Time Delay in Molecular Photoionization
TTime Delay in Molecular Photoionization
P. Hockett ∗ National Research Council of Canada, 100 Sussex Drive, Ottawa, K1A 0R6, Canada
E. Frumker
Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
D.M. Villeneuve, P.B. Corkum
Joint Attosecond Science Laboratory, National Research Council of Canadaand University of Ottawa, 100 Sussex Drive, Ottawa, K1A 0R6, Canada
Time-delays in the photoionization of molecules are investigated. As compared to atomic ioniza-tion, the time-delays expected from molecular ionization present a much richer phenomenon, witha strong spatial dependence due to the anisotropic nature of the molecular scattering potential. Weinvestigate this from a scattering theory perspective, and make use of molecular photoionization cal-culations to examine this effect in representative homonuclear and hetronuclear diatomic molecules,nitrogen and carbon monoxide. We present energy and angle-resolved maps of the Wigner delay timefor single-photon valence ionization, and discuss the possibilities for experimental measurements.
I. INTRODUCTION
The photoelectric effect – the emission of an electronfrom matter illuminated by light - is one of the most funda-mental phenomena in nature, which historically led to Ein-stein’s ground-breaking proposal of the quantization of light[1] and played a key role in the development of quantummechanics. In the early works, the electron emission wastacitly assumed to be instantaneous, following the absorp-tion of the excitation photon. However, more than half acentury ago, it was predicted theoretically that there shouldbe a time delay in the photoelectron emission process [2, 3],but it was only with the recent advances in attosecond sci-ence that direct measurements of electron dynamics withattosecond time resolution [4] required for the experimentalvalidation of this prediction could be realized. Time re-solved measurements of electron dynamics were reported[5–8] and the delay of photoemission was observed in con-densed matter [9] and atoms [10, 11] in the single photonweak-field regime. However, no measurements of photoe-mission time delays from molecular targets have been re-ported as yet. Here we discuss theoretical results of angleand energy resolved time delays in the photoionization ofmolecules, and the prospects for direct measurement of thisrich attosecond phenomena.In scattering theory the phase of the transmitted wave isa direct consequence of the interaction of the incident wavewith the scattering potential. Consequently, the scatter-ing phase can be associated with an advance or retardationof the transmitted wave caused by its interaction with thescattering potential V ( r, θ, φ ) , as measured in the asymp-totic limit. This phase-shift is always relative to the V = 0 case. A repulsive potential will lead to a negative phase,signifying an advance of the transmitted wave, while anattractive potential will lead to a positive phase, signify- ∗ Electronic address: [email protected] ing a retardation (or trapping) of the transmitted wave.These results are most simply derived in a stationary state(energy-domain) picture of scattering, but a wavepacket(time-domain) treatment yields the same essential features[12]. Hence, in a time-domain picture of photoionization,the scattering phase-shift and associated time delay can beviewed as a group delay of the outgoing photoelectron wave-packet, born at a time t within the ionizing laser pulse. Inthis case, the advanced wavepacket appears sooner than itwould for the V = 0 case, while the retarded wavepacketappears later than it would for V = 0 . This temporal re-sponse to the phase-shift is given by the Wigner delay, τ w ,which is determined by the energy-derivative of the scat-tering phase [2, 3].While the concept of the Wigner delay is well estab-lished [2, 3], interest has recently been rekindled due tothe experimental accessibility of the attosecond time do-main. Experiments using attosecond XUV pulse trains orisolated attosecond XUV pulses have been able to meas-ure the relative group delay of electron wavepackets fromatomic emission following single-photon absorption from aweak XUV field, with the measurements additionally re-quiring the interaction of the electron wavepacket with anIR field [10, 11]. The related possibility of determiningan absolute photoionization time t was discussed in thiscontext [10], and has also been explored in the strong-fieldregime via tunnel ionization with “attoclock” measurements[7], which employ pulses with rapidly changing instantan-eous polarization vector (e.g. circularly polarized light) toobtain high temporal resolution via angular streaking ofthe photoelectron wavepackets.In concert with these new experimental capabilities, nu-merous