# Electronic wavefunctions probed by all-optical attosecond interferometry

## Abstract

In photoelectron spectroscopy, the ionized electron wavefunction carries information about the structure of the bound orbital and the ionic potential as well as about the photoionization dynamics. However, retrieving the quantum phase information has been a long-standing challenge. Here, we transfer the electron phase retrieval problem into an optical one by measuring the time-reversed process of photoionization—photo-recombination—in attosecond pulse generation. We demonstrate all-optical interferometry of two independent phase-locked attosecond light sources. This measurement enables us to directly determine the phase shift associated with electron scattering in simple quantum systems such as helium and neon, over a large energy range. Moreover, the strong-field nature of attosecond pulse generation resolves the dipole phase around the Cooper minimum in argon through a single scattering angle. This method may enable the probing of complex orbital phases in molecular systems as well as electron correlations through resonances subject to strong laser fields.

## Main

Photoionization offers a unique insight into quantum mechanical phenomena in matter, by projecting bound wavefunctions into ionized electronic wavefunctions, which can then be imaged experimentally. The complex properties of the photoelectron wavefunction are dictated by all steps of the light–matter interaction: the initial state, the properties of the ionizing radiation as well as the complex phase accumulated by the departing electron as it interacts with the Coulomb binding force. Photoelectron spectroscopy provides a direct measurement of the amplitude of the ionized wavefunction (Fig. 1a), but does not directly resolve its phase.

To recover the phase information of the ionized wavefunctions, two main strategies have been applied. In the first approach, the angular dependence of the photoelectron spectra can be used to retrieve the phase information of the ionized wavefunction, at a given electron energy. These so-called ‘complete’ photoionization experiments require recording of photoelectron angular distributions for linear and circular ionizing radiation, and for fixed-in-space molecules using coincidence photoelectron photoion spectroscopy1,2,3. A recent two-dimensional momentum-resolved photoionization study enabled the isolation of a continuum wavefunction through an atomic resonance and, in addition, resolves its phase by interference with a reference wavefunction4. In the second approach, photoelectron interferometry methods such as the reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) and streaking techniques, based on nonlinear light–matter interaction, can be used to measure the variation of the spectral phase with energy5,6,7. These schemes were applied to reveal phase information in photoionization of atoms8,9,10,11, molecules12 or solids13. However, these approaches provide a partial measurement of the phase: they record the derivative of the phase with respect to energy (associated with the well known Wigner time delay14,15) while the absolute phase remains inaccessible.

We can convert the photoelectron phase retrieval problem into an optical one by means of photo-recombination. Photo-recombination, the time-reversed process of photoionization, is an inherent part of the mechanism of high-harmonic generation (HHG)16. Here, under the influence of a strong laser field, an electron is liberated by tunnelling ionization, propagates in the laser field and is driven back to the parent ion. Recombination of the electron with the ion leads to the emission of photons in the extreme ultraviolet (XUV) regime. Although strong-field-induced photo-recombination can be considered to be the time-reversed mechanism of photoionization, it introduces two fundamental differences (Fig. 1b). First, the photoionization process integrates over all initial orbitals while the tunnelling process selects specific initial orbitals, dictated by the strong field’s polarization17. We note that angle-resolved photoionization experiments can provide orbital selectivity as well18. Second, in photoionization the emitted photoelectron momentum spreads over a large angular range, whereas photo-recombination projects the returning electronic wavepacket in a well defined direction, thus providing a high degree of angular selectivity19.

We demonstrate a differential phase measurement of the photo-recombination process by all-optical linear XUV interferometry. Our measurement exploits both the state selectivity by tunnelling ionization and the angular resolution provided by the photo-recombination process, and we obtain a direct insight into the fundamental quantum properties of the photoelectron wavefunctions. A complete and accurate interferometric measurement is based on two fundamental components—an independent control over each arm’s signal and the ability to manipulate their relative delay with interferometric stability. Previous realizations of optical XUV interferometry based on two phase-locked HHG sources have been introduced within a single interacting medium, either by splitting the fundamental laser field into two focal spots in the medium20,21,22 or by driving HHG in a gas mixture23,24. The former method is limited to a single species owing to the proximity of the foci, whereas the latter method provides only a single point in the interferogram. We perform a complete interferometric measurement, schematically described in Fig. 1c. First, we fully decouple the two arms of the interferometer, allowing independent control over all aspects of the interaction—both the matter under scrutiny and the basic properties of the interacting light. Second, we measure a complete interferogram by scanning the relative delay between the two sources with attosecond precision, over a large energy range. Performing a time-resolved interferometric measurement is of great conceptual importance: the phase is extracted in the Fourier domain, allowing its decoupling from systematic or amplitude errors while reducing the statistical uncertainty of the absolute phase to values as low as 0.1 rad.

