Attosecond electron-spin dynamics in Xe 4d photoionization
Shiyang Zhong, Jimmy Vinbladh, David Busto, Richard J. Squibb, Marcus Isinger, Lana Neoričić, Hugo Laurell, Robin Weissenbilder, Cord L. Arnold, Raimund Feifel, Jan Marcus Dahlström, Göran Wendin, Mathieu Gisselbrecht, Eva Lindroth, Anne L'Huillier
AAttosecond electron–spin dynamics in Xe 4d photoionization
Shiyang Zhong, ∗ Jimmy Vinbladh, David Busto, Richard J. Squibb, Marcus Isinger, Lana Neoriˇci´c, Hugo Laurell, Robin Weissenbilder, Cord L. Arnold, Raimund Feifel, JanMarcus Dahlstr¨om, G¨oran Wendin, Mathieu Gisselbrecht, Eva Lindroth, and Anne L’Huillier Department of Physics, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden Department of Physics, University of Gothenburg, Origovgen 6B, SE-412 96 Gothenburg, Sweden Department of Microtechnology and NanoscienceMC2,Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
The photoionization of xenon atoms in the 70-100 eV range reveals several fascinating physicalphenomena such as a giant resonance induced by the dynamic rearrangement of the electron cloudafter photon absorption, an anomalous branching ratio between intermediate Xe + states separatedby the spin-orbit interaction and multiple Auger decay processes. These phenomena have beenstudied in the past, using in particular synchrotron radiation, but without access to real-time dy-namics. Here, we study the dynamics of Xe 4 d photoionization on its natural time scale combiningattosecond interferometry and coincidence spectroscopy. A time-frequency analysis of the involvedtransitions allows us to identify two interfering ionization mechanisms: the broad giant dipole reso-nance with a fast decay time less than 50 as and a narrow resonance at threshold induced by spin-fliptransitions, with much longer decay times of several hundred as. Our results provide new insightinto the complex electron-spin dynamics of photo-induced phenomena. The absorption of X-rays by matter has been used sincemore than a century ago to understand the structure ofits fundamental constituents [1]. An X-ray penetratinginside an atom triggers multiple electron dynamics. Theemission of an electron from an inner shell is accompa-nied by ultrafast rearrangement of the electronic cloud,which simultaneously modifies the potential seen by theelectron, sometimes leading to resonances in the emis-sion spectrum. An outer-shell electron may fill an innerhole, while another electron is emitted, a process calledAuger decay [2]. Finally, the electron rotational motion,i.e. spin, may be affected by the magnetic field inducedby the ultrafast orbital motion, giving rise to spin flip,which is forbidden for purely electric dipole transitions.All of this complex hole or electron motion occurs on arapid time scale, in the attosecond (1 as = 10 − s) range.The interaction of xenon atoms with photons in the70-100 eV range illustrates many aspects of the electrondynamics sketched above. Collective many-electron ef-fects in the 4 d shell [3–6] lead to a broad “giant dipole”resonance in the photoionization cross-section, which ismaximum at 100 eV [7, 8]. Photoionization is accom-panied by Auger decay, involving the 5 s and 5 p shells,leading to the formation of Xe ions [9] (see Fig. 1). Rel-ativistic (spin-orbit) effects can be observed at threshold(in the 70-75 eV region), with, in particular, an anoma-lous branching ratio between the D / and D / finalstates of the ion, separated by a spin-orbit splitting of 2eV [10–12].Attosecond pulses produced by high-order harmonicgeneration in gases [13, 14] enable measuring ultrafastelectron dynamics, as shown in a series of seminal ex- ∗ [email protected] FIG. 1.
