## 3 Results and Discussions
The electron density distribution of the coherent superposition electronic state in Equation (7) is described by,\[\mathcal{P}(\mathbf{r}) = |\psi_{0}(\mathbf{r},t)|^{2} \tag{10}\] \[= |c_{\sigma_{g}}(t)|^{2}|\psi_{\sigma_{g}}(\rho,z)|^{2}+|c_{\pi_{u}} (t)|^{2}|\hat{\psi}_{\pi_{u}}^{\pm}(\rho,z)|^{2}\] \[+2|c_{\sigma_{g}}(t)||c_{\sigma_{u}}(t)|\psi_{\sigma_{g}}(\rho,z) \hat{\Psi}_{\pi_{u}}^{\pm}(\rho,z)\cos(\Delta Et)\cos(\theta).\]  
The coherent electron dynamics is composed of two electronic state densities, \(\mathcal{P}^{(g)}(\mathbf{r})=|c_{\sigma_{g}}(t)|^{2}|\psi_{\sigma_{g}}( \mathbf{r})|^{2}\) and \(\mathcal{P}^{(u)}(\mathbf{r})=|c_{\pi_{u}}(t)|^{2}|\psi_{\pi_{u}}(\mathbf{r} )|^{2}\), and their interfering superposition \(\mathcal{P}^{(g,u)}(\mathbf{r},t)=2|c_{\sigma_{g}}(t)||c_{\pi_{u}}(t)|\psi_{ \sigma_{g}}(\mathbf{r})\psi_{\pi_{u}}(\mathbf{r})\cos(\Delta Et)\cos(\theta)\). After the pump pulse, the occupation coefficients \(c_{\sigma_{g}}(t)\) and \(c_{\pi_{u}}(t)\) are constant and \(\mathcal{P}^{(g)}(\mathbf{r})\) and \(\mathcal{P}^{(e)}(\mathbf{r})\) are symmetric and are insensitive to the time, whereas the coherent superposition term \(\mathcal{P}^{(g,u)}(\mathbf{r})\) is determined by the time \(t\) and the angle \(\theta\). Note that, for the linear polarization resonant excitation, the distribution of the coherent superposition term is mainly along the laser polarization direction with a function \(\cos(\theta)\). Figure 2 shows numerical results of electron density distributions \(\mathcal{P}(x,y)\) of the superposition state created by the \(\lambda_{pu}=70\) nm (\(\omega_{pu}=0.65\) a.u.) pump pulse as a function of the time \(t\), where \(\mathcal{P}(x,y)=\int dz|\psi^{2}(x,y,z)|^{2}\) in Equation (1). We use the pulse intensity \(I_{0}=1.0\times 10^{14}\) W/cm\({}^{2}\) (\(E_{0}=5.34\times 10^{-2}\) a.u. ), and durations \(T_{pu}=10\tau_{pu}=2.32\) fs (1.16 fs full width at half maximum, FWHM). Of note is that the occupation coefficients, \(c_{\sigma_{g}}(t)\) and \(c_{\pi_{u}}(t)\), are dependent on the intensity of the pump pulse. Varying \(I_{0}\) leads to a variance of the electron coherence in Equation (7). At such intensity \(I_{0}=1.0\times 10^{14}\) W/cm\({}^{2}\), one obtains the maximum interference effects. After the pump pulse, the occupation coefficients are constant. Figure 2 shows that the electron density distributions \(\mathcal{P}(x,y)\) are asymmetric with respect to the molecular center and oscillate with the time \(t\). The coherent electron wave packets move periodically in the left (\(x<0\)) and right (\(x>0\)) half planes along the laser \(x\) polarization axis. At time \(t=9.75\tau_{pu}\), the coherent electron wave packets are mainly localized in the left plane of the molecule. As the time \(t\) increases, the electron moves to the right. At time \(t=10.25\tau_{pu}\), the density distributions of the coherent electron wave packets lie in the right plane. As the time \(t\) increases further, the electron wave packets come back to the left plane. The evolution of the electron density distributions in Figure 2 describes the dependence of the attosecond electron coherence on the time \(t\) in charge migration, i.e., the superposition term \(\mathcal{P}^{(g,u)}(\mathbf{r})\) in Equation (10). The period of the oscillation is \(\tau=\tau_{pu}=232\) as for the time \(t\), in good agreement with the prediction in Equation (10). Since \(\mathcal{P}^{(g,u)}(\mathbf{r})\sim\cos(\Delta Et)\cos(\theta)\), the density distributions of the coherent electron wave packets are asymmetric along the \(x\)-axis and the extreme value occurs at \(\theta=0\) or \(\pi\) (180\({}^{\circ}\)). Varying the time \(t\) leads to a modulation of the coherent electron density distribution following a form \(\cos(\Delta Et)\). The time dependent electron density distributions reflects the electron coherence. To visualize the coherent electron wave packets in Figure 2, we present the process of time resolved photoelectron imaging by pump-probe pulses.