theoretical and computational studies have beenperformed. These can broadly be categorised as methodolo-gies based on (a) canonical scattering theory [13–16], or (b)fully-numerical approaches based on the time-dependentSchrodinger equation [17, 18]. In most cases Wigner delaysfrom the ionization of atomic targets have been of interest,and the angle-dependence of the process has not been in- a r X i v : . [ phy s i c s . a t o m - ph ] F e b vestigated; notable exceptions to this trend are the recentwork of Wätzel et. al. [19], who investigated the angle-dependence of the Wigner delay in detail for ionization ofneon and argon, and studies of H - the simplest molecularscatterer - from Serov et. al. [20], which includes someconsideration of the angle-dependence [57]. Conceptually,these methods are of course similar - one is seeking to solveequations that determine the continuum electron wavefunc-tions, and obtain scattering phase-shifts.The main distinction that can be drawn between theseapproaches is the generality of the method and the in-formation content of the results. A fully numerical treat-ment is, in principle, completely general, although in prac-tice may be limited by computational cost; nonetheless, ifperformed carefully, the "correct" final state wavefunctionshould be found for any given scattering system. A partic-ular strength of time-dependent numerical methods is theability to treat rapidly-varying scattering potentials, there-fore allowing the effects of strong laser-fields to be incor-porated into calculations. Such calculations have been em-ployed in order to model experiments incorporating strongfields [21–23], which cannot be treated adequately by atime-independent approach. More traditional scatteringtheory approaches are usually time-independent and mostsuited to the weak field regime, hence are appropriate forthe consideration of the intrinsic Wigner delay of the scat-tering system. Such approaches often use a partial-wave formalism, which allows separation into "geometric" and"dynamical" parts. In this case much progress can be madeanalytically, and a deep physical insight into the character-istics of the scattering can be gained (see, for example, ref.[24]). However, to obtain a complete solution to a complexscattering problem numerical methods are still ultimatelyneeded for the dynamical part, and a specific formalism forthe scattering system of interest is usually constructed inorder to yield tractable equations (see, for example, refs.[13, 20]); solving molecular scattering problems is thereforenon-trivial for even the simplest cases. This problem can,however, be addressed via the use of variational techniquesto solve the numerical part of the problem [25], allowingfor a methodology which retains the full physical insightsof scattering theory and the generality of fully-numericalapproaches, but at a significantly lower computational cost.In this work, we investigate Wigner delays from mo-lecular ionization based on this general approach. Weexplore the details of the time delay in the valence ion-ization of N and CO , based on calculations for single-photon ionization processes. The influence of the XUVfield on both the bound states and the continuum elec-tron are neglected, hence the results obtained correspondto the intrinsic Wigner delays of the photoemission pro-cess in the weak-field limit. We do not include any addi-tional continuum-continuum delays, which can be a signi-ficant contribution to the total observed delay in the caseof the XUV-IR measurements discussed above, but are de-pendent on the experimental technique [15] and not a fun-damental property of the ionizing system. In this limit, theeffect of the molecular potential on the energy and angle-resolved Wigner delay can be explored. This fundamental exploration forms the main thrust of the manuscript. Al-though the details are specific to valence ionization of N and CO , the results may be considered as prototypical formolecular ionization. As detailed below (Sect. III), wemake use of ePolyScat [26–28], a well-developed suite ofcodes from the scattering community, to solve the numer-ical integrals for arbitrary molecular potentials, thus ourmethodology is completely general and can be readily ap-plied to polyatomic molecules. We finish by discussing someattosecond metrology concepts which could provide deeperexperimental insight into ionization time delays in an angleand energy resolved manner. II. WIGNER TIME DELAY
