## 2 Numerical Methods  
We briefly describe the numerical and computational methods for simulating the photoelectron emission process in aligned H\({}_{2}^{+}\) by pump-probe pulses, as illustrated in Figure 1. The corresponding three-dimensional (3D) TDSE within static nuclear frames reads as,  
\[i\frac{\partial}{\partial t}\psi(\mathbf{r},t)=\left[-\frac{1}{2}\bigtriangledown _{\mathbf{r}}^{2}+V_{en}(\mathbf{r})+V_{L}(\mathbf{r},t)\right]\psi(\mathbf{r },t). \tag{1}\]  
Figure 1: Illustration of time-resolved photoelectron imaging of coherent electron excitation in molecules by an XUV pump pulse and a time delayed soft X-ray probe pulse. The molecule H\({}_{2}^{+}\) is aligned along the \(z\)-axis and the pump–probe pulses with their field vectors in the \((x,y)\) plane propagate along the \(z\)-axis. The time delay \(\Delta\tau\) is the time interval between the peak times of the pump and probe pulses. We describe the electron dynamics in cylindrical coordinates \(\mathbf{r}=(\rho,\theta,z)\) with \(x=\rho\cos\theta\) and \(y=\rho\sin\theta\). Then, the kinetic (Laplacian) operator gives  
\[-\frac{1}{2}\
abla_{\mathbf{r}}^{2}=-\frac{1}{2\rho}\frac{\partial}{\partial\rho }\left(\rho\frac{\partial}{\partial\rho}\right)-\frac{1}{2\rho^{2}}\frac{ \partial^{2}}{\partial\theta^{2}}-\frac{1}{2}\frac{\partial^{2}}{\partial z^{2 }}. \tag{2}\]  
\(V_{en}\) is the electron-nuclear potential. The molecule is aligned along the \(z\)-axis. The pump and probe laser pulses propagate along the \(z\)-axis, parallel to the molecular axis. The radiative interaction between the laser field and the electron \(V_{L}(\mathbf{r})=\mathbf{r}\cdot\mathbf{E}(t)\) is described in the length gauge. The total fields \(\mathbf{E}(t)=\mathbf{E}_{pu}(t)+\mathbf{E}_{pr}(t)\) have the forms,  
\[\mathbf{E}_{pu}(t) = \varepsilon_{x}E_{0}f(t)\cos(\omega_{pu}t), \tag{3}\]  
for a linearly polarized pump pulse with its field vector along the \(x\)-axis, and  
\[\mathbf{E}_{pr}(t) = E_{0}f(t)\left\{ \begin{array}{l}\delta_{x}\cos(\omega_{pr}t) \\ \varepsilon_{y}\xi\sin(\omega_{pr}t)\end{array} \right., \tag{4}\]  
for a circularly polarized probe pulse with its field vector in the \((x,y)\) plane. The symbol \(\xi=\pm 1\) denotes the helicity of the probe field, i.e., right (\(\xi=+1\)) or left (\(\xi=-1\)) handed circular polarization and \(\xi=0\) presents the linear polarization. A smooth \(\sin^{2}(\pi t/T_{pu/pr})\) pulse envelope \(f(t)\) for maximum amplitude \(E_{0}\), intensity \(I_{0}=I_{x}=I_{y}=c\varepsilon_{0}E_{0}^{2}/2\) and duration \(T_{pu/pr}=10\tau_{pu/pr}\) are used, where one optical cycle period \(\tau_{pu/pr}=2\pi/\omega_{pu/pr}\). This pulse satisfies the total zero area \(\int E_{pu/pr}(t)dt=0\) in order to exclude static field effects [9]. The 3D TDSE in Equation (1) for aligned H\({}_{2}^{+}\) is numerically solved by a second-order split operator method that conserves unitarity in the time step \(\delta t\) combined with a fifth order finite difference method and Fourier transform technique in the spatial steps \(\delta\rho\), \(\delta z\), and \(\delta\theta\)[51; 52]. The initial electron wavefunction \(\psi(\mathbf{r},t=0)\) is prepared in the ground \(1s\sigma_{g}\) state calculated by propagating an initial appropriate wavefunction in imaginary time using the zero-field TDSE in Equation (1). The time step is taken to be \(\delta t=0.01\) a.u. = 0.24 as. The spatial discretization is \(\delta\rho=\delta z=0.25\) a.u. for a radial grid range \(0\leq\rho\leq 128\) a.u. (6.77 nm) and \(|z|\leq 32\) a.u. (1.69 nm), and the angle grid size \(\delta\theta=0.025\) radian. One notes that reflections of the electron wave packet from the edges of the numerical grid can cause artificial effects [53; 54]. To prevent these unphysical effects, we multiply \(\psi(\rho,\theta,z,t)\) by a mask function or absorption potential in the radial coordinates \(\rho\) with the form \(\cos^{1/8}[\pi(\rho-\rho_{a})/2\rho_{\mathrm{abs}}]\). The important ultrafast electron dynamics in intense fields occur on a spatial volume close to the molecule.