Electron Vortices in Photoionization by Circularly Polarized Attosecond Pulses

J. M. Ngoko Djiokap,1 S. X. Hu,2 L. B. Madsen,3 N. L. Manakov,4 A. V. Meremianin,4 and Anthony F. Starace11Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588-0299, USA2Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623-1299, USA

3Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark4Department of Physics, Voronezh State University, Voronezh 394006, Russia

(Received 4 May 2015; revised manuscript received 1 July 2015; published 10 September 2015)

Single ionization of He by two oppositely circularly polarized, time-delayed attosecond pulses is shownto produce photoelectron momentum distributions in the polarization plane having helical vortex structuressensitive to the time delay between the pulses, their relative phase, and their handedness. Results areobtained by both ab initio numerical solution of the two-electron time-dependent Schrödinger equation andby a lowest-order perturbation theory analysis. The energy, bandwidth, and temporal duration of attosecondpulses are ideal for observing these vortex patterns.

DOI: 10.1103/PhysRevLett.115.113004 PACS numbers: 32.80.Fb, 02.70.Dh, 02.70.Hm, 03.75.-b

Ramsey interference [1] of laser-produced electron wavepackets has been investigated in both Rydberg states [2,3]and in the continuum [4] for the case of linearly polarizedlasers. In this Letter we investigate an unusual kind ofRamsey interference between photoelectron wave packetsproduced in ionization of the helium atom by a pair oftime-delayed, oppositely circularly polarized, attosecondlaser pulses. In the polarization plane, the photoelectronmomentum distributions exhibit helical vortex patterns,corresponding to Fermat spirals, whose handedness dependson whether the pair of attosecond pulses are right-left or left-right circularly polarized. Observation of the vortex patternsrequires the large bandwidth of few-cycle attosecond pulses.Moreover, the vortex patterns in the photoelectron momen-tum distributions that we predict have a counterpart in optics[5], in which similar vortex patterns have been produced byinterference of particular kinds of laser beams. Our predictedphotoelectron momentum distributions thus provide anexample of wave-particle duality, and our predicted angulardistributions demonstrate an unusual kind of control overphotoelectrons on an attosecond time scale.Consider the interaction of the helium atom in its 1Se

ground state with the electric field FðtÞ of a pair ofattosecond pulses having the same carrier frequency ωand smooth pulse envelope F0ðtÞ, but differing in theirpolarizations e1;2 and carrier envelope phases (CEPs) ϕ1;2,with the second pulse delayed in time by τ, i.e.,

FðtÞ ¼ F1ðtÞ þ F2ðt − τÞ≡ F0ðtÞRe½e1e−iðωtþϕ1Þ�þ F0ðt − τÞRe½e2e−iðωðt−τÞþϕ2Þ�: ð1Þ

For the jth pulse (j ¼ 1; 2), the polarization vector is

ej ≡ ðϵþ iηjζÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ η2j

q

, where ηj is the ellipticity

(−1 ≤ ηj ≤ þ1), ϵ and ζ ≡ k × ϵ are the major and minor

axes of the polarization ellipse, and k∥z is the pulse

propagation direction. The degrees of linear and circularpolarization of the jth pulse are, respectively, lj ≡ ðej ·ejÞ ¼ ð1 − η2jÞ=ð1þ η2jÞ and ξj ≡ Im½e�j × ej�z ¼ 2ηj=ð1þ η2jÞ, where l2

j þ ξ2j ¼ 1. We solve the two-electrontime-dependent Schrödinger equation (TDSE) for anelectric field FðtÞ of intensity 1014 W=cm2 and carrierfrequency ω ¼ 1.323 a:u:, which is greater than the Hebinding energy, Eb ¼ 0.9037 a:u: Each pulse has atemporal envelope F0ðtÞ ¼ F0cos2ðπt=TÞ with−T=2 ≤ t ≤ T=2, where T ≡ npð2π=ωÞ ¼ 344 as is thetotal pulse duration for np ¼ 3 optical cycles. This corre-sponds to a spectral width [6] Δω≃ 1.44ω=np ≈ 17 eV,and a FWHM in the intensity of 0.36T, or 1.1 cycles. Suchsingle-cycle attosecond pulses (having linear polarization)have been achieved experimentally [7–9], albeit with muchlower intensity. However, the vortices we predict involvesingle-photon ionization processes; thus, the vortex pat-terns will occur for lower intensity pulses, although thephotoelectron count rate will scale linearly with intensity.We obtain the wave packet ΨðtÞ by solving the