Excitation scheme. (a) Schematic illustrationof Xe 4 d photoionization (violet) and Auger decay processes(green) after absorption of XUV radiation. (b) Xe energy di-agram showing the Xe + intermediate and Xe final statesinvolved. periments [15–22]. Temporal information is obtainedby pump/probe techniques combining attosecond pulsesand a synchronized laser field. The reconstruction of at-tosecond beating by interference of two-photon transi-tion (RABBIT) technique [23], based on interferometry,allows the determination of the photoionization spectralamplitude in the complex plane. The temporal dynamicsis then obtained by Fourier transform or more generallyby time-frequency analysis [16, 24]. This technique hasbeen successfully used to measure photoionization timedelays, due to electron propagation in the potential fol-lowing the absorption of an extreme ultraviolet (XUV)photon. Most of these studies [25–28], however, haveconcentrated on relatively simple systems, ionized fromthe valence shells.In this work, we present measurements of photoioniza-tion time delays in the Xe 4 d shell for different ionic states a r X i v : . [ phy s i c s . a t o m - ph ] M a y FIG. 2.
Two-electron coincidence map.
The coincidence map using (a) XUV field only and (b) XUV+IR fields. Thenumber of measured two-electron pairs is indicated by the color code. The spots with constant slow-electron energy andvariable fast-electron energy correspond to photoelectrons created by absorption of consecutive harmonics (labelled as H57-H61in (a) as an example) and a given Auger electron [e.g 4 d − ( D / ) → s − ( S )]. Photoelectrons corresponding to absorptionor emission of an additional IR photon (sidebands, labelled as S58 and S60 as an example) appear in (b). The projection onthe fast electron energy axis (c) and (d) is the sum of the signal with slow electron energy from 10 eV to 10.4 eV, i.e. thephotoelectrons in coincidence with 4 d − ( D / ) → s − ( S ) Auger electrons. The projection on the slow electron energy axis(e) shows the sum of the signal for the different Auger processes indicated on the right, with (red) and without (blue) IR field.A RABBIT energy scheme is indicated at the top of (d). d − ( D / ) and 4 d − ( D / ), denoted 4 d / and 4 d / inthe following [see Fig. 1(b)]. Auger-photoelectron coinci-dence spectroscopy is used to disentangle electrons fromdifferent photoionization and decay channels [29]. TheRABBIT interferometric technique allows the extractionof a phase, or a time (or group) delay, from the photo-electron spectra. At high photon energy (between 80 and100 eV), both 4 d / and 4 d / photoelectrons are emit-ted with the same positive time delay. Close to the 4 d -ionization threshold (75-80 eV), the measured time de-lays differ by more than 100 as. Supported by relativisticrandom phase approximation (RRPA) theoretical calcu-lations [30, 31], we show that this difference is due to theinterference of the broad giant dipole resonance with anarrow threshold resonance due to relativistic spin-orbiteffects.The experiments were performed with attosecond pulsetrains generated in neon by a femtosecond Ti:sapphirelaser system, covering a spectral range from the 4 d ion-ization threshold to the maximum of the giant dipoleresonance (see Methods for details). A small fraction ofthe infrared (IR) laser beam was used as a probe witha variable time delay. The XUV and IR pulses were fo-cused into Xe gas and the created electrons were detectedby an electron spectrometer.Photoionization to different ionic states followed byAuger decay produces a complex electron spectrum, withtwo sets of photoelectrons separated by 2 eV [see Fig. 1].Single Auger decay from Xe + (e.g 4 d / ) to Xe (e.g. 5 s − p − ) leads to electrons at kinetic energies equal tothe difference between intermediate and final state en-ergies, spanning from 8.3 eV to 36.4 eV [32] and thusoverlapping with the photoelectrons ionized by 75-100 eVphotons [33]. Fig. 2 shows XUV-only (a) and XUV+IR(b) two-dimensional coincidence maps. For a given finalstate of Xe , Auger electrons detected in coincidencewith photoelectrons contribute to a stripe with discretespots related to absorption of different harmonics (withodd orders 53 to 63 in the figure), or absorption of har-monics and absorption or emission of an IR photon (side-bands 54 to 62). In addition, weak signals due to absorp-tion or emission of an IR photon by the Auger electron,are observed (see, e.g., the difference between the blueand red curves in Fig. 2(e) at 9.8 eV). This coincidencetechnique requires long acquisition times, but allows usto disentangle unambiguously the 4 d / and 4 d / pho-toelectrons by the energy of the Auger electron.Each sideband arises from the interference between twoquantum paths as illustrated at the top of Fig. 2(d). Thesideband signal oscillates as a function of the delay τ be-tween the attosecond pulse train and the probe IR field,according to, I SB = A + B cos(2 ωτ − φ ) , (1)where A and B are constants, ω is the IR frequency and φ is a phase offset, which can be extracted by fitting witha cosine function. The phase offset φ divided by the os-cillation frequency (2 ω ) can be written as the sum of two FIG. 3.