As discussed by Wigner [2], Smith [3] and, more recently,in some depth by various authors [29–31], the phase of thescattering wavefunction can be associated with a time delayof the outgoing wavepacket, Ψ g . In a partial-wave decom-position, Ψ g is expressed as a coherent sume over partial-waves, Ψ g = (cid:80) lm ψ lm . Here each component is defined bythe quantum numbers ( l, m ) , the electronic orbital angu-lar momentum and its projection onto a given quantizationaxis respectively, and each ( l, m ) pair defines a partial-wavescattering channel.The time delay in a given channel is simply the derivativeof the phase with respect to energy: τ w ( (cid:15) ) = (cid:126) dη lm ( (cid:15) ) d(cid:15) (1)where η lm = σ l + δ lm is the total scattering phase, com-prised of a central-potential (Coulombic) contribution σ l and non-central (non-Coulombic) contribution δ lm . For aCoulomb potential τ w can be obtained directly from σ l ,which can be determined analytically, but in the general(non-Coulombic) case the total phase η lm must be determ-ined numerically. (It is of note that this definition of theWigner delay does not include the full r -dependence of thephase of the outgoing wavefunction, which is divergent foran infinite-range Coulomb potential - a more general defin-ition incorporating the total phase is given below. For fur-ther discussion of this point, the reader is referred to ref.[31], for the specific case of Wigner delays, and ref. [32] fora more general discussion.)Similarly, the group delay of the outgoing electron wave-packet can be defined as the (coherent) sum over all con-stituent channels: τ gw ( (cid:15) ) = (cid:126) dη g ( (cid:15) ) d(cid:15) (2)Here η g represents the total (group) scattering phase, de-termined from Ψ g , hence from the coherent summation overthe partial-wave channels.The significance of τ w is as a time-domain manifestationof the scattering phase η lm . Both contain the same in-formation, namely the effect of the interaction potential onthe outgoing wave, expressed as either a phase or delay. Asnoted above, this definition means that τ w does not directlyexpress the “ionization time” in terms of the timescale ofthe interaction of the system with a photon (or perturb-ing electric field), rather it describes the time taken for theoutgoing wavepacket to leave the influence of the potential,as defined by an effective range beyond which free-particlebehaviour is assumed, and expressed relative to the timetaken for a free particle with the same asymptotic velocity.In this sense a true reference time, t , is only specified tobe within the duration of the ionizing radiation field.[58]In atomic ionization, the relatively simple nature of thescattering potential results in a continuum wavepacket withlittle spatial structure, which can often be described byjust two partial-wave channels. In molecular ionization,the anisotropic nature of the potential means that manymore partial-waves are required to describe the photoelec-tron wavepacket, and significant spatial and energy struc-ture is expected. In essence, the angular structure of thephotoelectron wavepacket is the result of the angular inter-ferences between the partial-waves at a given energy, whilethe difference in the dependence of the phase-shift of anygiven l -wave on the photoelectron kinetic energy results inthe strong energy-dependence of the photoionization cross-section and τ w .The consequence of the angular dependence is, naturally,different τ w as a function of angle, most clearly defined inthe ionizing or molecular frame. We can rewrite equation2 for this more general case: τ w ( k, θ, φ ) = (cid:126) d arg( ψ ∗ lm ( k, θ, φ )) d(cid:15) (3)In this case we explicitly write τ w as a function of thepartial-waves ψ lm ( k, θ, φ ) , labelled as a function of pho-toelectron momentum k , and polar and azimuthal angles ( θ, φ ) relative to the molecular axis. These wavefunctionscontain both the scattering phase η lm ( k ) plus an angularcontribution Y lm ( θ, φ ) . The complex conjugate is requiredhere because the scattering phase appears as e − iη lm in ψ lm (for a discussion of continuum wavefunctions in photoionza-tion, see ref. [32]). As before, this equation expresses τ w for each partial wave channel, and the group delay resultsfrom the sum over all ( l, m ) terms: τ gw ( k, θ, φ ) = (cid:126) d arg( (cid:80) l,m ψ ∗ lm ( k, θ, φ )) d(cid:15) (4)In this work we examine the form of the energy and angle-resolved group delay for two specific benchmark cases,valence ionization of the diatomic molecules N and CO ,and consider how the delay responds to the details of themolecular potential and the resulting continuum wavefunc-tion. Σ u Π g NN (a) Σ u continuum (b) Π g continuum E k /eVE k /eV τ ω /as τ ω /as Figure 1: Group delays for ionization of N ( σ g → kσ u , kπ g ).(a) Σ u continuum, (b) Π g continuum. The main plots showpolar surfaces, as a function of photoelectron kinetic energy andangle, with the topography defined by the photoionization crosssection and colour-map by the Wigner delay τ gw ; insets show thesame data as 2D polar colour-maps, upper plot for τ gw (samescale as main colour-map) and lower plot for photoionizationcross-sections (arb. units). III. GROUP DELAY IN THE MOLECULARFRAMEA. Numerical details
