New Method for Measuring Angle-Resolved Phases in Photoemission

PHYSICAL REVIEW X 10, 031070 (2020)

Daehyun You ,1 Kiyoshi Ueda,1,* Elena V. Gryzlova ,2 Alexei N. Grum-Grzhimailo ,2 Maria M. Popova ,2,3 Ekaterina I. Staroselskaya,3 Oyunbileg Tugs,4 Yuki Orimo ,4 Takeshi Sato ,4,5,6 Kenichi L. Ishikawa ,4,5,6 Paolo Antonio Carpeggiani ,7 Tamás Csizmadia,8 Miklós Füle,8 Giuseppe Sansone ,9 Praveen Kumar Maroju,9 Alessandro D’Elia ,10,11 Tommaso Mazza,12 Michael Meyer ,12 Carlo Callegari ,13 Michele Di Fraia,13 Oksana Plekan ,13 Robert Richter,13 Luca Giannessi,13,14 Enrico Allaria,13 Giovanni De Ninno,13,15 Mauro Trovò,13 ‡ Laura Badano,13 Bruno Diviacco ,13 Giulio Gaio,13 David Gauthier,13, Najmeh Mirian,13 Giuseppe Penco,13 † Primož Rebernik Ribič ,13,15 Simone Spampinati,13 Carlo Spezzani,13 and Kevin C. Prince 13,16, 1Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan 2Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia 3Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia 4Department of Nuclear Engineering and Management, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 5Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 6Research Institute for Photon Science and Laser Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 7Institut für Photonik, Technische Universität Wien, 1040 Vienna, Austria 8ELI-ALPS, ELI-HU Non-Profit Limited, Dugonics t´er 13, H-6720 Szeged, Hungary 9Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, 79106 Freiburg, Germany 10University of Trieste, Department of Physics, 34127 Trieste, Italy 11IOM-CNR, Laboratorio Nazionale TASC, 34149 Basovizza, Trieste, Italy 12European X-Ray Free Electron Laser Facility GmbH, Holzkoppel 4, 22869 Schenefeld, Germany 13Elettra-Sincrotrone Trieste S.C.p.A., 34149 Basovizza, Trieste, Italy 14INFN-Laboratori Nazionali di Frascati, 00044 Frascati, Rome, Italy 15Laboratory of Quantum Optics, University of Nova Gorica, Nova Gorica 5001, Slovenia 16Centre for Translational Atomaterials, Swinburne University of Technology, 3122 Melbourne, Australia

(Received 7 January 2020; revised 15 June 2020; accepted 31 July 2020; published 30 September 2020)

Quantum mechanically, photoionization can be fully described by the complex photoionization amplitudes that describe the transition between the ground state and the continuum state. Knowledge of the value of the phase of these amplitudes has been a central interest in photoionization studies and newly developing attosecond science, since the phase can reveal important information about phenomena such as electron correlation. We present a new attosecond-precision interferometric method of angle-resolved measurement for the phase of the photoionization amplitudes, using two phase-locked extreme ultraviolet pulses of frequency ω and 2ω, from a free-electron laser. Phase differences Δη˜ between one- and two- photon ionization channels, averaged over multiple wave packets, are extracted for neon 2p electrons as a function of the emission angle at photoelectron energies 7.9, 10.2, and 16.6 eV. Δη˜ is nearly constant for emission parallel to the electric vector but increases at 10.2 eV for emission perpendicular to the electric vector. We model our observations with both perturbation and ab initio theory and find excellent agreement. In the existing method for attosecond measurement, reconstruction of attosecond beating by interference of two-photon transitions (RABBITT), a phase difference between two-photon pathways

*Corresponding author. [email protected] † Corresponding author. [email protected] ‡ Present address: LIDYL, CEA, CNRS, Universit´e Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France.

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

2160-3308=20=10(3)=031070(14) 031070-1 Published by the American Physical Society DAEHYUN YOU et al. PHYS. REV. X 10, 031070 (2020)

involving absorption and emission of an infrared photon is extracted. Our method can be used for extraction of a phase difference between single-photon and two-photon pathways and provides a new tool for attosecond science, which is complementary to RABBITT.

DOI: 10.1103/PhysRevX.10.031070 Subject Areas: Atomic and Molecular Physics

I. INTRODUCTION The extracted value is related to the phase difference of the two-photon ionization amplitudes at the pair of energies. For The age of attosecond physics was ushered in by the two energy points separated by twice the IR photon energy, invention of methods for probing phenomena on a timescale the phase difference divided by twice the IR photon energy less than femtoseconds [1]. A phenomenon occurring on this can be regarded as a finite difference approximation to the timescale is photoemission delay. When the photon energy is energy derivative of phase of the two-photon ionization far from resonance, the photoemission delay for single- amplitude. The pulse duration requirements are relaxed: For photon ionization can be associated with the Wigner delay example, pulse trains and IR pulses of 30 fs duration may be experienced by an electron scattering off the ionic potential used [6]. Usually, the IR pulse is the fundamental of the odd [2]. Quantum mechanically, the photoionization process is harmonics in the pulse train, although Loriot et al. [22] fully described by the complex photoionization amplitudes report a variant using the second harmonic. Recent work on describing transitions between the ground state and the phase retrieval includes methods based on photorecombi- continuum state. The photoemission delay can be expressed nation [13,23], two-color, two-photon ionization via a as the energy derivative of the phase of the photoionization resonance [24], and a proposal to use successive harmonics amplitude, and, therefore, measuring the photoemission of circularly polarized light [25]. delay and the energy-dependent phase of the photoionization The phase of the photoionization amplitude depends on amplitude are practically equivalent. Their measurement is photoelectron energy ϵ, and it may also depend on the one of the central interests in attosecond science [3–14], electron emission direction. There is a physical origin for because they are a fundamental probe of the photoionization the directional anisotropy of the amplitude: An electron process and can reveal important information about, for wave packet may consist of two or more partial waves, with example, electron-electron correlations (see, e.g., Ref. [15]). different angular momenta and phases. There has been Currently, two methods are available to measure these significant theoretical work on the angle-dependent time quantities: streaking and reconstruction of attosecond beat- delay, for example, Refs. [26–32], but fewer related exper- ing by interference of two-photon transitions (RABBITT), imental reports [12,28,33], all using the RABBITT both of which require the use of an IR dressing field. We technique. The Wigner delay is theoretically isotropic for present a new interferometric method of angle-resolved single-photon ionization of He, but Heuser et al. [28] measurement for the photoionization phase, using two observed an angular dependence in photoemission delay, phase-locked extreme ultraviolet (XUV) pulses of fre- ω 2ω attributed to the XUV þ IR two-photon ionization process, quency and , from a free-electron laser (FEL), without inherent in RABBITT interferometry. a dressing field. In the present work, we demonstrate interferometric In attosecond streaking [16], an ultrafast, short-wave- measurements of the relative phase of single-photon and length pulse ionizes an electron, and a femtosecond IR pulse two-photon ionization amplitudes. The interference is acts as a streaking field, by changing the linear momentum created between a two-photon ionization process driven of the photoelectron. In this technique, one can extract the by a fundamental wavelength and a single-photon ioniza- photoemission delay difference between two photoemission lines at two different energies, arising, for example, from tion process driven by its phase-locked, weaker, second two different subshells [16] or the main line and satellites harmonic, in a setup like that demonstrated at visible [15]. Generally, time-of-flight electron spectrometers wavelengths [34]. Using short-wavelength, phase-locked located in the streaking direction (the direction of linear XUV light, we measure angular distributions of photo- polarization) are used, so that this method does not give electrons emitted from neon and determine the phase access to angular information. A related method is the difference for one- and two-photon ionization wave pack- attosecond clock technique [17–20], in which streaking by ets. The extremely short (attoseconds) pulses required for the circularly polarized laser pulse is in the angular direction. streaking or attosecond pulse trains for RABBITT are not The second technique for measuring photoemission needed, and, instead, access to the photoemission phase delays, RABBITT, is interferometric: It uses a train of with attosecond precision is provided by optical phase attosecond pulses dressed by a phase-locked IR pulse [21]. control with a precision of a few attoseconds, which is In the RABBITT technique, the phase difference between a available from the free-electron laser FERMI [35]. pair of two-photon pathways whose final energy is separated The rest of the manuscript is structured as follows: by multiples of an infrared photon energy is extracted. In Sec. II, we introduce the necessary notation and the basic