six-dimensional two-electron TDSE for the He atominteracting with two circularly polarized attosecond pulsesusing methods developed previously [6] for a singlearbitrarily polarized attosecond pulse. Our method usesthe finite-element discrete-variable representation com-bined with the real-space-product algorithm [10] as wellas Wigner rotation transformations at each time step fromthe laboratory frame to the frame of the instantaneouselectric field [11,12]. We extract the triply differentialprobability (TDP) [13] for single ionization of He toHeþð1sÞ from the wave packet ΨðT þ τÞ (i.e., after theend of the two pulses) by projecting ΨðT þ τÞ ontocorrelated field-free Jacobi matrix wave functions [14].The TDP, d3W=d3p≡Wξ1

ξ2ðpÞ, for single electron ioniza-

tion to the continuum with momentum p≡ ðp; θ;φÞ is

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d3Wd3p

≡ jhΘð−Þ1s ðr1; r2;pÞjΨðr1; r2; eϕ1

; eϕ2; T þ τÞij2; ð2Þ

where eϕj≡ eje−iϕj , with j ¼ 1; 2. We include four total

angular momenta (L ¼ 0 − 3), their azimuthal quantumnumbers jMj ≤ L, and all combinations of individualelectron orbital angular momenta l1; l2 ¼ 0–5. Regardlessof time delay τ, our numerical results show that only M ¼þL (respectively, M ¼ −L) states are populated for twoidentical pulses with ξ1 ¼ ξ2 ¼ þ1 (respectively, ξ1 ¼ξ2 ¼ −1). For two oppositely circularly polarized pulses(with ξ1 ¼ −ξ2 ¼ �1) only M ¼ �L states are populated.In all cases, 1Po states give the dominant contribution (by 2orders of magnitude) thus indicating our results stem fromsingle photon ionization processes.For our pulse parameters, first-order perturbation theory

(PT) applies. The PT analysis of the problem, given in theSupplemental Material [15], is similar to that for ionizationby a single pulse [19]. Below we present results of ourPT analysis for two cases: (i) the two pulses have the samepolarization e1 ¼ e2; (ii) the pulses are circularly polarizedin opposite directions, e1 ¼ e�2.For two identically polarized pulses, l1 ¼ l2 ¼ l and

ξ1 ¼ ξ2 ¼ ξ. The TDP d3W=d3p is thus (see SupplementalMaterial [15])

Wξξðp; θ;φÞ ¼

3Wp

2πð1þ l cos 2φÞsin2θcos2ðΦ=2Þ; ð3Þ

where the relative phase, Φ, is given by

Φ ¼ ðEþ EbÞτ þ ðϕ1 − ϕ2Þ≡ ðp2=2þ EbÞτ þ ϕ12: ð4Þ

The dynamical parameter Wp in Eq. (3) depends only onthe electron energy, E ¼ p2=2; it is independent of themomentum direction, p, and the pulse parameters e1, e2, τ,and ϕ12 ¼ ϕ1 − ϕ2. Consequently, the φ dependence ofthe TDP Eq. (3) is determined by the degree of linearpolarization l. For circularly polarized pulses η ¼ �1,ξ ¼ �1, and l ¼ 0. Hence the TDP Eq. (3) is independentof φ and its polar angle plots in the polarization plane(θ ¼ π=2) have circularly symmetric patterns, as shown inFig. 1 for two right-circularly polarized pulses with differ-ent CEPs for two time delays: τ ¼ 0 and τ ¼ 500 as.Owing to the dependence of the relative phase Φ on thetime delay τ [cf. Eq. (4)], for τ ¼ 0 there is no structure inthe momentum distribution. For τ ¼ 500 as, however, theRamsey interference of the two electronic wave packetshas a form similar to Newton’s rings, i.e., maxima andminima along the radial direction in momentum space.(The interference pattern in Fig. 1(b) is similar to that foundin interference of two identical optical beams [5].)Although the two pulses have different CEPs, owing tothe circular symmetry, varying the relative CEP ϕ12 onlychanges the magnitude of the differential probability

