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 longstanding challenge. Here, we transfer the electron phase retrieval problem into an optical one by measuring the timereversed process of photoionization—photorecombination—in attosecond pulse generation. We demonstrate alloptical interferometry of two independent phaselocked 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 strongfield 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 socalled ‘complete’ photoionization experiments require recording of photoelectron angular distributions for linear and circular ionizing radiation, and for fixedinspace molecules using coincidence photoelectron photoion spectroscopy^{1,2,3}. A recent twodimensional momentumresolved photoionization study enabled the isolation of a continuum wavefunction through an atomic resonance and, in addition, resolves its phase by interference with a reference wavefunction^{4}. In the second approach, photoelectron interferometry methods such as the reconstruction of attosecond beating by interference of twophoton transitions (RABBITT) and streaking techniques, based on nonlinear light–matter interaction, can be used to measure the variation of the spectral phase with energy^{5,6,7}. These schemes were applied to reveal phase information in photoionization of atoms^{8,9,10,11}, molecules^{12} or solids^{13}. 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 delay^{14,15}) while the absolute phase remains inaccessible.
We can convert the photoelectron phase retrieval problem into an optical one by means of photorecombination. Photorecombination, the timereversed process of photoionization, is an inherent part of the mechanism of highharmonic 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 strongfieldinduced photorecombination can be considered to be the timereversed 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 polarization^{17}. We note that angleresolved photoionization experiments can provide orbital selectivity as well^{18}. Second, in photoionization the emitted photoelectron momentum spreads over a large angular range, whereas photorecombination projects the returning electronic wavepacket in a well defined direction, thus providing a high degree of angular selectivity^{19}.
We demonstrate a differential phase measurement of the photorecombination process by alloptical linear XUV interferometry. Our measurement exploits both the state selectivity by tunnelling ionization and the angular resolution provided by the photorecombination 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 phaselocked HHG sources have been introduced within a single interacting medium, either by splitting the fundamental laser field into two focal spots in the medium^{20,21,22} or by driving HHG in a gas mixture^{23,24}. 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 timeresolved 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 copropagate and are refocused into the target gas by a curved twosegment mirror. The refocused infrared beam generates a second APT, phaselocked with the reference APT, which can be controlled in delay Δt and direction using the twosegment 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 delaydependent 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 information^{21,25}. 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 measurements^{26}.
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) = ϕ_{tar}(Ω) − ϕ_{ref}(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 Δϕ_{abs}(Ω) = ϕ_{Ne}(Ω) − ϕ_{He}(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 timeresolved 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 photorecombination step. Hence, as illustrated in Fig. 2c, we can express each harmonic phase as the sum of the strongfieldinduced phase ϕ^{SF} and the photorecombination dipole phase ϕ^{rec}. The differential phase measurement thus provides Δϕ_{abs}(Ω) = Δϕ^{SF}(Ω) + Δϕ^{rec}(Ω). The strongfield contribution Δϕ^{SF}(Ω) can be captured quantitatively by the well established strongfield approximation (SFA)^{27,28,29}. The solid green line in Fig. 3a represents the phase difference Δϕ^{SF}(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 photorecombination phase. For that, we employ the factorization of the HHG mechanism^{30,31,32} and subtract the SFA phase from the measured phase differences, isolating the photorecombination phase (see black dots in Fig. 3b). To experimentally justify the factorization of the total phase difference into the strongfield 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 photoionizationbased methods, stemming from the strongfield nature of the HHG mechanism. First, when an electron is ionized via tunnelling from a group of orthogonal orbitals (such as 2p_{x}, 2p_{y} and 2p_{z} in neon) of closedshell atoms, the orbital parallel to the electric field is much more efficiently ionized^{17,33}, as illustrated in Fig. 1b. Such initial orbital selectivity is generally not available in singlephoton photoionization experiments, where all the orthogonal orbitals will contribute to the dipole transition, as illustrated in Fig. 1a; in those experiments, only angleresolved detection in specific directions provides orbital selectivity^{18}. Fortunately, the initial state selectivity in the tunnelling ionization step translates into a final state selectivity of the photorecombination process because in HHG, which is a parametric process, the initial and final states are identical. Second, the strongfielddriven recollision process sets a strong spatial filter, allowing interaction through an extremely narrow recollision angle^{19}. Therefore, we avoid averaging over the angular distribution of the recombination dipole. Based on the state and angle selectivity, we calculate the photorecombination 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 partialwave expansion of the continuum wavefunctions in order to describe the dipole phases of helium and neon in terms of fundamental partialwave phase shifts σ_{k,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 amplitudes^{34}. Simple dipolar selection rules lead in helium to a dipole phase \(\phi _{{\mathrm{He}}}^{{\mathrm{rec}}}(k)\) directly linked to σ_{k,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 rule^{35}, 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 partialwave 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 partialwave 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.
