Angle-resolved time delays for shake-up ionization of helium
Stefan Donsa, Manuel Ederer, Renate Pazourek, Joachim Burgdörfer, Iva Březinová
AAngle-resolved time delays for shake-up ionization of helium
Stefan Donsa, ∗ Manuel Ederer, Renate Pazourek, Joachim Burgd¨orfer, and Iva Bˇrezinov´a Institute for Theoretical Physics, Vienna University of Technology,Wiedner Hauptstraße 8-10/136, 1040 Vienna, Austria, EU (Dated: May 27, 2020)Recent angle-resolved RABBITT experiments have shown that the photoionization time delaydepends on the emission angle of the photoelectron. In this work we demonstrate that for pho-toemission from helium accompanied by shake-up (correlation satellites), the angular variation ofthe time delay is dramatically enhanced by the dipolar coupling between the photoelectron and thehighly polarizable bound electron in the IR field. We show that the additivity rule for the timedelays due to the atomic potential, the continuum-continuum (cc) coupling by the IR field, and dueto this two-electron process remains valid for angle-resolved RABBITT. Our results are expected tobe also applicable to other multi-electron systems that are highly polarizable or feature a permanentdipole moment.
I. INTRODUCTION
Progress in the development of coherent light sourceshas fostered the development of new protocols to moni-tor and steer electronic and molecular motion [1]. Highharmonic generation [2–4] has enabled the generation ofisolated [5] and trains of light pulses [6–8] with dura-tion in the attosecond time domain. These developmentsenabled schemes to control valence electrons on their nat-ural time scale (attoseconds) [9] and to directly measurethe phase imprinted on the electronic wavepacket gen-erated by the photoelectric effect. Theoretical insightshave revealed that the phase of the ejected electronscan be directly related to the Eisenbud-Wigner-Smith(EWS) scattering time delay τ EWS [10]. The phase ofthe wavepacket and, thus, the photoionization time de-lay strongly depends on the underlying physical process.Scenarios investigated so far include electron emissionoriginating from atoms [11–18], molecules [19–21], andsolids [22], tunneling ionization [23], above-threshold ion-ization [24], and double ionization [25].First experiments and calculations investigating pho-toionization time delays have either been performed bythe attosecond streak camera [11, 22, 26] which recordsphotoelectrons along the polarization axis of the IRstreaking field, or by RABBITT (reconstruction of at-tosecond bursts by interference of two-photon transtions)[12, 14, 27, 28], measuring angle-integrated photoelec-tron spectra. Alternatively, angular streaking, employ-ing circularly polarized pulses has been used to extractionization phases and timing information from the elec-tron angular distribution, e.g. [23, 29–31]. More recently,measurements of time delays by RABBITT as a functionof the emission angle of the electrons with respect to thelaser polarization direction have become available [32–35], revealing a pronounced angle dependence for largeemission angles. Calculations have been able to repro-duce this dependence for different noble gas atoms [36– ∗ [email protected] τ EWS associ-ated with the XUV induced bound-free transition [39, 40]but rather to the additional time delay associated withthe absorption (emission) of the IR photon in a Coulombfield, the continuum-continuum delay τ CC , which is in-herent to the RABBITT protocol [12]. Recent resultsshow that the angle dependence of τ CC can be viewed asa result of the IR induced partial-wave interference andthe phase shift of the outgoing electron in the Coulombfield [34, 35, 42, 43].For several atomic species the time delay observed in pho-toionization was found to be the sum of EWS delay τ EWS and continuum-continuum delay τ CC (or, the Coulomb-laser coupling (CLC) delay for streaking) [10, 12, 26].For photoionization accompanied by shake-up of the sec-ond electron, the socalled correlation satellites, theoreti-cal predictions [13] and experiments [44] have shown thatthe coupling of the transient dipole moment of the boundelectron to the IR field imprints an additional phase shifton the correlated two-electron wave function which man-ifests itself as an additional correlation contribution τ e − e to the time delay of the emitted electron. So far, detec-tion of this observable was limited to the emission direc-tion along the polarization axis in the streaking geome-try [44]. RABBITT offers the opportunity to study theangular dependence of the time delay of these correla-tion satellites. Moreover, the superior energy resolutionof RABBITT allows to spectrally resolve [15] individualshake-up states He + ( n ) with n = 2 ,
3. While the cor-relation delay τ e − e for atoms is mostly due to transientpolarization of the target, molecules featuring a perma-nent dipole moment are expected to give rise to an evenstronger angle dependence of the time delay [21, 45].In this work we analyze the angular dependence of timedelays in the presence of shake-up excitation. We showthat the additivity of three distinct contributions to thetime delay, the Eisenbud-Wigner-Smith time delay τ EWS for bound-free transitions in the absence of the prob-ing IR field, the continuum-continuum contribution τ CC due to transitions induced by the IR field, and the two-electron correlation delay τ e − e remains valid for angle- a r X i v : . [ phy s i c s . a t o m - ph ] M a y resolved RABBITT. Disentangling the different contri-butions, we find that τ e − e provides by far the dominantcontribution to the angle dependence for moderate emis-sion angles θ relative to the polarization axis ( θ ≤ ◦ )where the emission probability is still large. Furthermore,we compare time delays extracted from angle-integratedRABBITT traces to time delays for angle-resolved RAB-BITT traces. We show that RABBITT spectra restrictedto forward direction ( θ = 0 ◦ ) yield the same delay asattosecond streaking. This demonstrates that attosec-ond streaking and RABBITT allow access to the samephysical quantity even for complex multi-electron sys-tems when electron-electron correlations are strong.The paper is organized as follows: In Sec. II we briefly re-view the RABBITT protocol for observing angle-resolvedtime delays. In Sec. III, we compare and discuss the an-gle dependence of the EWS delay τ EWS , the continuum-continuum delay τ CC , and of the dipole-induced correla-tion delay τ e − e in helium. Finally, a comparison betweenangle-integrated and angle-resolved delays for shake-upionization measured by RABBITT and those measuredby recent attosecond streaking experiments will be pre-sented in Sec. IV, followed by concluding remarks inSec. V. Numerical details of the simulations can be foundin the appendix. II. ANGLE-RESOLVED PHOTOIONIZATIONTIME DELAYS OBSERVED BY RABBITT
