## 3 Results
### Domain Model
Closer analyses of the eigenenergies defined as \(e(i)\), which appear in Figure 7c, indicate that the two peaks for \(\hbar\omega_{c}\) = 1.7 eV are assigned to \(e(2)-e(1)\) and \(e(3)-e(2)\). As is easily understood from the form of \(E(p)\), the present system is considered as a Stark ladder in a uniform field. Only one modification is the boundary condition at the shortest distance between the domain walls, which deforms the differences between the two consecutive eigenenergies at the lower edge from an equidistant structure. In contrast, excitations with higher energies fall within in the ladder more satisfactorily. The single frequency energy confirmed for \(\hbar\omega_{c}\) = 10.2 eV corresponds to the unique energy difference between the two consecutive eigenenergies in the high energy limit, although the former energy is deviated from the latter because of the resolution of the Fourier transform. Hereafter we define this energy difference as \(E_{d}\). Such a situation reminds us of a so-called Bloch oscillation that was first proposed for a particle in a crystal under a uniform field [20; 21; 22; 23]. In the case of a Bloch oscillation, a periodic motion in real space is expected due to a crystal effect, which is similar to the cases in Figure 6a except that for \(\Delta_{eff}\) = 0.27. For an infinite Stark ladder, it is well known that \(E_{d}\) coincides with the energy difference between the two consecutive sites [22]. Applying this property to the present case, \(E_{d}\) is evaluated as \((2V+\Delta_{eff}-U)\) for high-energy excitations. It is also known for the Stark ladder that the period and the amplitude of the oscillation are given as \(2\pi\hbar/E_{d}\)[20] and \(4t_{0}/E_{d}\)[23], respectively. Since the critical  
Figure 6: Dynamical behaviors in the domain model. All the results are obtained with \(\omega_{c}\) = 1.7 eV and \(\tau_{d}\) = 18 (\(\hbar/\)eV). (**a**) The time evolutions of the averaged domain size; (**b**) schematics of a real-space motion of a wave packet defined along the relative coordinate of the two domain walls. point is given by \((2V+\Delta_{eff}-U)=0\), both the period and the amplitude diverge toward the critical point. In particular, a more extended period, namely, a smaller oscillation frequency near the phase boundary pushes the signal within the optical gap, because the optical gap remains finite even on the phase boundary [29]. By the way, in the case of an ordinary Bloch oscillation, the wave packet almost keeps its initial form when the form of the initial wave packet is a smooth Gaussian [23]. In the present case, the wave packet gradually loses its original form defined at the pulse end (\(t=\tau_{d}\)), which tendency is attributed to the somewhat complicated form at the pulse end and the boundary condition. As the last remark, we mention the value of \(E_{d}\) itself. As is already mentioned, for an infinite Stark ladder, \(E_{d}\) is exactly the energy difference between the two consecutive sites. This fact suggests that the frequency of the oscillatory component in high energy excitations gives us the value of \((2V+\Delta_{eff}-U)\) reliably. This contrasts with general cases, where measured quantities are renormalized due to a finite amount of transfer, and it is difficult to estimate the bare values of parameters directly.