**3. Results**

#### *3.1. Separating a Pure Quadrupole Contribution to Photo-Double Ionization*

In quasifree photoionization, the photon is absorbed by the electron pair and the nucleus is only a spectator that receives no recoil momentum. By means of momentum and energy conservation, the ejection of two electrons back-to-back with equal energy is a strict consequence of a vanishing recoil momentum, if one neglects the photon momentum. For a dipole transition, this kinematic profile is forbidden for PDI of He and H2, whose ground-state wave functions both have the same 1*S* symmetry [4,30]. Hence, the QFM is a pure quadrupole contribution to PDI and it can be isolated particularly clearly in a differential cross section that shows the double-ionization yield as a function of the electron energy sharing *ε* = *Ee*1/(*Ee*1 + *Ee*2) and the angle enclosed by the two electron momentum vectors *ϑ*12 = cos<sup>−</sup><sup>1</sup>[*pe*<sup>1</sup> · *pe*2/(|*pe*<sup>1</sup>|×|*pe*<sup>2</sup>|)] (electron mutual angle). This is done in Figure 2 where we show the measured yield from the double ionization of H2 (A,B) and He (D,E) at 800 eV photon energy as a function of *ε* and *ϑ*12. Panels A and D show the full range of the two variables, but panels B and E show only the region indicated by the dashed black lines in A and D. A comparison between panels A and D points out a strong resemblance in the electron emission patterns of H2 and He double ionization, as expected from the similarities in the electronic ground states. A distinctive difference can be seen around equal energy sharing (*ε* = 0.5) and back-to-back emission (cos *ϑ*12 = −1) where there appears to be a noticeable signal in panel A, corresponding to QFM, but apparently a node in panel D. Panels C and D highlight this relevant region of the cross section. While the QFM is evident for H2, a logarithmic-scale display is required in panel D to make the weak relative contribution of QFM to the total double-ionization cross section of helium visible at all (see Ref. [19] for further discussions on this finding).

**Figure 2.** Measured electron distributions of H2 double ionization (**A**–**C**) and He double ionization (**D**–**F**) by a single 800 eV circularly polarized photon. (**A**,**D**) Measured electron yield as a function of the electron energy sharing *ε* and the electron mutual angle *ϑ*12. (**B**,**E**) Detailed section from panels A and D as indicated by the dashed black lines therein. Note the logarithmic-scale display in panel E. Contributions around *ε* = 0.5 and cos *ϑ*12 = −1 correspond to the QFM electrons. (**C**,**F**) Angular distributions of QFM electrons as a function of the polar angle enclosed by the electron momentum vector and the light propagation direction. The shown data are limited to 0.35 < *ε* < 0.65 and *ϑ*12 > 160◦, as indicated by the dashed black lines in panels B and E. For this selection, the dipole contribution to photoionization vanishes. The red lines represent |*<sup>Y</sup>*21|<sup>2</sup> and are normalized to the data points.

To highlight the quadrupole nature of QFM, we selected a subset of our data limited to 0.35 < *ε* < 0.65 and *ϑ*12 > 160◦, as indicated by the dashed black lines in Figures 2B,E, and present in Figures 2C,F—the measured electron yield as a function of the angle *ϑγ*, which is enclosed by the electron momentum vector and the light propagation direction. The angular-momentum transfer is an important physical difference between dipole and quadrupole transitions. By definition, a dipole transition transfers one unit of angular momentum to the two-electron final state due to the photon spin, while two units of angular momentum are available in a quadrupole transition. The angular momentum of the outgoing electron wave becomes observable in the angular distribution of the electron. The red lines in Figures 2C,F represent the square of the spherical harmonic for = 2 and *m* = 1, |*<sup>Y</sup>*21|<sup>2</sup> ∝ cos<sup>2</sup> *ϑγ* × sin<sup>2</sup> *ϑγ*, which describes the final-state angular distribution of electrons that result from a pure electric-quadrupole transition from any initial *s*-subshell. Here, we have chosen the photon propagation direction *k* **ˆ** *γ* as the quantization axis. The photon spin vector is (anti-)parallel to *k* **ˆ** *γ* and we ge<sup>t</sup> Δ = 1 and Δ*m* = 1 through the transfer of the spin angular momentum. The additional unit of orbital angular momentum— *kγ* × *re* = *h*¯— is oriented perpendicularly to the quantization axis. It increases the magnitude of the electron angular momentum but has no effect on its projection *m* onto *k* **ˆ** *γ*. The strong resemblances between the measured angular emission patterns and |*<sup>Y</sup>*21|2, as demonstrated in Figures 2C,F, underline that QFM electrons originate from a pure quadrupole contribution to photoionization. Note that this agreemen<sup>t</sup> is better for He, and we suspect this is simply