Ionization matrix elements, which include the full scat-tering phase, were calculated using the ePolyScat suite ofcodes, distributed by R. R. Lucchese (for further details seerefs. [26–28]). These calculations take input from stand-ard electronic structure codes (Gamess, Gaussian etc.) todefine the initial state of the system. Ionization is treatedas a 1-electron process, leading to an N − electron sys-tem and a free electron (hence there are no multi-electroneffects in the sense of core relaxation, polarization etc.).The continuum wavefunction is solved numerically in the N − electron potential, via a Schwinger variational pro-cedure [25], and ionization matrix elements (within the di- Σ Π CO (a) Σ continuum (b) Π continuum E k /eVE k /eV τ ω /as τ ω /as Figure 2: Group delays for CO ( σ → kσ, kπ ). (a) Σ con-tinuum, (b) Π continuum. The main plots show polar surfaces,as a function of photoelectron kinetic energy and angle, withthe topography defined by the photoionization cross section andcolour-map by the Wigner delay τ gw ; insets show the same dataas 2D polar colour-maps, upper plot for τ gw (same scale as maincolour-map) and lower plot for photoionization cross-sections(arb. units). pole approximation) are calculated as the spatial overlap ofthis wavefunction and the initial orbital wavefunction, fora given polarization of the light and at single photoelectronenergy. This approach has been shown to work well in theweak field regime [25], and also for calculation of recombin-ation matrix elements in HHG [33] although, in general, itis not an appropriate technique for the strong field regimeas the laser field is not included in the scattering calcula-tions.In this work, calculations were based on equilibrium geo-metries and electronic structure from Gamess calculations(run at a relatively low, but appropriate, level of theory:RHF/MP2/6-311G) [34], with equilibrium bond lengthsfound to be 1.07 Å ( N ) and 1.12 Å ( CO ). Continuumwavefunctions and dipole matrix elements were computedwith ePolyScat, for the highest-lying σ -orbitals in bothcases, for linearly polarized ionizing radiation in both paral- lel and perpendicular geometries, and for photoelectron en-ergies from 1 to 45 eV. The phase information from the rawmatrix elements, expressed in terms of angular momentumchannels, provides the full scattering phase-shift, and ap-plication of eqn. 3 provides τ w for each channel. Similarly,eqn. 4 provides the group, or photoelectron wavepacket,delay. In the calculations, radial integrals are evaluated for r max =
10 Å, defining an effective range to the interactionat which the total phase (hence delay) is defined. By cal-culating the photoionization matrix elements for a range ofphotoelectron energies, the energy dependence of the pro-cess can be mapped out, and the complete dependence ofthe Wigner delay τ w ( k, θ, φ ) obtained.In the following, we present and discuss these resultsfor the general reader. Supplementary materials, includ-ing additional technical details of the results, e.g. channel-resolved dipole matrix elements, which may be of interestto some readers, are available online via Figshare at http://dx.doi.org/10.6084/m9.figshare.2007486 . B. Results
The results for the group (channel-integrated) Wignerdelay, τ gw ( k, θ, φ ) , for nitrogen and carbon monoxide areshown in figures 1 and 2, and represent the main res-ults of this work. In the standard notation of ionizingorbital → continuum wave the 1-electron ionization channelsare given as σ g → kσ u , kπ g for N and σ → kσ, kπ for CO [25]. In the following discussion these cases are denotedby the overall N -electron symmetry ( Γ ion (cid:78) Γ electron ) andspecies, e.g. N (Σ u ) , CO (Π) etc.Due to the cylindrically symmetric nature of these mo-lecules, the φ -coordinate is redundant in these cases, andwe can show the complete results as polar surface plots, asa function of energy and angle θ (relative to the molecu-lar axis), without any loss of information. In these plotsthe surface topography follows the magnitude of the dipolematrix element (proportional to the square-root of the pho-toionization cross-section), while the colour-map shows theenergy and angle-resolved Wigner time. As an alternativepresentation of the results, which may be clearer in print,the insets show the same data as polar colour-maps. The Σ and Π continua shown correspond to parallel or perpen-dicular laser polarization in the molecular frame respect-ively. The difference in peak magnitude between the con-tinua is not shown in