031070-2 NEW METHOD FOR MEASURING ANGLE-RESOLVED PHASES IN … PHYS. REV. X 10, 031070 (2020) processes that may be active in the experiment; in Secs. III Iðθ; φ; ϕÞ ≐ Iðθ; ϕÞ Z and IV, we describe, respectively, the experimental and ∞n 2 2 theoretical methods used. In Sec. V, we present and ¼ cωðϵÞ þ c2ωðϵÞ þ 2cωðϵÞc2ωðϵÞ compare experimental and theoretical results. We discuss 0 o in Sec. VI the relationship between our data, namely, the × cos½ϕ − ΔηðϵÞ dϵ angular distribution of photoelectrons created by collinearly polarized biharmonics, and the time-delay studies ≡ A0 þ A cos½ϕ − Δηðϵ¯Þ; ð4Þ described in the introductory section. Section VII presents our summary and outlook, and the Appendix gives details where ϵ¯ is the average kinetic energy of the wave packet of the derivation of some equations. and ΔηðϵÞ ≡ ηωðϵÞ − η2ωðϵÞ is the phase of the two-photon ionization relative to the single-photon ionization. This treatment may be generalized to the case of multiple II. NOTATION AND BASIC PROCESSES wave packets, that is to say, with more than one magnetic We use Hartree atomic units unless otherwise stated and quantum number m of the residual ion. Wave packets with spherical coordinates r ¼fr; θ; φg relative to the direction each value of m interfere separately and then incoherently of polarization of the bichromatic field (linear horizontal in add. In particular, expressing the photoionization yield as in the experiment). We assume the electric dipole approxi- Eq. (4) mation, and the experiment is cylindrically symmetric Iðθ; ϕÞ¼A0 þ A cos ðϕ − Δη˜Þ about the electric vector, so that there is no dependence X on the azimuthal angle φ. The bichromatic electric field is ¼ ½A0;m þ Am cosðϕ − ΔηmÞ; ð5Þ described by m

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where summation is over the wave packets, leads to ω 2ω − ϕ EðtÞ¼ IωðtÞ cos t þ I2ωðtÞ cos ð t Þ; ð1Þ X X iΔη˜ iΔηm A0 ¼ A0;m;Ae¼ Ame : ð6Þ m m where ω and 2ω are angular frequencies, IωðtÞ and I2ωðtÞ ϕ ω 2ω are the pulse intensity envelopes, and denotes the - The second equation defines an average phase difference relative phase. Δη˜ of fΔηmg, weighted in terms of the corresponding We can consider the experimental sample as an ensemble phase factors. Equations (4) and (5) indicate that the yield of identical atoms of infinitesimal size, so we can reduce the of photoelectrons emitted by a bichromatic pulse in a theoretical treatment to that of a single atom centered at the particular direction oscillates sinusoidally as a function of ’ coordinates origin. The general form (omitting as implicit the optical phase ϕ. the dependence on θ; φ) of an electron wave packet sufficiently far away from the origin is III. EXPERIMENTAL METHODS AND SETUP Z ∞ pffiffiffiffi The experimental methods are described elsewhere [35], cðϵÞe−iϵtei½ 2ϵrþfðr;ϵÞþηðϵÞdϵ; ð2Þ and, here, we summarize the main aspects and the para- 0 meters used. The experiment is carried out at the low density matter beam line [36,37] of the FERMI free- where ϵ is the photoelectron kinetic energy, cðϵÞ the real- electron laser [38], using the velocity map imaging valued amplitude, and η ϵ the phase and the term f r; ϵ (VMI) spectrometer installed there. The VMI measures pffiffiffiffiffi pffiffiffiffiffi ð Þ ð Þ¼ ðZ= 2ϵÞ ln 8ϵr accounts for the Coulomb field of the the projection of the photoelectron angular distribution residual ion with charge Z. In our case, Z ¼þ1. (PAD) onto the planar detector (horizontal); the PAD is In the ω-2ω process, i.e., one driven by the field in obtained as an inverse Abel transform of this projection, Eq. (1), the wave packet can be expressed as using the BASEX (basis set expansion) method [39]. The images are divided into two halves along the line of the Z electric vector, labeled “left” and “right,” and analyzed ∞ pffiffiffiffi − ϵ 2ϵ ϵ separately. The PADs from the two halves agree generally, e i tei½ rþfðr; Þ 0 but the detector for the right half shows a small nonun-

iηωðϵÞ i½η2ωðϵÞþϕ iformity in detection efficiency. Therefore, the PADs × fcωðϵÞe þ c2ωðϵÞe gdϵ: ð3Þ are analyzed using the left half of the detector, denoted as 0°–180° below. The photoelectron yield as a function of optical phase ϕ The sample consists of a mixture of helium and neon, (we omit the spatial coordinates on the right-hand side) is and the helium PAD is used to calibrate the phase difference given by between the ω and 2ω fields. The atomic beam is produced

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created coherently in the two channels are identical, and electrons with the same linear momentum interfere [24]. The PAD Iðθ; ϕÞ is measured as a function of the phase ϕ; from the component oscillating with ϕ, the scattering phases are extracted, as shown in Sec. V. The wavelength is then changed and the measurement repeated. The relative phase of the two wavelengths is controlled by means of the electron delay line or phase shifter [35,41] used previously. It has been calculated that the two pulses have good temporal overlap with slightly different durations and only a small mean variation of the relative phase of two wavelengths within the full width at half maximum FIG. 1. Scheme of the experiment: Bichromatic, linearly of the pulses, for example, 0.07 rad for a fundamental polarized light (red and blue waves), with momentum kγ and photon energy of 18.5 eV [35]. electric vector Eγ, ionizes neon in the reaction volume. The The intensities of the two wavelengths for the experiments electron wave packets (yellow and magenta waves) are emitted are set as follows. With the last undulator open (that is, with electron momentum k. The m-averaged phase difference Δη˜ inactive), the first five undulators are set to the chosen between wave packets created by one- and two-photon ionization wavelength of the first harmonic. A small amount of spurious depends on the emission angle. The photoelectron angular second harmonic radiation (intensity of the order of 1% of the distribution depends on the relative (optical) ω-2ω phase ϕ. fundamental) is produced by the undulators [42],and,to Lower figures: Polar plots of photoelectron intensity at Ek ¼ absorb this radiation, the gas filter available at FERMI is 16 6 . eV for coherent harmonics (asymmetric, colored plot) and filled with helium. Helium is transparent at all of the incoherent harmonics (symmetric, gray plot). fundamental wavelengths used in this study. The two-photon photoelectron signal from the neon and helium gas sample is by a supersonic expansion and defined by a conical observed with the VMI spectrometer. The last undulator is skimmer and vertical slits. The length of the interaction then closed to produce the second harmonic, and the volume along the light propagation direction is approx- photoelectron spectrum of the combined beams is observed. imately 1 mm. In other experiments [5,7], the use of two The single-photon ionization by the second harmonic is at gases allows referencing of the photoemission delay of one least an order of magnitude stronger than the two-photon electron to that of another. In the present case, we use the ionization by the fundamental. The helium gas pressure in the admixture of helium to provide a phase reference. When the gas filter is then adjusted to achieve a ratio of the ionization free-electron laser wavelength is changed, the mechanical rates due to two-photon and single-photon ionization of 1∶2 settings of the magnetic structures (undulators) creating the for kinetic energies of 7.0 and 10.2 eV.For the kinetic energy light are changed. This change may introduce an unknown of 15.9 eV, the ratio is set to 1∶4. The bichromatic beam is phase error between fundamental and second harmonic focused by adjusting the curvature of the Kirkpatrick-Baez light. We have recently shown that the PAD of helium 1s active optics [43] and verified experimentally by measuring electrons can be used to determine the absolute optical the focal spot size of the second harmonic with a Hartmann phase difference between the ω and 2ω fields, with an input wavefront sensor. This instrument is not able to measure the of only a few theoretical parameters [40]. spot size of the beams at the fundamental wavelengths, so it is The light beam consists of two temporally overlapping calculated [44]. The measured spot is elliptical with a size harmonics with controlled relative phase ϕ [Eq. (1)] and of ð4.5 1Þ × ð6.5 1Þ μm2 (FWHM), and the estimated irradiates the sample, as shown schematically in Fig. 1. The pulse duration is 100 fs. intense fundamental radiation causes two-photon ioniza- Table I summarizes the experimental parameters: fun- tion, while the weak second harmonic gives rise to single- damental photon energy (ℏω), kinetic energy (Ek ¼ photon ionization. The energies of the photoelectrons 2ℏω − 21.6 eV) of the Ne photoelectrons emitted via

TABLE I. Experimental parameters. Ek is the kinetic energy of the Ne 2p photoelectrons. The pulse energies at the sample are calculated by multiplying the values at the source by the transmission of the beam line [37], which takes account of reflection and geometric losses. Pulse energies and irradiance are those of the fundamental.

