## 3 Results and Discussion
### Autoionizing 2s2p Resonance in Helium  
Pertaining to previous experimental results obtained at the Free-Electron Laser in Hamburg (FLASH) [16], we focus in the following on the impact of the detuning for the impulsive dressing of the 2s2p doubly excited state in helium, where the excited state couples to an ionization continuum. By repeating the numerical simulations for the autoionizing state (see H\({}_{\rm 3 lvl}\) in Equation (1)) for the same peak intensity and detuning values as used for the two-level system, the dipole phase shift and q parameter are again obtained from the same Fano fitting procedure as in the two-level case and are depicted in Figure 4. The literature asymmetry parameter q = \(-\)2.75 is reproduced for the lowest pulse intensity (see Figure 4b) and corresponds to an intrinsic dipole phase shift of \(\varphi_{0}\approx 0.7\) rad. To quantify only the field-induced dipole phase shift \(\Delta\varphi_{\rm D}^{\prime}\), we subtract this offset phase: \(\Delta\varphi_{\rm D}^{\prime}=\Delta\varphi_{\rm D}-\varphi_{0}\) for Figure 4a. With increasing pulse intensity, we observe the three following features in the dipole phase shifts: (i) there is no dipole phase shift and hence no modification of the absorption line shape approximately for \(\Delta\approx-0.2\) eV, which weakly depends on the laser intensity (violet line in Figure 4a, (ii) the dipole phase shift is negative for \(\Delta\lesssim-0.2\) eV and positive for \(\Delta\gtrsim-0.2\) eV and (iii) the magnitude of the dipole phase shift increases with the pulse peak intensity. Features (ii) and (iii) are very similar to the previous results of the two-level system, indicating that similar energy and phase shifts take place when the coupling to an ionization channel is explicitly included in contrast to the previous simple decay of an isolated state. However, now we observe a significant offset detuning \(\Delta\approx-0.2\) eV for which the measurable line-shape asymmetry is approximately unchanged for increasing laser intensity. Furthermore, the magnitude of the positive and negative dipole phase shift around this offset are also different in strength. We give a first qualitative explanation of these new observations through the initial (i.e., weak-field) asymmetric Fano line shape by invoking a similar idea as for the analytical and convoluted calculations of the two-level system: For the given q-parameter of \(-\)2.75, the weak-field absorption cross section of the 2s2p resonance has a peak at photon energies smaller than the resonance position and a minimum for higher photon energies (see Figure 1e). Hence, the spectral components of a resonant driving XUV pulse with lower photon energies are more likely to be absorbed and, thus, their contributions to the dipole phase shift are expected to be more pronounced than the contribution of the higher photon energy components of the pulse. The dipole phase shift, thereby, is not cancelled out for a resonantly tuned XUV pulse as is the case for a weak-field Lorentzian resonance, but rather it is cancelled out for a slightly red detuned pulse, which is in qualitative agreement with the finding of an unchanged line-shape asymmetry around \(\Delta\approx-0.2\) eV in Figure 4a. The observed trend, thus, seems to be a direct consequence of the configuration interaction and channel interference of a Fano resonance. We would like to note that the detuning offset (\(\Delta\approx-0.2\) eV) for this specific case may depend on the exact description of the internal configuration interaction between the excited state and the ionization continuum. Since we utilize a quasi-continuum state here, we do not expect this value to be quantitatively precise. Nevertheless, the exact model choice of the coupled configurations is not expected to change our main finding of an asymmetric line-shape modification as a function of detuning for the impulsive dressing of an autoionizing state.