the figures, which are independentlynormalised to emphasize the angular structure, but it is ofnote that the Σ continua dominate in both cases, with thepeak magnitude ratios of ∼ N (Σ u ) : N (Π g ) , and ∼ CO (Σ) : CO (Π) . The molecular structure andionizing orbital are also shown for reference, and the laserpolarization correlated with the different photoionizationcontinua accessed are indicated.These results present a complete, but complicated, pic-ture of the molecular photoionization event, and the asso-ciated Wigner delay for the outgoing photoelectron wave-packet. It is immediately apparent that there is a signi-ficant amount of structure observed, both as a functionof energy and angle, with τ gw values ranging from -200 to+200 as.In both cases, the ionizing orbital is the valence σ -bonding orbital, with lobes oriented along the molecularaxis. The choice of polarization of the ionizing radiation- either parallel or perpendicular to the molecular axis -defines the symmetry of the ionization continuum accessed,hence the symmetry of the continuum photoelectron wave-function. For both N and CO , this results in peaks inthe cross-section along the molecular axis ( θ = 0 ◦ , ◦ ) for the parallel case (figs. 1(a) and 2(a)), and orthogonalto the molecular axis ( θ = 90 ◦ , ◦ ) for the perpendic-ular case (figs. 1(b) and 2(b)). Weaker additional lobesare also observed in all cases, but are most pronounced inthe CO (Π) case, where they peak only around 20 % lowerthan the perpendicular features. Furthermore, the lack ofinversion symmetry in CO results in a significant differ-ence in the cross-sections between the oxygen ( θ = 0 ◦ ) andcarbon ( θ = 180 ◦ ) ends of the molecule, which is clearin both the Σ and Π continua, and again particularly pro-nounced in the additional lobes in the Π case, which domin-ate the cross-section around the carbon end of the molecule( θ = 140 ◦ , ◦ ) . IV. SCATTERING DYNAMICS
Physically, the peaks in the cross-section correspond tomaxima in the dipole integrals which define the couplingbetween initial orbital and final continuum wavefunctionsinduced by ionizing radiation, with an angular dependencegiven by the partial-wave interferences. For N (Σ u ) thispeak is the well-known shape-resonance [25, 35, 36], corres-ponding to an enhancement of the l = 3 partial-wave, whichcan be considered as a trapping of this part of the outgoingwavepacket due to the form of the molecular potential en-ergy surface. It is therefore not unexpected that the Wignerdelay is also long in this region. Less expected are the lobesalmost perpendicular to the molecular axis seen in fig. 1(a),and associated long delays. This can be physically ration-alized as a trapping of the outgoing wave in the bondingregion (i.e. the nitrogen-nitrogen triple bond), resultingin a long Wigner delay. For N (Π g ) the symmetry of theproblem results in a nodal plane along the molecular axis,so there is much reduced overlap between the main lobes ofthe ionizing orbital and the Π g continuum, as compared tothe Σ u case. Here the cross-section looks akin to scatteringthrough a slit, with a main feature and lower-intensity sidelobes, and the cross-section peak is significantly reduced ascompared to the Σ u case, as discussed above. The dipoleintegral peaks much closer to 0 eV, and it is only in the low-energy region that large Wigner delays are predicted. Formost of the energy and angular range the Wigner delay isclose to zero, consistent with a classical diffractive pictureof the ionization event, in which there is little trapping ofthe outgoing photoelectron wave.In the case of CO the picture is quite different. Herethe Wigner delays are predominantly negative, indicating aslight net repulsive effect from the molecular potential, and the results are highly asymmetric, consistent with the lossof inversion symmetry and the form of the ionizing orbitalfor a polar diatomic. The repulsive nature of the poten-tial is most significant at the oxygen end of the molecule,where the extent of the ionizing orbital is much reducedrelative to the carbon end. Chemically, the small extentof the orbital signifies the “electronegativity” of the oxygenatom, which will tend to acquire a slight negative chargerelative to the carbon atom. Based on chemical intuition,one might therefore expect to find a more repulsive po-tential than for the carbon end of the molecule, and thisis borne out in the Wigner delay results. At higher ener-gies, the Wigner delay at the carbon end becomes positiveand large. This can be understood by consideration of theradial part of the continuum wavefunction: at higher ener-gies the photoelectron wavelength becomes shorter, and thecontinuum function will become more penetrating relativeto the core