence is not accidental  
Figure 4: (**a**–**c**) Same as Figure 3, but for \(\Delta_{eff}\) = 0.50 eV and \(\hbar\omega_{c}\) = 1.5 eV; (**d**) weight distribution at the pulse end, i.e., \(t=\tau_{d}\).