**2. Methods**

The two experiments reported here have been performed at beamline P04 at the PETRA III synchrotron (DESY, Hamburg, Germany [20]) using circularly polarized light at 800 eV photon energy. We employed a cold target recoil ion momentum spectroscopy (COLTRIMS) reaction microscope [21–23] and intersected a supersonic gas jet of He or H2 with the photon beam at a right angle. Due to the supersonic expansion, the H2 were in their vibrational ground state. The charged reaction fragments were projected by an electric field and guided by a magnetic field to two time- and position-sensitive detectors with delay-line position readout [24,25]. The initial momenta after PDI were retrieved from the particles' times-of-flight and positions of impact. The concept of a COLTRIMS reaction microscope is illustrated in Figure 1. The experimental results reported in the present work were obtained from the same experimental runs as Refs. [18,19], where further experimental details can be found.

**Figure 1.** Concept of cold target recoil ion momentum spectroscopy (COLTRIMS). A supersonic jet (green) of a target gas is crossed with synchrotron light (violet) at a right angle. A homogeneous electric field *E*, generated by a spectrometer (copper plates), and a homogeneous magnetic field *B*, created by a Helmholtz pair (copper tubes), guide the charged reaction fragments (red trajectory: ion, blue trajectory: electron) towards time- and position-sensitive detectors. The initial three-dimensional momentum vector of each reaction fragment (blue and red arrows) was calculated from the time-offlight and the position of impact on the detectors (marked with a blue and a red cross).

The two-electron nondipole TDSE code used here was developed based on our previous dipole code for helium, which has been successfully applied in a series of studies on the two-photon double ionization of helium [26–28]. The form of the Hamiltonian with the nondipole corrections and the chosen parameters in the calculations can be found in Ref. [29]. After the end of the light pulse, the two-electron wave function was further propagated freely for a time of 10 a.u. Then, it was projected onto the uncorrelated symmetrical product of two single-electron scattering states to obtain the joint momentum distribution of the two ejected electrons *<sup>P</sup>*(*k*1, *k*2). The momentum spectrum of the ion *P*(*Q*) could then be obtained from *<sup>P</sup>*(*k*1, *k*2) by momentum conservation: *P*(*Q*) = *<sup>P</sup>*(*k*1, *<sup>k</sup>*photon − *k*1 − *Q*)*dk*1. For the case of the dipole approximation, *<sup>k</sup>*photon was set to 0. In the calculations, a linearly polarized light pulse along the *z* axis was adopted, with its propagation direction in the *x* axis. Therefore, the momentum spectrum of the ion in the propagation direction of the light pulse was given by *P*(*Qx*) = *dQzdQyP*(*Q*). For the distribution of *P*(*Q*) in the PDI of helium, previous studies showed that the majority of the events are located close to the surface of a sphere in the momentum space with a radius of *p*single = *ω* − *Ip*, where *ω* and *Ip*, respectively, represent the photon energy and ionization potential of helium. Such a behavior is closely related to the shake-off (SO) and the electron knock-out mechanism [8]. In order to show the effect of the QFM mechanism, we used a similar method as that in Ref. [8]. The integral interval in the light polarization direction of *Qz* was limited from −*p*single/4 to *<sup>p</sup>*single/4. Furthermore, the integral interval of *Qy* was limited by |*Qy*| ≤ *<sup>p</sup>*single/2.
