## 3 Results for a Model Atom
of the ATA peaks, dependent on the electronic structure of the atom (\(2\omega_{d}=0.20\) a.u. is missing in the table). Figure 5: \(|\sigma^{(1)}(\omega,\omega^{\prime})|\), i.e., the modulus of the Fourier transform of the XUV one-photon cross-section \(\sigma^{(1)}(\omega,\tau)\), calculated with Equation (24). One can see all these energies in the oscillations of a main peak of the ATA spectrum, except those that do not include this concrete state in the interference process, such as \(\omega^{\prime}=|E_{i^{-}}-E_{j^{-}}|\) for the absorption peak \(\omega=|E_{k^{+}}-E_{0^{+}}|\), where \(k\
eq i\) and \(k\
eq j\). The first three peaks of the Fourier spectrum of the XUV one-photon cross-section (marked with a green color in the projection onto the \(\omega^{\prime}\)-axis in Figure 5) correspond to energy differences \(|E_{i^{-}}-E_{j^{-}}|\), which appear due to coupling of two \(i^{-}\) states by the dressing field. This makes it possible for the atom to be excited into the state \(i^{-}\) with a following deexcitation from the \(j^{-}\) state. In the middle of Figure 5 there are the peaks corresponding to \(\omega^{\prime}=|E_{i^{-}}-E_{j^{+}}\pm\omega_{d}|\) energy differences. Oscillation energies marked in blue on the left in Figure 5 correspond to the coupling through or to the \(1^{+}\) state for \(2^{-}\) or \(3^{-}\): \(\omega^{\prime}=|E_{1^{+}}-E_{i^{-}}+\omega_{d}|\), \(i=2,3\), and through or to the \(2^{+}\) state for \(1^{-}\): \(\omega^{\prime}=|E_{2^{+}}-E_{1^{-}}-\omega_{d}|\). Analogously, in yellow are marked oscillation energies \(\omega^{\prime}=|E_{1^{+}}-E_{i^{-}}-\omega_{d}|\) for \(2^{-}\) and \(3^{-}\), and \(\omega^{\prime}=|E_{2^{+}}-E_{1^{-}}+\omega_{d}|\) for \(1^{-}\). The remaining energies of this type correspond to \(\omega^{\prime}=|E_{2^{+}}-E_{3^{-}}\pm\omega_{d}|\) are in the middle of the plot at the energies \(\omega^{\prime}=0.087\) a.u. and \(\omega^{\prime}=0.112\) a.u. The \(\omega^{\prime}=\omega_{d}=0.1\) a.u. oscillations appear due to the coupling of \(1^{-}\) and \(1^{+}\) or coupling of \(2^{-}\) and \(2^{+}\), whose energies were chosen rather close to each other. Oscillation at the energy \(\omega^{\prime}=2\omega_{d}=0.2\) a.u. is characteristic for all cross-section peaks. This feature was theoretically predicted and measured as subcycle fringes in laser-dressed helium atoms [52; 53; 54]. It is noteworthy that the oscillation energy of these LIS peaks depends only on the photon energy of the dressing pulse and not on the electronic structure of the atom. In contrast, the energy \(\omega^{\prime}=|E_{i^{-}}-E_{j^{-}}|\) of the main peaks' oscillations does not depend on the energy of the dressing pulse at all, which testifies that the XUV excitation creates a coherent superposition of the \(i^{-}\) states. The intensity of a Fourier peak strongly depends on the atomic parameters as well as the pulse parameters. An important atomic parameter is the transition dipole matrix element between two states. In Figure 5 one can see that the Fourier peaks of the transition into \(1^{-}\) are markedly weaker in comparison with \(2^{-}\) and \(3^{-}\), as the \(1^{-}\) transition dipole matrix element from the ground state is smaller (Table 2) and as the energy gap between \(1^{-}\) and the other two bright states is big. Moreover, our results obtained with Equation (24) show a sensitivity of ATA spectroscopy to the relative signs of the transition dipole matrix elements involved in the process. In Figure 6, we compare Fourier spectra of the oscillations of two different absorption peaks, where two cases are shown that differ from each other only through the sign of the transition dipole matrix element \((3^{-}|\hat{Z}|0^{+})\). One can see significant changes in the amplitude of the Fourier peaks not only in the spectrum at \(\omega=E_{3^{-}}-E_{0^{+}}\), which directly depends on parameters of the \(3^{-}\) state, but also in the spectrum at \(\omega=E_{2^{-}}-E_{0^{+}}\). This underscores the strong effective coupling between the \(3^{-}\) and \(2^{-}\) states.