## Results

### Interferometric measurement of scattering phase shifts in helium and neon

Our experimental setup consists of two main stages (Fig. 2a; for a detailed description see the Methods). In the first stage, a reference attosecond pulse train (APT) is generated by focusing an intense infrared laser pulse into a gas cell. This APT is common to all measurements, and provides a stable reference. The infrared and APT beams co-propagate and are refocused into the target gas by a curved two-segment mirror. The refocused infrared beam generates a second APT, phase-locked with the reference APT, which can be controlled in delay Δt and direction using the two-segment mirror. Finally, the XUV beams from the two APTs interfere on an XUV spectrograph. When both APT beams are temporally and spatially overlapping, we observe clear interference fringes in each of the high harmonics on the spectrograph. Scanning Δt leads to modulation of the fringe patterns over a large spectral range, directly visible in the spectrogram. A Fourier analysis (see Fig. 2b and Supplementary Fig. 4a) of the delay-dependent intensity reveals that each harmonic oscillates according to its fundamental frequency, indicating the linear nature of the XUV–XUV interference. The phase associated with each Fourier component provides a direct measurement of the relative spectral phase between the constant reference APT and the target APT for each harmonic order. Although the XUV spectrograph directly resolves the XUV interference for each harmonic, it is not strictly required. In principle, the Fourier transformation of the total XUV signal provides the same information21,25. We note that the Fourier analysis reveals the absence of coupling between the reference APT and the infrared pulse in the target gas medium (see Supplementary Fig. 4b). Such coupling would lead to oscillations of the signal at twice the infrared frequency, as observed for example, in attosecond transient absorption measurements26.

We perform a differential phase measurement by generating the reference APT in molecular nitrogen and interfering it with a target APT generated either in neon or helium under identical experimental conditions. In each measurement we scan the delay Δt and extract the Fourier phase of the signal oscillations as a function of harmonic order (see the Methods). The Fourier phase represents the relative spectral phase difference between the two arms of the interferometer: ϕ(Ω) = ϕtar(Ω) − ϕref(Ω), where Ω is the harmonic frequency. Alternating the target species while keeping the reference arm constant cancels the reference phase upon subtraction of the two interference phases. We are thus left with the absolute phase difference between neon and helium at each harmonic frequency Δϕabs(Ω) = ϕNe(Ω) − ϕHe(Ω) (see black dots in Fig. 3a). This phase measurement is fundamentally different from photoelectron interferometry approaches, such as RABBITT or streaking. Those methods record the spectral phase derivative, whereas our scheme resolves the absolute spectral phase difference between the harmonic radiation of the two atomic species. Our method provides a direct measurement of the phase through linear time-resolved XUV–XUV interferometry.

The phase of the emitted harmonics encodes all the steps of the interaction: ionization and propagation, both driven by the strong infrared field, along with the photo-recombination step. Hence, as illustrated in Fig. 2c, we can express each harmonic phase as the sum of the strong-field-induced phase ϕSF and the photo-recombination dipole phase ϕrec. The differential phase measurement thus provides Δϕabs(Ω) = ΔϕSF(Ω) + Δϕrec(Ω). The strong-field contribution ΔϕSF(Ω) can be captured quantitatively by the well established strong-field approximation (SFA)27,28,29. The solid green line in Fig. 3a represents the phase difference ΔϕSF(Ω) between neon and helium according to the SFA including a Coulomb correction (see Methods). Clearly, the measured phase differences show a large gap compared to the SFA phase calculation. Importantly, this gap does not affect the Wigner delay, and it is visible only owing to the ability to measure the absolute phase differences. Next, we extract the quantity of interest—the photo-recombination phase. For that, we employ the factorization of the HHG mechanism30,31,32 and subtract the SFA phase from the measured phase differences, isolating the photo-recombination phase (see black dots in Fig. 3b). To experimentally justify the factorization of the total phase difference into the strong-field and dipole parts, we performed a systematic study by measuring the phase differences of two atomic species for different infrared intensities. Once we have calibrated the SFA phase associated with each infrared intensity, the different measurements overlap, confirming the validity of the SFA calculation and enabling us to isolate the recombination dipole phase difference (see Supplementary Fig. 2).