Pulse energy, μJ Beam line Pulse energy, μJ Average irradiance 2 ℏω,eV Ek, eV (at source) Transmission (at sample) W=cm 14.3 7.0 45 0.10 4.5 2.3 × 1014 15.9 10.2 95 0.13 12.4 6.2 × 1014 19.1 16.6 84 0.23 19.3 9.5 × 1014

031070-4 NEW METHOD FOR MEASURING ANGLE-RESOLVED PHASES IN … PHYS. REV. X 10, 031070 (2020) single-photon (2ω) or two-photon (ω þ ω) ionization, ∂ΨðtÞ i ¼ Hˆ ðtÞΨðtÞ; ð7Þ average pulse energy of the first harmonic at the source ∂t and at the sample, beam line transmission, and average irradiance at the sample calculated from the above spot where the time-dependent Hamiltonian is sizes and pulse durations. The estimate of the pulse energy ˆ ˆ ˆ at ℏω ¼ 14.3 eV is indirect, since the FERMI intensity HðtÞ¼H1ðtÞþH2; ð8Þ monitors do not function at this energy, because they are based on ionization of nitrogen gas, and the photon energy with the one-electron part is below the threshold for ionization. The method employed is to first use the in-line spectrometer to measure spectra at XZ ˆ ˆ r 15.9 eV energy and, simultaneously, the pulse energies H1ðtÞ¼ hð i;tÞð9Þ 1 from the gas cell monitors, which gives a calibration of the i¼ spectrometer intensity versus pulse energy at this wave- and the two-electron part length. Then, spectrometer spectra are measured at 14.3 eV, and corrected for grating efficiency and detector sensitivity, XZ X 1 to yield pulse energies. ˆ H2 ¼ r − r : ð10Þ i¼1 j

031070-5 DAEHYUN YOU et al. PHYS. REV. X 10, 031070 (2020) can significantly reduce the computational cost without phase ϕ. Then, employing a fitting procedure very similar sacrificing the accuracy in the description of correlated to that used for the experimental data, we extract the phase multielectron dynamics. The equations of motion that shift difference Δη˜ between single-photon and two-photon describe the temporal evolution of the CI coefficients ionization at photoelectron energies 7.0, 10.2, and 16.6 eV. fCIg and the orbital functions fψ pg are derived by use The results are shown in Fig. 2. of the time-dependent variational principle [45]. The numerical implementation of the TD-CASSCF method C. Perturbation theory for atoms is detailed in Refs. [46,49]. In the experiment, the number of optical cycles in the pulse is of the order of 400 for the fundamental, B. Extraction of the photoelectron angular and, therefore, we can treat the field as having constant distribution and the phase shift difference amplitude and omit the initial phase of the field with respect From the obtained time-dependent wave functions, to the envelope (carrier-envelope phase). Within the per- we extract the angle-resolved photoelectron energy spec- turbation theory, we check that our final results with an trum (ARPES) by use of the time-dependent surface flux envelope including 100 optical cycles or more differ only (tSURFF) method [50]. This method computes the ARPES within the optical linewidth from those obtained with the from the electron flux through a surface located at a certain constant amplitude field. The bichromatic electric field is radius Rs, beyond which the outgoing flux is absorbed by then described by Eq. (1), with time-independent Iω and the infinite-range exterior complex scaling [49,51]. I2ω. The calculations described below are carried out for We introduce the time-dependent momentum amplitude 384 optical cycles and a peak intensity of 1 × 1012 W=cm2. k k apð ;tÞ of orbital p for photoelectron momentum , However, neither the absolute intensity nor the ratio of defined by intensities of the harmonics influences the calculated phase, as we show below. a ðk;tÞ¼hχkðr;tÞjuðR Þjψ ðr;tÞi We make two main assumptions: the dipole approxima- p Z s p tion for the interaction of the atom with the classically ≡ χ r ψ r 3r kð ;tÞ pð ;tÞd ; ð14Þ described electromagnetic field and the validity of the r>Rs lowest nonvanishing order perturbation theory with respect to this interaction. These approximations are well fulfilled where χkðr;tÞ denotes the Volkov wave function and uðR Þ s for neon in the FEL spectral range and intensities of interest the Heaviside function which is unity for r>Rs and vanishes otherwise. The use of the Volkov wave function here. We expand the amplitudes in the lowest nonvanishing implies that we neglect the effects of the Coulomb force order of perturbation theory in terms of matrix elements of from the nucleus and the other electrons on the photo- the operator of evolution [53]. The expansion implies that, in the second-order amplitude, all virtual intermediate electron dynamics outside Rs, which is confirmed to be a good approximation [52]. The photoelectron momentum states are taken into account. Excitations of the seven distribution ρðkÞ is given by lowest and most important intermediate dipole-allowed 2 5 3 4 3 X states originating from configurations ( p s, s, d) ˆ q are accounted for accurately within the multiconfiguration ρðkÞ¼ apðk; ∞Þaqðk; ∞ÞhΨðtÞjEpjΨðtÞi; ð15Þ pq intermediate-coupling approximation with relativistic Breit-Pauli corrections in the atomic Hamiltonian. All other P ˆ q † virtual states (of infinite number), including those in the with Ep ≡ σ aˆ qσaˆ pσ. One obtains apðk; ∞Þ by numeri- cally integrating: continuum, are accounted for by a variationally stable method [54,55] in the Hartree-Fock-Slater approximation. ∂ More details can be found in Ref. [56]. Further derivations − k χ ^ ψ i∂ apð ;tÞ¼h kðtÞj½hs;uðRsÞj pðtÞi within the independent particle approximation are given in t X ˆ q the Appendix. þ aqðk;tÞfhψ qðtÞjFjψ pðtÞi−Rpg; ð16Þ q V. RESULTS ˆ k2 2 A k q ψ ψ_ − ψ ˆ ψ where hs ¼ = þ ðtÞ · , Rp ¼ ih qj pi h qjhj pi We extract Δη˜ðθÞ from the measured PADs at three and Fˆ denotes a nonlocal operator describing the contri- combinations of ω and 2ω (corresponding to photoelectron bution from the interelectronic Coulomb interaction kinetic energies of 7.0, 10.2, and 16.6 eV), at each 5° [46,49]. The numerical implementation of tSURFF to interval of the polar angle. The spatial and temporal TD-CASSCF is detailed in Ref. [52]. symmetry properties of the system impose constraints on We evaluate the photoelectron angular distribution the oscillatory behavior of the two emission hemispheres. Iðθ; ϕÞ as a slice of ρðkÞ at the value of jkj corresponding Upon reflection in a plane perpendicular to the electric to the photoelectron peak and as a function of the optical vector (θ → π − θ), the electric field defined in Eq. (1) is

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(a) (b) Polar angle (deg) Panel A / B 75–80 / 100–105 60–65 / 115–120 45–50 / 130–135 Yield 30–35 / 145–150 15–20 / 160–165 0–5 / 175–180 0 0 90 180 270 360 450 540 0 90 180 270 360 450 540 Optical phase (deg)

300 (c)7.0 eV (d)10.2 eV (e) 16.6 eV 10.2 eV (II) 240

180

120

Phase shift difference (deg) 0 0 1530456075900 1530456075900 153045607590 Polar angle (deg)

