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Journal Article

Spin-orbit delays in photoemission

Author(s): Jordan, Inga; Huppert, Martin; Pabst, Stefan; Kheifets, Anatoli S.; Baykusheva, Denitsa; Wörner, HansJakob

Publication Date: 2017-01

Permanent Link: https://doi.org/10.3929/ethz-a-010890593

Originally published in: Physical Review A 95(1), http://doi.org/10.1103/PhysRevA.95.013404

Rights / License: In Copyright - Non-Commercial Use Permitted

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Spin-orbit delays in photoemission

I. Jordan,1 M. Huppert,1 S. Pabst,2, 3 A. S. Kheifets,4 D. Baykusheva,1 and H. J. Worner1, ∗

1Laboratorium fur Physikalische Chemie, ETH Zurich,Vladimir-Prelog-Weg 2, 8093 Zurich, Switzerland

2Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany3ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA

4Research School of Physical Sciences, The Australian National University, Canberra ACT 0200, Australia(Dated: December 16, 2016)

Attosecond delays between photoelectron wave packets emitted from different electronic shells arenow well established. Is there any delay between electrons originating from the same electronic shellbut leaving the cation in different fine-structure states? This question is relevant for all attosecondphotoemission studies involving heavy elements, be it atoms, molecules or solids. We answer thisfundamental question by measuring energy-dependent delays between photoelectron wave packetsassociated with the 2P3/2 and 2P1/2 components of the electronic ground states of Xe+ and Kr+.We observe delays reaching up to 33±6 as in the case of Xe. Our results are compared with twostate-of-the-art theories. Whereas both theories quantitatively agree with the results obtained forKr, neither of them fully reproduces the experimental results in Xe. Performing delay measurementsvery close to the ionization thresholds, we compare the agreement of several analytical formulae forthe continuum-continuum delays with experimental data. Our results show an important influenceof spin-orbit coupling on attosecond photoionization delays, highlight the requirement for additionaltheory development and offer a new precision benchmark for such work.

I. INTRODUCTION

In recent years, the measurement and interpretationof delays in photoemission has become one of the mostactive areas within attosecond science. All efforts haveso far concentrated on delays between photoelectronwave packets originating from different electronic shellsof atoms [1, 2], solids [3–5] and molecules [6] (see Ref.[7] for a review). These studies have revealed temporaldelays of a few tens of attoseconds between photoelec-trons associated with different final electronic states ofthe cation.

In this article, we add a new dimension to this field ofresearch by addressing the role of atomic fine structure onattosecond photoemission delays. This progress is madepossible by a considerably improved energy resolution inconjunction with advanced data-acquisition techniques.We measure the temporal delay between photoelectronwave packets associated with different final spin-orbitstates of the electronic ground states of Kr+ and Xe+.Whereas the delays are very small (≤ 8 as) in Kr, muchlarger delays up to 33±6 as are measured in the case ofXe. These results show that atomic fine-structure effectsare not, in general, negligible compared to electronic ef-fects. We expect this result to be extend from atoms,over molecules to solids and to be particularly impor-tant for heavy elements, since spin-orbit coupling scaleswith nuclear charge to the fourth power. We furthershow that our results point to significant shortcomingsin state-of-the-art theories. A detailed consideration ofthe possible origins of these discrepancies leads us to to

∗hwoerner@ethz.ch; www.atto.ethz.ch

identify an incomplete description of electron correlationas the most likely origin. Our results further offer a newapproach to investigating spin-orbit coupling, which isa ubiquitous phenomenon with a wide range of implica-tions in physics and chemistry. For example, spin-orbitcoupling is responsible for the creation of spin-polarizedelectrons in the photoionization of atoms and solids [8, 9]and for intersystem crossing in molecules, the importanceof which has been underestimated for a long time, par-ticularly in molecules containing relatively light elements[10].

