## 3 Results for a Model Atom
As there are three bright states \(i^{-}\) in our model atom, one can see three main peaks in Figure 3, which correspond to the XUV transitions \(0^{+}\to i^{-}\). The small additional peaks correspond to transitions to LIS. Numerical values for the transition energies are listed in Table 3.  
If dressing comes long after or before the excitation by the attosecond pulse, only the peaks associated with the bright states can be observed in the absorption spectrum (Figure 3a). When the time delay between the pulses gets shorter, optical dressing becomes possible, causing the appearance of LIS transition peaks as well as changes in the height of the bright-state transition peaks. New features can be positive or negative indicating whether the attosecond XUV beam is attenuated or amplified at the corresponding \(\omega\). In Figure 4 we show the difference \(\Delta_{\sigma^{(1)}}=\sigma^{(1)}(\omega,\tau)-\sigma^{(1)}(\omega,\infty)\) as a function of \(\omega\) and \(\tau\). Interference of different excitation paths from the ground state \(0^{+}\) to a final \(f\) state [see Equation (12)] gives rise to an oscillation in the XUV one-photon cross-section of all peaks [55], \begin{table}
\begin{tabular}{l c c c} \hline  & \(i=1\) & \(i=2\) & \(i=3\) \\ \hline \(E_{i^{-}}-E_{\Phi^{+}}-2\omega_{d}\) & 1.9889 & 2.0238 & 2.0361 \\ \(E_{i^{-}}-E_{\Phi^{+}}\) & 2.1888 & 2.2238 & 2.2360 \\ \(E_{i^{-}}-E_{\Phi^{+}}+2\omega_{d}\) & 2.3888 & 2.4238 & 2.4360 \\ \hline \(E_{i^{+}}-E_{\Phi^{+}}-\omega_{d}\) & 2.089 & 2.124 & \\ \(E_{i^{+}}-E_{\Phi^{+}}+\omega_{d}\) & 2.289 & 2.324 & \\ \hline \end{tabular}
\end{table} 
Table 3: Resonance energies (dressing photon energy \(\omega_{d}=0.10\) a.u.). which can be observed in Figure 4 in the overlap region between the optical dressing pulse and the attosecond XUV pulse. The oscillations become weaker after \(\tau\approx 2230\) a.u. as the time available to dress an excited state gets shorter and becomes comparable with the excited-state lifetime. Thus, the probability of interference with another transition path through a dressed state goes down. This \"ringing\" of the system driven by the excitation pulse can be observed even if the dressing pulse comes after the excitation and the pulses do not overlap, if the excited state can survive till the dressing comes. This can be seen in the region of negative \(\tau\) in Figure 4. If the attosecond pulse comes after the dressing pulse, the XUV one-photon cross-section remains constant and the height of the three main peaks at \(E_{i^{-}}-E_{0^{+}}\) does not change with the time delay \(\tau\), as the dressing pulse, by assumption, does not affect the ground state of the system. Studying the \"ringing\" allows us to reveal the real-time attosecond dynamics inside the atom. To this end, we take the Fourier transform of \(\sigma^{(1)}(\omega,\tau)\) with respect to the time delay, which gives an opportunity to study the reasons behind the oscillations observed. The absolute value of the resulting function \(\tilde{\sigma}^{(1)}(\omega,\omega^{\prime})\) [Equation (24)] is plotted in Figure 5.  
The oscillations of \(\sigma^{(1)}(\omega,\tau)\), caused by the excitation path interference, have the energies \(\Delta_{i}-\Delta_{j}\) (see Section 2.3). In accordance with the dipole selection rules, these are \(2\omega_{d}\), \(|E_{i^{-}}-E_{j^{-}}|\), \(|E_{i^{+}}-E_{j^{-}}\pm\omega_{d}|\) (Table 4). \begin{table}
\begin{tabular}{l c c c} \hline  & \(i\) = 1 & \(i\) = 2 & \(i\) = 3 \\ \hline \(|E_{i^{-}}-E_{(i+1)^{-}}|\) & 0.0350 & 0.0122 & 0.0472\({}^{*}\) \\ \hline \(|E_{1^{+}}-E_{i^{-}}+\omega_{d}|\) & 0.1000 & 0.0650 & 0.0528 \\ \(|E_{2^{+}}-E_{i^{-}}+\omega_{d}|\) & 0.1350 & 0.1000 & 0.0878 \\ \hline \(|E_{1^{+}}-E_{i^{-}}-\omega_{d}|\) & 0.0999 & 0.1350 & 0.1472 \\ \(|E_{2^{+}}-E_{i^{-}}-\omega_{d}|\) & 0.0650 & 0.0999 & 0.1122 \\ \hline \end{tabular} \({}^{*}\) Energy difference, \(E_{3^{-}}-E_{1^{-}}\).  
\end{table} 
Table 4: Oscillation energies \(\omega^{\prime}\) [a.u.]