## 3 Results
### Comparison of Streaking and RABBITT Delays with Classical Photoemission Times  
To compare the delay information that can be obtained from a streaking or RABBITT experiment at solid surfaces, streaking and RABBITT spectrograms were calculated for different center EUV energies. Again, the initial state is effectively localized at -5 A, and the potential specified in the Methods section is used as an example system. Spectra were calculated for \(E_{\mathrm{kin}}=\hbar\omega+\epsilon_{0}\), ranging from 5 to 90 eV. The IR pulse has a central energy of 1.55 eV, and pulse duration of 5 fs. The delay axis was sampled by 101 \(\tau_{\mathrm{IR-EUV}}\)-delay steps between \(-2.4\) and \(2.4\) fs. From the resulting spectrograms, the corresponding streaking and RABBITT delays were extracted following the procedures described in Appendix A. Resulting delays \(\tau_{\mathrm{RABBITT}}\) and \(\tau_{\mathrm{streaking}}\) are shown in Figure 5, and compared with \(t_{\mathrm{PE}}\) (Figure 5a). Photoemission times \(t_{\mathrm{PE}}\) were determined by calculating the classical transport time for a free electron at the same potentials propagating from emission depth (\(z=-5\) A) to the surface (\(z=0\) A). Streaking delays can be determined from simulated photoelectron spectra with an accuracy of \(\Delta\tau_{\mathrm{streaking}}=\pm 0.2\) as (Appendix A, Figure A1a). Due to the lack of an appropriate evaluation method for few-cycle RABBITT data (Appendix A, Figure A1b), RABBITT delays are subject to a larger systematic error, and can only be specified here with the accuracy \(\Delta\tau_{\mathrm{RABBITT}}=\pm 2\) as. \(\Delta\tau_{\mathrm{RABBITT}}\) decreases for longer IR pulses that would, however, significantly increase simulation computation time. Figure 5b shows the difference between classical photoemission times \(t_{\mathrm{PE}}\) and \(\tau_{\mathrm{RABBITT}}\) (red), as well as \(\tau_{\mathrm{streaking}}\) (green), respectively. Consistently with the quantum-classical correspondence principle, streaking and RABBITT delays from TDSE calculations approach the classical results with increasing kinetic energy. The difference becomes smaller than 1 as for \(E_{\mathrm{kin}}\geq 45\) eV, and one can conclude that the extracted delays from the spectrograms indeed reveal photoemission time. Note that  
Figure 5: Comparison of streaking and RABBITT delays with classical photoemission transport times \(t_{\mathrm{PE}}\). (**a**) Variation of streaking (green crosses) and RABBITT delays (red dots), as well as classical photoemission times (black) as a function of (vacuum) kinetic energy. For the streaking data, error bars are not shown because they are smaller than the symbol size. Streaking and RABBITT delays were extracted from spectrograms, calculated by solving the one-dimensional time-dependent Schrödinger equation (TDSE) (Equation (6)) for potential properties described in the main text. (**b**) Difference between RABBITT and streaking delays and classical photoemission times. (**c**) Difference between RABBITT and streaking delays. deviation from classical behavior is not significant for \(E_{\rm kin}\geq 25\) eV in the RABBITT case because of larger systematic error. For smaller energies, streaking and RABBITT results exhibit oscillatory deviations from the classical case toward smaller delays. This is a quantum mechanical interference effect caused by partial reflections of the excited state wave packet at the inhomogeneities of the potential. In TDSE calculations, the actual shape of the potential composed of the potential step at the surface and the Yukawa potential determine energy-dependent transmission and reflection coefficients. Therefore, parts of the wave packet undergo reflections before being emitted through the surface. The corresponding interference pattern that emerges in the spatial region between the initial position and the surface step is more pronounced for low kinetic energies. This results in modulated photoelectron energy distribution.