FIG. 2. Upper: Typical photoelectron yields Iðθ; ϕÞ as a function of optical phase ϕ at intervals of polar angles θ. The signal is integrated over the 5° intervals shown on the right. The curves are not shifted; thus, their vertical position reflects directly the value A0 in Eq. (4). The photoelectron kinetic energy is 7.0 eV. Circles are experimental results; lines are sinusoidal fits of the experimental results. Lower: Extracted phase shift differences as a function of the polar angles, for four datasets and three photoelectron kinetic energies: left (c), 7.0 eV; middle (d), 10.2 eV; right (e), 16.6 eV. Circles are experimental results; shaded areas show their uncertainties. Dashed lines, perturbation theory; solid lines, real-time ab initio theory. Note that the curves in (a) and (b) oscillate in antiphase, because they correspond to emission directions on opposite sides of the photon propagation direction. inverted: EðtÞ → −EðtÞ, and the ω-2ω relative phase theory and real-time ab initio methods (see Sec. IV and becomes ϕ þ π. From the arguments above, Eq. (5) the Appendix), and both theories reproduce well the becomes observed behavior; see Figs. 2(c)–2(e). The perturbation theory result at the intermediate angles is very sensitive to

Iðπ − θ; ϕ þ πÞ¼A0 þ A cos ½ϕ þ π − Δη˜ðπ − θÞ the contribution of the p-d-f two-photon ionization path, which may not be accurately reproduced by the local- − ϕ − Δη˜ π − θ ¼ A0 A cos ½ ð Þ; ð17Þ potential approximation in summation over the Rydberg and continuum d states. where we omit the argument ϵ¯ and include explicitly the Figure 3 shows the theoretical dependence of Δη˜ðθÞ on argument θ. Comparison with Eq. (4) indicates that the electron kinetic energy and polar angle θ, calculated using intensities at the two opposite angles oscillate in antiphase, perturbation theory. There is a single-photon 2p → 3s that is, Δη˜ðπ − θÞ¼Δη˜ðθÞþπ. It can be seen in Figs. 2(a) resonance of the fundamental wavelength at 16.7 eV and 2(b) that the experimental data do indeed oscillate in photon energy (12 eV kinetic energy for the two-photon antiphase for each angular interval over θ. Since this is a or second harmonic). The behavior of Δη˜ in the region of symmetry constraint, it is imposed in the analysis of the data. the resonance is complicated: We can clearly see that Δη˜ðθÞ In Figs. 2(c)–2(e), it can be seen that there is a significant at θ ∼ 90° increases near the resonance around 12 eV and increase of Δη˜ðθÞ at approximately 90°, especially for then returns to a value similar to that at approximately 7 eV. 10.2 eV. The angular-dependent variations of Δη˜ðθÞ at 7.0 This result indicates that the large phase shift difference and 16.6 eV are similar. We perform calculations for the observed at 10.2 eV in Fig. 2(d) is due to the influence of phase shift differences Δη˜ðθÞ using both perturbation the resonance at 12.0 eV [32,33] and suggests that future

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of atoms, such as 1s2 of Ne. Single-photon ionization leads to a continuum state with angular momentum p, while two- photon ionization leads to two final quantum states s and d. Then, the PAD IeðθÞ is described by

iηs iðηpþϕÞ IeðθÞ¼jcse Y00ðθ; φÞþcpe Y10ðθ; φÞ

iηd 2 þ cde Y20ðθ; φÞj ; ð18Þ

where cs, cp, and cd are real-valued partial-wave amplitudes and ηs, ηp, and ηd are the corresponding arguments. IeðθÞ can also be expressed as X4 θ 2 2 2 1 β θ Ieð Þ¼ðcs þ cp þ cdÞ þ lPlðcos Þ ; ð19Þ l¼1

where Plðcos θÞ are the Legendre polynomials describing Δη˜ FIG. 3. Phase shift differences of two-photon ionization the angular distributions and βl are the corresponding relative to single-photon ionization, calculated by perturbation asymmetry parameters. After some algebra, we have [40] θ theory, as a function of the polar angle and photoelectron ffiffiffiffiffi kinetic energy. The large variation near 12.0 eV kinetic energy is p 6 15c c cos ð−η þ η þ ϕÞ due to the 2p → 3s resonance. β d p d p 3 ¼ 5 2 2 2 ðcd þ cp þ cs Þ ≡ β ϕ − η − η experiments should explore this region in fine detail, to ½ 30 cos ½ ð d pÞ; ð20Þ observe the predicted rapid changes in Δη˜. Both theories pffiffiffi 2 2 3 η − η ϕ reproduce this behavior well, with the time-dependent cpcs cos ð p s þ Þ β1 − β3 ¼ ab initio method exhibiting excellent agreement, validating 3 c2 þ c 2 þ c2 d p s the present experimental method. 2 ≡ β − β ϕ − η − η We show in the Appendix that the method is independent 1 3 3 cos ½ ð s pÞ; ð21Þ of the relative intensities of the fundamental and second 0 harmonic radiation; see Eqs. (A5)–(A9). This advantage is 2 where ½β30 and ½β1 − β30 are constants. Thus, if we considerable from an experimental point of view, as it is not 3 record PADs as a function of ϕ and extract β (l 1–4), we necessary to measure precisely the intensity and focal spot l ¼ can directly read off η − η and η − η from the oscil- shape. Furthermore, there are no effects due to volume d p s p β β − 2 β averaging over the Gaussian spot profile or over the lations of 3 and 1 3 3 using Eqs. (20) and (21). Let us τ duration of the pulses. We verify this conclusion exper- recall that the Wigner delay of each partial wave, l, imentally for the kinetic energy of 16.6 eV [Fig. 2(d)], corresponds to the energy derivative of the argument of θ φ τ ϵ where the ratio of ionization rates is 1∶4 (rather than 1∶2 the amplitude [note that Yl0ð ; Þ are real], lð Þ¼ η ϵ ϵ η − η η − η used for the other energies), and the experiment and theory ðd lð Þ=d Þ [2]. By measuring d p and s p as a agree well. function of the energy, one can take the energy derivative and obtain the Wigner delay differences τlðϵÞ − τpðϵÞ with l ¼ s and d. In simple models, like the Hartree-Fock VI. DISCUSSION approximation, ðdηlðϵÞ=dϵÞ¼ðdδlðϵÞ=dϵÞ, where δlðϵÞ In this section, we elucidate the relationship of our is the scattering phase, while in more complicated cases, data, i.e., photoelectron angular distributions created by an extra energy-dependent phase may be acquired by the collinearly polarized biharmonics, to time-delay studies partial amplitude [27]. described in the introduction. We limit ourselves to the case We now group the s and d waves as a two-photon- where any discrete state in the continuum (autoionizing ionization wave packet. Then, the photoelectron wave state) or in the discrete spectrum lies outside the bandwidth packet in a given direction θ sufficiently far from the of the pulses (<0.02 eV in the present case). These nucleus and the corresponding PAD are expressed as conditions are well fulfilled in our experiments. The Eqs. (3) and (4), respectively. The energy derivative of resonant case is discussed elsewhere [10]. ΔηðθÞ ≡ ηωðθÞ − η2ωðθÞ is a difference between the group We first consider the simple situation of photoionization delays of the two wave packets, generated by two- from a spherically symmetric orbital s. The present method and single-photon ionization, respectively. In the original can be extended straightforwardly to inner shell ionization photoemission delay experiment [16] with attosecond