Wξξðp; π=2;φÞ but has no effect on the pattern in the

polarization plane. Averaging Eq. (3) over p, we obtain thesingle differential probability dW=dE¼pWp½2cosðΦ=2Þ�2.Hence, the ionization probability for fixed ϕ12 can becontrolled by varying the time delay τ [cf. Eq. (4)].For oppositely circularly polarized pulses, e�1 ¼ e2,

l1 ¼ l2 ¼ 0, and ξ1 ¼ −ξ2 ¼ �1. The TDP d3W=d3ptakes the form (see Supplemental Material [15]):

Wξ1ξ2ðp; θ;φÞ ¼ 3Wp

2πsin2θcos2ðΦ=2 − ξ1φÞ; ð5Þ

where ξ1 ¼ −ξ2 ¼ �1 corresponds to a right-left (þ) or aleft-right (−) pair of oppositely circularly polarized pulses.In contrast to the case of identical pulses, the angle-averaged probability, dW=dE ¼ 2pWp, depends neitheron the time delay, τ, nor the relative CEP, ϕ12. Thephotoelectron angular distribution, Eq. (5), in the polari-zation plane (θ ¼ 90°) has the form of a two-start or two-arm spiral structure, as may be seen from the followingconsiderations. From Eq. (5), the TDP Wξ1

ξ2is maximal for

Φ=2 − ξ1φ ¼ Φ=2þ ξ2φ ¼ πn and is zero for Φ=2−ξ1φ¼Φ=2þξ2φ¼ð2nþ1Þπ=2, where n¼ 0;�1;�2;…,and 0 ≤ φ ≤ 2π. Using Eq. (4), the p dependence of thepolar angles φ at these maximum and zero values ofWξ1

ξ2ðp; θ;φÞ is

φmaxn ðpÞ ¼ ξ2½πn − ðτEb þ ϕ12Þ=2 − τp2=4�;

φzeron ðpÞ ¼ ξ2½π=2þ πn − ðτEb þ ϕ12Þ=2 − τp2=4�: ð6Þ

Equations (6) define Fermat (or Archimedean) spirals (orhelixes) in the ðp;φÞ plane. As φmax

n ðpÞ and φzeron ðpÞ,

shifted by the angle π=2 with respect to each other, varywith energy p2=2 (through possibly many 2π cycles,depending upon τ), they trace out the maxima and thezeros of the TDP. Since jξ2j ¼ 1, the pattern is a two-armhelical spiral, corresponding to n ¼ 0; 1, as other values of

px(a.u.)

py(

a.u

.)

-2

-1

0

1

2

0.001 0.002 0.003 0.005 0.010

Right/Right

(a)

τ=0 asφ1=0 φ2=π/2

Τ=344 as

px(a.u.)

py(

a.u

.)

-2 -1 0 1 2 -2 -1 0 1 2-2

-1

0

1

2

0.001 0.001 0.002 0.003 0.007

Right/Rightτ=500 as φ2=π/2φ1=0

(b) Τ=344 as

FIG. 1 (color online). Triply differential probability (TDP)d3W=d3p [see Eqs. (2), (3), and (4)] in the polarization planefor ionization of He by two right-circularly polarized pulsesdelayed in time by (a) τ ¼ 0; and (b) τ ¼ 500 as. Each pulse hascarrier frequency ω ¼ 36 eV, np ¼ 3 cycles with total durationT ¼ 344 as, intensity I ¼ 1014 W=cm2, and CEPs of ϕ1 ¼ 0 andϕ2 ¼ π=2. The magnitudes in a.u. of the TDPs are indicated bythe color scales in each panel.

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113004-2

n replicate the same lines. Pulses with ξ1 ¼ −ξ2 ¼ �1correspond to right- (þ) or left- (−) handed spirals. TheFermat spirals become wound more densely as τ increases.Our numerical results for these PT predictions for two

oppositely circularly polarized few-cycle attosecond pulsesare shown in Figs. 2 and 3, where we plot the photoelectronmomentum distributions in the polarization plane forvarious CEPs and time delays. Figures 2(a) and 2(b) arefor zero time delay between the pulses and two relativeCEPs ϕ12. For τ ¼ 0, superposing two oppositely circularlypolarized pulses gives a linearly polarized pulse and weobserve in Fig. 2(a) the expected symmetric dipole pattern[∝ cos2ðξ1φÞ] of the ionized electron momenta along thelinear polarization axis, which for ϕ12 ¼ 0 is the px axis(φ ¼ 0; π). A similar result is shown in Fig. 2(b) except thathere ϕ12 ¼ −π=2 so that the linear polarization axis isrotated clockwise by φ ¼ π=4, giving an angular distribu-tion ∝ cos2ðϕ12=2 − ξ1φÞ. For ϕ12 ≠ 0, a change in sign ofξ1 will change the angular distribution, unlike whenϕ12 ¼ 0. This sensitivity to the helicity of ξ1 is essentialfor producing vortices when the time delay is nonzero.For a time delay of τ ¼ 500 as and ϕ12 ¼ −π=2, we

