**5. Results**

The two-photon matrix elements for the absorption, *Ma*, and emission, *Me*, paths are calculated as indicated in Equation (50) and then the atomic delay for electrons emitted in the direction of the laser field polarization is obtained from Equation (41). The Wigner delays are calculated as in Equation (34).

#### *5.1. A Light Element: Argon*

Results for ionization of argon atoms to the outermost *p* doublet <sup>3</sup>*p*<sup>−</sup><sup>1</sup> 1/2 and <sup>3</sup>*p*<sup>−</sup><sup>1</sup> 3/2, are shown in Figure 2. The two curves for the atomic delays are, more or less, indistinguishable. The negative atomic delay peak at 50 eV is due to the 3*p*-Cooper minimum in the cross section of argon. A slight shift of the negative atomic delays peaks of a few meV is observed. The similarity of the two fine-structure split channels is expected for such a light system with Δ*EAr*:3*pj* FS = 0.18 meV. The Wigner delays from the two fine-structure channels are also mostly indistinguishable. Just below the threshold for release from the 3*s*-orbital, ∼30 eV, there are narrow resonances that are not fully resolved in the present calculation.

**Figure 2.** The atomic and Wigner delay calculated in length gauge for ionization from Ar <sup>3</sup>*pj*, for electrons emitted along the polarization axis. The figure shows the region in the vicinity of the Cooper minimum. The thick dashed blue line shows the atomic delay for electrons ionized from 3*p*3/2. It is hardly distinguishable from the dashed–dotted red line that shows the atomic delay for electrons ionized from 3*p*1/2. The thin dashed green and solid grey lines show the Wigner delay for electrons ionized from 3*p*1/2 and 3*p*3/2 respectively. Dirac–Fock orbital energies have been replaced with experimental ionization energies: For 3*p*3/2 the binding energy is 15.76 eV, and for 3*p*1/2 it is 15.94 eV.

#### *5.2. Heavy Elements: Krypton and Xenon*

Atomic and Wigner delays for ionization to the outermost *p*-doublet in krypton and xenon are shown in Figures 3 and 4, respectively. Here the delay differences between the electrons ionized to the <sup>4</sup>*p*<sup>−</sup><sup>1</sup> 1/2 and <sup>4</sup>*p*<sup>−</sup><sup>1</sup> 3/2 (5*p*<sup>−</sup><sup>1</sup> 1/2 and <sup>5</sup>*p*<sup>−</sup><sup>1</sup> 3/2) in krypton (xenon) show that relativistic effects are important. Differences between the delays are clearly visible on the order of a few eV at the Cooper minima. Such shifts can be expected because the finestructure shifts are Δ*E*4*pj* FS = 0.67 eV for krypton (Δ*E*5*pj* FS = 1.3 eV for xenon). Furthermore, a difference between the doublet channels is observed at low energies, where xenon shows the largest delay difference that exceeds 10 as.

**Figure 3.** The atomic and Wigner delay calculated in length gauge for ionization from Kr <sup>4</sup>*pj*, for electrons emitted along the polarization axis. The thick dashed blue line shows the atomic delay for electrons ionized from <sup>4</sup>*p*3/2, and the dotted–dashed red line shows the atomic delay for electrons ionized from <sup>4</sup>*p*1/2. The thin dashed green and solid grey lines show the Wigner delay for electrons ionized from <sup>4</sup>*p*1/2 and <sup>4</sup>*p*3/2 respectively. Dirac–Fock orbital energies have been replaced with experimental ionization energies: For <sup>4</sup>*p*3/2 the binding energy is 14.00 eV, and for <sup>4</sup>*p*1/2 it is 14.67 eV [94]. Dirac–Fock orbital energies are used for the deeper lying orbitals.

**Figure 4.** The atomic delay calculated in length gauge for ionization from Xe <sup>5</sup>*pj*, for electrons emitted along the polarization axis. The thick dashed blue line shows the atomic delay for electrons ionized from 5*p*3/2, and the dotted–dashed red line shows the atomic delay for electrons ionized from 5*p*1/2. The thin dashed green and solid grey lines show the Wigner delay for electrons ionized from 5*p*1/2 and 5*p*3/2 respectively. Dirac–Fock orbital energies have been replaced with experimental ionization energies. For 5*p*3/2, the binding energy is 12.13 eV, and for 5*p*1/2 it is 13.44 eV [94]. Dirac–Fock orbital energies are used for the deeper lying orbitals.

#### *5.3. Study of Continuum–Continuum Delay*

The difference between the atomic and the Wigner delay is plotted for argon, krypton, xenon, and radon in Figure 5. For all the elements, and all fine-structure components, the results are very similar. This is in accordance with earlier findings, using non-relativistic calculations [41,42,44], and the corresponding numerical continuum–continuum delay: *τ*MBPT *cc* , is shown as a dotted line in Figure 5 for comparison with the relativistic results. Thus, the contribution from the second photon depends on the kinetic energy and the long-range potential, but only weakly, or not at all, on the structure of the remaining ion, or its angular momentum, for photoelectrons emitted along the polarization axis.