031070-8 NEW METHOD FOR MEASURING ANGLE-RESOLVED PHASES IN … PHYS. REV. X 10, 031070 (2020) streaking, for example, Ne 2s and 2p electrons are ionized We now turn to photoionization from a p orbital, which by an attosecond pulse to different final kinetic energies. includes the present case of Ne 2p ionization and is more As a result, the more energetic photoelectron from 2p complicated. The complexity arises from two sources. We arrives at the detector much earlier than that from 2s, have three incoherent contributions from m ¼ 0 and 1 for regardless of the measured delay. The situation is similar the magnetic sublevels of the remaining ion core Neþ and for subsequent measurements using streaking and four contributions of partial waves s, p, d, and f in the RABBITT. In great contrast, in the present case, both photoelectron wave packet. Detailed derivations of the single- and two-photon ionization result in the same equations describing the PADs are given in the Appendix, photoelectron energy. Therefore, the single- and two- and, here, we describe only the results relevant to the photon-ionization wave packets actually reach a given present discussion. For m ¼1, we have a pair of two- distance with a relative (group) delay given by ∂Δη=∂ϵ. photon pathways via d intermediate states, i.e., p → d → p By comparing Eqs. (4) and (18), we can describe the and p → d → f, together with the single-photon pathway iΔηðθÞ phase factor e with ΔηðθÞ ≡ ηωðθÞ − η2ωðθÞ being the p → d. Thus, only three partial waves are involved, and, angle-resolved phase difference between the two-photon therefore, a discussion similar to that above for s ioniza- and single-photon ionization amplitudes as tion holds. For m ¼ 0, we have two-photon pathways via s and d AðθÞeiΔηðθÞ intermediate states, leading to p and f final states, together with single-photon ionization that leads to s and d final η −η η −η 2 θ φ θ φ ið d pÞ θ ið s pÞ ¼ cpY10ð ; Þ½cdY20ð ; Þe þ csY00ð Þe : states. Thus, there are four partial waves involved. ð22Þ Although we can derive equations similar to Eqs. (20) and (21) [see Eqs. (A14)–(A17)], what we can extract from the measurement is only a vectorial average of phase Thus, the phase factor eiΔηðθÞ is the coherent (i.e., with respect differences ηl − ηl0 between even and odd different partial i η −η to amplitudes) average of e ð d pÞ (d-p interference) and waves l and l0. We can define the angle-resolved phase iðηs−ηpÞ e (s-p interference) with the relative weight difference ΔηmðθÞ for each m [see Eq. (5)], which is also a vectorial average of ηl − ηl0 . Similar to ionization from the s c Δη θ WðθÞ¼BgðθÞ;B¼ d ; state, the energy derivative of mð Þ may be regarded as cs an angle-resolved group delay between single- and two- pffiffiffi θ φ 5 photon wave packets for each m. θ Y20ð ; Þ 3 2θ 1 gð Þ¼ θ φ ¼ 4 ½ cosð Þþ : ð23Þ In the experiment, we measure an (incoherently) Y00ð ; Þ weighted average Δη˜ of angle-resolved phase differences Δη of different m as defined in Eqs. (5) and (6). One can Δη θ m In other words, ð Þ can be regarded as a vectorial introduce the energy derivative of the weighted average η − η η − η average of s p and d p with the relative weight phase difference Δη˜ðθÞ and may call it generalized delay, WðθÞ. Equivalently, ΔηðθÞ may be presented as but this definition of time delay is different from that commonly employed for the time delay of an incoherent cosðη − η ÞþWðθÞ cosðη − η Þ sum of wave packets. Usually, the phase of each wave tan ΔηðθÞ¼ s p d p : ð24Þ sinðηs − ηpÞþWðθÞ sinðηd − ηpÞ packet is first differentiated with respect to energy and then averaged over m [57], while, in this study, ðdΔη˜=dϵÞðθÞ The energy derivative of ΔηðθÞ does not give us additional first averages the wave packet phase over m and then information about the photoionization amplitudes but differentiates it with respect to the photoelectron energy. provides us with the group delay and may enhance the sensitivity to the energy-dependent behavior of the two- VII. SUMMARY AND OUTLOOK photon ionization amplitudes, as described below. In this work, we have described a new method to Note two important characteristics of ΔηðθÞ: (i) ΔηðθÞ determine the angle-resolved relative phase between single- exhibits a quasicosine shape and monotonic dependence on and two-photon ionization amplitudes and used it to θ due to the geometric factor gðθÞ [see Figs. 2(c)–2(e) and measure the 2p photoionization of Ne. Our approach the Appendix), and (ii) ΔηðθÞ is sensitive to the two-photon allows us to explore the phase difference between different ionization dynamics due to the dynamical factor B.For ionization pathways, e.g., those of odd and even parities, example, if the two-photon pathways are close to an with the same photoelectron energy. intermediate discrete resonance (but still well outside the The method is based on FEL radiation, so that it can be bandwidth), the group delay difference ð∂Δη=∂ϵÞðθÞ is extended to shorter wavelengths, eventually to inner shells, sensitive to it through rapid change in B, while dηs=dϵ, which lie in a wavelength region where optical lasers have dηp=dϵ, and dηd=dϵ are small individually, as can be seen reduced pulse energy. This method is an important addition in Fig. 3. to the armory of techniques available to attosecond science

031070-9 DAEHYUN YOU et al. PHYS. REV. X 10, 031070 (2020) and gives access to the phase difference between single- Advancement of Theoretical Physics and Mathematics (odd-parity) and two-photon (even-parity) transition ampli- “BASIS.” D. Y. acknowledges supports by JSPS tudes or the energy variation of the phase of two-photon KAKENHI Grant No. JP19J12870, and a Grant-in-Aid of ionization amplitudes affected by the intermediate reso- Tohoku University Institute for Promoting Graduate Degree nances, as seen in the Ne 2p photoionization. For ns2 Programs Division for Interdisciplinary Advanced Research subshells of atoms, e.g., 1s2 of He, 1s2 and 2s2 of Ne, etc., and Education. We acknowledge the support of the Alexander in particular, one can extract the eigenphase differences for von Humboldt Foundation (Project Tirinto), the Italian s, p, and d partial waves of electron-ion scattering, and Ministry of Research Project FIRB No. RBID08CRXK their energy derivatives correspond to the Wigner delay and No. PRIN 2010 ERFKXL 006, the bilateral project difference of the partial waves. This method is also CNR JSPS Ultrafast science with extreme ultraviolet free applicable to molecules. electron lasers, and funding from the European Union While it does not yet appear to be feasible with present Horizon 2020 research and innovation program under the high-harmonic generation (HHG) sources, it may become Marie Sklodowska-Curie Grant Agreement No. 641789 possible in the future, but there are many technical MEDEA (Molecular Electron Dynamics investigated by challenges. Since HHG sources produce a frequency comb, IntensE Fields and Attosecond Pulses), G. S. acknowledges the chief technical challenges are to filter the beam to funding from the Deutsche Forschungsgemeinschaft under achieve bichromatic spectral purity, maintain attosecond the Project No. 429805582. E. V.G., A. N. G., M. M. P., and temporal resolution, and provide enough pulse energy at E. I. S. acknowledge funding from the Russian Foundation the fundamental wavelength to initiate two-photon ioniza- for Basic Research (RFBR) under the Project No. 20-52- tion. Furthermore, HHG sources have not yet demonstrated 12023. T. M. and M. M. acknowledge support by Deutsche the level of phase control which we have at our disposal. Forschungsgemeinschaft Grant No. SFB925/A1. We thank Given the rapid progress in HHG sources, these conditions the machine physicists of FERMI for making this experiment may eventually be met, in which case our method will possible by their excellent work in providing high-quality become more widely accessible. FEL light. The information obtained by this method is complementary to that of streaking and RABBITT methods, in the APPENDIX: PERTURBATION THEORY. sense that different phase differences are measured. We DERIVATION OF EQUATIONS IN THE have directly measured the angle-resolved average phase INDEPENDENT PARTICLE MODEL difference Δη˜ðθÞ of two-photon amplitude relative to the single-photon ionization amplitude. The basic physics In addition to the approximations described in Sec. IV C giving rise to its angular dependence is related to interfer- (the dipole approximation, the validity of the lowest ence between photoelectron waves emitted in one- and two- nonvanishing order perturbation theory), here, we add photon ionization, consisting of partial photoelectron the LS-coupling approximation within the independent waves with opposite parities. We have shown that the particle model. The photoelectron angular distribution θ; ϕ 2 overall shape of Δη˜ðθÞ versus the angle can be understood Ið Þ of a Ne p electron can be derived by standard qualitatively. methods [58] in the form