obtain the vortices shown in Figs. 2(c) and 2(d) for right-left and left-right circularly polarized pulses, respectively.As discussed above, these are two-start Fermat spiralpatterns with opposite handedness, i.e., counterclockwisein Fig. 2(c) and clockwise in Fig. 2(d). (For a 3D plot of thevortex pattern in Fig. 2(c), see Fig. 1 of the SupplementalMaterial [15].) As predicted by PT Eq. (6), the number and

locations of the maxima and minima of the differentialprobabilityWξ1

ξ2ðp; π=2;φÞ in the polarization plane depend

on the time delay τ, as shown in Fig. 3 for four additionalvalues of τ. The results in Fig. 3 show that time delays ofseveral hundred attoseconds are necessary to observe well-defined vortex patterns. As shown in the SupplementalMaterial [15], the broad bandwidth of attosecond pulsesis necessary to observe the spiral patterns clearly.Vortices in the probability distribution for a physical

process are quite general phenomena [20,21]. The con-nection between a zero of the probability distribution and azero of the system wave function is given by the so-called“imaging theorem” [21]. Velocity field vortices have beenintensively studied for collision processes involving ion-ization of atoms and molecules by electron impact [21–23],proton impact [24], positron impact [25], ion impact [26],and both antiproton and photon impact [27]. Also, vorticesassociated with population transfers to excited and con-tinuum states have been studied in the electron probabilitydensity of an atom subject to short linearly-polarized electricfields [27,28]. However, as noted by I. Bialynicki-Birulaet al. [20], “We have to admit that vortex lines associatedwith wave functions are rather elusive objects.” In particular,vortex lines in impact ionization amplitudes are not explicitlyseen as vortices in the angular distributions. In contrast,for ionization by two oppositely circularly polarized pulses,the TDP vanishes along the two Fermat spiral arms in theðpx; pyÞ plane [cf. Eq. (6) for φzero

n ðpÞ for n ¼ 0; 1]. Thesecurves are thus a pair of vortex lines in the velocity fieldof the photoelectron wave function [21].

px(a.u.)

py(

a.u

.)

-2

-1

0

1

2

0.002 0.003 0.006 0.010 0.018

(a)

φ1=0 φ2=0Right/Left

τ=0 as

Τ=344 as

px(a.u.)

py(

a.u

.)

-2

-1

0

1

2

0.002 0.004 0.006 0.011 0.019

(b)

Right/Leftφ1=0 φ2=π/2τ=0 as

Τ=344 as

px(a.u.)

py(

a.u

.)

-2

-1

0

1

2

0.002 0.004 0.006 0.011 0.019

φ2=π/2φ1=0

(c)

Right/Leftτ=500 as

Τ=344 as

px(a.u.)

py(

a.u

.)

-2 -1 0 1 2 -2 -1 0 1 2

-2 -1 0 1 2 -2 -1 0 1 2-2

-1

0

1

2

0.002 0.004 0.006 0.011 0.019

Left/Right

(d)

φ2=π/2φ1=0 τ=500 as

Τ=344 as

FIG. 2 (color online). Triply differential probability (TDP)d3W=d3p [see Eqs. (2), (4), and (5)] in the polarization planefor He ionization by (a),(b),(c) right-left and (d) left-rightcircularly polarized attosecond pulses. Top row results: timedelay τ ¼ 0; bottom row results: τ ¼ 500 as. In (a), ϕ1 ¼ ϕ2 ¼ 0;in (b), (c), (d), ϕ1 ¼ 0, ϕ2 ¼ π=2. In all panels ω ¼ 36 eV,np ¼ 3 cycles, I ¼ 1014 W=cm2, and the magnitudes in a.u. ofthe TDPs are indicated by the color scales.

px (a.u.)

py

(a.u

.)

