## 2 Gaussian Pulse
### Hydrogen Atom  
Let us now verify the above qualitative idea for the case of a hydrogen atom, for which an analytical expression of the Coulomb phase shift \(\delta_{sc,l}\) for an azimuthal quantum number \(l\) is at hand as,  
\[\delta_{sc,l}=\arg\Gamma(l+1-i/k) \tag{9}\]  
where \(\Gamma(z)\) denotes the gamma function, and \(k\) the wave number. In principle, one could calculate \(c_{S}\) and \(c_{D}\) using the analytical expression for the dipole transition matrix elements in Equation (2), but it is not trivial to perform an integration over the continuum intermediate states, since the [PERSON]'s integral has a long tail (\(\sim 1/x\)). Instead, we take an approach in the time domain, namely, direct numerical solution of the time-dependent Schrodinger equation (TDSE) in the length gauge,  
\[i\frac{\partial\Phi(\mathbf{r},t)}{\partial t}=\left[-\frac{1}{2}\
abla^{2}- \frac{1}{r}-zf(t)\sin\omega t\right]\Phi(\mathbf{r},t) \tag{10}\]  
where we have assumed that the field is linearly polarized in the \(z\) direction. Equation (10) is numerically integrated using the alternating direction implicit (Peaceman-Rachford) method [9; 65; 66; 67; 68; 69; 70; 71; 72; 73]. Sufficiently long (typically a few times the pulse width) after the pulse has ended, the ionized wave packet moving outward in time is spatially well separated and clearly distinguishable from the non-ionized part remaining around the origin. We calculate \(\beta_{2}\) and \(\beta_{4}\) by integrating the ionized part of \(|\Phi(\mathbf{r})|^{2}\) over \(r\) and \(\phi\), from which one obtains \(W\) and \(\delta\) by solving Equation (8). We have monitored the convergence of the results by calculating the PAD at different times. The calculation has been done for a Gaussian pulse envelope with a peak intensity of \(10^{10}\,\mathrm{W/cm}^{2}\), at which we have confirmed that the interaction is still in the perturbative regime. The pulse-width dependence of \(\delta\) and \(W\) for \(\hbar\omega=10.2\,\mathrm{eV}\) corresponding to the \(2p\) resonance is shown in Figure 2 (a). The calculations have been done at different values of full-width-at-half-maximum (FWHM) pulse width \(T_{1/2}=2\sqrt{\ln 2}\,T\) between 1 and 21 fs. As expected, both \(\delta\) and \(W\) substantially change with pulse width, especially when the pulse is shorter than 10 fs. \(\delta\) approaches \(|\delta_{sc}|\equiv|\delta_{sc,0}-\delta_{sc,2}|=\frac{\pi}{2}\) for a larger pulse width. Accordingly, the PAD also varies as shown in Figure 3 (a). One finds that the distribution to the direction perpendicular to the laser polarization, _i.e._, \(\theta\approx 90^{\circ},270^{\circ}\) decreases as the pulse is shortened, which is more prominent for He as we will discuss later [Figure 3 (b)]. This can be understood as follows: roughly speaking, \(\delta\) changes from \(\sim\frac{\pi}{2}\) to \(\sim\pi\) as \(T_{1/2}\) varies from 21 fs to 1 fs. Thus, \(c_{s}/c_{d}\) is approximately real and negative in the short-pulse limit, which leads to the cancellation between \(Y_{00}(\theta,\varphi)\) and \(Y_{20}(\theta,\varphi)\) around \(\theta=\frac{\pi}{2}\). In Figure 1, we plot the photon energy dependence of the phase difference \(\delta\) and the amplitude ratio \(W\) for 7 fs and 14 fs pulse widths. By noting that the spectral width is 0.26 eV and 0.13 eV for each pulse width, respectively, we can identify several photon energy ranges. First, if the pulse is not resonant with any level and the first term in Equation (3) is negligible, Equation (4) can be approximated as,  
\[\frac{c_{S}}{c_{D}}=\frac{\sum_{m}\mu_{Sm}\mu_{mi}/\Delta_{m}}{\sum_{m}\mu_{Dm} \mu_{mi}/\Delta_{m}} \tag{11}\]  
which is real. In our plots, we define the value of \(\delta\) within the range \([0,\pi]\).