We compare our results with two of the most ac-curate currently available ab initio theories, i.e. thetime-dependent configuration-interaction singles (TD-CIS) theory and the relativistic random-phase approxi-mation (RRPA). The TDCIS theory is an explicitly time-dependent multi-electron method that we use to directlysimulate our experiment, without additional approxima-tions. The RRPA represents the state of the art inatomic photoionization, particularly of heavy elements,achieving near quantitative agreement with photoioniza-tion cross sections and angular distributions [11, 12]. TheRRPA is a time-independent method that we couple withan analytical treatment of the continuum-continuumtransitions to describe our experiments based on attosec-ond interferometry [13].

Our work addresses the novel topic of the fine-structuredependence of photoemission delays. This dependenceprovides access to the manifestation of relativistic effectsin photoionization delays. It therefore represents a newtesting ground for theory that we exploit in two com-plementary ways. First, by performing measurementsat high photon energies (≥ 25 eV), where the contribu-tions of the Coulomb singularity and the differences of thecontinuum-continuum delays between the closely-spaced

2

−1 0 1

diff.PE-signal (arb. units)

H19 H17

SB18

Kr 1S0

Kr+ 2P3/2 14.00 eVKr+ 2P1/2 14.67 eV

~~ ~~

0 2 4 6

5

10

15

20

delay (fs)

elec

tron

kine

tic e

nerg

y (e

V)

-1 0 1

differencePE counts

0 0.5 1

norm.PE counts

0 2 4 60

1

delay (fs)

norm

.PE

coun

ts

0 2 4 6

10

12

14

delay (fs)

eKE

(eV

)

-1 0 1

differencePE counts

0 0.5 1

norm.PE counts

10

12

14

eKE

(eV

)

0 2 4 6

10

12

14

delay (fs)

eKE

(eV

)Residuals (x2)

Fit

H17 H15

SB16

Xe 1S0

Xe+ 2P3/2 12.13 eVXe+ 2P1/2 13.44 eV

~~ ~~

(a) Kr (b) Xe

FIG. 1: (color online) Measurement of delays between photoelectron wave packets associated with the 2P3/2 and 2P1/2 final

states of Kr+ and Xe+. An attosecond pulse train consisting of H11-H27, superimposed with an IR pulse centered at 800 nm isused to ionize the neutral atoms. Photoelectron spectra are acquired in the presence and absence of the IR field and subtractedon a single-shot basis. In the case of Xe, the two-dimensional fitting procedure of the difference spectrograms is illustrated.

spin-orbit levels are both negligible, our measurementsprobe the influence of doubly-excites states and the effectof relativistic phase shifts between the photoelectron con-tinua accessed at the one-photon level. These phase shiftshave never been measured before because they are inac-cessible to static photoelectron measurements, which areonly sensitive to phase shifts between continua belongingto the same ionization threshold. Our experimental re-sults are in significant disagreement with theory, demon-strating the need for further development. Second, per-forming measurements in the vicinity of the ionizationthreshold, where the different ionization potentials of thetwo spin-orbit states completely dominate over the rela-tivistic effects on the delays, we test the performance ofseveral analytical formulae for the continuum-continuumtransitions derived for the hydrogen atom in describingexperimental data for heavier atoms.

II. EXPERIMENTS

The experimental setup consists of an actively-stabilized attosecond beamline [14] and a magnetic-bottlephotoelectron spectrometer [15]. The photoelectron res-olution is sufficient to fully resolve the spin-orbit split-tings of Kr+ (0.66 eV) and Xe+ (1.31 eV) beyond thehighest kinetic energies reported in this work. High-harmonic generation in a gas cell filled with 10 mbarargon is driven by a 1.5 mJ, 30-fs laser pulse centered at800 nm. These laser pulses are generated by an amplifiedtitanium:sapphire laser system (Femtopower Pro V CEPby Femtolasers). The generated attosecond extreme-ultraviolet (XUV) pulse train is separated from the resid-ual infrared (IR) pulse under vacuum by means of a per-