Phaseresolved 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 minimum^{36}. 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 πshift of the dipole phase in one of the angular momentum ionization channels. As the timereversed process of photoionization, photorecombination is subject to the same effect, leading to a local minimum in the HHG spectrum of argon^{37,38}. 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 HFXα framework with the experimental results shows a good agreement. Surprisingly, this agreement worsens as the Random Phase Approximation with Exchange approach^{39} is employed to compute the phases. This approach is a postHF method and accounts for multielectron correlations and interchannel coupling. Its behaviour is presented and discussed in the Supplementary Information. In Fig. 4a, we compare our measurement with an angleintegrated HFXα 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 angleaveraged (dashed line) calculations. Again, the angle and state selectivity are found to be crucial to probe the underlying structure of argon. Recent experiments^{40} 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 (3s3p^{6}np), 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 resonanceinduced 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 angleresolved photoionization time delays were measured in a weakfield environment by RABBITT^{42,43}. 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 shape^{44,45}. 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 work^{44}. Accordingly, the deviation of the measured phase from the singleelectron 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 multielectron nature of the resonant process occurring in the strongfield environment. Finally, we note that traces of 2s2p^{6}np autoionizing resonances of neon^{46} should similarly arise in our phase measurements (Figs. 3b and 4a) at harmonics 29–31. However, those resonances have a significantly lower crosssection than do their argon counterparts^{47}, 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 timeresolved 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 speciesresolved partialwave 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 strongfieldinduced 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 multielectron 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 phaselocked 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 1kHz repetition rate delivers approximately 23fs pulses at a central wavelength of 792 nm. Focusing the beam into a continuousflow gas cell (filled with nitrogen or argon or neon) generates the reference APT. We spatially separate the copropagating infrared and APT beams with a thin aluminium filter (200 nm thickness). Both beams are then refocused by a curved twosegment mirror (750 mm focal length) into the target gas (continuousflow 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 copropagating APT beams are spectrally resolved by a flatfield aberrationcorrected concave grating and recorded by a microchannel plate detector, imaged by a chargecoupled 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 signaltonoise ratio and are located in regions where the shorttrajectory HHG signal dominates. We then calculate the experimental phase differences Δϕ_{abs}(Ω) = ϕ_{Ne}(Ω) − ϕ_{He}(Ω) of neon and helium or Δϕ_{abs}(Ω) = ϕ_{Ne}(Ω) − ϕ_{Ar}(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 frequencydependent 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 tdistribution with 90% confidence. In both the neonminushelium and the neonminusargon 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 dispersion^{48}, we set an upper limit of 0.05 rad and 0.1 rad, for neonminushelium and neonminusargon, 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.
Strongfield theory model
The strongfield theory model is based on the stationary phase approximation applied to the SFA^{27,28} and includes a spinorbit correction^{49} and optionally a Coulomb correction^{50}. Without these corrections, the strongfield phase ϕ^{SF} of a harmonic with frequency Ω can be expressed in atomic units as
where t_{0}(Ω) and t_{1}(Ω) are the (complex) ionization and recollision times, respectively, and p(Ω) is the canonical momentum of the outgoing electron. A(t) is the timedependent vector potential of the infrared laser field (here assumed to be a continuous wave) and I_{p} 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 t_{0}, t_{1} and p, which are calculated numerically for the short trajectory branch. For the details on the Coulomb correction and the additional spinorbit correction necessary for argon and neon, see the Supplementary Information.
HartreeFock/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 photorecombination dipole in the strongfield driven zdirection is then
where ϕ_{0}(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 = k^{2}/2 to yield the harmonic photon of energy Ω = E + I_{p} through recombination, I_{p} being the ionization potential of atom X. ϕ_{0} is obtained by means of Hartree–Fock (HF) calculations, using the quantum chemistry package GAMESSUS^{51} with a largescale tripleζ augccpVTZ underlying Gaussian basis^{52}. \(\Psi _k^ +\), normalized on the kscale, is expanded onto spherical partial waves ψ_{klm}(r) with definite (l, m)symmetry as
where \({\mathrm{Y}}_l^m\) are usual spherical harmonics and σ_{kl} is the phase shift for electron wavevector k and angular momentum l. The radial part R_{kl}(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 \(V(r) = V_{{\mathrm{e  n}}}(r) + V_{{\mathrm{e  e}}}^{{\mathrm{(d)}}}(r) + V_{{\mathrm{e  e}}}^{{\mathrm{(ex)}}}(r)\). V_{e−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 closedsubshell atomic systems, the total electron density n of the multielectron 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 socalled 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 tables^{54}. 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 socalled Latter tail −1/r from r_{0}, such that V(r_{0}) = −1/r_{0}, onwards^{55}. The σ_{kl} phase shift encodes both the Coulombian asymptotic behaviour and the shortrange 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 function^{56}, and \(\sigma _{kl}^{\mathrm{S}}\) is determined by matching the computed radial wavefunction R_{kl}(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 partialwave expansion (3) into the expression (2) of the dipole leads to
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 partialwave phase shift \(\sigma _{k1}\) according to
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
The phase of these dipoles is
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 rule^{35}. 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
taking into account that \({\cal R}_{k2} > 0\). Under such approximation, the Ne–He dipole phase difference reduces to a difference of partialwave 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 σ_{k2} of argon in the measurement of the dipole phase \(\phi _{{\mathrm{Ar}}}^{{\mathrm{rec}}}({\mathit{\Omega }})\) out from the energy range where (3s3p^{6}np) 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 ANR14CE320014 MISFITS.
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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.
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Azoury, D., Kneller, O., Rozen, S. et al. Electronic wavefunctions probed by alloptical attosecond interferometry. Nature Photon 13, 54–59 (2019). https://doi.org/10.1038/s4156601803034
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