Atomic time delays and branching ratio.
Ab-solute atomic time delays for (a) 4 d / and (b) 4 d / ; (c)Difference between these delays. The experimental data aregiven in red dots. The black lines indicate the results of two-photon RRPA calculations. The estimation of the error barsis the standard error of the weighted mean, detailed in Meth-ods section. In (d), the calculated branching ratio of the 4 d / over 4 d / cross sections is shown in black. The experimentaldata in blue dots is from [38]. delays, τ XUV + τ A . The first one is the group delay of theattosecond pulses, while the second, called atomic timedelay, arises from the two-photon ionization process. Asshown in previous work [34, 35] and as discussed in moredetails in the Supplemental Material (SM, see Fig. S1),the variation of the atomic time delay τ A , as a functionof XUV photon energy or between two spin-orbit splitfinal states, reflects, to a large extent, one-photon ion-ization dynamics. To remove the influence of τ XUV inour time delay measurements, we alternate experimentsin Xe and Ne, and measure the time delay difference.Atomic time delays in Ne 2 p can be measured and cal-culated with good accuracy. They are very small in theenergy range considered [34], so that the time delay dif-ferences between Xe and Ne are, to a very good approx-imation, absolute time delays in Xe (see SM, Fig. S2).The sidebands corresponding to the same photoelectronbut different Auger final state are found to oscillate inphase within our error bar, which allows us to averagethe time delays over the different Auger decay channels.Fig. 3 presents the absolute atomic delays for 4 d / (a) and 4 d / (b) as a function of photon energy. Wealso present theoretical calculations obtained by solvingthe Dirac equation in RRPA (see Methods). At highphoton energy ( >
80 eV), both 4 d / and 4 d / exhibitvery similar positive time delay, about 40 as at 80 eV,slightly decreasing with energy. At low photon energy( <
80 eV), the delays show a rapid variation with energy,opposite for the two final states. Theory and experimentare in good agreement at high photon energy. Althoughthe agreement in the threshold region is not as good, themain features of the experiment are reproduced. Ourtheoretical calculations are in excellent agreement with
FIG. 4.
Transition matrix elements. (a,d) Modulus, (b,e)phase and (c,f) time delay of the transition matrix element D i ( E ) as a function of photon energy for the coupled channels(a-c) 4 d / → (cid:15)f / (blue), 4 d / → (cid:15)f / (red), 4 d / → (cid:15)f / (brown) and eigenchannels (d-f) P (black), P (magenta), D (orange), extracted from [12]. We exclude the Coulombphase-shift in (b,e) and (c,f). A π phase shift has been addedto the phase of 4 d / → (cid:15)f / for better comparison. Thetime delay is the energy derivative of the phase. the predictions of Mandal and coworkers [36, 37].Fig. 3(c) shows the difference between 4 d / and 4 d / time delays. This difference is close to zero at high energybut jumps to more than 100 as at 75 eV. Unfortunately,we could not reliably extract time delays below 75 eV,due to the low cross section and the overlap with doubleAuger electrons below 5 eV kinetic energy. The RRPAcalculation predicts a strong decrease of the time delaydifference towards the threshold. It also shows a strongdeviation of the branching ratio between 4 d / and 4 d / cross sections from the statistical prediction in the sameenergy range, reproducing well experimental results [38,39] [see Fig. 3(d)].To understand