## 3 Results and Discussion
### Analytical Approximation  
In the following, we will relate these findings to the well-known description of a strongly coupled two-level system in the generalized Rabi formalism [1; 2]. To understand the connection between the splitting and shifting of the involved energy levels to the modified absorption profile, we propose the comparison to the following analytical approximation: In the presence of the intense XUV pulse, the excited state of the system is split into two states \(\mathrm{E}^{\pm}\), which are the eigenstates of the laser-dressed two-level system, and which are energetically shifted with increasing field strength. For a temporally Gaussian-shaped XUV pulse envelope, the dressed states also shift energetically along a Gaussian curve in time (see Figure 1c). Hereby, one dressed state emerges directly from the atomic excited  
Figure 2: Results for the two-level system obtained by fitting a generalized Fano line shape (Equation (4)) to the simulated optical density. (**b**) dipole phase shift \(\Delta\varphi_{\mathrm{D}}\) obtained after applying Equation (5) to the direct fit results of \(\mathrm{q}\) shown in (**c**) as a function of detuning and intensity, with corresponding (**a**) horizontal lineouts as a function of intensity for different detunings and (**d**) vertical lineouts as a function of detuning for different intensities. (**c**) the fitted \(\mathrm{q}\) -parameter as a function of detuning and intensity. state, whereas the other one is created from the ground state with the absorption of an XUV photon. It depends on the detuning \(\Delta\) of the XUV pulse, which of the two bare states transforms into the energetically upshifted state \(\mathrm{E}^{+}\), and which transforms into the downshifted state \(\mathrm{E}^{-}\) of the dressed two-level system. For a red-detuned XUV pulse, i.e., when the central photon energy of the pulse is smaller than the resonance transition energy, the excited state translates into \(\mathrm{E}^{+}\), whereas for a blue-detuned XUV pulse it translates into \(\mathrm{E}^{-}\). After the interaction with the XUV pulse, both bare states have collected a phase shift:  
\[\Delta\varphi_{\mathrm{g,e}}=\int\Delta\mathrm{E}_{\mathrm{g,e}}(\mathrm{t}) \leavevmode\
obreak\ \mathrm{dt}, \tag{7}\]  
due to the energy difference between the laser- dressed and bare states with respect to the weak-field coupling. The energy shifts are thus given by:  
\[\Delta\mathrm{E}_{\mathrm{g}}(\mathrm{t})=\left\{ \begin{array}{ll}\mathrm{E }^{-}(\mathrm{t})-\mathrm{E}_{\mathrm{g+\omega}}\leavevmode\
obreak\,&\Delta<0\\ \mathrm{E}^{+}(\mathrm{t})-\mathrm{E}_{\mathrm{g+\omega}}\leavevmode\
obreak\,&\Delta>0 \end{array} \right.,\leavevmode\
obreak\ \Delta\mathrm{E}_{\mathrm{e}}(\mathrm{t})=\left\{ \begin{array}{ll}\mathrm{E}^{+}(\mathrm{t})-\mathrm{E}_{\mathrm{e}} \leavevmode\
obreak\,&\Delta<0\\ \mathrm{E}^{-}(\mathrm{t})-\mathrm{E}_{\mathrm{e}}\leavevmode\
obreak\,&\Delta>0 \end{array} \right., \tag{8}\]  
where \(\mathrm{E}_{\mathrm{g+\omega}}=\mathrm{E}_{\mathrm{g}}+\hbar\omega_{\mathrm{c}}\) stands for the ground state energy plus the XUV photon energy. With the dipole phase \(\varphi_{\mathrm{D}}=\arg\!\left(\mathrm{e}_{\mathrm{g}}\mathrm{c}_{\mathrm{e} }^{\mathrm{c}}\right)\) according to Equation (2), this directly translates into a phase shift of the resulting dipole emission:  
\[\Delta\varphi_{\mathrm{D}}=\Delta\varphi_{\mathrm{g}}-\Delta\varphi_{\mathrm{e}}. \tag{9}\]  
For a monochromatic driving field of constant electric-field amplitude \(\mathrm{E}_{\mathrm{o}}\), we can use the generalized Rabi formulas and obtain  
\[\mathrm{E}^{\pm}=\mathrm{E}_{\mathrm{e}}+\frac{\Delta}{2}\pm\frac{1}{2}\sqrt{ \Delta^{2}+\left|\hbar\Omega\right|^{2}} \tag{10}\]  
for the energies of the laser-dressed states with the Rabi frequency \(\Omega=\mathrm{d}_{\mathrm{ge}}\mathrm{E}_{\mathrm{o}}/\hbar\).