Up to now, the dipole-induced correlation delay τ e − e inphotoionization accompanied by shake-up in helium wascalculated and observed within an attosecond streakingprotocol [44]. Therefore, the accessible information wasrestricted to the emission in forward direction ( θ = 0 ◦ )relative to the polarization axis of the streaking IR field.The RABBITT protocol provides an attractive alterna-tive as it offers two advantages: higher spectral selectivitythereby enabling the resolution of nearby shake-up lines[15] as well as both angle-integrated and angle-resolvedtime delay measurements [12, 14, 32]. We thereforebriefly review the underlying concepts of angle-resolvedtime delays as detected by RABBITT for the well-knownEWS and cc delays before analyzing in detail the shake-up specific correlation time delay τ e − e .The interferometric RABBITT technique relies on theinterference between two different two-photon pathwaysto the same continuum final state involving the absorp-tion of one XUV photon from an attosecond pulse train(APT) and the absorption or stimulated emission of oneIR photon with frequency ω IR from a time-delayed weakreplica of the IR field which generated the APT. Thepeaks in the photoelectron spectrum reached via thesetwo-photon transitions, commonly called sidebands, atenergies E , P ( E ) ∝ (cid:12)(cid:12)(cid:12) A (2) f ← i ( E ) (cid:12)(cid:12)(cid:12) , (1) show a characteristic oscillation as a function of the timedelay ∆ t between the APT and IR fields P ( E ) = A + B cos (2 ω IR ∆ t − ∆Φ( E )) (2)with an energy dependent phase offset ∆Φ( E ). For lowIR intensities the oscillations of the sidebands as a func-tion of ∆ t are well described employing second-order per-turbation theory [12, 42, 46, 47]. The functional formof Eq. (2) is independent of the emission angle θ of thephotoelectron relative to the polarization direction of thecolinear IR and APT field. The amplitude A (2) f ← i for thetransition to the final state | f (cid:105) corresponding to the side-band H n can be described [16] as a superposition of twopaths, namely absorption of one photon of the harmonicbelow the sideband ( H n − ), followed by absorption ofone photon of the fundamental IR field A (2) H n − + ω IR , andabsorption of one photon of the harmonic above the side-band ( H n +1 ) followed by stimulated emission of one IRphoton A (2) H n +1 − ω IR . Accordingly, the transition ampli-tude is given by A (2) f ← i ( E = H n ) = A (2) H n − + ω IR + A (2) H n +1 − ω IR . (3)The phase offset of the 2 ω IR beating in RABBITT traces,∆Φ, can be obtained from the phase difference betweenthese two partial amplitudes∆Φ ( H n ) = arg (cid:104) A (2) H n +1 − ω IR (cid:105) − arg (cid:104) A (2) H n − + ω IR (cid:105) . (4) A. EWS and cc delays
When each of the involved intermediate states( H n − , H n +1 ) and the final state involve structure-less continuum states, experiment and theory [12, 42, 47]have shown that the acquired phase ∆Φ can be ap-proximated as the sum of two distinct phases: the one-photon half-scattering phase of the XUV triggered tran-sition from the initial bound state | i (cid:105) to the interme-diate continuum states H n − or H n +1 , and the ad-ditional scattering phase acquired by the IR field in-duced continuum-continuum transitions ( H n − → H n )or ( H n +1 → H n ). In addition to these atomic phases,also phase differences between adjacent harmonic peaksof the APT may contribute [8, 46, 48, 49]. As this XUVpulse related phase does not depend on the emission an-gle, we will omit this contribution in the following.The additivity of the phase difference∆Φ ( θ ) = Φ ( H n +1 , θ ) − Φ ( H n − , θ ) , (5)i.e. ∆Φ ( θ ) = ∆Φ EWS ( θ ) + ∆Φ CC ( θ ) , (6)for any electron emission angle θ directly translates intothe additivity of the corresponding angle-dependent timedelays τ given by the finite-difference approximation tothe spectral derivative τ ( θ ) = ∆Φ( θ )2 ω IR (7)with τ ( θ ) = τ EWS ( θ ) + τ CC ( θ ) . (8)It should be noted that while both time delays in Eq. (8)are atomic EWS-type time delays for half-scattering,only the delay associated with the bound-continuum (bc)transition is conventionally referred to as EWS delay τ EWS = τ bc . Recently [35] the delay associated with thecontinuum-continuum transition was also identified as anEWS-type delay. For convenience, we adhere in Eq. (8)to the standard convention. The finite-difference approx-imation to the delay within RABBITT differs from thestreaking protocol where the spectral derivative τ is di-rectly observable.The first term in Eq. (8), the one-photon bound-continuum delay τ EWS ( θ ) = ∂∂E arg {(cid:104) α, E, θ | ˆ z | i (cid:105)} = ∂∂E η α ( E, θ ) , (9)is associated with the half scattering of the outgoingelectron, transferred by the XUV pulse from the bound(ground) state | i (cid:105) to the (intermediate) continuum state | α, E, θ (cid:105) in the atomic potential. Here, E and θ denotethe energy and emission angle of the departing electron, α comprises all other quantum numbers of the ion andelectron, and η α is commonly called scattering phase.If the state α corresponds to a single partial wave with an-gular momentum (cid:96) (e.g. ionization by a dipole transitionof an initial s electron to a p wave) τ EWS has no intrin-sic angle dependence. If, however, the intermediate