the underlying physics behind the vari-ation of the time delays, we examine the behavior of theRRPA transition matrix elements involved in Xe 4 d sin-gle photon ionization. In the energy range consideredin this work, photoionization is dominated by 4 d → (cid:15)f transitions (Fig. 4). The behavior of the weaker 4 d → (cid:15)p transitions is shown in the SM (Fig. S3). The asymp-totic phase for a given channel is the sum of the Coulombphase and a phase due to the short-range potential. TheCoulomb phase is removed in the phases displayed inFig. 4, as well as in the calculation of the time delays,in order to focus on the short range effects (see SM,Fig. S1). Fig. 4(a) shows that photoionization is dom-inated by 4 d / → (cid:15)f / and 4 d / → (cid:15)f / , especiallyat high photon energy, in the region of the giant dipoleresonance. In the threshold region, the 4 d / → (cid:15)f / channel contributes significantly. This transition is ac-companied by a spin flip, which points out the role ofthe spin-orbit interaction. The phases and time delaysfor the three channels [Fig. 4 (b,c)] coincide above 80 eVphoton energy, showing the first half of a π phase varia-tion across the giant dipole resonance with a time delayof ∼
40 as. Below 80 eV, the three quantities plottedin Fig. 4 (a-c) show a strong, oscillating, channel depen-dence, indicating a quantum interference phenomenon.The dynamics behind this effect can be unravelled bycalculating the Wigner representation [24, 40], definedas the Fourier transform of the auto-correlation functionof the transition matrix elements, D i ( E ), i denoting thechannel, and E the electron kinetic energy. W ( E, t ) = 12 π Z ∞−∞ d(cid:15)D i (cid:16) E + (cid:15) (cid:17) D ∗ i (cid:16) E − (cid:15) (cid:17) e − i(cid:15)t ~ , (2)where ~ is the reduced Planck constant. The results areshown in Fig. 5(a) for the 4 d / → (cid:15)f / channel (seeSM, Fig. S4, for the other two channels). All three chan-nels show similar features. (i) A broad resonance witha maximum around 100 eV and a short decay of a fewtens of attosecond, which can be interpreted as the giantdipole resonance; (ii) A sharp resonance at low energy,around 75 eV, with a long decay of a few hundreds ofattoseconds; (iii) Interferences between these resonances,leading to rapid oscillations of the Wigner distribution.To interpret the sharp spectral feature at 75 eV, weutilize the theoretical analysis performed in the semi-nal work of Cheng and Johnson [12]. Using a multi-channel quantum defect theory approach [41], resultsobtained within RRPA, similar to those of the presentwork, were analyzed and eigenchannel solutions were ex-tracted. These eigenchannel solutions are completely de-coupled from each other and can be used as a basis to de-scribe coupled-channel transitions. They are labelled us-ing the closest corresponding LS-coupled channel [( d f )and ( d p ) P, P, D]. Neglecting the weak contributionof the 4 d → (cid:15)p transitions, the 4 d → (cid:15)f transitions aresuperpositions of ( d f ) P, P and D eigenchannels (Inthe following, we drop the d f label).Fig. 4 (d-f) present the modulus, phase and time delayof these three eigenchannels. The behavior of the threecurves is much simpler than those in Fig. 4 (a-c). For eacheigenchannel, a single resonance feature can be identified,with a peak for the modulus and time delay and a π phase variation across the resonance. While the broadfeature, maximum at 100 eV, obviously represents thegiant dipole resonance ( S → P), the narrow peaks at 75
FIG. 5.