-2

-1

0

1

2

0.002 0.004 0.006 0.011 0.019

Right/Leftτ=50 as

(a)

φ1=0 φ2=π/2

Τ=344 as

px (a.u.)

py

(a.u

.)

-2

-1

0

1

2

0.002 0.004 0.006 0.011 0.019

Right/Leftτ=150 as

(b)

φ1=0 φ2=π/2

Τ=344 as

px(a.u.)

py(

a.u

.)

-2

-1

0

1

2

0.002 0.004 0.006 0.011 0.019

Right/Leftτ=373 asφ1=0 φ2=π/2

(c) Τ=344 as

px(a.u.)

py(

a.u

.)

-2 -1 0 1 2 -2 -1 0 1 2

-2 -1 0 1 2 -2 -1 0 1 2-2

-1

0

1

2

0.002 0.003 0.006 0.011 0.019

Right/Leftτ=976 as φ2=π/2φ1=0

(d) Τ=344 as

FIG. 3 (color online). Photoelectron momentum distributionsd3W=d3p [see Eqs. (2), (4), and (5)] in the polarization plane forionization of He by right-left circularly polarized attosecondpulses delayed in time by (a) τ ¼ 50 as; (b) τ ¼ 150 as;(c) τ ¼ 373 as; and (d) τ ¼ 976 as. Pulse parameters are thesame as in Figs. 2(b) and 2(c). The magnitudes in a.u. of the TDPsare indicated by the color scales.

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The photoelectron vortex patterns in our TDPs in Figs. 2and 3 are similar to the interference fringes of optical beamscarrying orbital angular momentum of unity [5]. In bothcases the interference patterns are two-start Fermat spiralswhose orientation is determined by a relative phase differ-ence. In our case, this phase difference Φ is determined bythe energy ðp2=2þ EbÞ, time delay τ, and ϕ12, while in theoptical case the relative phase is determined by the wavefront curvature difference [5] (see Supplemental Material[15]). In neither case is the appearance of helical fringescaused by the polarization of either the optical or electronicwaves. Indeed, in the optical case both light beams werelinearly polarized [5], and in our electronic case, theelectron states with Lz ¼ �1 are essentially equally popu-lated upon ionization by the pair of oppositely circularlypolarized attosecond pulses.Figure 4 shows the time-delay periodicity of the photo-

electron angular distributions in the polarization plane.Results of our numerical solutions of the TDSE inFig. 4(a) are compared with the PT results in Fig. 4(b) fora fixed electron kinetic energy, E1 ¼ ω − Eb. For a fixedrelative CEP ϕ12 between the two pulses, the angulardistribution is unchanged for time delays of τn ¼ nπ=ðEþEbÞ with n an even integer, as expected from the PT Eqs. (4)and (5). Figure 4(b) shows this PT result to be valid: theangular distributions for τ0 and τ10 are identical. Ournumerical results in Fig. 4(a) for these two time delaysare nearly the same. This indicates the PTassumption that thesecond pulse sees the same initial state as the first pulse (i.e.,the He ground state), instead of the state resulting frominteraction of the He ground state with the first attosecondpulse, is valid to a very good approximation (especially giventhe greater sensitivity of differential probability results totheoretical approximations as compared to results for total

probabilities). For time delays τn with odd integer n, the PTEqs. (5) and (4) predict the angular distributions to be shiftedby π=2 with respect to those for even integers n, as shown inFigs. 4(a) and 4(b). This sensitivity of the angular distribu-tions to the time delay implies the ability to control thedirection of ionization of electrons by adjusting the timedelay between the two attosecond pulses.Experimental observation of these vortex patterns in the

photoelectron momentum distributions requires a pair ofoppositely circularly polarized attosecond pulses with evenlow intensity but with control of the relative CEP and thetime delay between the two pulses. The spectral widthΔω ofeach pulse should span the energies of several spiral fringes,i.e., 2π=τ < Δω [see Eqs. (4), (5), and (6)]. Generation ofcircularly polarized harmonics in the extreme ultravioletenergy regime has recently been achieved [29]. Velocity mapimaging or reaction microscope techniques can be usedto measure the photoelectron momentum distribution. Thesensitivity of these vortex patterns to the parameters of thepair of attosecond pulses makes them an ideal means ofcharacterizing these pulses and of timing ultrafast processes.Finally, the He atom and other light atoms such as H, Li, andBe are ideal targets owing to the fact that they have onlyoccupied subshells of l ¼ 0 electrons, thus obviating (in thecase of heavier atoms) effects of ionization of two ormore subshells having different angular momenta by broadbandwidth attosecond pulses.