X X 2 ACKNOWLEDGMENTS θ; ϕ m ϕ θ φ Ið Þ¼I0 ξCξ ð ÞYlξmð ; Þ ; ðA1Þ 0 1 This work was supported in part by the X-ray Free m¼ ; Electron Laser Utilization Research Project and the X-ray where m is the magnetic quantum number of the initial 2p Free Electron Laser Priority Strategy Program of the Ministry electron, Yl ðθ; φÞ is a spherical harmonic in the Condon- of Education, Culture, Sports, Science, and Technology of m Shortley phase convention, and I0 is a normalization factor Japan (MEXT) and the IMRAM program of Tohoku irrelevant to further discussion; note that the dependence on University, and the Dynamic Alliance for Open φ cancels out. The complex coefficients CmðϕÞ depend on Innovation Bridging Human, Environment and Materials ξ ionization amplitudes, and the index ξ denotes the ioniza- program. K. L. I. gratefully acknowledges support by the tion path. For single-photon ionization, ξ ¼ lξ, where lξ is Cooperative Research Program of the Network Joint Research Center for Materials and Devices (Japan), Grant- the orbital momentum of the photoelectron with possible l ξ l l0 in-Aid for Scientific Research (Grants No. 16H03881, values ξ ¼ s, d. For two-photon ionization, ¼f ξ; ξg, l0 No. 17K05070, No. 18H03891, and No. 19H00869) from where ξ is the orbital momentum of the virtual inter- MEXT, JST COI (Grant No. JPMJCE1313), JST CREST mediate state, with possible combinations ξ ¼ ps, pd, fd. (Grant No. JPMJCR15N1), MEXT Quantum Leap Flagship After applying the Wigner-Eckart theorem [59] to factor Program (MEXT Q-LEAP) Grant No. JPMXS0118067246, out the dependence on the projection m, the coefficients m and Japan-Hungary Research Cooperative Program, JSPS Cξ ðϕÞ may be expressed as (for brevity, we omit the and HAS. E. V.G. acknowledges the Foundation for the argument ϕ when writing the coefficients)

031070-10 NEW METHOD FOR MEASURING ANGLE-RESOLVED PHASES IN … PHYS. REV. X 10, 031070 (2020) rffiffiffiffiffi 1 2 1 Equations (A2)–(A10) define Δη˜, provided the reduced C0 ¼ −pffiffiffiD eiϕ;C0 ¼ D eiϕ;C1 ¼ pffiffiffiffiffiD eiϕ; s 3 s d 15 d d 10 d matrix elements (A4) are calculated. The intensities of the fundamental and of the second harmonic are factored out in ðA2Þ 0 the coefficients Zλ; therefore, they cancel out in Eqs. (A7) 1 and (A8), and the phases Δη˜ are independent of the 0 1 Cps ¼ − Dps;Cps ¼ 0; intensities of the harmonics. 3 Δη˜ 2 1 Note that the angle-resolved average phase difference C0 − D ;C1 − D ; between one- and two-photon ionization implies not less pd ¼ 15 pd pd ¼ 10 pd pffiffiffi than two ionization channels, which is reflected in the 2 2 nonvanishing sum over channels in Eq. (A10). Therefore, C0 ¼ pffiffiffi D ;C1 ¼ pffiffiffiffiffi D : ðA3Þ fd 5 7 fd fd 5 21 fd Δη˜ and its energy derivative, or as we called it, generalized delay, is always angle dependent. Here, The coefficients (A10) are directly related to the anisotropy parameters βλ in the angular distribution of iηξ Dξ ¼ dξe ðA4Þ photoelectrons (A1) written in the form

are complex reduced matrix elements, independent of m, X4 W0 with magnitude dξ ¼jDξj and phase ηξ. Note that one- (first- θ; ϕ 1 β θ Ið Þ¼ 4π þ λPλðcos Þ ; ðA12Þ order) and two-photon (second-order) matrix elements (A4), λ¼1 both marked by a single index ξ, are, respectively, propor- tional to the square root of intensity and to the intensity of the where associated field. Equation (A1) can be readily cast into the form (4), Zλ where βλ ¼ ðA13Þ Z0 X I0 θ A0 ¼ 4π ZλPλðcos Þ; ðA5Þ λ¼0;2;4 and W0 ¼ I0Z0. Substituting Eqs. (A2), (A3), and (A10) into (A13), one can express the anisotropy parameters in I0 terms of reduced matrix elements (A4): A ¼ 4π N; ðA6Þ X pffiffiffi −1 0 1 12 3 cos Δη˜ ¼ N ReZλPλðcos θÞ; ðA7Þ β1 ¼ dddfd cosðηfd − ηd − ϕÞ λ¼1;3 15Z0 5 pffiffiffi X 17 2 Δη˜ −1 0 θ − η − η − ϕ sin ¼ N ImZλPλðcos Þ; ðA8Þ 5 dddpd cosð pd d Þ λ¼1;3 pffiffiffi − 4 2d d cosðη − η − ϕÞ X d ps ps d 0 N ¼ Z Pλðcos θÞ ; ðA9Þ þ 4d d cosðη − η − ϕÞ λ s pd pd s λ¼1;3 10 η − η − ϕ X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ dsdps cosð ps s Þ ; ðA14Þ −1 m m m 2l 1 2l 1 Zλ ¼ ð Þ Cξ Cξ¯ ð ξ þ Þð ξ¯ þ Þ m¼0;1 ξ¼s;d;ps;pd;fd ¯ ξ¼s;d;ps;pd;fd 1 32 49 140 β 70 2 2 2 2 × ðlξm; lξ¯ − mjλ0Þðlξ0; lξ¯0jλ0Þ; ðA10Þ 2 ¼ dd þ dfd þ dpd þ dps 210Z0 5 15 3 pffiffiffi where ðj1m1;j2m2jjmÞ are Clebsch-Gordan coefficients − 140 2ddds cosðηd − ηsÞ 0 pffiffiffi [59] and Zλ ¼ Zλjϕ¼0. In particular, 48 6 − dfddpd cosðηfd − ηpdÞ 1 4 17 10 5 2 2 2 2 2 pffiffiffi Z0 ¼ 10ðd þ dsÞþ d þ d þ dps 30 d 5 fd 15 pd 3 − 12 6dfddps cosðηfd − ηpsÞ 8 112 η − η þ d d cosðη − η Þ ; ðA15Þ þ 3 dpddps cosð pd psÞ : ðA11Þ 3 pd ps pd ps

031070-11 DAEHYUN YOU et al. PHYS. REV. X 10, 031070 (2020)

terms of the ionization amplitudes are used. Moreover, it holds for circularly polarized collinear photon beams (except for chiral targets), provided the angle θ is measured from the direction of the beam propagation. Expression (A18) may be written in an equivalent form in terms of θ-independent “average partial” TPI-SPI phase differences Δη˜λ (λ ¼ 1, 3). Indeed, we can write Eq. (A12) explicitly in the form [see Eqs. (5) and (6)]

−1 4πW0 Iðθ; ϕÞ¼½1 þ β2P2ðcos θÞþβ4P4ðcos θÞ

þ½γ1 cosðϕ − Δη˜1ÞP1ðcos θÞ FIG. 4. Plot of 5 cos 2θ − 1 =4 as a function of θ. ð Þ þ γ3 cosðϕ − Δη˜3ÞP3ðcos θÞ

pffiffiffi ¼ A0ðθÞþAðθÞ cos½ϕ − Δη˜ðθÞ: ðA19Þ 1 16 3 β3 ¼ dddfd cosðηfd − ηd − ϕÞ 30Z0 5 Here, the prefactors γ1 and γ3 and phases Δη˜1 and Δη˜3 are pffiffiffi θ ϕ 6 2 independent of and . It follows from Eq. (A19) that Δη˜ θ “ ” Δη˜ − dddpd cosðηpd − ηd − ϕÞ ð Þ can be viewed as a vectorial average of 1 and 5 ˜ pffiffiffi Δη3 with weights γ1P1ðcos θÞ and γ3P3ðcos θÞ, in the sense − 12 2d d cosðη − η − ϕÞ that Δη˜ðθÞ is the directional angle of vector d ps ps d pffiffiffi − 4 6dsdfd cosðηfd − ηs − ϕÞ ; ðA16Þ ½γ1P1ðcos θÞ cos Δη˜1