Wigner representation and effective poten-tials. (a) Wigner representation W ( E, t ) for the 4 d / → (cid:15)f / channel. The amplitude is indicated by the color codeon the right. (b) Illustration of one-electron potentials for the S → P, D transitions (red) and S → P (black). Dashedlines suggest possible electron trajectories in the two cases. and 76 eV exist because of the spin-orbit interaction thatenables singlet to triplet mixing. The maximum of thetime delay varies from a few tens ( P) to a few hundreds( D and P) of attoseconds, in agreement with the resultsin Fig. 5(a).The difference in time delays can be further interpretedby examining the effective potential experienced by theescaping f photoelectron. We represent in Fig. 5(b) amean-field average potential (red), as well as the poten-tial modified by S → P dipole polarization (screening)effects (black), which are included in the random phaseapproximation with exchange (RPAE) approach. Theseeffects lead to an effective high and narrow potential bar-rier and therefore to a broad resonance, with a maximumat high energy, and a short decay time (see black dashedline). In contrast, an electron emitted in the triplet chan-nels does not feel these dipole polarization effects andsees essentially the potential indicated in red, with a rel-atively low barrier only due to angular momentum anda long decay time (red dashed line). The time delay isdirectly related to the resonance lifetime, being equal toit at the maximum of the resonance [42]. Fig. 5(b) evensuggests that the increase of the temporal width of thebroad resonance in Fig. 5(a) towards low energy mightbe due to the influence of the long tail of the screenedpotential (black).The rapid variation of the amplitude, phase and delaysof the three 4 d → (cid:15)f channels [Fig. 4(a-c)] at thresh-old can therefore be interpreted as a quantum interfer-ence effect between the “direct” dipole-allowed S → Ptransition and the spin-orbit-induced S → P, D tran-sitions, which have similar amplitudes in this region.This interference explains the difference in time delaysfor 4 d / and 4 d / , as well as the anomalous branchingratio [Fig. 3(c,d)].In conclusion, we have measured photoionization timedelays in Xe using attosecond interferometry, giving ushigh temporal resolution, and coincidence spectroscopy,which allows us to avoid spectral congestion and to obtaina high spectral resolution. These time delays are positiveand similar for the two spin-orbit split Xe + states over alarge energy range (up to 100 eV photon energy), exceptat threshold (75 eV) where they differ by 100 as. Withthe help of RRPA calculations for one- and two-photonionization, we attribute this difference to the interferenceof several channels coupled by the spin-orbit interaction.A time-frequency analysis of the dominant transition ma-trix elements, allows us to unravel two main ionizationprocesses, with very different time and energy scales: thebroad giant dipole resonance, dominated by the S to Ptransition and the narrow resonances due to the S to Pand D transitions, which are enabled by the spin-orbitinteraction. While the former takes place over a few tensof attoseconds, the latter, involving spin flip, occurs overseveral hundreds of attoseconds. Our experimental ap-proach, which adds temporal information to traditionalspectral studies, provides increased understanding of thecomplex electron dynamics taking place in Xe 4 d pho-toionization. METHODSExperimental method
The experiments were performed with 40-fs longpulses, centered at 800 nm with 1-kHz repetition ratefrom a Ti:sapphire femtosecond laser system. The laserwas focused into a 6-mm long gas cell filled with Ne togenerate high-order harmonics. A 200-nm thick Zr fil-ter was used to filter out the infrared (IR) and most ofthe harmonics below the Xe 4 d threshold (67.5 eV for4 d / ). The filtered harmonics thus span the 4 d ioniza-tion region from the threshold to the maximum of thegiant dipole resonance (100 eV). A small fraction (30%)of the IR beam, split-off before generation, is used asa probe with a variable time delay. The XUV and IRpulses were focused in an effusive Xe gas jet. The elec-trons were detected by a magnetic bottle electron spec-trometer, which combines high collection efficiency andhigh spectral resolution up to E/ ∆ E ∼ Data analysis
Each data point in Fig. 3 is the arithmetic meanweighted with the uncertainty estimated from the cosinefitting to Eq. 1. In each measurement, we average thetime delays of electron pairs corresponding to the samephotoelectron but different Auger decay. For N measure-ments yielding N data points: τ , τ , . . . , τ N with corre-sponding uncertainties: σ , σ , . . . , σ N , the weighted av-erage can be calculated as: τ = P Ni =1 w i τ i P Ni =1 w i , (3) where w i = 1 /σ i is the weight. The uncertainty for eachmeasurement is estimated from the fit of the RABBIToscillation to a cosine function. The uncertainty on thetime delay difference, τ A − τ B , can be expressed as: σ = q σ A + σ B . (4)The error bars of the experimental results indicate thestandard error of the weighted mean and can be calcu-lated as: σ τ = s N ( N − ω s X ω i ( τ i − τ ) , (5)where ω s = P ω i . Theoretical method
Theoretical calculations consisted in calculating one-photon and two-photon matrix elements within lowest-order perturbation theory for the radiation fields, us-ing wavefunctions obtained by solving the Dirac equa-tion, and including electron correlation effects within theRPAE for one-photon XUV photoionizaton. Continuum–Continuum transitions were computed within an effectivespherical potential for the final state.