stateis a superposition of partial waves with different angularmomenta (e.g. ionization of an electron out from the 2 p state in neon to a superposition of an s and d wave) thescattering delay associated with the XUV transition itselfbecomes angle dependent [38]. For a statistical mixtureof inital states (e.g. 2 p and 2 p m = ± , of the 2 p subshell),the angle dependent EWS delay is given by the ensembleaverage (cid:104) τ EWS (cid:105) α ( θ ) = (cid:80) i σ α i ( θ ) τ ( α i ) EWS ( θ ) (cid:80) i σ α i ( θ ) , (10)weighted by the cross sections σ α i ( θ ) = |(cid:104) α i , ε, θ | ˆ z | i (cid:105)| .The second contribution to Eq. (8), the continuum-continuum delay τ CC , is caused by the transition be-tween continuum states in the atomic potential inducedby absorption or emission of an IR photon [12]. Un-like for streaking where the IR field employed is, typ-ically, stronger, the continuum-continuum transition inthe RABBITT protocol is well described by lowest-orderperturbation theory. Employing an asymptotic expan-sion for large kr of the outgoing Coulomb wave and ne-glecting partial wave interferences, a simplified analytic asymptotic expression for the phase shift φ asym CC and, inturn, τ asym CC has been derived [42] τ asym CC ( E, ω IR ) = (11) φ asym CC ( E, E + ω IR ) − φ asym CC ( E, E − ω IR )2 ω IR . In this limit τ asym CC is independent of the emission angle.Only when numerically including non-asymptotic contri-butions from smaller values kr of the Coulomb wave tothe cc transition matrix elements and partial wave inter-ferences, a θ -dependent τ CC ( E, ω IR , θ ) emerges [33, 47]. B. Two-electron delay for correlation satellites
An additional contribution to the phase shift of thewavepacket and, thus, to the time delay originates fromtrue two-electron processes beyond the direct ionizationdiscussed above. A prototypical case is the photoioniza-tion of helium accompanied by shake-up of the residualelectron, (cid:126) ω + He(1 s ) → He + ( n(cid:96)m ) + e, (12)often referred to as correlation-satellite lines in the pho-toelectron spectrum. For this process, the two-electronwavepacket acquires an additional dynamical phase andtime delay within a streaking or RABBITT protocol, be-yond the bound-continuum EWS and the cc contribution.The IR field polarizes the (quasi) degenerate n manifoldof the bound electron formed in the correlated ionizationprocess, and the resulting dynamical Stark shift imprintsan additional phase on the two-electron wave functionwhich, in turn, contributes to the scattering phase of theentangled photoelectron. The resulting two-electron timedelay can be analytically estimated as [13, 44] τ e − e ( θ ) = 1 ω IR atan (cid:18) − ω IR d · ˆ π IR k · ˆ π IR (cid:19) = 1 ω IR atan (cid:18) − ω IR d z k cos ( θ ) (cid:19) , (13)where d is the dipole moment of the bound electron, ˆ π IR is the polarization direction of the IR field, k is the mo-mentum of the outgoing electron, and we have assumedthat the IR field is linearly polarized along the ˆ z di-rection. A similar contribution to the time delay hasoriginally been predicted for molecules with a permanentdipole moment [45]. For streaking experiments, wherethe emitted photoelectrons are only measured in forwarddirection, the analytic estimate has been shown to beaccurate for the prototypical example of shake-up ioniza-tion in helium [13, 44].Eq. (13) extends this estimate to angles θ (cid:54) = 0 ◦ which be-come accessible by angle-resolved RABBITT. Assumingthat the additivity of the time delay holds also for τ e − e ,the total angle-dependent time delay τ ( t ) ( θ ) is given by τ ( t ) ( θ ) = τ EWS ( θ ) + τ CC ( θ ) + τ e − e ( θ ) . (14) E W S [ a s ] E=16.5 eV
H: 1 s Ne: 2 p Ne: 2 p ±1 Ne: 2 p FIG. 1. Bound-free scattering time delays τ EWS for a finalphotoelectron energy of 16.5 eV computed numerically forionization of the 1 s ground state of hydrogen and the 2 p m =0 and the 2 p m = ± states of neon represented by a model po-tential [50] yielding the experimental ionization energy. Thesubshell average (cid:104) τ EWS (cid:105) p is obtained using Eq. (10). In the following we will numerically test Eq. (14). Prob-ing τ e − e ( θ ) by RABBITT is of particular importance asthe resolution of different correlation satellite lines withincreasing n requires high spectral resolution as achievedby such a protocol [15] but is difficult to realize by anattosecond streaking camera. III. NUMERICAL RESULTS FOR THEANGULAR VARIATION OF THE TIME DELAY
We explore in the following the relative importanceof the different contributions to the total time delay[Eq. (14)] with the help of numerical results for a fewprototypical cases before presenting detailed results forthe helium shake-up satellites.
A. Angle dependence of the EWS delay
The partial-wave interference as origin of the angle de-pendence can be demonstrated for prototypical cases,the ionization of the 1 s state of hydrogen and of the2 p and 2 p ± states of neon described by a model po-tential [50] (Fig. 1). For the hydrogen ground state,the outgoing electron wavepacket created by the absorp-tion of a linearly polarized XUV photon is formed solelyby a p wave, | E, (cid:96) = 1 , m = 0 (cid:105) , resulting in an angle-independent delay. By contrast, photoionization of the | p, m = 0 (cid:105) state of neon creates a wavepacket contain-ing a superposition of an s -wave, | E, (cid:96) = 0 , m = 0 (cid:105) , anda d -wave, | E, (cid:96) = 2 , m = 0 (cid:105) . Cross sections and spec-tral phases of these two partial waves are different and,thus, τ EWS shows a characteristic dependence on θ [38].Photoionization of the Ne | p, m = ± (cid:105) states, on theother hand, gives access only to a single partial wave -700-600-500-400-300-200-1000 0 10 20 30 40 50 60 70 80 90 τ ( θ ) − τ ( θ = ◦ ) [ a s ] Emission angle θ [deg] θ c (a)-20-100 50 60 70 80 90757881 14 16 18 20 22 24 26 θ c [ d e g ] Electron energy [eV](b)CoulombYukawa CoulombYukawa
