## 3 Results for a Model Atom  
By focusing on a simple few-level atom, we perform in this section an analysis of the ATA signal dependence on the time decay between the attosecond XUV pulse and the optical dressing pulse. The model considered is shown in Figure 2.  
The corresponding energy levels and transition dipole matrix elements, which underlie the numerical results shown in the following, are presented in Tables 1 and 2, respectively. The numbers employed do not correspond to any real atom. To illustrate the basic features of the kind of ATA spectroscopy considered in this paper, we employ an artificial few-level system characterized by level spacings matching XUV and optical energies. A key assumption is that parity is a good quantum number, which allows us to cleanly distinguish between bright and dark states (with respect to excitation from the ground state). The model in Figure 2 is intended as a pedagogical tool for identifying generic features of \(\sigma^{(1)}(\omega,\tau)\) and \(\tilde{\sigma}^{(1)}(\omega,\omega^{\prime})\). As a consequence of its large spectral bandwidth, the attosecond XUV pulse can excite the atom from its ground state \(0^{+}\) to any of the bright states \(i^{-}\), \(i=1,2,3\), which have a negative parity (see Figure 2). The dressing pulse with photon energy \(\omega_{d}\) cannot, by assumption, affect the ground state, but can couple a bright state \(i^{-}\) to a dark state \(j^{+}\), by exchanging with the atom one dressing-laser photon (energy change by \(\pm\omega_{d}\)) or to a bright state \(j^{-}\) by exchanging with the atom two dressing-laser photons (energy change by \(\pm 2\omega_{d}\)). Because \(0^{+}\) and the dark states \(j^{+}\), \(j=1,2\), have the same parity (see Figure 2), the latter cannot be excited via one-photon absorption from the ground state (i.e., the corresponding transition dipole moments are zero); but in the presence of the dressing laser, they can give rise to light-induced states [52; 53; 54]. As a function of the XUV photon energy \(\omega\), the following transition energies can be observed in the transmitted spectrum of the attosecond pulse: \(E_{i^{-}}-E_{0^{+}}\) for a transition to a bright state \(i^{-}\), and \(E_{i^{+}}\pm\omega_{d}-E_{0^{+}}\) and \(E_{i^{-}}\pm 2\omega_{d}-E_{0^{+}}\), respectively, for light-induced states (LIS). Excited states of the atom have a finite decay rate \(\Gamma=\Gamma_{f}\) and relax through fluorescence or Auger decay, which leads to a finite width of the absorption peaks. In our model, the decay rate \(\Gamma\) is assumed to be the same for all excited states \(i^{+}\) and \(i^{-}\) (\(\Gamma=1/263\) a.u.). In Figure 3, the XUV one-photon cross-section \(\sigma^{(1)}(\omega,\tau)\) [Equation (12)] for our model atom is presented for three different time delays \(\tau\). \begin{table}
\begin{tabular}{l|c c c c} \hline \(\mathbf{E_{0^{+}}}\) & \(\mathbf{E_{1^{-}}}\) & \(\mathbf{E_{2^{-}}}\) & \(\mathbf{E_{3^{-}}}\) & \(\mathbf{E_{1^{+}}}\) & \(\mathbf{E_{2^{+}}}\) \\ \hline
0 & 2.1888 & 2.2238 & 2.2360 & 2.1889 & 2.2238 \\ \hline \end{tabular}
\end{table} 
Table 1: Energy levels of the model atom. \begin{table}
\begin{tabular}{l|c c c} \hline \(\tilde{\mathbf{Z}}\) & \(|\mathbf{0^{+}}\rangle\) & \(|\mathbf{1^{+}}\rangle\) & \(|\mathbf{2^{+}}\rangle\) \\ \hline \(\langle\mathbf{1^{-}}|\) & 0.25 & 0.33 & 0.33 \\ \(\langle\mathbf{2^{-}}|\) & 0.42 & 0.33 & 0.33 \\ \(\langle\mathbf{3^{-}}|\) & 0.33 & 0.33 & 0.33 \\ \hline \end{tabular}
\end{table} 
Table 2: Transition dipole matrix elements \(\langle i^{-}|\tilde{2}|j^{+}\rangle\). Figure 2: Level structure of the few-level atom employed to illustrate ATA spectroscopy. XUV transitions from the ground state are shown in blue. Red arrows indicate laser dressing of the excited states. The superscripts represent the parity of each state. In all figures below we use a dressing pulse with amplitude \(A_{d}=0.06\) a.u., photon energy \(\omega_{d}=0.10\) a.u. and duration \(t_{2}=3141\) a.u. \(\approx 11.95/\Gamma\) a.u., unless stated otherwise.