*3.3. Time Delay*

Results of the time delay calculations by taking the energy derivative of the SO phase Equation (2) are displayed in Figure 3. The horizontal axis in the figure denotes the slow photoelectron energy. It is assumed that the photon energy is very large and nearly all of it is carried away by the second fast photoelectron. The three panels of this figure display the time delay results for the He 1*s*, Be 2*s*, Ne 2*s* and Mg 3*s* (left), H− 1*s* (center) and Ar 3*<sup>s</sup>*, Cl− 3*s* and Xe 5*s* (right). We observe a very small SO time delay in He 1*s* not exceeding a few attoseconds. A similarly small time delay is found in the intra-shell SO process in Be 2*s* and Mg 3*s* as well as the inter-shell SO process in Ne 2*s*. Incidentally, the time delay in the inner 1*s* shell of Be is much smaller than that in the valence 2*s* shell, the reason being the Coulomb field of the bare nucleus, which makes many-electron correlation and associated time delay negligible.

The SO time delay in H− 1*s* is markedly higher by nearly an order of magnitude. Time delay grows further in Ar 3*s* and Xe 5*<sup>s</sup>*, while it exceeds the 100 as mark in the negative Cl− ion. We relate this growth of time delay to much bolder shake-up and shake-off processes in these targets, which are strongly affected by many-electron correlation.

Another indication of this effect is presented in Table 2. Here we compare the summary displacement of the main and shake-up satellite lines in the photoelectron spectra of He 1*s* and Ar 3*s* with the corresponding time delay integral Equation (8). We observe in this table that the many-electron correlation causes a much stronger line displacement in Ar than in He. This is matched by a much larger SO time delay in Ar in comparison with He.

**Figure 3.** Shake-off time delay (in attoseconds) in He 1*s*, Be 2*s*, Ne 2*s* and Mg 3*s* (**left**), hydrogen H− 1*s* (**center**) and Ar 3*<sup>s</sup>*, Cl− 3*s* and Xe 5*s* (**right**), calculated by taking the energy derivative of the SO phase Equation (2).


**Table 2.** The energies of the shake-up satellites in the photoelectron spectra of He 1*s* and Ar 3*s* as calculated in the HF approximation *Ek* and as the poles of the SHGF *<sup>ε</sup>k*. The sum ∑*k*(*Ek* − *<sup>ε</sup>k*) in each target is compared with the corresponding time delay integral Equation (8).

## **4. Summary and Outlook**

In the present work, we demonstrated that the SO process is prevalent in DPI at large photon energies exceeding significantly the double ionization threshold. Under this condition, the two-electron energy sharing is highly asymmetric with the primary photoelectron taking nearly all of the photon energy, while the secondary SO electron is rather slow. The slow electron can be delayed by repeated interaction with the ionic core. In the intra-shell shake-off process, this interaction is confined to the same shell and is typically rather quick. The intra-shell SO processes in He 1*s*, Be 2*s* and Mg 3*s* take no more than several attoseconds to complete. The marked exception is the intra-shell SO in the H− ion, which may take several tens of attoseconds. We attribute this effect to a strongly correlated nature of H− which will not bind in the absences of correlation. The inter-shell SO process are more involved and take typically longer time. We observe a considerable delay in the SO of Ar 3*s* and Xe 5*<sup>s</sup>*. The Cl− ion, which is iso-electronic to Ar, demonstrates a very significant SO delay exceeding 100 as. All these targets are prone to strong final-state correlation and display intense shake-up satellite spectra with a strong line displacement relative to the corresponding HF energies. The only exception is the inter-shell SO in Ne, which is still rather quick.

It is instructive to compare the SO time delay in DPI with the analogous characteristic of single photon one-electron ionization (single photoionization—SPI). The energy derivative of the SPI amplitude is known as the Wigner time delay. Similarly to electron elastic scattering [39], it characterizes the photoelectron group delay in the dispersive potential of the ionic core. This potential includes an exchange with the core electrons [40]. In addition, the Wigner time delay in SPI is strongly affected by inter-shell correlation [41]. All these characteristics of the Wigner time delay are present in DPI. The SO adds an extra component of the time delay which is specific to DPI.

To resolve the SO process in time, one needs to use various laser-based interferometric techniques which introduce an additional component to the measurable time delay. This component, known commonly as the Coulomb laser coupling (CLC) [42] or the continuum– continuum (CC) correction [43], depends on laser frequency and the asymptotic Coulomb charge *Z* acting on the departing photoelectron. For the fast primary photoelectron, this

charge *Z* = 1 for the neutral targets and *Z* = 0 for the negative ions. In the former case, the fast photoelectron does not experience any CLC correction, while in the latter case, this correction is relatively small because the photoelectron is sufficiently fast. The slow electron sees the asymptotic charge *Z* = 2 for the neutral targets and *Z* = 1 for the negative ions. So while the CLC correction still affects the slow SO elecrtron, it will be relatively weaker.

**Funding:** This research received no external funding.

**Data Availability Statement:** The numerical data reported in the present work are available on request from the author.

**Acknowledgments:** Nearly 40 years on, the legacy of the seminal work initiated by Miron Amusia is still alive and produces fruitful results. Some of these results are reported in the present paper.

**Conflicts of Interest:** The author declares no known conflict of interest.
