## 3 Results  
### Extended Hubbard model  
In Figure 3, We show the results of time-dependent calculations for \(\Delta_{eff}=0.28\) eV. These results correspond to the case slightly apart from the phase boundary, and we expect that the domain grows most effectively in the absence of dissipation. In other words, this is a typical PIPT, in which a macroscopic part of the sample is converted into another phase [40; 41; 42]. In Figure 3a, the electron site density at the odd-numbered site (odd site, hereafter) is depicted. Here, \(\tau_{d}\) and \(\hbar\omega_{c}\) are 30 (\(\hbar/\text{eV}\)) and 1.0 eV, respectively. The strength of the amplitude of the electric field is set at \(eaE_{0}=0.2\) eV throughout this subsection. Since the initial state is in the neutral phase, the odd site is initially the charge-rich site. The decrease in the density at the odd site consequently indicates the transition from the neutral state to the ionic state. In experiments, the time dependency is typically observed for the photoinduced reflectivity (\(\Delta R/R\)) at a fixed frequency, and the result is transformed into a spectrum via wavelet analyses. Here, we avoid direct calculation of reflectivity and focus on the site density because the latter affects the former very sensitively. By the way, the present system size is limited to 16 sites, which is the reason that the curve of the conversion seems to be rather saturated despite the initial state just on the phase boundary. In addition to the overall decrease in the site density, we observe an oscillatory component. To analyze the oscillatory component, we fit the smooth background to a function expressed by the green curve and subtract it from the original curve. The extracted component is applied to wavelet analysis, giving a spectrogram appearing in Figure 3b. Finally, we integrate this over the observation time and obtain the frequency distribution (red curve) in Figure 3c. For comparison, we also show the ordinary optical conductivity spectrum (blue curve) and find that they are similar to each other. This is understood in the following way. The general form of the time-dependent state vector is \(|\Psi(t)\rangle=\sum_{\mu}a_{\mu}e^{-i\omega_{\mu}t}|\mu\rangle\), where \(|\mu>\) and \(\omega_{\mu}\) are the eigenstate and its eigenenergy, respectively. Note that the coefficient \(a_{\mu}\) are constant after the pulse end, namely, \(t\geq\tau_{d}\). When we take the matrix element as \(\langle\Psi(t)|n_{l}|\Psi(t)\rangle\), the terms like \(a_{\mu}^{*}a_{\mu}e^{-i(\omega_{\mu}-\omega_{\mu^{\prime}})t}\) appear, and the combinations of \(|\mu\rangle\) and \(|\mu^{\prime}\rangle\) as \(|g\rangle\) and any excited state give the oscillations corresponding to the optical conductivity spectrum. Here, we make two remarks on the peaks around zero energy in the optical conductivity. Firstly, the enhancement at zero energy originates from the prefactor of \(1/\omega\) in the optical conductivity formula in Equation (5). Secondly, the peaks at less than 0.25 eV are attributable to the spin excitations from the state of the ionic phase. As was mentioned in Ref. [13], the state of the ionic phase exists energetically slightly higher than that of the neutral phase, that is, the ground state of this case. According to a past study [43], the spin excitations from the ionic state are optically active due to the site-alternate potential appearing as the last term of Equation (1). Anyway, the appearance of such low energy peaks is due to a size effect, and we neglect them in the following. Meanwhile, we carefully inspect Figure 3c and notice an oscillatory component that is not seen in the optical conductivity, which is located around 0.40 eV (hatched region), slightly below the lowest peak of the optical conductivity. We consider this contribution as originating from the term as \(|\mu\rangle\) and \(|\mu^{\prime}\rangle\) being the first and second optically excited states that correspond to the first and second peaks in the optical conductivity, respectively.