þ γ3P3ðcos θÞ cos Δη˜3; γ1P1ðcos θÞ sin Δη˜1 pffiffiffi 4 1 2 β 2 − ffiffiffi η − η þ γ3P3ðcos θÞ sin Δη˜3; ðA20Þ 4 ¼ dfd p dfddpd cosð fd pdÞ 35Z0 5 5 3 pffiffiffi 2 2 satisfying ffiffiffi − p dfddps cosðηfd − ηpsÞ : ðA17Þ 3 γ θ Δη˜ γ θ Δη˜ Δη˜ θ 1P1ðcos Þ sin 1 þ 3P3ðcos Þ sin 3 tan ð Þ¼γ θ Δη˜ γ θ Δη˜ : The terms containing cosine functions describe the 1P1ðcos Þ cos 1 þ 3P3ðcos Þ cos 3 contribution from the phase difference between two-photon ðA21Þ and single-photon ionization channels. For example, “ ” cosðηfd − ηd − ϕÞ in Eq. (A14) corresponds to the phase There are simple relations between the average partial difference between p → d → f two-photon-ionization TPI-SPI phase differences and parameters of Eq. (A18): (TPI) and p → d single-photon-ionization (SPI) channels 0 0 as well as the relative phase of the harmonics ϕ. Note that Δη˜ ImZλ Δη˜ ReZλ λ 1 3 sin λ ¼ 0 ; cos λ ¼ 0 ð ¼ ; ÞðA22Þ β2 and β4 do not depend on ϕ. jZλj jZλj It immediately follows from Eqs. (A7) and (A8) that and also 0 0 ImZ1P1ðcos θÞþImZ3P3ðcos θÞ tan Δη˜ 0 −1 0 −1 ¼ 0 0 γ1 Z ; γ3 Z : A23 ReZ1P1ðcos θÞþReZ3P3ðcos θÞ ¼j 3j ¼j 1j ð Þ ImZ0 þ fðθÞImZ0 ¼ 1 3 ; ðA18Þ As stated above, we can use the fact that the parity of 0 θ 0 n ReZ1 þ fð ÞReZ3 Legendre polynomials obeys Pnð−xÞ¼ð−1Þ PnðxÞ,so that the vector defined by Eq. (A20) changes sign upon θ 1 5 2θ − 1 where the function fð Þ¼4 ½ cosð Þ is displayed performing the substitution θ → π − θ; i.e., the two halves in Fig. 4. This result qualitatively explains the quasi- of the VMI image oscillate in antiphase: Δη˜ðθÞ¼ Δη˜ θ cosine shape and monotonic dependence of on Δη˜ðπ − θÞþπ. 0 [Figs. 2(c)–2(e)]. Zλ are independent of ϕ, so that Δη˜ and the generalized delay are independent of the relative phase between the harmonics. The functional form of Eq. (A18) is very general and [1] F. Krausz and M. Ivanov, Attosecond Physics, Rev. Mod. valid, within the perturbation theory and the dipole approxi- Phys. 81, 163 (2009). mation, for randomly oriented atoms and molecules, pro- [2] E. P. Wigner, Lower Limit for the Energy Derivative of the vided corresponding expressions for the coefficients Zλ in Scattering Phase Shift, Phys. Rev. 98, 145 (1955).

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[3] S. Pabst and J. M. Dahlström, Eliminating the Dipole Phase [17] P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, in Attosecond Pulse Characterization Using Rydberg Wave H. G. Muller, M. Büttiker, and U. Keller, Attosecond Packets, Phys. Rev. A 94, 013411 (2016). Ionization and Tunneling Delay Time Measurements in [4] K. Klünder, J. M. Dahlström, M. Gisselbrecht, T. Fordell, M. Helium, Science 322, 1525 (2008). Swoboda, D. Guénot, P. Johnsson, J. Caillat, J. Mauritsson, [18] P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, A. Maquet, R. Taïeb, and A. L’Huillier, Probing Single- M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, Photon Ionization on the Attosecond Time Scale, Phys. Rev. Attosecond Angular Streaking, Nat. Phys. 4, 565 (2008). Lett. 106, 143002 (2011). [19] A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, [5] D. Guénot, D. Kroon, E. Balogh, E. W. Larsen, M. Kotur, M. Abu-samha, L. B. Madsen, and U. Keller, Attoclock M. Miranda, T. Fordell, P. Johnsson, J. Mauritsson, M. Reveals Natural Coordinates of the Laser-Induced Gisselbrecht, K. Varjú, C. L. Arnold, T. Carette, A. S. Tunnelling Current Flow in Atoms, Nat. Phys. 8,76 Kheifets, E. Lindroth, A. L’Huillier, and J. M. Dahlström, (2012). Measurements of Relative Photoemission Time Delays in [20] A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Dörner, and U. Noble Gas Atoms, J. Phys. B 47, 245602 (2014). Keller, Timing the Release in Sequential Double Ionization, [6] D. Guénot, K. Klünder, C. L. Arnold, D. Kroon, J. M. Nat. Phys. 7, 428 (2011). Dahlström, M. Miranda, T. Fordell, M. Gisselbrecht, P. [21] P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Johnsson, J. Mauritsson, E. Lindroth, A. Maquet, R. Taïeb, Balcou, H. G. Muller, and P.Agostini, Observation of a Train A. L’Huillier, and A. S. Kheifets, Photoemission-Time-Delay of Attosecond Pulses from High Harmonic Generation, Measurements and Calculations Close to the 3s-Ionization- Science 292, 1689 (2001). Cross-Section Minimum in Ar, Phys. Rev. A 85, 053424 [22] V. Loriot, A. Marciniak, G. Karras, B. Schindler, G. Renois- (2012). Predelus, I. Compagnon, B. Concina, R. Brédy, G. Celep, C. [7] C. Palatchi, J. M. Dahlström, A. S. Kheifets, I. A. Ivanov, Bordas, E. Constant, and F. Lépine, Angularly Resolved D. M. Canaday, P. Agostini, and L. F. DiMauro, Atomic RABBITT Using a Second Harmonic Pulse, J. Opt. 19, Delay in Helium, Neon, Argon and Krypton, J. Phys. B 47, 114003 (2017). 245003 (2014). [23] D. Azoury, O. Kneller, S. Rosen, B. D. Bruner, A. Clergerie, [8] A. S. Kheifets, Time Delay in Valence-Shell Photoioniza- Y. Mairesse, B. Fabre, B. Pons, N. Dudovich, and M. tion of Noble-Gas Atoms, Phys. Rev. A 87, 063404 (2013). Krüger, Electronic Wavefunctions Probed by All-Optical [9] S. Saha, A. Mandal, J. Jose, H. R. Varma, P. C. Deshmukh, Attosecond Interferometry, Nat. Photonics 13, 54 (2019). A. S. Kheifets, V. K. Dolmatov, and S. T. Manson, [24] D. M. Villeneuve, P. Hockett, M. J. J. Vrakking, and H. Relativistic Effects in Photoionization Time Delay near Niikura, Coherent Imaging of an Attosecond Electron Wave the Cooper Minimum of Noble-Gas Atoms, Phys. Rev. A Packet, Science 356, 1150 (2017). 90, 053406 (2014). [25] S. Donsa, N. Douguet, J. Burgdörfer, I. Brezinová,ˇ and L. [10] L. Argenti, A. Jiménez-Galán, J. Caillat, R. Taïeb, A. Argenti, Circular Holographic Ionization-Phase Meter, Maquet, and F. Martín, Control of Photoemission Delay Phys. Rev. Lett. 123, 133203 (2019). in Resonant Two-Photon Transitions, Phys. Rev. A 95, [26] J. M. Dahlström and E. Lindroth, Study of Attosecond 043426 (2017). Delays Using Perturbation Diagrams and Exterior [11] D. Busto, J. Vinbladh, S. Zhong, M. Isinger, S. Nandi, S. Complex Scaling, J. Phys. B 47, 124012 (2014). Maclot, P. Johnsson, M. Gisselbrecht, A. LHuillier, E. [27] J. Wätzel, A. S. Moskalenko, Y. Pavlyukh, and J. Berakdar, Lindroth, and J. M. Dahlström, Fanos Propensity Rule in Angular Resolved Time Delay in Photoemission, J. Phys. B Angle-Resolved Attosecond Pump-Probe Photoionization, 48, 025602 (2015). Phys. Rev. Lett. 123, 133201 (2019). [28] S. Heuser, Á. Jimenez Galán, C. Cirelli, C. Marante, R. [12] J. Vos, L. Cattaneo, S. L. Patchkovskii, T. Zimmermann, C. Sabbar, M. Boge, M. Lucchini, L. Gallmann, I. Ivanov, Cirelli, M. Lucchini, A. Kheifets, A. S. Landsman, and U. A. S. Kheifets, J. M. Dahlström, E. Lindroth, L. Argenti, Keller, Orientation-Dependent Stereo Wigner Time Delay F. Martín, and U. Keller, Angular Dependence of and Electron Localization in a Small Molecule, Science Photoemission Time Delay in Helium, Phys. Rev. A 94, 360, 1326 (2018). 063409 (2016). [13] D. Azoury, O. Kneller, M. Krüger, B. D. Bruner, O. Cohen, [29] A. Mandal, P. C. Deshmukh, A. S. Kheifets, V. K. Dolmatov, Y. Mairesse, and N. Dudovich, Interferometric Attosecond and S. T. Manson, Angle-Resolved Wigner Time Delay in Lock-in Measurement of Extreme-Ultraviolet Circular Atomic Photoionization: The 4d Subshell of Free and Dichroism, Nat. Photonics 13, 198 (2019). Confined Xe, Phys. Rev. A 96, 053407 (2017). [14] J. Fuchs, N. Douguet, S. Donsa, F. Martin, J. Burgdrfer, [30] I. A. Ivanov and A. S. Kheifets, Angle-Dependent Time L. Argenti, L. Cattaneo, and U. Keller, Time Delays from Delay in Two-Color XUV þ IR Photoemission of He and One-Photon Transitions in the Continuum, Optica 7, 154 Ne, Phys. Rev. A 96, 013408 (2017). (2020). [31] A. W. Bray, F. Naseem, and A. S. Kheifets, Photoionization [15] M. Ossiander, F. Siegrist, V. Shirvanyan, R. Pazourek, A. of Xe and Xe and Xe@C60 from the 4d Shell in RABBITT Sommer, T. Latka, A. Guggenmos, S. Nagele, J. Feist, J. Fields, Phys. Rev. A 98, 043427 (2018). Burgdrfer, R. Kienberger, and M. Schultze, Attosecond [32] S. Banerjee, P. C. Deshmukh, V. K. Dolmatov, S. T. Manson, Correlation Dynamics, Nat. Phys. 13, 280 (2017). and A. S. Kheifets, Strong Dependence of Photoionization [16] M. Schultze et al., Delay in Photoemission, Science 328, Time Delay on Energy and Angle in the Neighborhood of 1658 (2010). Fano Resonances, Phys. Rev. A 99, 013416 (2019).

