## 3 Results and Discussion
### Analytical Approximation
We assume a rectangular pulse profile of duration \(\mathrm{T}=2.82\) fs for the XUV pulse so that the dipole phase shift can now be analytically written as:  
\[\Delta\varphi_{\mathrm{D}}=\frac{\left(\Delta\mathrm{E}_{\mathrm{g}}-\Delta \mathrm{E}_{\mathrm{e}}\right)}{\hbar}\mathrm{T}=\frac{1}{\hbar}\left\{ \begin{array}{ll}\left(-\Delta-\sqrt{\Delta^{2}+\left|\Omega\right|^{2}} \right)\mathrm{T}\leavevmode\
obreak\,&\Delta<0\\ \left(-\Delta+\sqrt{\Delta^{2}+\left|\Omega\right|^{2}}\right)\mathrm{T} \leavevmode\
obreak\,&\Delta>0\end{array} \right.\leavevmode\
obreak. \tag{11}\]  
We note, that this approximation is only valid in the impulsive limit (\(\mathrm{T}\ll 1/\Gamma_{\mathrm{r}}\)). The choice of \(\mathrm{T}=2.82\) fs was made to match the integrated intensity -the fluence- of the rectangular pulse with the one of the \(\mathrm{T}=3\) fs Gaussian pulse used previously. This analytical calculation of the dipole phase shift already explains some of the previously obtained trends (see Figure 3a): The dipole phase shift increases monotonically with peak intensity and assumes opposite sign for positive/negative detuning. However, we now observe a sharp jump from a maximal to a minimal phase shift across \(\Delta=0\) instead of a smooth transition through \(\Delta\varphi_{\mathrm{D}}=0\) (compare with Figure 2b). This discontinuity in fact arises from the initial assumption of monochromatic light fields and thus well-defined detunings. Note that for zero detuning Equation (11) does not apply and thus the dipole phase shift is set to zero in this illustration. For temporally short and, thus, spectrally broad pulses, all spectral components of the pulse, i.e., photons with a detuning within the spectral pulse band width, contribute to this energy- and phase-shifting mechanism at the same time. Therefore, the resulting dipole phase shift is expected not to be determined by the central photon energy of the driving pulse alone, but rather by the average over its spectrum. In Figure 3b, we convolute (result shown in violet) the analytical expectation (shown in blue) of the detuning-dependent dipole phase shift for lowest intensities, \(\mathrm{I}_{0}=0.1\times 10^{15}\leavevmode\
obreak\ \mathrm{W}/\mathrm{cm}^{2}\), with the spectrum of the 3 fs Gaussian driving pulse, which is also a Gaussian profile with an intensity FWHM of 0.61 eV (shown in red) and compare it to the numerical results from Figure 2d (shownin black). Additionally, we scale the magnitude of the convoluted dipole phase shift with a factor of \(\sim\)0.53 for a better qualitative comparison to the numerical results. One possible interpretation for this rescaling with a factor of approximately one half could be that the \"effective\" phase shift of the dipole emission is mainly induced by the trailing edge of the pulse, while its leading edge is mainly responsible for the excitation of the system. Since the dipole phase shift (see Equation (11)) is of equal magnitude but changes sign for positive/negative detuning, it evaluates to zero after convolution for a resonant driving pulse. In Figure 3c,d, we plot the detuning-dependent dipole phase shift from Figure 3a in the same manner as in Figure 3b, but for higher peak intensities, \(\mathrm{I}_{0}=2.4\times 10^{15}\) W/cm\({}^{2}\) and \(\mathrm{I}_{0}=4.7\times 10^{15}\) W/cm\({}^{2}\), respectively. For the result of this convolution, we find excellent agreement of the analytically calculated and convoluted phase shifts with the numerically obtained phase shifts, when we set the FWHM of the Gaussian convolution function to 0.45 eV and 0.3 eV and rescale the convoluted dipole phase shifts by \(\sim\)0.64 and \(\sim\)0.65, respectively. The finding of an intensity-dependent FWHM of the convolution function indicates a nonlinear process, which needs further investigation.