As discussed in the introduction, our approach provides a twofold advantage over photoionization-based methods, stemming from the strong-field nature of the HHG mechanism. First, when an electron is ionized via tunnelling from a group of orthogonal orbitals (such as 2px, 2py and 2pz in neon) of closed-shell atoms, the orbital parallel to the electric field is much more efficiently ionized17,33, as illustrated in Fig. 1b. Such initial orbital selectivity is generally not available in single-photon photoionization experiments, where all the orthogonal orbitals will contribute to the dipole transition, as illustrated in Fig. 1a; in those experiments, only angle-resolved detection in specific directions provides orbital selectivity18. Fortunately, the initial state selectivity in the tunnelling ionization step translates into a final state selectivity of the photo-recombination process because in HHG, which is a parametric process, the initial and final states are identical. Second, the strong-field-driven recollision process sets a strong spatial filter, allowing interaction through an extremely narrow recollision angle19. Therefore, we avoid averaging over the angular distribution of the recombination dipole. Based on the state and angle selectivity, we calculate the photo-recombination phase of both atomic species using a Hartree–Fock (HF)/Xα approach (see Methods). Figure 3b shows striking agreement between the experimentally extracted dipole phase differences and the calculated ones (indicated by the black line) over a large energy range between harmonics 17 to 35.

A deeper insight into the information encoded in our measurement can be obtained by using partial-wave expansion of the continuum wavefunctions in order to describe the dipole phases of helium and neon in terms of fundamental partial-wave phase shifts σk,l, where k and l are the photoelectron momenta and angular momentum quantum numbers, respectively (see Methods). Experimental access to these phase shifts on an absolute scale is of paramount importance because they characterize the underlying ionic potential and further dictate the magnitude of scattering amplitudes34. Simple dipolar selection rules lead in helium to a dipole phase $$\phi _{{\mathrm{He}}}^{{\mathrm{rec}}}(k)$$ directly linked to σk,1 according to $$\phi _{{\mathrm{He}}}^{{\mathrm{rec}}}({\mathit{\Omega} }) = \sigma _{k,1} - \frac{\uppi }{2}$$. On the other hand, the well known (Fano) propensity rule35, which favours l → l + 1 transitions, yields $$\phi _{{\mathrm{Ne}}}^{{\mathrm{rec}}}(\Omega ) \simeq \sigma _{k,2} - \uppi$$ in neon. We can therefore interpret our dipole phase difference as a direct measurement of the partial-wave phase shifts, that is $$\mathrm{\Delta }\phi _{{\mathrm{Ne}} - {\mathrm{He}}}^{{\mathrm{rec}}}({{\mathit{\Omega} }}) \simeq \sigma _{k^\prime ,2} - \sigma _{k,1} - \frac{\uppi }{2}$$, where k′ and k are related by the ionization potential difference of neon and helium. The phase shift difference is illustrated in Fig. 3b by the red dashed line. We note that this is an absolute measurement of the phase, so no vertical offset was artificially added between experiments and theory. The close proximity between the phase shift difference, $$\sigma _{k^\prime ,2} - \sigma _{k,1} - \frac{\uppi }{2}$$, and the dipole phase difference, $${\mathrm{\Delta }}\phi _{{\mathrm{Ne}} - {\mathrm{He}}}^{{\mathrm{rec}}}({\mathit{\Omega }})$$, confirms our interpretation—indeed, our measurement provides an extremely accurate and sensitive probe of the partial-wave phase shifts. In addition, the total error of measured phases (about ±0.4 rad for harmonic 25) then sets an upper boundary for the difference in ionization dipole phase, associated with the first step of the HHG mechanism, between the two atomic species.

### Phase-resolved Cooper minimum in argon

We have shown that our interferometer can reveal phase shifts between the wavefunctions ionized from the two lightest noble gases. In the following step we investigate the possibility of probing a more complex quantum mechanical system: the argon atom, whose photoionization produces a characteristic structural feature in the energy range of our measurement—a Cooper minimum36. This minimum, located around 50 eV, is induced by the shape of the atomic potential. In our study, we demonstrate the ability to directly resolve the absolute phase associated with it via XUV–XUV interferometry.