forated off-axis parabolic mirror that simultaneously rec-ollimates the IR beam. The XUV beam is refocused bya toroidal mirror and is recombined with the IR beamby means of a second perforated off-axis parabolic mir-ror. The delay between XUV and IR is varied by tuningthe length of the IR beam path and is actively stabi-lized during the measurement to a residual jitter of ∼30 as [14]. We note that this jitter does not affect theaccuracy of the measured delays, because we measurerelative delays. Photoelectrons generated from the over-lapping XUV and IR pulses are collected and energy ana-lyzed using a ∼0.9 m-long magnetic-bottle spectrometerequipped with a permanent magnet. Electron time-of-flight spectra are acquired on a single-shot basis using adigitizer card (Agilent U1071A). A chopper wheel is usedin the IR beam path to block every other laser shot andto record two-color (XUV/IR) and one-color (XUV-only)spectra in immediate temporal sequence. This approachsubstantially improves the signal-to-noise ratio and gen-erates high-fidelity difference spectrograms that are usedin the subsequent analysis.

III. EXPERIMENTAL RESULTS

Figure 1 illustrates the experimental results and dataanalysis. Large positive delays correspond to the XUVpulse train preceding the IR pulse. In the case of Kr(panel a) all photoelectron lines and side bands are fullyresolved. The relative phases of the side-band oscilla-tions φq3/2 − φq1/2, for a given side-band order q, were de-

termined by averaging the complex-valued Fourier trans-form of the difference spectrogram over the width of theside bands and the width of the oscillation frequency in

3

(a) Kr

(b) Xe

4s4p6np

4s24p4nℓn′ℓ′

12 14 16 18 20 22 24 26

-20

0

20

40

sideband ordersp

in-orbitde

lay(as)

τ 3/

2−

τ 1/

2

20 25 30 35 40photon energy (eV)

5s5p6np

5s25p4nℓn′ℓ′

-20

0

20

40

spin-orbitde

lay(as)

τ 3/

2−

τ 1/

2

Exp.TDCISRRPA + ccP

1FIG. 2: (color online) Delays (τ tot3/2−τ tot1/2) between photoelec-

trons associated with the 2P3/2 and 2P1/2 final states of (a)

Kr+ and (b) Xe+. The experimental results (circles with er-ror bars) are compared to TDCIS calculations (squares) andRRPA calculations with added continuum-continuum contri-bution according to Eq. (3) (diamonds). The green (full)lines represent singly-excited states, whereas the grey (dashedor dotted) lines represent doubly-excited states given in Ref.[16]. The dashed grey lines represent states assigned to spe-cific series and the dotted lines in panel b unassigned states.

the Fourier domain. The proximity of the IR-photon en-ergy (1.55 eV) to the spin-orbit splitting in the 2P groundstate of Xe+ (1.31 eV) causes partial overlap between the2P3/2 sideband of a given order and the 2P1/2 photoelec-tron signal generated by the next-higher harmonic order(see Fig. 1b). We overcome this challenge by analyzingthe difference spectrogram (XUV+IR minus XUV-only),acquired on a single-shot basis, with a two-dimensionalfitting procedure using common oscillation frequencies.The two sidebands of the same harmonic order, and thetwo high-harmonic peaks with which they partially over-lap, are analyzed simultaneously to extract the measuredphase difference φq3/2 − φq1/2. The success of this data-

analysis approach is demonstrated by the weakness ofperiodic structures in the residuals. This novel analysisstrategy will be particularly useful to perform attosecondinterferometry on systems with complex photoelectronspectra [6].

Figure 2 shows the measured spin-orbit delays ∆τ q =(φq3/2 − φq1/2

)/(2ω), where ω is the IR-laser angular fre-

quency, between photoelectron wave packets associated

with the 2P3/2 and 2P1/2 spin-orbit components of the

electronic ground states of Kr+ and Xe+. In the case ofKr, all delays are smaller than 8 attoseconds in magni-tude. In the case of Xe, the measured delays are larger,reaching from -9±4 as at 21.7 eV to +33±6 as at 33.4 eV.All error bars represent 95 % confidence intervals basedon 48 measurements in the case of Kr and 94 in the caseof Xe.