FIG. 2. (a) Angle dependence of the relative time delay τ ( θ ) − τ ( θ = 0 ◦ ), at the sideband centered at E =16.5 eVfor the Yukawa potential (blue circles) and hydrogen (redsquares). (b) Critical angle θ c at which the jump in ∆Φ and τ occurs, shown for the Yukawa potential and hydrogen. Thewavelength of the IR is λ IR = 740 nm. | E, (cid:96) = 2 , m = ± (cid:105) . Consequently, τ EWS is again inde-pendent of θ . The ensemble average (cid:104) τ EWS (cid:105) p over the 2 p subshell yields only a weak θ dependence (Fig. 1). Thisis due to the fact that the strong variation with angle of τ EWS for the 2 p state is confined to the region where thecross section is small and where ionization from the 2 p ± states dominates. Overall, the observed angular varia-tion of τ EWS is rather small. Further, for large electronemission angles it is orders of magnitude smaller than theangular variation in RABBITT experiments [32, 33] andsimulations [39, 40] in this case. It is also much smallerthan the variation of the two-electron delay τ e − e ( θ ) dis-cussed in the following. B. Angle dependence of the continuum-continuumdelay
The angle dependence of the continuum-continuum de-lay τ CC was recently observed for the direct ionization ofhelium [32] and argon [33, 34]. Moreover, also the an-gular momentum dependence of the phase acquired bythe IR photon absorption or emission in the continuum-continuum transition was analyzed [33, 35]. In order todisentangle the influence of long-range Coulomb interac-tion from partial-wave interference effects contributing tothe θ dependence of τ CC we compare RABBITT simula-tions for the 1 s state of hydrogen with that for a modelatom bound by a short-ranged Yukawa-like potential V Y ( r ) = − . r e − r , (15)with the same ground-state binding energy E s = − . (cid:96) = 0. The simulation employs thepseudo-spectral method [26, 51] (see appendix for numer-ical details). Previous simulations for a streaking setting[26] have shown that for short-ranged potentials the to-tal time delay is given by τ EWS , i.e. the cc contributionis negligible. Differently, in the presence of long-rangedinteraction the coupling of the IR field to the Coulombcontinuum gives an additional Coulomb-laser contribtion τ CLC which has been found to closely agreee with τ CC from RABBITT settings [10, 52].We explore now the θ dependence of τ CC in a RABBITTsetting for a long-ranged and short-ranged potential inphotoionization by an APT of the odd harmonics from15 to 31 of an IR field with λ IR = 740 nm. Focusing onthe sideband centered at 16.5 eV (the other sidebandsshow a very similar behavior), the time delay obtainedfor the Yukawa potential (Fig. 2a) shows almost no an-gle dependence up tp θ ≈ ◦ where a phase jump of π occurs [corresponding to a time delay change of ap-proximately 617 as for this specific IR wavelength, seeEq. (7)]. The time delay for hydrogen closely mimicsthis behavior, however, smooths the phase jump and re-duces the jump height from π to 0.92 π . The reductionin jump height can be traced to the angular momentumdependence of φ CC [47] and to the propensity rule thatfor photoabsorption the increase while for photoemissionthe decrease in angular momentum is preferred [43, 53].The critical angle where the jump occurs, θ c (cid:39) ◦ , isidentical for both potentials.In order to identify the origin of this rapid phase jump, weemploy second-order perturbation theory for absorptionof one XUV photon and subsequent absorption (emis-sion) of an IR photon to the wavepacket following [34, 42]for the short-ranged potentialΨ ( E, ∆ t, θ ) = (cid:88) L =0 , e i π Y L ( θ ) (cid:110)(cid:12)(cid:12)(cid:12) A (+) pL (cid:12)(cid:12)(cid:12) e i [ ω IR ∆ t + η ( − ) p ] + (cid:12)(cid:12)(cid:12) A ( − ) pL (cid:12)(cid:12)(cid:12) e i [ − ω IR ∆ t + η (+) p ] (cid:111) . (16)The two-photon transition amplitude for absorption ofone XUV photon to a p wave (angular momentum (cid:96) = 1)and subsequent absorption ( A (+) pL ) or emission ( A ( − ) pL ) ofan IR photon to the final angular momentum L , A ( ± ) pL ,is assumed to have three phase contributions: the phasedue to half-scattering at the centrifugal potential, (cid:96)π/ η ( ∓ ) (cid:96) = η (cid:96) ( E ∓ ω IR ), and the additional phase due tothe XUV-IR pump probe delay ± ω IR ∆ t . For a short-ranged “atomic” potential, the IR induced continuum-continuum transition of the outgoing wavepacket doesnot generate a significant additional phaseshift, unlikethe φ CC phase in the Coulomb potential.The angular resolved photoelectron spectrum for a given -3-2.5-2-1.5-1-0.50 50 60 70 80 90 ∆ Φ ( θ ) − ∆ Φ ( θ = ◦ ) [ r a d ] Emission angle θ [deg](a) 30 40 50 60 70 80 90Emission angle θ [deg](b)-60-40-20 0 10 20 30 40 50 60 70 80 τ cc [ a s ] Emission angle θ [deg](c)E=13.2 eVE=16.6 eVE=19.9 eVE=23.3 eVE=26.6 eV λ IR = 500 nm λ IR = 740 nm λ IR = 1480 nmE=19.9 eVE=36.7 eV FIG. 3. Angle dependence of the RABBITT phase ∆Φfor (a) different sideband energies E obtained for ionizationof hydrogen ( λ IR = 740 nm), (b) three different APTs andfundamental IR fields. The sideband energies are 16.15 eV( λ IR = 500 nm) and 16.5 eV ( λ IR = 740 nm, and 1480 nm).(c) Angle dependence of τ CC for ionization of hydrogencomparing the asymptotic prediction [Eq. (11)] (dashed)and numerical evaluation (symbols) for two energies and λ IR = 740 nm (for details see text). sideband with energy E = H n is proportional to P ( E = H n , θ ) ∝ | Ψ ( E, ∆ t, θ ) | (17)= A ( E, θ ) + 2 (cid:12)(cid:12)(cid:12) A (+) p (cid:12)(cid:12)(cid:12) B ( E, θ ) cos (2 ω IR ∆ t − ∆ η p ) , with ∆ η p = [ η p ( E + ω IR ) − η p ( E − ω IR )], and A ( E, θ ) = Y ( θ ) (cid:20)(cid:12)(cid:12)(cid:12) A (+) p (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) A ( − ) p (cid:12)(cid:12)(cid:12) (cid:21) (18)+ Y ( θ ) (cid:20)(cid:12)(cid:12)(cid:12) A (+) p (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) A ( − ) p (cid:12)(cid:12)(cid:12) (cid:21) + 2 Y ( θ ) Y ( θ ) (cid:104)(cid:12)(cid:12)(cid:12) A (+) p (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) A (+) p (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) A ( − ) p (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) A ( − ) p (cid:12)(cid:12)(cid:12)(cid:105) is independent of the XUV-IR delay ∆t. The amplitudeof the oscillation ∼ cos (2 ω IR ∆ t − ∆ η p ) is governed by B ( E, θ ) = aY ( θ ) + bc Y ( θ ) + Y ( θ ) Y ( θ ) c (1 + ab )(19)with a = (cid:12)(cid:12)(cid:12) A ( − ) p (cid:12)(cid:12)(cid:12) / (cid:12)(cid:12)(cid:12) A (+) p (cid:12)(cid:12)(cid:12) , (20) b = (cid:12)(cid:12)(cid:12) A (+) p (cid:12)(cid:12)(cid:12) / (cid:12)(cid:12)(cid:12) A ( − ) p (cid:12)(cid:12)(cid:12) , (21) c = (cid:12)(cid:12)(cid:12) A ( − ) p (cid:12)(cid:12)(cid:12) / (cid:12)(cid:12)(cid:12) A (+) p (cid:12)(cid:12)(cid:12) . (22) FIG. 4. Angular-resolved RABBITT phase ∆Φ obtained forionization from different initial states: (a) 2 p m states of neondescribed by a model potential [50], and (b) 3 d m states boundby a model potential V = − (cid:0) e − r + 14 e − . r (cid:1) /r . Thefinal electron energy in the sideband considered is 18.8 eV in(a) and 18.7 eV in (b), and λ IR =740 nm. As Y ( θ ) changes sign at the “magic” angle θ = 54 . ◦ , B ( E, θ ) may eventually change sign at the critical angle θ c > . ◦ given by the condition aY ( θ c ) + bc Y ( θ c ) = c (1 + ab ) Y ( θ c ) (cid:12)(cid:12) Y ( θ c ) (cid:12)(cid:12) . (23)This sign change results in a phase jump of the retrievedphase ∆Φ, observed in Fig. 2a for the Yukawa potential.The angle at which B ( E, θ ) changes sign, varies with thefinal photoelectron energy, and depends on the parame-ters a, b, c [Eq. (20)–Eq. (22)].Our simulations show that the critical angle increasesmonotonically with the sideband energy (Fig. 2b) whichis in line with previous results for helium [32, 39], neon[39], and argon [33, 34]. Further, the height of the phasejump for hydrogen, which is smaller than π due to thedependence of φ CC on the intermediate and final angu-lar momenta, approaches asymptotically π as the energyof the photoelectron is increased (Fig. 3a). Similarly, thephase jump approaches π when increasing the wavelengthof the IR field while keeping the sideband energy fixed(Fig. 3b). This is, again, an effect of the angular momen-tum dependence of the φ CC phase in the Coulomb field.Further, the emission angle at which the phase jump oc-curs increases with the IR wavelength, which is due tothe dependence of (cid:12)(cid:12)(cid:12) A ( ± ) pL (cid:12)(cid:12)(cid:12) on the IR wavelength. Ournumerical results for hydrogen allow the determinationof the angle dependence of τ CC by applying Eq. (8), i.e.subtracting τ EWS from the simulation results to obtain τ CC ( θ ) (Fig. 3c). Comparing these numerical results tothe asymptotic prediction τ asym CC [Eq. (11)], we find al-most perfect agreement for θ < ◦ . The rapid jumpobserved for higher emission angles is not reproduced bythe asymptotic expansion. As expected we find that forincreasing electron energy, the asymptotic τ asym CC agreeswith the simulation results up to larger emission angles.As the appearance of the phase jump is a direct conse-quence of the IR field induced partial wave interference, Y i e l d [ a r b . u .] Electron energy [eV](a) 20 25 30 35 40 00.511.52 Y i e l d [ a r b . u .] Electron energy [eV](b)20 25 30 35 40 00.511.52 Y i e l d [ a r b . u .] Electron energy [eV](b)20 25 30 35 40 00.511.52 Y i e l d [ a r b . u .] Electron energy [eV](b)He + ( n = 1) He + ( n = 2) [ × He + ( n = 3) [ × He + ( n = 4) [ × He + ( n = 2) He + ( n = 3) [ × FIG. 5. Electron spectra for different He + ( n ) correlationsatellites for photoionization of helium by (a) a 300 as FHWMduration XUV pulse with E XUV = 100 eV. (b) RABBITTspectrum for the n = 2 (blue) and n = 3 (red) channels and∆ t = 0. The high spectral resolution provided by RABBITTallows to spectrally select the sidebands (marked by arrows).The spectra for the different channels are scaled to enhancevisibility. its position and shape is sensitively dependent on theangular momentum of the initial bound state to be ion-ized. Starting from different initial (cid:96) , the two-photonprocess gives access to different superpositions of partialwaves. This is illustrated for ionization from the 2 p shellof neon, and from the 3 d shell bound by a model po-tential V = − (cid:0) e − r + 14 e − . r (cid:1) /r (Fig. 4). In allcases, a single-active electron approximation is employed.A simple systematic pattern emerges: all initial stateswith the largest angular momentum quantum number ofa given shell (cid:96) = n − | m | = n − s , 2 p ± ,3 d ± ) display a phase jump at a critical angle θ c ≥ ◦ .In each case, only two partial waves are accessible bythe final IR transition with only one of which featuringa spherical harmonic with a node for θ ∈ (0 ◦ , ◦ ), i.e. (cid:0) Y , Y , Y (cid:1) , giving rise to a change of sign. A qual-itatively different shape appears for those initial states(e.g. 2 p , 3 d ± ) which give rise to an interference amongthree partial waves. Here the phase increases by almost π at angles θ ≤ ◦ before reverting back close to zeronear 90 ◦ . For initial states where even more interferingpathways lead to the same final state (e.g. four inde-pendent pathways to three partial waves for ionizationof 3 d ) the peak in the phase excursion moves to muchsmaller θ ( (cid:39) ◦ ). Finally, after performing the subshellaverage (cid:104) ∆Φ (cid:105) n(cid:96) over all m states, the angular variationof the phase change reduces for all n(cid:96) to the simple phasejump of the node-free initial 1 s . This is a consequenceof the small cross sections in the range of large phase ex-cursions. These qualitative features are governed by thepartial-wave distribution in the continuum final state andare only marginally affected by the long-range tail of theCoulomb field. This non-Coulombic contribution to thecontinuum-continuum phase of RABBITT is key to un-derstand the dependence on emission angle. FIG. 6. Angle dependence of the time delay of (1 s ) directionization (blue) and for the 2 s correlation satellite (green).Shown are the total time delays τ ( t )1 s (blue squares) and τ ( t )2 s (green triangles) obtained from the simulations, as well astheir EWS (solid lines), cc (dotted lines), and e-e contribu-tions (dashed-dotted line). For the analytic prediction τ ana (dashed lines) using the additivity rule [Eq. (14)] we take τ asym CC [Eq. (11)]. τ CC ( θ ) is extracted from RABBITT simu-lations of helium in the single-active electron approximationfor final electron energies of 69.2 eV (1s) and 28.4 eV (2s). C. Angle dependence of the two-electron delay forshake-up ionization in helium