We performed a differential measurement by alternating the target source between neon and argon. For the reference APT, we used argon or neon as a source for measurements between harmonics 15–29, and harmonics 23–39, respectively. We exploit the relatively flat dipole phase of neon to resolve the dipole phase associated with the more complex structure of the argon atom. Figure 4a shows the measured phase difference between neon and argon after the subtraction of the SFA phase contribution, as in the neon and helium measurement. The black and red symbols correspond to measurements with argon or neon as the source for the reference APT, respectively. We found complete consistency in the common energy range (harmonics 23–29; see Supplementary Fig. 3), indicating the reliability of the absolute phase difference measurement. The solid line corresponds to the calculated dipole phase differences, as we described above, which shows a very good agreement with the experimental results for harmonics 21–39. The phase deviation at harmonics 15–19 will be discussed further below.

We now focus on the role of the atomic structure in argon that is manifested in the Cooper minimum in photoionization spectra. This extensively studied phenomenon is created by a π-shift of the dipole phase in one of the angular momentum ionization channels. As the time-reversed process of photoionization, photo-recombination is subject to the same effect, leading to a local minimum in the HHG spectrum of argon37,38. Here, we directly measure the spectral phase over this energy range (harmonics 27–33 in Fig. 4a). In particular, we use the differential scheme to isolate the phase difference with respect to the structureless dipole phase of neon. The dipole phase calculation (solid line in Fig. 4a) takes into account only one scattering angle (parallel to the strong field polarization) and accounts for both angular momentum recombination channels, s → p and d → p. Comparing the phase calculated within the HF-Xα framework with the experimental results shows a good agreement. Surprisingly, this agreement worsens as the Random Phase Approximation with Exchange approach39 is employed to compute the phases. This approach is a post-HF method and accounts for multi-electron correlations and interchannel coupling. Its behaviour is presented and discussed in the Supplementary Information. In Fig. 4a, we compare our measurement with an angle-integrated HF-Xα calculation (see Supplementary Information). Clearly, such an integration does not exhibit the spectral fingerprint of the Cooper minimum. Figure 4b shows the group delay associated with the measured phases, along with the group delay for the single scattering angle (solid line) and the angle-averaged (dashed line) calculations. Again, the angle and state selectivity are found to be crucial to probe the underlying structure of argon. Recent experiments40 showed that HHG together with photoelectron interferometry measurements can be used to measure the group delay (spectral derivative) across the energy range of the Cooper minimum without angle averaging. The results in Fig. 4b agree well with their measurements of the group delay. In this study, the combination of the high angular selectivity provided by the HHG process, together with the linear nature of XUV–XUV interferometry, allows us to directly measure the absolute dipole phase of the Cooper minimum in argon.

In the energy range of harmonics 15–19 the measured phase differences show clear deviation with respect to the dipole calculation. We trace back the root of this deviation to the autoionization resonances of argon (3s3p6np), located approximately within a photon energy range of 26.6 eV (harmonic 17 in our experiment) to 29.3 eV (ref. 41). Our dipole phase calculation, based on a single active electron approximation, cannot capture any resonance-induced phase shifts associated with electron–electron correlation. In photoionization measurements, these resonances induce narrow dips in the photoelectron spectrum. The spectral phase of the argon resonance as well as angle-resolved photoionization time delays were measured in a weak-field environment by RABBITT42,43. Recent works have revealed the influence of a strong laser field on autoionization resonances and identified both broadening as well as substantial spectral modifications of the resonance line shape44,45. The broad spectral response we observe, ranging over harmonics 15–19, may originate from the high intensity we apply, two orders of magnitude higher than in a previous work44. Accordingly, the deviation of the measured phase from the single-electron phase calculation (Fig. 4a) can be interpreted as the phase shift induced by the strongly perturbed (dressed) resonances. We note that in the vicinity of resonances the factorization of HHG may require some modifications or might become invalid. We therefore checked the validity of this assumption by performing an intensity scan. Our experimental results did not show any pronounced intensity dependence, suggesting that in this case the validity of the SFA holds (see Supplementary Fig. 2b). We note that a quantitative analysis of this spectral response requires an advanced calculation that would account for the multi-electron nature of the resonant process occurring in the strong-field environment. Finally, we note that traces of 2s2p6np autoionizing resonances of neon46 should similarly arise in our phase measurements (Figs. 3b and 4a) at harmonics 29–31. However, those resonances have a significantly lower cross-section than do their argon counterparts47, so they are not visible in our phase measurements, as indicated by the agreement of the neon–helium results with the single electron calculations.

