## 2 Gaussian Pulse
### Hydrogen Atom
Hence, the value of \(\delta\) is either \(|\delta_{sc}|\) [thin black line in Figure 1(a)] or \(\pi-|\delta_{sc}|\) (thin dashed black line), depending on the sign of  
Figure 3: Pulse-width dependence of the photoelectron angular distribution for (**a**) H and \(\hbar\omega=10.2\,\mathrm{eV}\); and (**b**) He and \(\hbar\omega=21.2\,\mathrm{eV}\). Figure 2: Pulse-width dependence of the TDSE-derived apparent phase shift difference (relative phase) \(\delta\) (left axis) and \(W=|c_{s}/c_{d}|\) (right axis). (**a**) The target atom is hydrogen, and \(\hbar\omega=10.2\,\mathrm{eV}\). The thin horizontal line denotes the value (\(=\pi/2\))) of the Coulomb phase-shift difference; (**b**) The target atom is helium, and \(\hbar\omega=21.2\,\mathrm{eV}\). The thin horizontal line denotes the value (1.511 [63]) of the intrinsic scattering phase-shift difference \(\delta_{sc}\) for \(\hbar\omega=21.2\,\mathrm{eV}\). \(c_{S}/c_{D}\). This situation corresponds to \(\hbar\omega\lesssim 9.8\,\mathrm{eV}\) and \(10.6\,\mathrm{eV}\lesssim\hbar\omega\lesssim 11.6\,\mathrm{eV}\). The relative phase \(\delta\) between the \(S\) and \(D\) partial waves does not depend on the pulse width [Figure 4(a)], except for the case of very short pulses where the broadened spectrum begins to be resonant with a \(2p\) or \(3p\) level, which belong to the second category. We can also see from Figure 1 that the sign of \(c_{S}/c_{D}\) changes, e.g., around \(\hbar\omega=10.9\,\mathrm{eV}\) and \(\hbar\omega=11.7\,\mathrm{eV}\), where \(W\) sharply peaks or approaches to zero [Figure 1(b)]. The jump in \(\delta\) is not completely step-function-like, since the contribution from the resonant path can be neglected no longer there. Second, if the pulse is resonant with a single excited level (near \(\hbar\omega=10.2\,\mathrm{eV}\) and \(12.1\,\mathrm{eV}\) for the \(2p\) and \(3p\) resonance, respectively), the competition between the resonant and nonresonant ionization paths leads to the value of \(\delta\) that deviates from its intrinsic value (\(|\delta_{sc}|\) or \(\pi-|\delta_{sc}|\)) and varies with pulse width, as we have already seen above [Figure 2(a)] and can also see in Figure 4(b). It tends to either \(|\delta_{sc}|\) or \(\pi-|\delta_{sc}|\) for the longer pulse width. Interestingly, at \(\hbar\omega=11.7\,\mathrm{eV}\) [blue line in Figure 4(b)], \(\delta\) first decreases with increasing pulse width, approaching \(|\delta_{sc}|=1.40\) (_i.e._, the resonant path is dominant), but then, for \(T_{1/2}\gtrsim 8\,\mathrm{fs}\), increases again, tending to \(\pi-|\delta_{sc}|=1.74\) (_i.e._, the nonresonant paths are dominant). Similar features can be observed for 9.0 and 11.0 eV in Figure 4(a) as well. Then, above the \(3p\) resonance, the pulse is always resonant with one or, more often, multiple levels. In this case, again, the value of \(\delta\) deviates from its intrinsic value and varies with pulse width [see the lines for 12.75 and 13.0 eV in Figures 4(a) and 4(c)], while its photon-energy dependence is somewhat complicated. A new feature is that \(\delta\) is roughly constant at \(T_{1/2}\lesssim 4\,\mathrm{fs}\) and 7 fs for 12.75 and 13.0 eV, respectively, which we will discuss below. Finally, when the pulse excites a Rydberg manifold (\(\hbar\omega\gtrsim 13.0\) and \(13.3\,\mathrm{eV}\) for 7 and 14 fs, respectively), \(\delta\) varies smoothly with photon energy even across the ionization threshold (13.6 eV), indicating that the Rydberg manifold behaves similarly to the continuum for the case of an ultrashort pulse (note that a similar smooth transition can also be seen in the 2 PI photoelectron spectrum [73]). Moreover, the relative phase is different from its intrinsic value, but surprisingly constant as a function of \(T_{1/2}\) within the pulse width range investigated here [Figure 4(c),(d)]. These features, seen also for He [Figure 4(g)] and presumably general for all the atoms, can be understood as follows.