ACKNOWLEDGEMENT
The authors acknowledge support from the SwedishResearch Council, the European Research Council (ad-vanced grant PALP-339253), the Knut and Alice Wallen-berg Foundation and Olle Engkvist’s Foundation.
AUTHOR CONTRIBUTIONS
S.Z., D.B., R.J.S., M.I., L.N., H.L., R.W. and C.L.Aperformed the experiment. R.J.S. and R.F. provided partof the experimental setup. J.V., J.M.D. and E.L. per-formed RRPA calculations. S.Z., D.B., J.M.D., G.W.,M.G., E.L. and A.L. worked on the analysis and the the-oretical interpretation. S.Z. and A.L. wrote the mainpart of the manuscript. All authors gave feedback on themanuscript.
COMPETING FINANCIAL INTERESTS
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Theoretical Atomic Physics (Springer-Verlag Berlin Heidelberg, 1998). upplemental Material: Attosecond electron–spin dynamics in Xe4d photoionization Shiyang Zhong, ∗ Jimmy Vinbladh, David Busto, Richard J. Squibb, Marcus Isinger, Lana Neorii, Hugo Laurell, Robin Weissenbilder, Cord L. Arnold, Raimund Feifel, Jan Marcus Dahlstrm, GranWendin, Mathieu Gisselbrecht, Eva Lindroth, and Anne LHuillier Department of Physics, Lund University,P.O. Box 118, SE-221 00 Lund, Sweden Department of Physics, Stockholm University,AlbaNova University Center, SE-106 91 Stockholm, Sweden Department of Physics, University of Gothenburg,Origovgen 6B, SE-412 96 Gothenburg, Sweden Department of Microtechnology and NanoscienceMC2,Chalmers University of Technology, SE-412 96 Gothenburg, Sweden ∗ [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] M a y . ONE- AND TWO-PHOTON IONIZATION TIME DELAYS In the main article, we state that the atomic time delays measured through the RABBITtechnique reflect the dynamics of one-photon ionization. In this section, we explain therelation between the atomic delays as measured in the experiment to the photoionizationtime delays shown in Fig. 4. We select the 4 d / → (cid:15)f / channel to describe this relationship.Fig. S1 shows the time delay already presented in Fig. 4(c) of the main manuscript (blueline). First, we add the effect of the Coulomb phase, as shown by the red line. The delayobtained through the RABBIT technique for a single intermediate channel can be expressedwith good approximation as a sum of two contributions: τ A = τ + τ cc [1], where the firstterm is the delay discussed previously and the second term, τ cc , shown in black line, is acorrection term, due to IR laser induced continuum-continuum transitions, which is channel-independent over a large energy range. Fig. S1(a) presents the variation of τ A (green line).As can be observed, the Coulomb phase and the τ cc correction largely compensate eachother.In the RABBIT technique, the atomic delay is extracted by the phase difference dividedby 2 ω (2 ~ ω = 3 . d / → (cid:15)f / channel. Similar agreement is obtained for the 4 d / → (cid:15)f / channel.Therefore neglecting the influence of the d → p channels for simplicity as we did in themanuscript is reasonable. In addition, we demonstrate that the dynamics is governed byone-photon ionization. 2IG. S1. (a) Wigner delay with (red) and without (blue) the influence of the Coulombphase for 4 d / → (cid:15)f / . τ cc is shown as black line. The atomic delay for 4 d / → (cid:15)f / (red+black) is shown in green line. (b) Angular-integrated atomic delay for 4 d / (black,including all channels) , the same as in Fig. 3(a) in the manuscript. It is compared withthe single channel atomic delay with (magenta) and without (green) the finite differenceapproximation. II. MEASUREMENTS OF ABSOLUTE TIME DELAYS
