Double-slit photoelectron interference in strong-field ionization of the neon dimer

Wave-particle duality is an inherent peculiarity of the quantum world. The double-slit experiment has been frequently used for understanding different aspects of this fundamental concept. The occurrence of interference rests on the lack of which-way information and on the absence of decoherence mechanisms, which could scramble the wave fronts. Here, we report on the observation of two-center interference in the molecular-frame photoelectron momentum distribution upon ionization of the neon dimer by a strong laser field. Postselection of ions, which are measured in coincidence with electrons, allows choosing the symmetry of the residual ion, leading to observation of both, gerade and ungerade, types of interference.


Supplementary Figures
Supplementary Figure 1. Sketch of the experimental setup. COLTRIMS meets matter wave diffraction. Neon dimers are produced in the supersonic expansion of the neon gas through 5 µm nozzle into the vacuum. The transmission grating with a period of 100 nm is used to deflect neon dimers towards the laser focus, where dimers get ionized. The electron and the neon ion are guided by weak electric and magnetic fields towards two position sensitive detectors. Measuring the impact position as well as time a charged particle needs to reach the detector allows to deduce the 3D momentum of a particle after ionization.
Supplementary Figure 2. Dependence of the dissociation time on the initial internuclear distance. The time the neon dimer needs to reach an internuclear distance of 10 Å during direct dissociation along the II(1/2)g potential curve. Supplementary

Supplementary Note 1. Dissociation times
We have estimated dissociation times by classical propagation of a particle with reduced mass of Ne2 ( =10 amu) along the potential curve II(1/2)g from ref. 1 . The times the dimer needs to reach an internuclear distance of 10 Å (taken as a reference) during dissociation starting at different internuclear distances according to the ground state probability distribution are shown in Supplementary Figure 2.
Thus, in the worst case, when dimer starts to dissociate along II(1/2)g at an internuclear distance of According to our estimations, the indirect dissociation happens even faster: an internuclear distance of 10 Å is reached within ca. 200-300 fs. Qualitative explanation for this is that the initial movement on the I(1/2)u potential should be accomplished within a laser pulse (40 fs, FWHM in intensity), otherwise the vibrational wave packet will not be lifted up to the dissociative II(1/2)g potential curve, leaving dimer ion bound. The subsequent movement on the II(1/2)g potential curve is faster than in case of direct dissociation, since the starting internuclear distance is much shorter. The same dissociation path was used for explanation of the 200 fs-long signal depletion in the pump-probe experiment with Ar2 2 . Since the potential energy curves of Ne2 are similar to those of Ar2, but the reduced mass is only a half of Ar2, the dissociation dynamics of Ne2 + should be even faster.
Supplementary Note 2. Advanced theory: active electron coupled to a twolevel ion The purpose of this section is a further theoretical investigation of Ne2 strong-field ionization using an extended model which allows us to include the hole dynamics in the remaining ion and its interaction with the laser field as well as the correlated interaction of the outgoing electron with the parent ion.
We describe the Ne2 + molecular ion as a two-level system where the states (1,0) and (0,1) correspond to the hole being located at the neon atom on the negative or positive -axis, respectively. The ionic Hamiltonian is where = /2 = 2.93 a.u. is half the equilibrium distance between the two neon atoms in the neutral dimer and 2 = 0.00870 a.u. is tuned to obtain the energy splitting between the gerade and ungerade states of the molecular ion at this distance. An active electron is coupled to the two-level system such that a state of For the electron-ion interaction we use the same potential as in the main text. The interaction with the neutral neon atom is given by a neutral-neon scattering potential 3 . The weak energy dependence of the latter potential is neglected and the potential is evaluated at zero energy. We follow a similar pseudopotential procedure as in the main text and remove the singularity by matching to an s-wave scattering state at momentum = 1 a.u. at a distance 1 a.u. from the singularity.
The initial state is an eigenstate of the field-free Hamiltonian and consists of two p-type orbitals for and located at the respective ions. It has total gerade symmetry, where the symmetry operation in the coupled system inverts the coordinates in the wave function of the active electron and exchanges the two states of the two-level system, i.e. ( ) ↔ (− ). Starting from this bound state, the TDSE is solved with the split-operator method. Here, the short-time propagator is split as All parameters are chosen as in the main text. We also account for the different possible orientations of the dimer with respect to the polarization plane. After the time evolution, the final state is projected onto the gerade and ungerade states of the molecular ion, leading to From these single-electron wave functions we obtain the photoelectron momentum distributions for the two types of interference patterns shown in Supplementary Figure 3. In contrast to the calculations from the main text that had perfect contrast by construction, these distributions show a significant decrease in contrast. In the presence of a second atom in the case of the dimer, the two pathways in strong-field ionization are not completely equivalent anymore. This is different from XUV photoionization where perfect contrast has been observed 4 .
Another source that reduces the contrast are false coincidences (see Supplementary Note 4). However, they account for only one-half of the background in the case of the indirect dissociation, and are rather irrelevant for the direct breakup channel.
As seen in Supplementary Figure 3 the advance theory produces some additional features in the photoelectron momentum distribution that are not observed in the experiment. The reason for this might be an insufficient accuracy of the model potential for the neutral atom that was used in the TDSE calculations. The model potential for the neutral Ne atom is taken from Supplementary Reference 3. The quality of the potential in this reference was checked by comparing the computed electron scattering cross-sections based on the proposed potential with the experimental ones. This comparison was done however for electrons with energies higher than 50 eV (a momentum of 1.9 a.u.). Moreover, it was stated that the experimental scattering cross-section is not well reproduced by the simulation based on the model potential in the low energy range. Since electrons upon tunneling have even lower energies (E<3.4 eV or p<0.5 a.u.), as was measured from the experimental momentum distribution perpendicular to the polarization plane (where the laser field is zero), one might assume that the accuracy of the model potential in the desired electron energy region is not that high. Another possible source of error relates to the fact that the potential of the neutral atom had to be converted into a pseudopotential to make the calculation feasible.
The idea, however, behind using the advance theory was to give qualitative explanation of the finite contrast in the measured interference pattern.