wavefunction. Consequently, the spatial overlapintegral will incorporate more bound-state density closer tothe core, which is effectively more strongly bound due tothe slightly positive overall charge over the carbon atom,and will thus be delayed relative to bound-state density farfrom the core. At the oxygen end, the same change in over-lap has the opposite effect, and continues to result in largenegative Wigner delays due to the repulsive nature of themolecular potential over a large spatial region.In order to visualize this behaviour, figure 3 shows themolecular electrostatic potentials V ( r, θ ) for both (neut-ral) molecules [37, 38]. In the figure, a cut through thecylindrically symmetric potentials are shown by both acolour-map and contours. The ranges plotted are chosento highlight the long-range part of the potential which ismost structured, and largely responsible for the complexityof the scattering problem. The short-range, highly pos-itive, part of the potential, within which the majority ofthe bound electronic population resides, therefore appearsstructureless in these figures. Here it is clear that the negat-ive, repulsive part of the potential is much more significantfor CO than for N , and most significant around the oxygenend of the molecule, thus leading to the most pronouncednegative Wigner delays for wavepackets which experiencethis region. Conversely, the primarily attractive or neutralnature of the scattering potential for N , is responsible forthe positive Wigner delays observed in the calculations.Visualization of the scattering wavefunctions providesadditional physical insight into the dynamics of the pro-cess. Figure 4 shows a selection of continuum wavefunc-tions at different photoelectron energies, chosen to repres-ent the evolution of the scattering wavefunctions towardsthe peak in the cross-sections (shape-resonance), with sym-metries concomitant with ionization parallel to the molecu-lar frame ( N (Σ u ) and CO (Σ) ). At the highest energyshown, the far-field wave-front (approximately establishedat length-scales as short as several Å [39]) shows little ob-vious angular structure correlated with the core, save for abasic 2-centre scattering pattern. In contrast, at the twolower energies the angular structure is more complex, withthe nodal planes more pronounced. This change in angularstructure, for N , is exactly the shape-resonance effect dis- CO NN (a) N (b) CO δ− δ+ Figure 3: Molecular electrostatic potentials. (a) N , (b) CO .Contours show the long-range part of the molecular poten-tial, with the colour scale indicating slightly positive ( δ + ) andslightly negative ( δ − ) regions. cussed above, with the observed continuum structure cor-responding to the rise and fall of the l = 3 partial-wavecomponent over this energy range, including a significantchange in the magnitude of the wavefunction in the coreregion which has a strong effect on the overall ionizationyield. For CO the effect is slightly less clear, since thecontinuum structure is more complicated, but the generaltrend in complexity of the angular structure with energy issimilar, and has been labelled by other authors as a shaperesonance analogous to the N case [25]. In all cases, theasymptotic phase-shift of the waves is approximately estab-lished at the length-scales shown ( r max =
10 Å), and phasedifferences can be observed in the plots. The lack of inver-sion symmetry in the far-field phases for CO is clear, withthe phase shift between the carbon and oxygen ends of themolecule apparent in the intensity at the 10 Å cut-off. N N CO N CO Å Å Å Figure 4: Continuum wavefunctions | ψ ( θ, φ ) | for scattering from N (Σ u ) and CO (Σ) at E = 8 , , eV. Each plot is normal-ized to the peak of the wavefunction to highlight the spatialstructure. Most generally, the complex structures observed for thesetwo, relatively simple, diatomics might be regarded as in-dicative of molecular photoionization from valence orbit-als, which invariably involves spatially diffuse, highly struc-tured wavefunctions. The nature of the molecular poten-tial, which is responsible for the shape of the bound-stateorbitals, will similarly result in a continuum scattering wavewhich is highly sensitive to angle and energy. In this par-ticular set of results, the effect of symmetry-breaking alongthe molecular axis is very clear, and in general larger mo-lecules with lower symmetry may be expected to show sim-ilar, asymmetric, highly structured photoionization delays.The angular-sensitivity of the results points to the import-ance of angle-resolved measurement (or, equivalently, theloss of information inherent in angle-integrated measure-ments) for the investigation of molecular photoionizationdelays, and we consider this further in the following sec-tion.