## Discussion

We have thus presented a phase measurement scheme using time-resolved interferometry of two fully independent broadband XUV sources, allowing for a differential phase measurement over distinct atomic species with temporal resolution of about 6 as. We were able to measure the absolute phase difference of the harmonic emission of neon and helium atoms over a large energy range and isolate their dipole phase difference. In particular, our results provide a direct measurement of the species-resolved partial-wave phase shift of the electronic wavefunction in the continuum. Moreover, we used our scheme to probe the argon atom, and demonstrated a direct measurement of its complex dipole phase. The strong-field-induced spatial confinement allowed us to follow the rapid variation of the dipole phase over the energy range of the Cooper minimum, visualizing the robustness of the angular momentum propensity rule in the vicinity of an atomic structure.

Looking forward, the interferometric control over two fully independent XUV sources opens the door to new directions in both attosecond metrology and control. In molecular systems, a strong laser field can initiate multi-electron dynamics, leading to population transfer, electron rearrangement during ionization or charge migration, all of which leave a clear spectral fingerprint. A direct measurement of the spectral phase associated with such phenomena will serve as a sensitive probe of the different internal channels, and will allow their temporal evolution to be followed. An extension of our method to Fourier transform spectroscopy is straightforward, removing the need for a spectrograph. Our interferometer can also be extended straightforwardly from a measurement to a control scheme. Manipulating the interference between two independent phase-locked sources opens up a new route for shaping the spectral amplitude or polarization state of an attosecond beam.

## Methods

### Experimental setup and measurement scheme

Figure 2 shows a schematic sketch of the experimental setup for collinear XUV–XUV interferometry of two independent sources. An amplified Ti:sapphire laser system operated at a 1-kHz repetition rate delivers approximately 23-fs pulses at a central wavelength of 792 nm. Focusing the beam into a continuous-flow gas cell (filled with nitrogen or argon or neon) generates the reference APT. We spatially separate the co-propagating infrared and APT beams with a thin aluminium filter (200 nm thickness). Both beams are then refocused by a curved two-segment mirror (750 mm focal length) into the target gas (continuous-flow glass nozzle, orifice of approximately 10 μm) to produce the target APT, which interferes with the reference APT. The position of the target source with respect to the infrared focus is adjusted to produce short trajectories of HHG. The inner part of the focusing mirror reflects the APT in the spectral range 17–51 eV. A piezo stage controls the temporal delay Δt between the reference APT and target APT with a step size of 6.7 as and an accuracy of about 1 as. The infrared intensity at the target gas can be adjusted independently by means of a motorized iris. The co-propagating APT beams are spectrally resolved by a flat-field aberration-corrected concave grating and recorded by a micro-channel plate detector, imaged by a charge-coupled device (CCD) camera.

In each scan, we varied Δt over a range of 6.7 fs (about 2.5 infrared cycles) and recorded the spectrum at each step. We applied the differential scheme by repeating the delay scan multiple times, alternating the target source every two scans. By carefully monitoring the individual gas pressure of each target source while alternating the atomic species we ensured the repeatability of the interaction conditions in the target source. We overcome the macroscopic averaging by performing a spatial analysis. For each scan, we extracted the Fourier phase ϕ(Ω) of selected pixels in each harmonic. In our analysis we selected only the pixels that exhibit a significant signal-to-noise ratio and are located in regions where the short-trajectory HHG signal dominates. We then calculate the experimental phase differences Δϕabs(Ω) = ϕNe(Ω) − ϕHe(Ω) of neon and helium or Δϕabs(Ω) = ϕNe(Ω) − ϕAr(Ω) of neon and argon, for each differential dataset and for each pixel (see Supplementary Fig. 5 for more details of the spatial analysis). Finally, we averaged over all pixels in each harmonic and determined the error of the mean of the phase differences from averaging over all differential datasets. To estimate the slow thermal drift of the piezo (up to 50 as per hour), we determined the temporal drift between pairs of identical measurements of every target source. We corrected for the drift, resulting in a systematic error in the experimental group delay (or equivalently, a linear phase shift). Typically, this error amounts to about 5 as, translating into a frequency-dependent error in the phase differences of N 0.012 rad (where N is the harmonic number). We calculated the group delay from the dipole recombination phase differences as $${\mathrm{\Delta }}\tau _{N + 1} = \left( {{\mathrm{\Delta }}\phi _{N + 2}^{{\mathrm{rec}}} - {\mathrm{\Delta }}\phi _N^{{\mathrm{rec}}}} \right)/2\omega$$, where N and N + 2 are two neighbouring odd harmonics and ω is the infrared frequency. The data points are assigned to the even harmonic N + 1, in analogy to RABBITT. The associated error represents the error of the mean corresponding to the 90% confidence interval. For the black points in Fig. 4b, for instance, the error is given by the standard deviation of the group delays evaluated from the six intensity measurements, divided by the square root of the number of measurements and multiplied by the corresponding factor derived from a t-distribution with 90% confidence. In both the neon-minus-helium and the neon-minus-argon differential measurements, we measured the absorption of the reference APT in each target gas in order to evaluate the neutral gas dispersion difference. Using tabulated values of gas dispersion48, we set an upper limit of 0.05 rad and 0.1 rad, for neon-minus-helium and neon-minus-argon, respectively, which is below our experimental error. In addition we measured the influence of the pressure in the target gas on the phase of argon HHG (see Supplementary Fig. 1), which yields an upper limit of 0.1 rad. Therefore, we conclude that in our measurements electron plasma dispersion and neutral gas dispersion induced phase distortion can be neglected.

