## 2 Model and Method  
We employ the same model that we used in past studies. This model has been used for describing TTF-CA [12; 24; 25; 26; 27], and its model hamiltonian in the absence of lattice deformation is  
\[H=-t_{0}\sum_{l\sigma}(e^{-i(\alpha/\hbar c)A(t)}C_{l+1\sigma}^{\dagger}C_{l \sigma}+H.c.)+U\sum_{l}n_{l\uparrow}n_{l\downarrow}+V\sum_{l}n_{l}n_{l+1}+ \Delta_{eff}\sum_{l=even}n_{l}\, \tag{1}\]  
where \(C_{l\sigma}\) is the electron operator at the \(l\)th site with spin \(\sigma\), and \(n_{l\sigma}\) is its number operator with \(n_{l}=n_{l\uparrow}+n_{l\downarrow}\). We adopt the periodic boundary condition. Here, we place TTF (CA) molecules at the odd- (even-) numbered sites. The effect of light is incorporated via the Peierls phase, namely, the extra phase factor in the first term, with \(A(t)\), \(-e\), and \(a\) being the vector potential of light, the electronic charge (\(e=|e|\)), and the lattice constant, respectively. TTF -CA is well known to have two fundamental phases, which are the neutral phase and the ionic phase, as illustrated in Figure 2[28; 29; 30; 31; 32; 33; 34; 35]. In the neutral phase, each molecule remains electrically neutral, while a charge transfer occurs from the donor (D) to the acceptor (A), leading to a chain of D\({}^{+}\) and A\({}^{-}\). From here on, we assume that the initial state is in the neutral phase. In this case, the excitation energies in the localized limit are \(V+n_{ex}(2V+\Delta_{eff}-U)\) with \(n_{ex}\) being the number of electron excitations, which are nearly degenerate around the phase boundary, i.e., \(2V+\Delta_{eff}=U\). The parameters appropriate for TTF-CA are estimated as \(U=2.4080\) eV, \(V=1.070\) eV, and \(t_{0}=0.17\) eV [13; 36; 37]. The remaining parameter, \(\Delta_{eff}\), takes a value close to 0.27 eV, which is the value on the phase boundary. In this article, we mainly assume much larger values because we are interested in cases that are rather apart from the phase boundary. In the actual case, lattice deformations such as the molecular dimerization are known to occur in the course of the neutral-to-ionic transition, although we do not argue them here because we are interested in the early stage of dynamics before the lattice deformations occur. To discuss dynamical behaviors, namely, time evolutions after photoexcitation, we first determine the ground state \(|g\rangle\) by exact diagonalization and next perform exact time-dependent calculation, using a \(N\)-site ring. For the extended Hubbard model, we choose \(N=16\). The vector potential along the chain direction, \(A(t)\), is expressed by the electric field along the same direction, \(E(t)\), as  
\[A(t)=-c\int_{0}^{t}dt^{\prime}E(t^{\prime}). \tag{2}\]  
Figure 2: Schematics for a neutral-ionic-transition system. (**a**) The neutral phase and (**b**) the ionic phase. D and A mean a donor and an acceptor, which correspond to TTF and CA, respectively, for TTF-CA. The green oval indicates molecular dimerization. In this work, we assume a step function for the envelope of \(E(t)\), as  
\[E(t)=E_{0}\,\theta(t)\theta(\tau_{d}-t)\sin(\omega_{c}t)\, \tag{3}\]  
where \(E_{0}\), \(\omega_{c}\), and \(\tau_{d}\) are the amplitude of the uniform electric field, the central frequency, and the duration time of the pulse, respectively, and \(\theta(t)\) is the step function. Using Equations (2) and (3), the explicit functional form of \(A(t)\) for \(t<\tau_{d}\) is  
\[A(t)=cE_{0}(\cos(\omega_{c}t)-1)/\omega_{c}\, \tag{4}\]  
and \(A(t)=A(\tau_{d})\) for \(t\geq\tau_{d}\). Compared with other pulse forms, the present form has an advantage that the check of the calculation is easy because the total energy is conserved after the pulse, i.e., \(t\geq\tau_{d}\). The method is based on a [PERSON] expansion of the time evolution operator [38]. We include the terms up to 10 th, which preserve the norm of the state vector with practically sufficient precision during the time region treated in this article. Throughout this article, the unit of time is \(\hbar/\text{eV}\sim 0.66\) fs.