We turn now to the two-electron induced delay forphotoionization of helium accompanied by shake-up ex-citation of the bound electron [Eq. (12)]. For electronemission along the polarization direction, the time delayfor these correlation satellites has been theoretically [13]and experimentally investigated [44]. Here we presentthe first study of the angular variation of the time delayfor correlation satellites. We compare and contrast thebehavior of the θ dependence of τ e − e with that of τ EWS and τ CC .During shake-up ionization of a multi-electron system theionized electron interacts with the residual electron viaelectron-electron interaction and thus can promote exci-tations of the ion. Consequently, the kinetic energy of theemitted electron is reduced compared to the direct ion-ization where the residual ion stays in the ground state(“main line”). For the prototypical system of helium thisleads to the well-known correlation satellites in the pho-toelectron spectrum at energies E n = E XUV − I p, − (cid:0) E He + ( n ) − E He + (1) (cid:1) (24)= E XUV − I p, − n − n .E XUV is the energy of the absorbed XUV photon, I p, isthe first ionization potential for helium (0.904 a.u.), and E He + ( n ) = − n a.u. is the energy of the He + ( n ) residualion. If the spectral width of the ionizing pulse is narrowenough the peaks do not overlap (Fig. 5b). The spec- FIG. 7. Time delays for the n = 2 shake-up channels for thelowest energetic sideband (E=28.4 eV). (cid:104) τ (cid:105) n =2 ( θ ) for the full n = 2 shell, and resolved for the angular momentum eigen-states of the ion 2 s and 2 p . Electron correlation effects alsopopulate the ionic states 2 p ± contributing to the delay ofthe 2 p subshell (cid:104) τ (cid:105) p . Dashed lines represent the result fromEq. (14). tral overlap between different shake-up channels, how-ever, is an inherent challenge to attosecond streaking be-cause two effects limit spectral sensitivity. The spectralwidth of the single attosecond pulse of a few hundred at-toseconds duration used in the streaking protocol is, gen-erally, too broad to resolve different satellites for n ≥ E (∆ t ) = E n + √ E n A IR (∆ t ) by the moderately strongIR streaking field may become comparable to the spacingbetween adjacent satellites. This renders the analysis ofstreaking spectra in the presence of overlapping shake-upchannels quite difficult as was seen, e.g., for photoioniza-tion of neon [11, 54]. The RABBITT protocol offeringsimultaneously time- and energy resolution [15] promisesimproved access to time delay information of correlationsatellites. To analyze the shake-up delays for atomic he-lium accessible by RABBITT we employ time-dependent ab initio simulations [55, 56]. We choose an IR pulsewith wavelength λ IR = 740 nm, and an APT consistingof the 55 th , 57 th , 59 th , 61 st , and 63 rd harmonic. For moredetails see appendix. Due to the high spectral resolutionof RABBITT the electron spectra associated with themain line ( n = 1) for direct ionization and the correla-tion satellites ( n = 2 ,
3) are well resolved.For direct ionization the photoionization time delays arein very good agreement with single-active electron calcu-lations showing that electronic correlations do not have asignificant influence on the photoionization time delay inthis case [10, 44, 57]. For shake-up channels a drasticallydifferent picture emerges (Fig. 6). Here a single-activeelectron approximation fails and τ e − e plays an impor-tant role. The total time delay τ ( t )2 s becomes strongly an-gle dependent for the shake-up state at angles well belowthe critical angle θ c where the partial wave interferenceinduced phase jump occurs. For the 2 s state the criti-cal angle θ c is around ∼ ◦ and for the 1 s state it is > ◦ . Thus, the angular variation of τ ( t )2 s for θ < ◦ is exclusively due to the τ e − e ( θ ) contribution absent indirect ionization. For both direct ionization as well asshake-up ionization the additivity rule [Eq. (14)] appliesas confirmed by comparison between the full numericalsolution (symbols) and the analytic prediction (dashedlines) [Eqs. (11), (13)]. Note that we use the asymptoticprediction to calculate τ CC entering the additivity rulesince for ionization from s states the θ dependence of thecontinuum-continuum delay is negligible for θ < ◦ inthe investigated energy range.The n = 2 correlation satellite comprises 4 degenerateionic final states 2 s, p , p ± . Due to electron correla-tion the final residual ion can not only be in the He + (2 p )state, but also in the He + (2 p ± ) state, as only the to-tal magnetic moment of the atom, M = m + m , butnot the individual magnetic momenta of the electrons,( m , m ), is conserved for linearly polarized XUV andIR fields. Thus, the RABBITT traces for He(2 p ) con-tain an incoherent sum over these substates (Fig. 7). Weobserve that τ ( t ) for the full 2 p shell coincides with thetime delay of the 2 p state for low emission angles wherethe 2 p ± states have a nodal line at θ = 0 ◦ . For largeremission angles, however, the latter states become moreimportant and the averaged time delay increasingly dif-fers from that of the He + (2 p ) state.We find that the analytic prediction for τ ( t ) ( θ ) [Eq. (14)](dashed lines in Fig. 7) coincides quite well with the resultobtained from the simulation for all n = 2 final states.For small emission angles the agreement is almost per-fect. Separating the analytic prediction for the full n = 2shell into its different components, we find that for anglesbelow θ c the by far dominant contribution to the angu-lar variation is given by τ e − e ( θ ). For the cc phase weuse the analytic approximation for τ asym CC [Eq. (11)] [42].We speculate that the residual differences between theanalytic prediction and the full numerical result is dueto the asymptotic approximation to τ CC which neglectsthe residual angle dependence of continuum-continuumtransitions in the Coulomb field. The simulations lie sys-tematically below the analytic prediction for θ ≥ ◦ , inline with our observation of the angle dependence of τ CC for ionization of a p -shell electron and recent results pre-sented in literature [32, 34, 39]. The overall good agree-ment between the simulation and the analytic prediction,nevertheless, offers several qualitative insights. First, thephotoionization delay obtained by RABBITT for polar-izable targets can, similarly to attosecond streaking, beseparated into three different contributions. Second, thetwo-electron delay τ e − e is the by far dominant contribu-tion to the angle dependence of τ ( t ) for angles θ < ◦ where the cross section is still sizeable. Third, the angledependence of τ e − e is well captured by Eq. (13).The same qualitative trends can be observed for the n = 3shake-up channels (Fig. 8). Similar to the He + ( n = 2) FIG. 8. Time delays for the n = 3 shake-up channels forthe lowest energetic sideband (E=20.9 eV). (cid:104) τ (cid:105) n =3 ( θ ) for thefull n = 3 shell, and resolved for the angular momentum sub-shells (cid:104) τ (cid:105) p and (cid:104) τ (cid:105) d . Dashed lines represent the result fromEq. (14). shell, the retrieved time delays for the n = 3 shake-upchannels decrease monotonically with increasing emis-sion angle θ . This qualitative trend is also reproducedby the analytic prediction. With increasing n the mag-nitude of the delay substantially increases. For n = 3values of the order of 200 as are reached at intermedi-ate angles well below θ c . Unlike for n = 2, we observesystematic deviations between the approximate analyticpredictions and the numerical results. Most notable arethe differences for the 3 d shake-up channel already in for-ward direction ( θ = 0 ◦ ). One possible explanation is theenergetic proximity of the He + ( n = 3) shake-up chan-nels to the He + ( n = 4) channels (∆ E ≈ . τ e − e [Eq. (13)] in-cludes, however, only the intrashell coupling to the IRfield. Nevertheless, the qualitative trend of the angledependence of τ ( t ) is well captured by the analytic pre-diction for the averaged n = 3 shell and the 3 p shell, aswell as for the 3 s shell for θ < ◦ . Again, τ e − e is foundto be the by far dominant contribution to τ ( t ) . IV. FROM ANGULAR RESOLVED TO ANGLEINTEGRATED TIME DELAYS
Until recently, RABBITT experiments were mostlyconducted by collecting all emitted photoelectrons with amagnetic-bottle spectrometer and, thus, integrating overall emission angles. To connect these results to the an-gular resolved time delays investigated in this work, wepresent now photoionization time delays for RABBITTtraces partially integrated over angles up to an openingangle Θ max for atomic helium. We note that in manystreaking experiments electrons are collected in a conewith opening angle Θ streak around the IR polarizationdirection [21, 44], which resembles RABBITT spectra in- -40-30-20-1001020 20 40 60 80 100 120 140 160 180 T i m e d e l a y τ ( t ) [ a s ] Θ max [deg] n=1n=2n=3 FIG. 9. Total time delays as a function of the maximumcollection angle Θ max for direct ( n = 1) ionization of heliumand ionization accompanied by shake-up to n = 2 and n = 3.The final electron energy is ∼ + ( n = 1)channel, ∼ + ( n = 2) channel and ∼ + ( n = 3) channel. tegrated up to that angle.Time delays for direct ionization show no angle depen-dence for θ < ◦ . The steep phase drop for θ > ◦ ,however, is associated with a vanishingly small cross sec-tion. Therefore the n = 1 time delays for partially in-tegrated RABBITT traces are constant as a function ofΘ max (Fig. 9). A drastically different picture emerges forthe correlation satellites. The partially angle integratedspectra show a pronounced dependence on the maximumangle of integration Θ max . While τ ( t ) decreases mono-tonically up to Θ max (cid:39) ◦ for n = 2 and 3, its valueincreases for larger Θ max approaching at Θ max = 180 ◦ thevalue at Θ max = 90 ◦ . The reason for this is that for suf-ficiently long APTs and IR pulses RABBITT traces areforward-backward symmetric with respect to the electronemission angle (i.e. for θ → π − θ ), due to the interferencebetween partial waves with the same parity in the side-bands. If very short pulses or APTs consisting of evenand odd harmonics were used [58] this symmetry wouldbe broken.Partially integrated time delays from a RABBITT pro-tocol also allow a direct comparison with time delaysextracted from a streaking protocol. In general thesetwo protocols give access to different observables. At-tosecond streaking is strongly directional collecting onlyelectrons within an emission cone with typical openingangle, Θ streak < ◦ , about the forward direction colin-ear with the polarization axis. For delays that are Θ max independent such as for the direct ionization of helium(Fig. 9) streaking and angle-integrated RABBITT yieldthe same result for τ ( t ) . For correlation satellites signif-icant differences are expected. The difference betweenthe time delay for shake-up ( n ≥
2) and direct ionization( n = 1), τ ( n ≥ − τ ( n =1) , considerably varies with the ex-traction protocol utilized (Fig. 10). The angle-integratedand angle-resolved ( θ = 0 ◦ ) values differ by (cid:39)