IV. THEORETICAL RESULTS

This section briefly describes the TDCIS and RRPAtheories to which our experimental results will be com-pared. The TDCIS method is an ab initio time-dependent multi-electron theory [17, 18], which has beenapplied to Kr and Xe in Refs. [19–21]. Here, we use gridparameters [34] similar to those in Ref. [21]. The XUVand IR fields are represented by Gaussian envelopes of 30fs duration. Photoelectron spectra are calculated usingthe splitting method described in Refs. [22, 23]. Photo-electron sidebands are calculated for photoemission par-allel to the common polarization axis of XUV and IRfields and are integrated over their energy width to ob-tain the predicted spin-orbit delays.

The RRPA is an ab initio time-independent multi-electron theory [11, 12]. This theory has been used topredict photoionization delays [24, 25] and has also beencompared to experiments in various noble gases [26–28].For the present work, we have included the followingphotoionization channels and all mutual channel inter-actions: 4p−1, 4s−1 and 3d−1 in the case of Kr and5p−1, 5s−1, 4d−1 and 4p−1 in the case of Xe. In bothcases all fine-structure components of all channels wereincluded. In what follows, we use the atomic units unlessotherwise specified. Within the dipole approximation,the photoionization amplitudes contributing to photoe-mission along the polarization direction of the ionizingradiation are given by [29]

Tm=1/2np1/2

= 1√6Y00Dnp1/2→εs1/2 + 1√

15Y20Dnp1/2→εd3/2

Tm=1/2np3/2

= 1√6Y00Dnp3/2→εs1/2 − 1

5√6Y20Dnp3/2→εd3/2

− 15

√32Y20Dnp3/2→εd5/2 , (1)

where Y`m represent the values of the spherical harmonicsY`m(θ = 0, φ = 0) and Dnpj→ε`′j′

are the reduced pho-

toionization matrix elements defined in Ref. [11] [35].Since our measurements are integrated over the emissiondirection of the photoelectron by virtue of the magnetic-bottle configuration, possible effects of this angular av-eraging on the observed delays must be considered. Wehave therefore calculated the delays for different emissionangles using the complete off-axial expressions given inRef. [29]. In the case of Kr, the spin-orbit delays varyby less than 3 as as a function of angle and the variationis even smaller in the case of Xe. We therefore neglected

4

(a) Kr

(b) Xe

RRPA τ3/2RRPA τ1/2RRPA τ3/2-τ1/2

12 14 16 18 20 22 24 26

-100

0

100

200

300

sideband orderatom

icde

lay(as)

20 25 30 35 40photon energy (eV)

RRPA τ3/2RRPA τ1/2RRPA τ3/2-τ1/2

-100

0

100

200

atom

icde

lay(as)

1FIG. 3: (color online) Atomic delays (τ3/2 and τ1/2) be-

tween photoelectrons associated with the 2P3/2 and 2P1/2 fi-

nal states of (a) Kr+ and (b) Xe+ obtained from the RRPAcalculations.

the angle dependence in our further analysis and compareour measurements with calculations carried out for pho-toemission parallel to the polarization of the XUV field.The atomic contribution to the delay of a photoelectronwave packet associated with the 2PJ state of Kr+ or Xe+

is calculated according to:

τj =∂

∂Earg(Tnpj

), (2)

which is illustrated in Fig. 3.