### Strong-field theory model

The strong-field theory model is based on the stationary phase approximation applied to the SFA27,28 and includes a spin-orbit correction49 and optionally a Coulomb correction50. Without these corrections, the strong-field phase ϕSF of a harmonic with frequency Ω can be expressed in atomic units as

$$\phi ^{{\mathrm{SF}}}({\mathit{\Omega }}) = {\mathrm{Re}}\left\{ {{\mathrm{\varOmega }}t_1({\mathrm{\varOmega }}) - {\int}_{t_0({\mathrm{\varOmega }})}^{t_1({\mathrm{\varOmega }})} \left( {\frac{{[p({\mathit{\Omega }}) - A(t)]^2}}{2} + I_{\mathrm{p}}} \right){\mathrm{d}}t} \right\}$$
(1)

where t0(Ω) and t1(Ω) are the (complex) ionization and recollision times, respectively, and p(Ω) is the canonical momentum of the outgoing electron. A(t) is the time-dependent vector potential of the infrared laser field (here assumed to be a continuous wave) and Ip is the ionization energy of the atom. The integral contains both the classical action of the electron trajectory and the phase evolution of the ground state. The trajectory is defined by the parameters t0, t1 and p, which are calculated numerically for the short trajectory branch. For the details on the Coulomb correction and the additional spin-orbit correction necessary for argon and neon, see the Supplementary Information.

### Hartree-Fock/Xα calculation of the dipole phase

Atomic units are used throughout this section unless otherwise stated. We assume that the whole HHG process is dictated by the dynamics of only one outer valence electron, with all other electrons remaining frozen throughout the interaction. The photo-recombination dipole in the strong-field driven z-direction is then

$$d_{\mathrm{X}}^{{\mathrm{rec}}}({\mathit{\Omega }}) = \langle \phi _0|z|{\mathrm{\varPsi }}_{\mathbf{k}}^ + \rangle$$
(2)

where ϕ0(r) is the outer valence bound orbital of atom X and $$\varPsi _k^ + ({\mathbf{r}})$$ is the outgoing scattering state associated with the electron returning to the core with energy E = k2/2 to yield the harmonic photon of energy Ω = E + Ip through recombination, Ip being the ionization potential of atom X. ϕ0 is obtained by means of Hartree–Fock (HF) calculations, using the quantum chemistry package GAMESS-US51 with a large-scale triple-ζ aug-cc-pVTZ underlying Gaussian basis52. $$\Psi _k^ +$$, normalized on the k-scale, is expanded onto spherical partial waves ψklm(r) with definite (l, m)-symmetry as

$${\mathrm{\varPsi }}_{{k}}^ + ({\mathbf{r}}) = \frac{1}{k}\mathop {\sum}\limits_{l = 0}^\infty \mathop {\sum}\limits_{m = - l}^l {\mathrm{i}}^l{\mathrm{e}}^{{\mathrm{i}}\sigma _{kl}}\psi _{klm}({\mathbf{r}}){\mathrm{Y}}_l^{m \ast }\left( {{\hat{\mathbf k}}} \right)$$
(3)