13 as ata photon energy of 90 eV. We also find a striking differ-ence of approximately 7 as between the angle-integrated -30-25-20-15-10-50 95 100 105 110 T i m e s h i f t τ ( n ≥ ) − τ ( n = ) [ a s ] Photon energy [eV]Exp. streaking [44]Sim. streaking [44]Sim. streaking ( n = 2) [13]Sim. RABBITT ( n = 2) Sim. RABBITT θ = 0 ◦ ( n = 2) FIG. 10. Relative photoionization time delay τ ( n ≥ − τ ( n =1) between electrons in the shake-up channels [He + ( n > + ( n = 1)] for streaking and RABBITT.In RABBITT the shake-up channel He + ( n = 2) is spectrallywell isolated. Angle-integrated RABBITT traces (open greentriangles), simulations for n = 2 streaking (black dots), RAB-BITT traces evaluated in forward direction, i.e. θ = 0 ◦ (filledpurple triangles). RABBITT data and the streaking results (both, theoryand experiment) [44]. This difference is not caused bythe slightly different IR wavelength used by Ossianderet al. [44] (800 nm), which alters the time delay causedby the IR transition ( τ CC or τ CLC ) by less than 1 as.Rather the difference can be attributed to two distincteffects. First, due to the spectral width of the ionizingXUV pulse in streaking, it is experimentally not pos-sible to completely separate the contributions from theHe + ( n = 2) and He + ( n >
2) channels, see Fig. 5a. Thisadmixture lowers the effective time shift τ ( n ≥ − τ ( n =1) compared to τ ( n =2) − τ ( n =1) [13]. Second, the retrievedtime delay is strongly θ dependent for the shake-up chan-nels (see also Fig. 9) and, thus, a difference between thetime delay obtained from an angle-integrated RABBITTtrace and a streaking trace evaluated in forward direc-tion ( θ = 0 ◦ ) has to be expected. Consequently, onlythe time delay difference τ ( n =2) R − τ ( n =1) R obtained fromRABBITT traces evaluated in forward direction ( θ = 0 ◦ )agrees with streaking calculations for the same quantity,i.e. for the isolated n = 2 shake-up channel [13]. Eventhough completely disentangling the different shake-upchannels is possible for streaking only in the simulation,the agreement between the independent RABBITT andstreaking calculations confirms that the two methods do,indeed, accurately measure the same quantity for a com-plex multi-electron system. V. CONCLUDING REMARKS
We have shown that the RABBITT protocol is wellsuited to analyze the time delay in photoionization asa function of the emission angle of the ejected electronrelative to the polarization direction for shake-up ion-ization of helium. The exquisite spectral resolution al-0lows spectral selection of different correlation satelliteswith residual ionic states He + ( n ) for n = 2 ,
3. We findthat the angular variation of the delay is much more pro-nounced for shake-up channels than for the main line ofdirect ionization. This is due to the two-electron con-tribution to the time delay, τ e − e , by which the dipolarinteraction of the shaken-up polarizable bound electronwith the IR field imprints an additional phase on thetwo-electron wave function which manifests itself as anadditional phaseshift of the ionized electron. The vari-ation of τ e − e with θ dominates over the angle variationof the EWS or cc delays at angles where the cross sec-tion is still sizeable. This contribution therefore leaves itsmark on the angle-integrated time delay. Our numericalsimulations confirm that the additivity rules for differ-ent time delays extends to the two-electron contribution.Our present results are expected to be applicable to morecomplex multi-electron systems, in particular moleculesfeaturing permanent dipole moments [21, 45]. ACKNOWLEDGMENTS
We wish to acknowledge helpful discussions withFabian Lackner, Luca Argenti, Jan Marcus Dahlstr¨om,and David Busto. We gratefully acknowledge funding bythe FWF-DK 1243, the WWTF via project MA14-002,and the COST Action CA18222 - Attosecond Chemistry.S. Donsa thanks the International Max Planck ResearchSchool of Advanced Photon Science for financial support.The calculations have been performed on the Vienna Sci-entific Cluster (VSC).
Appendix A: Numerical details
In this appendix we briefly provide information on thenumerical details of the calculations shown in the maintext. For more details see [52].
1. Single-active electron calculations
We employ a pseudo-spectral method [26, 51] to solvethe time-dependent Schr¨odinger equation (TDSE) inlength gauge, and expand the three-dimensional wavefunction into spherical harmonics. The maximal size ofthe radial box was 4417 a.u., where the radial degreeof freedom is discretized using the finite-element discretevariable representation (FEDVR) using up to 15 finite el-ements for each FEDVR element spanning 4 a.u. close tothe core and 5 a.u. for r >
24 a.u. An absorbing bound-ary was used to avoid reflections of the wave function atthe boundary. We achieved converged results when in-cluding angular momenta up to L max = 8.The short-ranged Yukawa potential is given by V Y ( r ) = − . r e − r , (A1) with an ionization potential of 0.5 a.u. For the he-lium and neon single-active electron calculations we use apseudo-potential which correctly describes the ionizationpotential [50]. To analyze the angle dependence of thecontinuum-continuum delay τ CC for the ionization of a d shell electron we design a single-active electron potential,where the initial 3 d state is energetically well separatedfrom all other bound states. The latter potential is givenby V ( r ) = − e − r + 14 e − . r r . (A2)The energetically lowest bound states of this potentialare given in Tab. I.The full-width at half maximum duration of the IR(XUV) pulses was chosen to be 20 (15) fs and the cor-responding peak intensities were well in the perturba-tive regime, i.e. I IR = 2 × W/cm and I XUV < × W/cm . n s p d f
2. Parameters for the ab initio helium calculations
For the helium shake-up calculations we use two-electron calculations from first principles [55, 56]. Briefly,we solve the six-dimensional TDSE for atomic helium us-ing the time-dependent close-coupling expansion and dis-cretizing the radial wave functions on a spatial FEDVRgrid. For the temporal propagation we employ the short-iterative Lanczos algorithm with adaptive time-step con-trol. Spectral information is extracted by projecting thesix-dimensional wave function onto products of uncorre-lated Coulomb wave functions. We use an asymmetricbox where the bigger (smaller) radial size is 3857 a.u.(37 a.u.) with 11 basis function for every finite elementspanning 4 a.u. close to the core and 5 a.u. for r > L max = 3, (cid:96) = (cid:96) = 9.We choose an IR pulse with wavelength λ IR = 740 nm,FWHM duration of 20 fs and peak intensity of 2 × W/cm . 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