A. F. S. and J. M. N. D. gratefully acknowledge usefuldiscussions with Cornelis Uiterwaal on optical vortices.Research of A. F. S. and J. M. N. D. is primarily supportedby the U.S. Department of Energy (DOE), Office ofScience, Basic Energy Sciences (BES), under AwardNo. DE-FG03-96ER14646; research of S. X. H. is sup-ported by the DOE National Nuclear SecurityAdministration under Award No. DE-NA0001944, theUniversity of Rochester, and the New York State EnergyResearch and Development Authority; research of L. B. M.is supported by ERC-StG (Project No. 277767—TDMET)and the VKR Center of Excellence, QUSCOPE; research ofN. L. M. and A. V.M. is supported by Russian Foundationfor Basic Research Grant No. 13-02-00420 and by Ministryof Education and Science of the Russian Federation Project1019. Computations were carried out using the Sandhills,Tusker, and Crane computing facilities of the HollandComputing Center at the University of Nebraska-Lincoln,as well as the Stampede supercomputer at the TexasAdvanced Computing Center under NSF Grant No. TG-PHY-120003.

[1] N. F. Ramsey, A molecular beam resonance method withseparated oscillating fields, Phys. Rev. 78, 695 (1950).

[2] L. D. Noordam, D. I. Duncan, and T. F. Gallagher, Ramseyfringes in atomic Rydberg wave packets, Phys. Rev. A 45,4734 (1992).

0

30

60

90

120

150

180

210

240

270

300

330

0 2 4 6

τ0

τ17

τ10

(a) TDSE

0

30

60

90

120

150

180

210

240

270

300

330

0 2 4 6

τ0

τ17τ10

(b) PT

FIG. 4 (color online). Angular distributions d3W=d3p (in unitsof 10−3 a:u:) for a fixed photoelectron energy E1 ¼ ω − Eb,produced by a pair of right-left circularly polarized attosecondpulses. Results in panel (a) are obtained by ab initio TDSEcalculations [see Eq. (2)]; results in panel (b) are obtained usingthe PT formula (5) in which the dynamical parameter Wp isdetermined from our TDSE results for a single right-circularlypolarized attosecond pulse including only the 1Po final stateamplitude. In each panel, results are shown for three time delays:τ0, τ10, and τ17, where τn ¼ nπ=ω.

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[3] M. Strehle, U. Weichmann, and G. Gerber, Femtosecondtime-resolved Rydberg wave-packet dynamics in the two-electron system calcium, Phys. Rev. A 58, 450 (1998).

[4] M. Wollenhaupt, A. Assion, D. Liese, Ch. Sarpe-Tudoran,T. Baumert, S. Zamith, M. A. Bouchene, B. Girard, A.Flettner, U. Weichmann, and G. Gerber, Interferences ofUltrashort Free Electron Wave Packets, Phys. Rev. Lett. 89,173001 (2002).

[5] M. Harris, C. A. Hill, and J. M. Vaughan, Optical helicesand spiral interference fringes, Opt. Commun. 106, 161(1994).

[6] J. M. Ngoko Djiokap, N. L. Manakov, A. V. Meremianin,S. X. Hu, L. B. Madsen, and A. F. Starace, NonlinearDichroism in Back-to-Back Double Ionization of He byan Intense Elliptically Polarized Few-Cycle ExtremeUltraviolet Pulse, Phys. Rev. Lett. 113, 223002 (2014).

[7] G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi,R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta,S. Stagira, S. De Silvestri, and M. Nisoli, Isolated single-cycle attosecond pulses, Science 314, 443 (2006).

[8] E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev,J. Gagnon, M. Uiberacker, A. L. Aquila, E.M. Gullikson,D. T. Atwood, R. Kienberger, F. Krausz, and U. Kleineberg,Single-cycle nonlinear optics, Science 320, 1614 (2008).

[9] K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z.Chang, Tailoring a 67 attosecond pulse through advanta-geous phase-mismatch, Opt. Lett. 37, 3891 (2012).