Fig. 3 (a,b) shows differences between Xe 4 d and Ne 2 p atomic time delays. A Ne RAB-BIT scan is taken consecutively after each Xe scan with the same experimental parametersexcept the gas species, allowing us to remove the influence of the attosecond pulses in thismeasurement. Fig. S2 (a) presents the atomic time delays for the Ne 2 p shell in the rangeof 70 eV to 100 eV photon energy [2]. Because this energy region is well above the 2 p threshold (21.56 eV), with no sharp spectral features like autoionization or Cooper mini-mum, τ A [Ne(2 p )] remains small, less than ten attoseconds, and can be neglected with respectto the Xe 4 d time delays. Fig. S2 (b) shows τ A [Xe(4 d / )] (solid line) and τ A [Xe(4 d / )]- τ A [Ne(2 p )] (dashed line), in addition to the experimental results (dots). The differencebetween the two theoretical results is negligible, which justifies our approximation.3IG. S2. (a) Calculated Ne 2 p atomic delays in ionization. (b) Calculated (dashed line)and experimental (dot) τ A [Xe(4 d / )]- τ A [Ne(2 p )] and calculated τ A [Xe(4 d / )] (solid line)using the RRPA. III. INFLUENCE OF THE d → (cid:15)p TRANSITIONS
In the main article, we concentrate on the 4 d → (cid:15)f transitions which dominate over4 d → (cid:15)p in the vicinity of the giant dipole resonance. Fig. S3 compares the modulus ofthe photoionization dipole matrix element and time delay for all transitions. Indeed, thetransition strengths for the 4 d → (cid:15)p channels are generally much weaker than for 4 d → (cid:15)f .The 4 d → (cid:15)p time delays vary weakly with energy and remain small and negative. Theyare dominated by the IR correction τ cc (see below), and converge towards 0 at high energy,similarly to Ne 2 p . In contrast, the 4 d → (cid:15)f time delays, which can be interpreted as thetime that the f electron spends in the potential barrier of the shape resonance, are muchlonger. The delay variation observed at low energies present oscillations which probablyindicate the influence of spin-flip transitions as for the d → f channels, but these are muchweaker. 4IG. S3. (a) Modulus and (b) time delay for the following transitions: 4 d / → (cid:15)p / (black), 4 d / → (cid:15)p / (purple), 4 d / → (cid:15)f / (blue), 4 d / → (cid:15)p / (green), 4 d / → (cid:15)f / (red), 4 d / → (cid:15)f / (brown). We removed the Coulomb phase-shift in (b). IV. WIGNER REPRESENTATION
In Fig. S4, we show Wigner representations for the three channels discussed in this article.They exhibit similar features, (i) a broad resonance with a maximum around 100 eV anda short decay; (ii) a sharp resonance at low energy, around 75 eV, with a long decay;(iii) interferences between these resonances. Since the relativistic 4 d / → (cid:15)f / channelcontributes only weakly to the giant dipole resonance at photon energies larger than 80 eV,the relative amplitude of the narrow feature at 75 eV is much stronger than for the othertwo channels. We chose to present this result in the main article in order to emphasize thedynamics of relativistic threshold effects. 5IG. S4. Wigner representation W ( E, t ) for the (a) 4 d / → (cid:15)f / (b) 4 d / → (cid:15)f / and (c)4 d / → (cid:15)f / channels. The amplitude is indicated by the color code on the right handside. [1] Dahlstrm, J. M., Carette, T. & Lindroth, E. Diagrammatic approach to attosecond delays inphotoionization. Phys. Rev. A , 061402 (2012).[2] Isinger, M. et al. Photoionization in the time and frequency domain. Science , 893-896(2017), 893-896(2017)