## 3 Pulse Shapes Typical of Experiments
### High-Harmonic Pulses Containing Multiple Harmonic Orders  
We simulate the 2 PI of He by an attosecond pulse train composed of the ninth (H9) to the seventeenth (H17) harmonics of a fundamental wavelength of 808.7 nm, whose fifteenth harmonic (H15, 23.0 eV) is resonant with the \(1s3p\,^{1}P\) state. We use the relative intensity of each harmonic component reported in Figure 1 of [82], neglecting inter- and intra-order chirps usually observed in experiments [83]. Each component is expressed as a Gaussian pulse with a common pulse width \(T_{ov}\), referred to as overall pulse width hereafter. We show the pulse shape and intensity spectrum for \(T_{ov}=7\,\mathrm{fs}\) in Figure 10. The whole pulse has a 7 fs envelope, while composed of many individual pulses with a pulse width \(\sim 0.3\) fs [Figure 10(a)]. The pulse spectrum, though composed of discrete harmonic peaks, is so broad as a whole that it spans from 13.8 eV (H9) to 26.1 (H17) [Figure 10(b)]. This latter manifests itself as a fact that each pulse in the pulse train [Figure 10(a)] has a sub-femtosecond duration, while the overall pulse width corresponds to the spectral width of each harmonic component.  
The triangles in Figure 11 plot the calculated relative phase \(\delta\) between the \(S\) and \(D\) partial waves for several values of \(T_{ov}\). Although \(\delta\) changes with \(T_{ov}\) especially at \(T_{ov}<4\) fs, its variation is much smaller than for the case of a Gaussian pulse centered at 23.0 eV (squares in Figure 11). This is probably because the H9-H13 (nonresonant) and H17 (in the continuum) components would lead to a constant phase if alone, and that only resonant excitation by H15 contributes to the variation.  
As can be seen from Figure 10(b), H15 is the most intense harmonic component in the spectrum, resonant with the \(1s3p\,^{1}P\) state. The corresponding 2 PI photoelectron energy is 21.4 eV, while the other photoelectron peaks are separated by 3.07 eV from each other. Then, in order to highlight the contribution from H15, let us focus ourselves on the wave packet between 20 and 23 eV, for which the values of \(\delta\) are plotted as circles in Figure 11. Only the results at \(T_{ov}\geq 2\) fs are plotted, since the pulse is so short at \(T_{ov}<2\) fs that the photoelectron peaks are not well separated from each other anymore. As expected, we can see that \(\delta\) now varies with \(T_{ov}\) as rapidly as for the case of the Gaussian pulse, though its value is somewhat shifted due to an additional contribution from the H13+H17 and H17+H13 processes.  
Figure 10: (**a**) Pulse shape and (**b**) intensity spectrum of a 7 fs high-harmonic pulse (attosecond pulse train) composed of multiple harmonic orders used in the present study (see text).  
The change in PAD corresponding to the wave packet between 20 and 23 eV, shown in Figure 12, is not so large as one might expect from the variation in \(\delta\), again probably due to the relatively small variation with pulse width below 10 fs even for the case of Gaussian pulses. Nevertheless, it is clearly different from what would be expected from the scattering phase shift difference [see the curve for 21 fs in Figure 6(c)], well indicating the presence of the competition between the resonant and nonresonant ionization paths.  
Figure 11: Pulse-width dependence of the phase shift difference \(\delta\) for high-harmonic pulses composed of multiple harmonic orders (see text). The target atom is helium. Black triangles: derived from the whole 2 PI photoelectron wave packets. Red circles: derived from the 2 PI photoelectron wave packets within the 20–23 eV energy range. Green squares: for Gaussian pulses with a photon energy of 23 eV. The thin horizontal line denotes the value (1.410 [63]) of the intrinsic scattering phase-shift difference \(\delta_{sc}\) for \(\hbar\omega=23.0\,\mathrm{eV}\).  
Figure 12: Pulse-width dependence of the photoelectron angular distribution for high-harmonic pulses. The target atom is helium.