Supplementary Note 3. Molecular-frame transformation
In the natural molecular-frame transformation, the ion momentum vector, not its projection to the polarization plane, defines the ∥ direction. The electron momentum vector is then projected onto the ∥ -plane to get the molecular frame photoelectron momentum distributions shown in Supplementary Figure 4. This projection conserves • but the momentum distributions show a node along the molecular axis due to the vanishing volume element.

Supplementary Note 4. Determination of the bond length
From the interference pattern The accurate measurement of the bond length would require a more advanced theoretical model than the very simple one used for simulations in Fig. 4 b,d, since the ionization weighting discussed in the paper changes the shape of the interference fringes. We could partially reduce this weighting by dividing the experimental momentum distribution by the corresponding spectrum of the monomer, however, it was not possible to completely remove it. Moreover, the ionization process depends on internuclear distance due to change in the ionization potential. This dependence should be considered as well for accurate estimation of the bond length distribution. Despite these arguments, we have done estimation of a bond length by fitting the fringe distributions corresponding to different ion momenta (Fig. 4a,c) to a function based on eq. (1) from the main text: where is the internuclear distance, ∆ is a phase that is 0 for the indirect channel and π for the direct one, σ is the doubled variance of the residual Gaussian distribution (ionization weighting). The Gaussian distribution was used only for the indirect channel, since the direct one has only two fringes that are symmetric with respect to ∥ = 0. A and B are the amplitude and the background, respectively. The typical fits for the direct and indirect fragmentation are shown in Supplementary  Figure 5.

From ion momentum by potential mapping
In order to obtain the bond length from the measured ion momenta we have inverted the II(1/2)g and I(1/2)u energy potentials for the direct and indirect dissociation channels, respectively. The potentials have been taken from ref. 1 . The initial potential energies (with respect to the dissociation energy of 21.6 eV) of the ion upon ionization have been calculated as following: where and are measured ion momenta for the direct and indirect dissociation channels, respectively. = 1.59 is the photon energy corresponding to a central wavelength of 780 nm.
= 0.1 is the spin-orbit coupling. After calculating the energies and , the corresponding internuclear distances and (both labeled as in Fig. 5) have been found from II(1/2)g and I(1/2)u energy potentials.

Supplementary Note 5. False coincidences
The momentum distributions of the Ne2 breakup into Ne + <> Ne 0 with the corresponding energy distributions are shown in Supplementary Figure 6. As one can see on the left, apart from the direct, indirect breakup channels and monomer ionization, there is also unwanted background coming from false coincidences. The reason for this is that no selection based on momentum conservation can be applied here, since one particle of the reaction is uncharged (Ne 0 ) and, thus, is not detected in experiment. Another unwanted source of false coincidences is single ionization of 22 Ne, which resides very close to the indirect dissociation channel. The events related to single ionization of 22 Ne have been cut by the following momentum conditions [in a.u.]: -1<px<1 && -4<py<4 && -38<pz<-30 and not used during further analysis (see the right of Supplementary Figure 6). The photoelectron momentum distribution in the molecular frame corresponding to the background shows no interference (see Supplementary Figure 7). We have estimated the background influence on the fringe contrast for the indirect channel. For this, we have plotted the component of the photoelectron momentum that is parallel to the molecular axis ( ∥ ) for both the indirect breakup channel and the background (see Supplementary Figure 8). The background was chosen by requiring the ion momentum to be within a window of 18-35 a.u. The indirect channel corresponds to the ion momenta in the range of 37-46 a.u. In addition all ion momenta were restricted by |px|<12 a.u. The volume of the indirect breakup channel in the momentum space was thus by about 0.83 times smaller than that of the background channel. Assuming the constant density of the false coincidences in the momentum space, one get the following estimation for the background signal in the indirect breakup channel caused by false coincidences: 175(red line)*0.83≈145(green lines) counts. This background makes up about a half of the indirect channel background, as seen in the ∥ -projection on the left of Supplementary Figure 8. The influence of the false coincidences on the direct breakup channel is negligible because of its tiny volume in the ion momentum space and large amount of measured events (the ratio of the useful events to false coincidences is very high, which can also be seen in the energy distributions in Supplementary Figure 6).

Supplementary Note 6. The interference fringe contrast
We have calculated the fringe contrast by making use of the projections shown in Supplementary  Figure 8. The contrast was found to be 970/330≈2.9 and 580/245≈2.4 for the direct and indirect fragmentation channels, respectively (Supplementary Figure 9).