The total delay accessible to attosecond interferome-try is then obtained by adding the contribution of thecontinuum-continuum (cc) transition: τ totj = τj + τcc,j .Several formulae have been given to calculate cc delays(τcc) based on the phase of the cc transition matrix el-ements [30]. The simplest approximation, obtained byconsidering only the long-range phase in the asymptoticpart of the wave functions is

φ(P )cc (k, κ) ≡ arg

((2κ)iZ/κ

(2k)iZ/kΓ[2 + iZ(1/κ− 1/k)]

(κ− k)iZ(1/κ−1/k)

),

(3)where k and κ are the photoelectron momenta before andafter interaction with the IR field, respectively. Thesecontinuum-continuum delays are displayed in Fig. 4.

The next better approximation consists in correctingEq. (3) for the long-range variation of the amplitudes ofthe scattering wave functions:

φ(PA)cc (k, κ) ≡ φ(P )

cc (k, κ) + α(k, κ), (4)

(a) Kr

(b) Xe

τccP, 2P3/2

τccP, 2P1/2

τccP, 2P3/2 −2 P1/2

12 14 16 18 20 22 24 26

-300

-200

-100

0

100

sideband order

cc-d

elay

(as)

20 25 30 35 40photon energy (eV)

τccP, 2P3/2

τccP, 2P1/2

τccP, 2P3/2 −2 P1/2-400

-300

-200

-100

0

100

cc-d

elay

(as)

1FIG. 4: (color online) Continuum-continuum delays (τccP,3/2,τccP,1/2), and their difference, pertaining to photoelectron

wavpackets associated with the 2P3/2 and 2P1/2 final states

of (a) Kr+ and (b) Xe+ obtained from Eq. (3).

where

α(k, κ) = arg

(1 +

iZ

2

(1

κ2+

1

k2

)κ− k

1 + iZ(1/κ− 1/k)

).

(5)A further improvement of the cc delays was obtainedby regularizing the wave functions at the origin by sub-stituting the radial coordinate with a complex variable.This approach provides quasi-exact results in the case ofatomic hydrogen, as can be judged from the comparisonwith exact numerical calculations [30].

V. DISCUSSION

We now compare the results of the experiment with theTDCIS and RRPA results in Fig. 2. In the case of kryp-ton, quantitative agreement is obtained between experi-ment and the RRPA theory for most of the data points,when we use Eq. (3) to evaluate the cc delays. Themissing points between 25 and 29 eV are a consequenceof the presence of singly-excited resonances convergingto the 4s−1 threshold of Kr at 27.51 eV. This regionwas partially left out of the RRPA calculation becausethe spin-orbit delays display very large variations in thevicinity of the resonances. We have however very care-fully studied the possible influence of these autoionizingresonances on the measured delays in our experiment.We have systematically varied the central wavelengthof the Ti:Sa laser system and have used emission linesfrom a discharge-generated plasma source to calibratethe XUV-photon spectrometer. Although we have ob-

5

served clear shifts of the high-harmonic photon energies,sufficient to fulfill the resonance conditions with singly-excited states, no systematic dependence of the measuredphotoionization delays on the photon energies could bedetermined. Therefore, all measurements were averagedtogether, yielding the data points shown in Fig. 2a. Thisis an interesting observation, because the effects of res-onances on the phase of photoelectron side-band oscilla-tions [31] and electronic photoionization delays [32] havebeen observed and reproduced by theory on other atomicsystems.

The experimental results in Fig. 2a also agree well withthe TDCIS calculations. The calculated delays lie withinthe 95% confidence intervals of our measurements forall energies that are not affected by auto-ionizing res-onances. The data points between 24.8 eV (SB16) and31.0 eV (SB20) agree with the TDCIS delays within twotimes the 95% confidence intervals. The possible effectsof the singly-excited resonances were further investigatedby comparing the results of TDCIS calculations in whichonly the 4p−1 channel was active, with calculations inwhich both the 4p−1 and the 4s−1 channels were activeand inter-channel coupling was included. The differencebetween the two calculations was systematically smallerthan 3 as, confirming our experimental conclusion to theinsignificant influence of the singly-excited auto-ionizingresonances on the spin-orbit delays.