where $${\mathrm{Y}}_l^m$$ are usual spherical harmonics and σkl is the phase shift for electron wavevector k and angular momentum l. The radial part Rkl(r) of the continuum states $$\psi _{klm}({\mathbf{r}}) = R_{kl}(r){\mathrm{Y}}_l^m\left( {\hat {\mathbf{r}}} \right)$$ is obtained by solving numerically the Schrödinger equation $$\left[ {\frac{1}{{r^2}}\frac{\partial }{{\partial r}}\left( {r^2\frac{\partial }{{\partial r}}} \right) - \frac{{l(l + 1)}}{{r^2}} + 2(E - V(r))} \right]R_{kl}(r) = 0$$, where the potential V is split as $$V(r) = V_{{\mathrm{e - n}}}(r) + V_{{\mathrm{e - e}}}^{{\mathrm{(d)}}}(r) + V_{{\mathrm{e - e}}}^{{\mathrm{(ex)}}}(r)$$. Ve−n is the electron–nucleus potential and $$V_{{\mathrm{e - e}}}^{{\mathrm{(d)}}}$$ and $$V_{{\mathrm{e - e}}}^{{\mathrm{(ex)}}}$$ are the direct (Hartree) and exchange parts of the electron–electron interaction, respectively. For closed-subshell atomic systems, the total electron density n of the multi-electron atom in its ground state is radial so that $$V_{{\mathrm{e - e}}}^{{\mathrm{(d)}}} = {\int} {\mathrm{d}}{\mathbf{r}}\prime \frac{{n(r)}}{{|{\mathbf{r}} - {\mathbf{r}}\prime |}}$$. We employ for $$V_{{\mathrm{e - e}}}^{{\mathrm{(ex)}}}$$ the so-called Xα statistical form $$V_{{\mathrm{e - e}}}^{{\mathrm{(ex)}}}(r) \equiv V_{{\mathrm{e - e}}}^{{\mathrm{(ex)}}}({\mathbf{r}}) = - \frac{3}{2}\alpha \left( {\frac{{3n(r)}}{\uppi }} \right)^{1/3}$$(ref. 53). Optimized values for the parameter α are taken from spectroscopic tables54. The total potential V(r) defined in this way does not present the expected −1/r asymptotic behaviour, so we switch from the computed V(r) to the so-called Latter tail −1/r from r0, such that V(r0) = −1/r0, onwards55. The σkl phase shift encodes both the Coulombian asymptotic behaviour and the short-range distortion according to $$\sigma _{kl} = \sigma _{kl}^{\mathrm{C}} + \sigma _{kl}^{\mathrm{S}}$$. $$\sigma _{kl}^{\mathrm{C}} = {\mathrm{arg}}\,{\mathrm{\varGamma }}(l + 1 - {\mathrm{i}}/k)$$, where Γ is the Gamma function56, and $$\sigma _{kl}^{\mathrm{S}}$$ is determined by matching the computed radial wavefunction Rkl(r) to the expected asymptotic behaviour $$\sqrt {2/\uppi } {\mathrm{sin}}\left( {kr - l\pi /2 + \sigma _{kl}^{\mathrm{C}} + \sigma _{kl}^{\mathrm{S}}} \right)/r$$ at r → ∞.

Once the bound and continuum wavefunctions are computed, the insertion of the partial-wave expansion (3) into the expression (2) of the dipole leads to

$$d_{{\mathrm{He}}}^{{\mathrm{rec}}}({\mathit{\Omega }}) = \frac{{ - {\mathrm{i}}}}{{\sqrt {12\pi } k}}{\mathrm{e}}^{{\mathrm{i}}\sigma _{k1}}{\cal R}_{k1}$$
(4)

for X ≡ He, taking into account that $$\phi _0({\mathbf{r}}) = R_0(r){\mathrm{Y}}_0^0\left( {\hat {\mathbf{r}}} \right)$$ and $$\hat {\mathbf{k}}||z$$. $${\cal R}_{k1}$$ is the radial integral $${\cal R}_{k1} = {\int}_0^\infty R_0(r)R_{k1}(r)r^3{\mathrm{d}}r$$. The phase of the dipole $$\phi _{{\mathrm{He}}}^{{\mathrm{rec}}}({\mathit{\Omega }})$$ then reduces to the fundamental partial-wave phase shift $$\sigma _{k1}$$ according to

$$\phi _{{\mathrm{He}}}^{{\mathrm{rec}}}({\mathit{\Omega }}) = \sigma _{k1} - \uppi /2$$
(5)