031070-13 DAEHYUN YOU et al. PHYS. REV. X 10, 031070 (2020)

[33] C. Cirelli et al., Anisotropic Photoemission Time Delays [46] T. Sato, K. L. Ishikawa, I. Brezinová, F. Lackner, S. Nagele, Close to a Fano Resonance, Nat. Commun. 9, 955 (2018). and J. Burgdörfer, Time-Dependent Complete-Active- [34] M. Shapiro and P. Brumer, Principles of the Quantum Space Self-Consistent-Field Method for Atoms: Application Control of Molecular Processes (Wiley-Interscience, to High-Order Harmonic Generation, Phys. Rev. A 94, Hoboken, NJ, 2003). 023405 (2016). [35] K. C. Prince et al., Coherent Control with a Short- [47] K. L. Ishikawa and K. Ueda, Competition of Resonant Wavelength Free-Electron Laser, Nat. Photonics 10, 176 and Nonresonant Paths in Resonance-Enhanced Two- (2016). Photon Single Ionization of He by an Utlrashort Extreme- [36] V. Lyamayev et al., A Modular End-Station for Atomic, Ultraviolet Pulse, Phys. Rev. Lett. 108, 033003 (2012). Molecular, and Cluster Science at the Low Density Matter [48] K. L. Ishikawa and K. Ueda, Photoelectron Angular Beamline of FERMI@Elettra, J. Phys. B 46, 164007 (2013). Distribution and Phase in Two-Photon Single Ionization [37] C. Svetina et al., The Low Density Matter (LDM) Beamline of H and He by a Femtosecond and Attosecond Extreme- at FERMI: Optical Layout and First Commissioning, Ultraviolet Pulse, Appl. Sci. 3, 189 (2013). J. Synchrotron Radiat. 22, 538 (2015). [49] Y. Orimo, T. Sato, A. Scrinzi, and K. L. Ishikawa, [38] E. Allaria et al., The FERMI Free-Electron Lasers, Implementation of the Infinite-Range Exterior Complex J. Synchrotron Radiat. 22, 485 (2015). Scaling to the Time-Dependent Complete-Active-Space [39] V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Self-Consistent-Field Method, Phys. Rev. A 97, 023423 Reisler, Reconstruction of Abel-Transformable Images: The (2018). Gaussian Basis-Set Expansion Abel Transform Method, [50] L. Tao and A. Scrinzi, Photo-electron Momentum Spectra Rev. Sci. Instrum. 73, 2634 (2002). from Minimal Volumes: The Time-Dependent Surface Flux [40] M. Di Fraia et al., Complete Characterization of Phase and Method, New J. Phys. 14, 013021 (2012). Amplitude of Bichromatic XUV Light, Phys. Rev. Lett. 123, [51] A. Scrinzi, Infinite-Range Exterior Complex Scaling as a 213904 (2019). Perfect Absorber in Time-Dependent Problems, Phys. Rev. [41] B. Diviacco, R. Bracco, D. Millo, and M. M. Musardo, A 81, 053845 (2010). Phase Shifters for the FERMI@Elettra undulators,in [52] Y. Orimo, T. Sato, and K. L. Ishikawa, Application of Proceedings of the 2nd International Particle Accelerator the Time-Dependent Surface Flux Method to the Conference, San Sebastián, Spain (EPS-AG, Spain, 2011), Time-Dependent Multiconfiguration Self-Consistent-Field p. 3278. Method, Phys. Rev. A 100, 013419 (2019). [42] L. Giannessi, E. Allaria, K. C. Prince, C. Callegari, G. [53] A. Messiah, Quantum Mechanics (Dover, New York, 1961). Sansone, K. Ueda, T. Morishita, C. N. Liu, A. N. Grum- [54] B. Gao and A. F. Starace, Variational Principle for Grzhimailo, E. V. Gryzlova, N. Douguet, and K. Bartschat, High-Order Perturbations with Application to Multiphoton Coherent Control Schemes for the Photoionization of Neon Processes for the H Atom, Phys. Rev. A 39, 4550 (1989). and Helium in the Extreme Ultraviolet Spectral Region, Sci. [55] E. I. Staroselskaya and A. N. Grum-Grzhimailo, A Varia- Rep. 8, 7774 (2018). tionally Stable Method in the Problem of Two-Photon [43] M. Zangrando, D. Cocco, C. Fava, S. Gerusina, R. Gobessi, Atomic Ionization, Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. N. Mahne, E. Mazzucco, L. Raimondi, L. Rumiz, and C. 5, 45 (2015) [Moscow Univ. Phys. Bull. 70, 374 (2015)]. Svetina, Recent Results of PADReS, the Photon Analysis [56] E. V. Gryzlova, A. N. Grum-Grzhimailo, E. I. Staroselskaya, Delivery and REduction System, from the FERMI FEL N. Douguet, and K. Bartschat, Quantum Coherent Control Commissioning and User Operations, J. Synchotron Radiat. of the Photoelectron Angular Distribution in Bichromatic- 22, 565 (2015). Field Ionization of Atomic Neon, Phys. Rev. A 97, 013420 [44] L. Raimondi et al., Microfocusing of the FERMI@Elettra (2018). FEL Beam with a KB Active Optics System: Spot Size [57] F. T. Smith, Lifetime Matrix in Collision Theory, Phys. Rev. Predictions by Application of the WISE Code, Nucl. 118, 349 (1960). Instrum. Methods Phys. Res., Sect. A 710, 131 (2013). [58] M. Y. Amusia, Atomic Photoeffect (Plenum, New York, [45] T. Sato and K. L. Ishikawa, Time-Dependent Complete- 1990). Active-Space Self-Consistent-Field Method for Multielec- [59] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, tron Dynamics in Intense Laser Fields, Phys. Rev. A 88, Quantum Theory of Angular Momentum (World Scientific, 023402 (2013). Singapore, 1988).

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