In the case of xenon (Fig. 2b), significant discrepan-cies are found between experiment and theory. Althoughthe first and third data points agree with the RRPA cal-culations combined with Eq. (3), the three other datapoints lie far away from the theoretical results. The re-gion between 20.5 and 23.5 eV is affected by auto-ionizingresonances in the vicinity of the 5s−1 threshold of Xe+

at 21.77 eV and is therefore left out of the RRPA cal-culations. As in the case of Kr, we have systematicallyinvestigated the possible influence of the singly-excitedresonances converging to the 5s−1 threshold of Xe+ at23.40 eV, on the delay measured at 21.7 eV (SB14) buthave not found any systematic effect. The delays calcu-lated by the TDCIS method in Fig. 2b are closer to zerothan the RRPA results at low energies, but agree wellwith the RRPA results at higher energies. As in the caseof Kr, TDCIS calculations including only the 5p−1 chan-nel or both the 5p−1 and the 5s−1 channel differ by lessthan 4 as.

What are possible origins for the large discrepancy be-tween experiment and theory in the case of xenon? First,although RRPA and TDCIS fully include the singly-excited states, such as the auto-ionizing resonances con-verging to the 5s−1 threshold (full green lines in Fig. 2),neither RRPA nor TDCIS include doubly-excited states,i.e. states that are dominated by a two-electron-two-holesconfigurations (grey lines in Fig. 2). The interaction be-tween singly- and doubly-excited states is entirely me-diated by electron correlation, which makes such effectsvery interesting from a fundamental point of view. Theabsence of doubly-excited states from RRPA has indeed

been identified as the dominant origin of inaccuracies inthe calculation of photoelectron-asymmetry parametersin argon [33]. The high density of doubly-excited statesin the region of 26-33 eV in Xe is therefore a likely expla-nation for the observed deviations between experimentand theory (both TDCIS and RRPA). In other words,the incomplete treatment of electron correlation in bothRRPA and TDCIS, is the most likely explanation for theobserved discrepancy. In further support of this conclu-sion, we note that in the region above ∼25 eV, the con-tribution of the continuum-continuum delays is negligible(< 10 as in Xe and < 8 as in Kr, Fig. 3), which enables usto attribute the observed deviations to the atomic partτj of the photoionization delays.

Second, the treatment of spin-orbit coupling in TDCISis approximate. Although spin-orbit coupling is fully in-cluded in the angular-momentum algebra underlying theTDCIS code, the radial electronic wave functions are thesame for both spin-orbit channels. Relativistic effects arehowever expected to affect the radial wave functions ofthese two channels differently with the largest differencesexpected for low kinetic energies. This fact is likely toexplain why the spin-orbit delays are underestimated inthe TDCIS calculation of the lowest two data points inXe (see Fig. 2), whereas the highest three points agreewell with the RRPA calculations.

Third, a possible reason for the deviation between ex-periment and theory at 27.9 and 31.0 eV is the sensi-tivity of spin-orbit delays to the phase shifts betweencontinuum channels associated with the 2P3/2 state on

one hand and the 2P1/2 state on the other. These phaseshifts play little or no role in any traditional observableof photoelectron spectroscopy, such as cross sections, an-gular distributions or spin-polarization parameters [8, 9].Measurements of photoionization delays between photo-electrons leaving the ion in one of the two spin-orbitstates are, however, mainly sensitive to these phase shifts.Therefore, it is conceivable that both theories are not suf-ficiently accurate in predicting these phase shifts.

Since our measurements probe delays between photo-electrons associated with close-lying ionization thresholdsat low kinetic energies they additionally offer the possibil-ity of testing the performance of several analytical expres-sions for the hydrogenic continuum-continuum delays inthe case of heavier atoms. The atomic part of the relativedelays τ3/2 and τ1/2, with τj given by Eq. (2) and shownin Fig. 3, increase to large positive values towards theionization threshold because they are dominated by thecontribution of the Coulomb potential. The differenceτ3/2− τ1/2 decreases to large negative values because the2P3/2 ionization threshold lies below the 2P1/2 threshold.The large negative relative atomic delays τ3/2 − τ1/2 arecompensated by large positive differences of the cc delaysat low kinetic energies (Fig. 4). Therefore, the total rela-tive delays at low kinetic energies must be very sensitiveto each of the two contributions.