due to larger momentum uncertainty in case of H2. In our experiment, one of the two QFM electron momentum vectors is reconstructed by means of momentum conservation. This is less accurate for H2 because the ions' center-of-mass momentum has to be retrieved from two protons instead of being measured via the doubly charged nucleus. Thus, the momentum uncertainty propagating to the calculated electron is larger in the case of H2 than for He (see Refs. [15,19] for further details). While such a four-fold symmetry in the angular emission pattern of QFM electrons has already been shown for He PDI at 1100 eV photon energy [18], the results shown in Figure 2C for H2 PDI further support our current understanding of quasifree photoionization.

#### *3.2. Transfer of Photon Momentum*

Nondipolar photoionization induces a forward-backward asymmetry in light propagation direction into the momentum distributions of the reaction fragments. This is due to the nonzero linear photon momentum *kγ* and the interference between electric dipole and higher multipole contributions to the photoionization process. The question of which fragments obtain the photon momentum after the reaction has been investigated since the early days of photoionization studies [31,32]. In the case of photo-single-ionization, momentum and energy conservation dictate that the center of mass—which is essentially the residual photo-ion—obtains the photon momentum [21]. Additional degrees of freedom allow for a more intricate sharing of the photon momentum between the reaction fragments in photo-double ionization. However, a recent experiment–theory collaboration investigated He PDI up to 1100 eV photon energy and showed that the momentum distribution of helium nuclei after double ionization exhibits the same forward-backward asymmetry as helium nuclei from single ionization [29]. In this photon–energy range, He PDI is dominated by the shake-off process while the quasifree mechanism is negligible in absolute terms. Apparently, the shake-off process, where the second electron is emitted due to electron–electron correlation [33,34], treats the photon momentum similarly to how single ionization does, and the photon momentum is imparted onto the doubly charged helium nucleus. However, the quasifree mechanism proceeds without involvement of the nucleus, as the photon couples directly to the two electrons. Hence, one could expect that the photon momentum is not imprinted onto the photo-ion.

In order to test this assumption, we inspect the momentum distributions of photo-ions after He PDI at 800 eV photon energy in Figure 3 for SO (A) and QFM (B) by selecting subsets of our data.

In panel A, the measured photo-ion momenta accumulate on a semicircle with a radius that equals the maximum electron momentum and which is off-centered to the right by the magnitude of the photon momentum (*kγ* = 0.215 au), indicating the transfer of the photon momentum onto the photo-ion. Note that this is the same behavior that has previously been shown for He single ionization [35]. In panel B, however, the measured average momentum of QFM photo-ions in light propagation direction is much closer to zero. This is even more apparent in panel C, where we projected the distribution shown in panel B onto the light propagation direction and determined the center through a Gaussian fit (solid blue line). Our experimental results speak in favor of the assumption that QFM photo-ions do not receive the photon momentum in the double-ionization process.

Further proof are the results of nondipole TDSE calculations for He double ionization at 800 eV photon energy, and linearly polarized light that are shown in Figure 3D. The local maximum of the nondipole curve (red line) that corresponds to QFM is exactly at zero, while the outer local maxima are shifted to the right by the magnitude of the photon momentum.

**Figure 3.** Photo-ion distributions of He double ionization by a single 800 eV photon. (**A**) The experimental data shown are limited to *ε* < 0.005 or *ε* > 0.995 and correspond to double ionization through the shake-off process. The blue semicircle is shifted to the right by the photon momentum. The SO photo-ions accumulate on this semicircle. (**B**) The experimental data shown are limited to 0.35 < *ε* < 0.65 and *ϑ*12 > 160◦. They resemble double ionization via QFM. (**C**) Projection of data from B onto the *x* axis (black) and Gaussian fit (blue) to obtain the center of the momentum distribution of QFM photo-ions along the light propagation direction (red, *μGauss*). Note that the indicated error is the standard deviation of the mean value from the Gaussian fit estimated as the square root of the respective diagonal element of the covariance matrix. The green line indicates the photon momentum for comparison. (**D**) Dipole (green) and nondipole (red) TDSE calculations for helium double ionization with 800 eV linearly polarized photons. The polarization vector is parallel to the *z* axis and the data shown are limited to *pi*,*<sup>y</sup>* = 0 ± *p*single 2 & *pi*,*<sup>z</sup>* = 0 ± *p*single 4 . Experiment and theorysugges<sup>t</sup>thatthephotonmomentumisnotimprintedontoQFMphoto-ions.
