## 2 Gaussian Pulse
### Helium Atom
When the pulse is resonant (\(\Delta_{r}T\ll 1\)) and sufficiently long (\(T\gg a_{S},a_{D}\)) at the same time, assuming that the resonant excitation is not saturated, one can approximate the extra phase shift as  
\[\delta_{ex}\approx(a_{D}-a_{S})/\sqrt{\pi}\,T \tag{18}\]hence, it is proportional to the spectral width, which can be confirmed in Figure 3 of [44]. On the other hand, if we plot \(\delta\) as a function of spectral width, \(\delta\) tends to an asymptotic value in the wide-spectrum, _i.e._, short-pulse limit. Correspondingly, the PAD does not change much with the pulse width for \(T_{1/2}\lesssim 1\) fs [Figures 3(b) and 6(b),(c)]. This is because the pulse becomes resonant with multiple levels; the spacing between the \(1s2p\) and \(1s3p\) is 1.9 eV. As we have shown in the previous subsection, when many neighboring states are resonantly excited by the pulse, the extra phase shift difference \(\delta_{ex}\) does not much depend on the pulse duration. Similarly to the case of the hydrogen atom, this especially applies when the photon energy lies in the Rydberg manifold, and exceeds the ionization potential (24.59 eV), _i.e._, in the case of above-threshold two-photon ionization. Figure 6: Pulse-width dependence of the photoelectron angular distribution for **(a)**\(\hbar\omega=20.3\,\mathrm{eV}\); **(b)**\(\hbar\omega=21.3\,\mathrm{eV}\); **(c)** 23.0 eV; **(d)** 23.9 eV; **(e)** 24.3 eV; and **(f)** 25.0 eV. The target atom is helium. In Figures 4(e)-(g) and 6 we compare the pulse-width dependence of \(\delta_{ex}\) and the PAD, respectively, for different values of \(\hbar\omega\), where we find trends similar to the case of H. While \(\delta\approx\delta_{sc}\) (\(\delta_{ex}\approx 0\)) and the PAD are nearly independent of \(T_{1/2}\) for nonresonant pulses (\(\hbar\omega=20.3\,\mathrm{eV}\), \(T_{1/2}\gtrsim 3.5\,\mathrm{fs}\)), when the pulse is close to resonance with an excited level (\(\hbar\omega=21.2,21.3\), and \(23.0\,\mathrm{eV}\)), \(\delta_{ex}\) and PAD rapidly change with \(T_{1/2}\). On the contrary, \(\delta_{ex}\) is finite and nearly constant for \(\hbar\omega=24.3,24.6\), and \(25.0\,\mathrm{eV}\) [Figure 4(g)]; accordingly, the PAD hardly varies with pulse width. At \(T_{1/2}\lesssim 1\) fs, the spectrum is so broad that \(\delta_{ex}\) restarts to change slightly. One also sees that the transition across the ionization potential is smooth [Figure 4(g)], as has been seen for a hydrogen atom in Figures 1 and 4(c),(d). It should be pointed out that the extra phase shift difference due to free-free transitions plays a significant role in the recently observed time delay in photoemission by attosecond EUV pulses [75, 76, 77, 78, 79]. Figure 7: Photon-energy dependence of (**a**) the relative phase (phase-shift difference) \(\delta\) and (**b**) the amplitude ratio \(W\) between the \(S\) and \(D\) wave packets for a pulse width of 7 fs. The target atom is helium. Thin sold and dashed lines in (**a**): \(|\delta_{sc}|\) and \(\pi-|\delta_{sc}|\), respectively. Vertical dotted lines: positions of \(1s2p^{1}P\) to \(1s6p^{1}P\) resonances. Figure 7 shows the photon energy dependence of the phase difference \(\delta\) and the amplitude ratio \(W\) for a pulse width of 7 fs. Although the number of data points is limited, we can see features basically similar to those in Figure 1 for H.