## 2 Gaussian Pulse
### Helium Atom  
Let us now turn to a helium atom. As we have emphasized in Section 1, He is much more relevant to experiments than H, since it is much easier to handle experimentally, and its excitation energies lie well within the typical wavelength range of HHG and EUV FEL sources. We use direct numerical solution of the full-dimensional two-electron TDSE in the length gauge [54]:  
\[i\frac{\partial\Phi(\mathbf{r}_{1},\mathbf{r}_{2},t)}{\partial t}=[H_{0}+H_{I}( t)]\Phi(\mathbf{r}_{1},\mathbf{r}_{2},t) \tag{15}\]  
with the atomic and interaction Hamiltonian,  
\[H_{0} =-\frac{1}{2}\
abla_{1}^{2}-\frac{1}{2}\
abla_{2}^{2}-\frac{2}{r_ {1}}-\frac{2}{r_{2}}+\frac{1}{|\mathbf{r}_{1}-\mathbf{r}_{2}|} \tag{16}\] \[H_{I}(t) =(z_{1}+z_{2})f(t)\sin\omega t \tag{17}\]  
We solve Equation (15) using the time-dependent close-coupling method [50, 51, 52, 53, 54]. The numerically obtained excitation energies for the \(1s2p\,^{1}P\) and \(1s3p\,^{1}P\) states are 21.220 and 23.086 eV, respectively, in fair agreement with the experimental values (21.218 and 23.087 eV [48], respectively). Similarly to the case of a hydrogen atom, sufficiently long after the pulse has ended, we calculate \(\beta_{2}\) and \(\beta_{4}\) by integrating the ionized part of \(|\Phi(\mathbf{r}_{1},\mathbf{r}_{2})|^{2}\) over \(r_{1},r_{2},\theta_{2},\phi_{1},\phi_{2}\), from which one obtains \(W\) and \(\delta\) by solving Equation (8). We use the values of \(\delta_{sc}\) from [63] to calculate \(\delta_{ex}=\delta-\delta_{sc}\). The calculation has been done for a Gaussian pulse envelope with a peak intensity of \(10^{11}\,\mathrm{W/cm}^{2}\), at which we have confirmed that the interaction is in the perturbative regime. Our preliminary investigation indicates that the interaction begins to deviate slightly from this regime around \(10^{12}\,\mathrm{W/cm}^{2}\) if we increase the intensity, but the correction is still small below \(10^{13}\,\mathrm{W/cm}^{2}\), which is the typical focal intensity of SCSS [40]. The pulse-width dependence of \(\delta\) and \(W\) for \(\hbar\omega=21.2\,\mathrm{eV}\) close to the \(1s2p\) resonance (21.218 eV) is shown in Figure 2(b). The calculations have been done at different values of pulse width \(T_{1/2}\) between 500 as and 21 fs. As expected from the discussions in Sections 2.1 and 2.2, and similarly to the case of a hydrogen atom (Subsection 2.3), both \(\delta\) and \(W\) substantially change with pulse width, especially when the pulse is shorter than 10 fs. Accordingly, the PAD also varies as shown in Figure 3(b). One finds that the distribution to the direction perpendicular to the laser polarization again decreases as the pulse is shortened. As stated earlier, strictly speaking, Equation (4) is applicable only to \(\omega_{f}=\omega_{i}+2\omega\), and the actual PAD involves integration over \(\omega_{f}\). Nevertheless, the results in Figure 2(b) can well be described by Equation (4) [44], except for \(\delta\) in the ultrashort pulse regime \(T_{1/2}\lesssim 1\,\mathrm{fs}\), where the spectrum becomes broader than the level spacing. If we compare Figure 2(a) for H and (b) for He, both for the \(2p\) resonance, the phase-shift difference \(\delta\) has a similar pulse-width dependence. As for the amplitude ratio \(W=|c_{s}/c_{d}|\), it is smaller than unity, _i.e._, \(|c_{d}|>|c_{s}|\) regardless of pulse width for H, while its variation is larger for He; \(|c_{d}|>|c_{s}|\) in the short-pulse limit, and \(|c_{d}|<|c_{s}|\) in the long-pulse limit. Accordingly, the variation in PAD is more prominent for He than for H (Figure 3). It is interesting at this stage to examine some limiting cases. With increasing pulse duration, \(\delta\) approaches the scattering phase shift difference \(\delta_{sc}\), and the PAD changes only slowly with \(T_{1/2}\) [Figure 3(b)].