# The Role of Electron Trajectories in XUV-Initiated High-Harmonic Generation
## 4 Discussion
### Frequency Shifts of Long Trajectory Harmonics  
Figure 4a reveals that some HHG signals are strongly shifted with respect to the harmonic frequencies (marked by \"A\" in Figure 4a,b). Particularly in H19, the effect is very pronounced, with a total shift up to half a harmonic order. At negative delays, we find a strong blue shift, whereas at positive delays a red shift dominates. Converting the frequency shifts to \(\alpha\) parameters (Equation (1))  
Figure 4: Delay-dependent frequency shifts around harmonics H17 to H23. (**a**) Experimental data. We normalize the signal for each harmonic order separately. The arrows indicate characteristic features in the spectrum: A, strongly frequency-shifted spectral components, with positive (negative) shifts for negative (positive) XUV–IR delays; B, sub-cycle temporal delay of the oscillations of the shifted spectral components with respect to the odd harmonic signal. (**b**) TDSE calculation results. (**c**) The Coulomb-corrected three-step model (CCTSM) \(\alpha\) parameter for H19 as a function of intensity. The black (blue) circles indicate downhill (unphill) short trajectories and the red (green) circles indicate downhill (unphill) long trajectories. (**d**) The same as frame (c), but for H21. (**e**) CCTSM results overlaid with the experimental color plot. The black (blue) curves indicate downhill (unphill) short trajectories and the red (green) curves indicate downhill (unphill) long trajectories. (**f**) CCTSM results overlaid with the TDSE color plot. (**g**) Zoom into H21 from frame (a). The XUV–IR oscillations of the short trajectories are lagging behind those of the much fainter blue-shifted long trajectories by about 450 as.  
yields \(\alpha\approx 24\pm 8\times 10^{-14}\,\mathrm{cm}^{2}\,\mathrm{W}^{-1}\), reflecting the contribution of long trajectories [13; 16]. Indeed, the \(\alpha\) parameters and frequency shifts of long trajectories calculated by the CCTSM model are in good agreement with experiments and the TDSE (see Figure 4c-f). Here, downhill and uphill long trajectories yield similar values, with downhill long trajectories creating a slightly larger frequency shift than uphill long trajectories.  
The qualitative agreement between the experiments and the CCTSM also suggests that the strongly frequency-shifted feature found at H16.4 for \(\Delta t\)\(\sim\)\(+7\) IR cycles is due to a long trajectory (see Figure 4e). The experimental \(\alpha\) parameter extracted at \(\Delta t\)\(\sim\)6.2 IR cycles is \(34\times 10^{-14}\,\mathrm{cm}^{2}\,\mathrm{W}^{-1}\). A similar feature is found in the TDSE calculation (see Figure 4b), albeit at smaller delays. The IR intensity at \(\Delta t\)\(\sim\)6.2 IR cycles in our model pulse is about \(2\times 10^{13}\,\mathrm{W}\,\mathrm{cm}^{-2}\), which means that the excited states of helium are still intact and cannot be described as a quasi-continuum as in the CCTSM. We propose that the mechanism of XUV-initiated HHG is slightly different in this case. Here, the XUV field transfers a population to a discrete excited state of helium, for example the \(2p\) state, which is in resonance with H15. Subsequently, the electron is accelerated from the excited state and recombines to the ground state under the influence of the IR field. This process is known as excited-state HHG and has been observed in IR-driven HHG from argon [38].