Only in the vicinity of Cooper minima, or close to resonances, is there are a deviation from this general trend. We stress that non-relativistic deviations, of a few attoseconds, have also been found for Ar3*p* at the Cooper minimum by using the effective one-body potential for the final state [44]. In that case, however, the complete 2P2C-RPAE method was used to show that these deviations could be reduced, as shown Figure 9b of ref. [44]. Thus, we may speculate that the present relativistic deviations at the Ar3*pj* Cooper minima could be reduced by turning to 2P2C-RRRA theory. On the other hand, the correlation-induced 3*s*-minimum was shown to be non-separable by using the 2P2C-RPAE method, as shown in Figure 9a of [44]. Obviously, fast photoelectrons are also well described by the analytical cc-delay in ref. [33], but more importantly, Figure 5 demonstrates that a universal behaviour extends to much lower energies than expected from the asymptotic theory (>20 eV) [33], in good agreemen<sup>t</sup> with non-relativistic 2P2C-RPAE matrix elements [44].

**Figure 5.** The difference between the atomic delay and the Wigner delay for the two outermost orbitals in Ar, Kr, Xe, and Rn calculated in length gauge and for for electrons emitted along the polarization axis. The red dotted line shows the non-relativistic result calculated for Ne 2*p*, i.e., the numerically obtained continuum–continuum delay discussed in the introduction.

## *5.4. Comparison with Experiments*

The delay difference between photoelectrons originating from the outermost *p*3/2 and *p*1/2 orbitals in krypton and xenon have been studied in refs. [71,72] by using the interferometric RABBITT technique. In Figure 6, this difference, as calculated here, is shown for xenon. The experiment from ref. [71] includes one data point at 18.6 eV and one at 24.8 eV which are in qualitative agreemen<sup>t</sup> both with the calculation presented here, and with accompanying calculations in ref. [71], based on the Wigner delay from RRPA

augmented with the the cc-delay from ref. [33]. Three other data points, on the other hand, differ markedly from both theoretical results. Especially striking are the large measured delays for higher energies (around 30 as at 30 eV), where the calculated result is very small. This might be due to resonances, not fully accounted for in the calculations, as discussed in ref. [71].

Moreover, a higher energy region has been explored. Ref. [72] measured the the delay difference for the xenon 5*p* fine-structure components for the sideband at 90 eV (with IR photon energies of 1.55 eV) to *<sup>τ</sup>A*(<sup>5</sup>*p*3/2) − *<sup>τ</sup>A*(<sup>5</sup>*p*1/2) = 14.5 ± 9.3 as. Moreover, the calculated delay is much smaller, around 2 as (not shown in the figures). We note that the cross section to produce photoelectrons in the <sup>5</sup>*j* channels at around 90 eV photon energy is comparable to those for 4*d* and shake-up satellites [95]. Because shake-up channels can have significantly larger delays [12], this region might need a more careful investigation of all competing channels.

**Figure 6.** The delay difference between photoelectrons originating from the 5*p*3/2 and 5*p*1/2 orbitals in xenon. The dashed blue line shows the atomic delay, and the solid red the Wigner delay.

Figure 7 shows finally the atomic and Wigner delay for photoelectrons released from the xenon 4*d* orbitals. The result agrees within error bars with the measurement, from threshold up to ∼100 eV, in ref. [74]. It is interesting to note the large difference between the two channels, defined by the two fine-structure components, in the region just above the 4*d* thresholds at 67.5 and 69.5 eV, and the rapid variation of the delay with photon energy. The experiments in refs. [96,97] have shown that also the cross-section branching ratio (for leaving the ion with 4*d*−<sup>1</sup> 3/2 or 4*d*−<sup>1</sup> 5/2) shows a rapid variation in this region. In both cases, this behaviour can be traced back to the presence of two resonances close to threshold. They are of 3D and 3P character and cannot be populated by one-photon absorption in a non-relativistic description. The spin-orbit-induced singlet-triplet mixing opens, however, the route to ionization through these resonances, and thus for a population transfer from one final channel to the other. This has been further discussed in refs. [74,98]. We note that although the resonances in argon, mentioned above, are just unresolved in the calculation, the reason that these xenon resonances are not seen directly is not a question of resolution. The cross section in this region is completely dominated by the so-called giant resonance of 1P character and the spin-orbit-induced resonances can simply not be seen against this background. Still, their mark in the more sensitive observables, such as atomic delays and branching ratios, is clearly seen.

Also for xenon 4*d* ref. [72] gives a value at 90 eV: *<sup>τ</sup>A*(4*d*5/2) − *<sup>τ</sup>A*(4*d*3/2) = −4.0±4.1 as, which agrees with our value of −1.2 as.

**Figure 7.** The thick dashed blue line shows the atomic delay for electrons ionized from <sup>4</sup>*d*3/2, and the dotted–dashed red line shows the atomic delay for electrons ionized from <sup>4</sup>*d*5/2. The thin dashed green and solid grey lines show the Wigner delay for electrons ionized from <sup>4</sup>*d*3/2 and <sup>4</sup>*d*5/2 respectively. Dirac–Fock orbital energies have been replaced with experimental ionization energies. For <sup>4</sup>*d*5/2 the binding energy is 67.5 eV, and for <sup>4</sup>*d*3/2 it is 69.5 eV [99].