Figure 5 compares the experimental measurements andTDCIS results to the sum of the atomic delay from RRPA

6

(a) Kr

(b) Xe

12 14 16 18 20 22 24 26

-10

0

10

sideband ordersp

in-orbitde

lay(as)

τ 3/

2−

τ 1/

2

RRPA + ccP Exp.

RRPA + ccPA TDCISRRPA + reg. cc

20 25 30 35 40photon energy (eV)

-20

0

20

40

spin-orbitde

lay(as)

τ 3/

2−

τ 1/

2

1FIG. 5: (color online) Comparison of the measured delays(τ3/2 − τ1/2) with RRPA calculations combined with differ-ent versions of the continuum-continuum delays. The version”ccP” refers to Eq. (3), ”ccPA” refers to Eq. (4) and ”reg”refers to the cc delays calculated on the basis of the regular-ized radial wave functions (see text and Ref. [30] for details).

calculations and the analytical cc delays. The largest ef-fects are seen at low photon energies, as expected. In thecases of both Kr and Xe, the cc delays calculated accord-ing to Eqs. (3) and (5) yield the best results. Using Eq.(4) markedly deteriorates the agreement with the lowestexperimental points. In the case of Kr, the agreement be-tween the results of Eqs. (3) and (5) is quantitative withmost of the experimental data points, as well as with theTDCIS results. A similar situation occurs in xenon. Thedata points at 18.6 and 24.8 eV agree best with the pre-dictions of Eq. (3), followed by Eq. (5), and finally byEq. (4), for which the worst agreement with experimentis obtained. Concerning the data point at 21.7 eV, we re-call the proximity of multiple doubly-excited states, suchthat one cannot expect quantitative agreement for this

data point.

VI. CONCLUSIONS

In conclusion, we have realized precision measurementsof delays between photoelectron wave packets associatedwith the spin-orbit components of the electronic groundstates of Kr+ and Xe+. In the case of krypton, the spin-orbit delays were found to be small, amounting to lessthan 8 attoseconds in the investigated range of photonenergies. This result quantitatively agrees with state-of-the-art theories. In the case of xenon, much largerspin-orbit delays were measured, reaching up to 33±6 asat a photon energy of 33.4 eV. These results demon-strate the importance of fine-structure effects, such asspin-orbit coupling, in photoemission time delays. Thisis particularly important for photoemission from all sam-ples in the gas or condensed phases containing heavy el-ements [3–5]. Even more importantly, our results sug-gest significant shortcomings in state-of-the-art theoriesof atomic photoionization which were known to achievenear-quantitative accuracy in traditional photoemissionexperiments. The most likely explanation of this defi-ciency is an incomplete treatment of electron correlation,arising from the absence of doubly-excited configurationsin TDCIS and RRPA. Our work therefore contributes todemonstrate that attosecond metrology has now reachedthe level to challenge well-established theories and there-fore has the potential for driving major progress in ourunderstanding of electron correlation in multi-electronsystems.

Acknowledgments

We thank U. Keller and her group for discussions. Wegratefully acknowledge funding from an ERC StartingGrant (contract 307270-ATTOSCOPE) and the NCCR-MUST, a funding instrument of the Swiss National Sci-ence Foundation. S.P. is funded by the Alexander vonHumboldt Foundation and by the NSF through a grant toITAMP A. K. acknowledges support from the AustralianResearch Council under Discovery grant DP120101805.

I. J. and M. H. contributed equally to this work.

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[35] We note that the authors of Ref. [29] found the necessityto add an extra parity factor to the expressions of Ref.[11] to make them compatible with the Wigner-Eckhardttheorem.