[10] S. X. Hu, Optimizing the FEDVR-TDCC code for exploringthe quantum dynamics of two-electron systems in intenselaser pulses, Phys. Rev. E 81, 056705 (2010).

[11] H. G. Muller, An efficient propagation scheme for the time-dependent Schrödinger equation in the velocity gauge,Laser Phys. 9, 138 (1999).

[12] T. K. Kjeldsen, L. A. A. Nikolopoulos, and L. B. Madsen,Spectral and partial-wave decomposition of time-dependentwave functions on a grid: Photoelectron spectra of H and Hþ

2

in electromagnetic fields, Phys. Rev. A 75, 063427 (2007).[13] L. B. Madsen, L. A. A. Nikolopoulos, T. K. Kjeldsen, and J.

Fernández, Extracting continuum information from ΨðtÞ intime-dependent wave-packet calculations, Phys. Rev. A 76,063407 (2007).

[14] J. M. Ngoko Djiokap, S. X. Hu, W.-C. Jiang, L.-Y. Peng,and A. F. Starace, Enhanced asymmetry in few-cycle atto-second pulse ionization of He in the vicinity of autoionizingresonances, New J. Phys. 14, 095010 (2012).

[15] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.115.113004 for a moredetailed derivation of our PT equations and for an illustrationof the time-delay sensitivity of the photoelectron energy

distributions; the Supplemental Material includesRefs. [16–18].

[16] V. B. Berestetskii, E.M. Lifshitz, and L. P. Pitaevskii, Quan-tum Electrodynamics, 2nd ed. (Pergamon Press, Oxford,1982).

[17] A. S. Davydov, Quantum Mechanics, 2nd ed. (PergamonPress, Oxford, 1976).

[18] A. F. Starace, Theory of Atomic Photoionization, in Hand-buch der Physik, edited by W. Mehlhorn, Vol. 31 (Springer-Verlag, Berlin, 1982), pp. 1–121.

[19] E. A. Pronin, A. F. Starace, M. V. Frolov, and N. L.Manakov, Perturbation theory analysis of attosecond photo-ionization, Phys. Rev. A 80, 063403 (2009).

[20] I. Bialynicki-Birula, Z. Bialynicka-Birula, and C. Sliwa,Motion of vortex lines in quantum mechanics, Phys. Rev. A61, 032110 (2000).

[21] S. J. Ward and J. H. Macek, Effect of a vortex in the triplydifferential cross section for electron-impact K-shell ioniza-tion of carbon, Phys. Rev. A 90, 062709 (2014).

[22] J. H. Macek, J. B. Sternberg, S. Y. Ovchinnikov, and J. S.Briggs, Theory of Deep Minima in (e; 2e) Measurements ofTriply Differential Cross Sections, Phys. Rev. Lett. 104,033201 (2010).

[23] J. M. Feagin, Vortex kinematics of a continuum electronpair, J. Phys. B 44, 011001 (2011).

[24] J. H. Macek, J. B. Sternberg, S. Y. Ovchinnikov, T.-G. Lee,and D. R. Schultz, Origin, Evolution, and Imaging ofVortices in Atomic Processes, Phys. Rev. Lett. 102,143201 (2009).

[25] F. Navarrete, R. Della Picca, J. Fiol, and R. O. Barrachina,Vortices in ionization collisions by positron impact, J. Phys.B 46, 115203 (2013).

[26] L. Ph. H. Schmidt, C. Goihl, D. Metz, H. Schmidt-Böcking,R. Dörner, S. Yu. Ovchinnikov, J. H. Macek, and D. R.Schultz, Vortices Associated with the Wave Function of aSingle Electron Emitted in Slow Ion-Atom Collisions, Phys.Rev. Lett. 112, 083201 (2014).

[27] S. Y. Ovchinnikov, J. H. Macek, and D. R. Schultz, Hydro-dynamical interpretation of angular momentum and energytransfer in atomic processes, Phys. Rev. A 90, 062713 (2014).

[28] S. Y. Ovchinnikov, J. B. Sternberg, J. H. Macek, T.-G. Lee,and D. R. Schultz, Creating and Manipulating Vortices inAtomic Wave Functions with Short Electric Field Pulses,Phys. Rev. Lett. 105, 203005 (2010).

[29] O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D.Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A.Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, Gen-eration of bright phase-matched circularly polarized extremeultraviolet high harmonics, Nat. Photonics 9, 99 (2014).

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