For neon and argon, for which $$\phi _0({\mathbf{r}}) = R_0(r){\mathrm{Y}}_0^1\left( {\hat {\mathbf{r}}} \right)$$, one obtains in a similar way

$$d_{{\mathrm{Ne,Ar}}}^{{\mathrm{rec}}}({\mathit{\Omega }}) = \frac{1}{{\sqrt {12\pi } k}}\left( {{\mathrm{e}}^{{\mathrm{i}}\sigma _{k0}}{\cal R}_{k0} - 2{\mathrm{e}}^{{\mathrm{i}}\sigma _{k2}}{\cal R}_{k2}} \right)$$
(6)

The phase of these dipoles is

$$\phi _{{\mathrm{Ne,Ar}}}^{{\mathrm{rec}}}({\mathit{\Omega }}) = {\mathrm{tan}}^{ - 1}\left[ {\frac{{{\cal R}_{k0}{\mathrm{sin}}\sigma _{k0} - 2{\cal R}_{k2}{\mathrm{sin}}\sigma _{k2}}}{{{\cal R}_{k0}{\mathrm{cos}}\sigma _{k0} - 2{\cal R}_{k2}{\mathrm{cos}}\sigma _{k2}}}} \right]$$
(7)

and this is the one which is used to compute the phase differences $${\mathrm{\Delta }}\phi _{{\mathrm{Ne}} - {\mathrm{He}}}^{{\mathrm{rec}}}({\mathit{\Omega }})$$ and $${\mathrm{\Delta }}\phi _{{\mathrm{Ne}} - {\mathrm{Ar}}}^{{\mathrm{rec}}}({\mathit{\Omega }})$$ displayed in terms of continuous lines in Figs. 3b and 4a, respectively.

However, dipolar interaction generally favours l → l + 1 transitions, which is known as the (Fano) propensity rule35. This translates into our computations as $${\cal R}_{k2} \gg {\cal R}_{k0}$$, but over distinct energy ranges for neon and argon. The propensity shows up over the whole energy range considered in the experiment for neon, so that $$d_{{\mathrm{Ne}}}^{{\mathrm{rec}}}({\mathrm{\varOmega }})$$ can be approximated by

$$\phi _{{\mathrm{Ne}}}^{{\mathrm{rec}}}({\mathit{\Omega }}) \simeq \sigma _{k2} - {\mathrm{\uppi}}$$
(8)

taking into account that $${\cal R}_{k2} > 0$$. Under such approximation, the Ne–He dipole phase difference reduces to a difference of partial-wave phase shifts, represented by the dashed line in Fig. 3b. The propensity does not hold in argon for harmonic orders greater than about 21 because of the occurrence of the Cooper minimum around E = 50 eV. This impedes the isolation of σk2 of argon in the measurement of the dipole phase $$\phi _{{\mathrm{Ar}}}^{{\mathrm{rec}}}({\mathit{\Omega }})$$ out from the energy range where (3s3p6np) autoionizing resonances come into play.

## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank S. Patchkovskii, C. Ott and A. Harth for discussions. N.D. is the incumbent of the Robin Chemers Neustein Professorial Chair. N.D. acknowledges the Minerva Foundation, the Israeli Science Foundation, the Crown Center of Photonics and the European Research Council for financial support. M.K. acknowledges financial support by the Minerva Foundation and the Koshland Foundation. B.P., A.C., B.F. and Y.M. acknowledge financial support from the French National Research Agency through grant ANR-14-CE32-0014 MISFITS.

## Author information

Authors

### Contributions

N.D. and M.K. supervised the study. D.A. and M.K. designed and built the experimental setup. D.A., M.K. and O.K. carried out the measurements and analysed the data. B.P., A.C. and B.F. conceived and performed the theoretical calculations. D.A., M.K., N.D., B.P., B.F. and Y.M. interpreted the experimental and theoretical results. B.D.B. supported the operation of the laser system. All authors discussed the results and contributed to the final manuscript.

### Corresponding author

Correspondence to Nirit Dudovich.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary notes and figures.

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Azoury, D., Kneller, O., Rozen, S. et al. Electronic wavefunctions probed by all-optical attosecond interferometry. Nature Photon 13, 54–59 (2019). https://doi.org/10.1038/s41566-018-0303-4

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