V. MEASUREMENT
Recent measurements in atomic ionization using atto-second pulses have shown how τ w can be measured inthe time-domain. Ionization with attosecond XUV pulses,probed via streaking measurements [10], and side-bandmeasurements [11] have been demonstrated. In both casesthe effect of the IR probe field on the measurement is sig-nificant, and its effect on the photoelectron must be takeninto account in order to model (or extract) and understandthe measured delays. A series of theory papers have alsodiscussed this issue (for example refs. [15, 17, 21–23, 40]),most recently considering the angle-dependence of the time-delays in atomic ionization [16, 19], and left-right asym-metry in molecular ionization for CO [17].In essence, the measurements work by mixing the elec-tron wavepackets with the IR field, creating a spectrogramwith modulations referenced to the carrier-envelope phaseof the IR pulse. The streaking experiments used a strongIR field and FROG reconstruction of the resulting spec-trogram in order to determine the delay between photo-electrons emitted from different initial states (and at dif-ferent energies). The side-band measurement is based onsingle-photon absorption or emission in order to interferephotoelectrons from the same initial state, but created atdifferent energies via different harmonic orders in the pumppulse train. This is effectively the RABBIT technique [41],but implemented to obtain photoelectron scattering phasesinstead of optical phase information as per its original con-ception [59]. For example, ref. [42] investigated the effectof ionization resonances on the phases obtain from RAB-BIT studies of molecular nitrogen. In essence, these typesof measurement rely on phase differences between the in-terfering photoelectron wavepackets, so are sensitive to thedifference in the group delays between different photoelec-tron energies, however they are angle-averaged over thephotoelectron emission direction in the lab frame, and allpartial-wave components. It is of particular note that theangle-resolved cross-section will weight the angle-integratedmeasurement towards the Wigner delays of the main angu-lar features. [60]The scattering phases of individual partial waves, ata single energy, can be determined by measurements ofphotoelectron angular distributions. These are usuallytermed “complete” photoionization experiments, and havebeen successfully demonstrated for a range of atomic andmolecular ionization process (see refs. [43, 44] for ex-ample, for more comprehensive reviews see refs. [45, 46]),and most recently for multi-photon ionization with femto-second pulses, including electronic dynamics [47]. However,these measurements are typically not able to ascertain thephase structure with respect to energy, so can only determ-ine η lm for a given set of partial-waves with one of thewaves serving as a reference. These types of measurementtherefore provide detailed information on the angular partof the problem, including the phases of the contributingpartial-waves, but do not directly provide a full mapping of τ w ( k, θ, φ ) [61]. The possibility of such experiments in theatto-second regime has also yet to be explored, although it is feasible that the broad energy bandwidths availablewould allow for phase structure as a function of energy toalso be determined.Ultimately, a combination of these techniques would becapable of measurements of the full τ gw ( k, θ, φ ) . An angle-resolved RABBIT methodology would provide the energyand angular dependence of τ gw [62], and measurements inthis framework have very recently been investigated foratomic ionization [18, 48–50]. A detailed analysis of thephotoelectron angular distributions - possibly from thesame measurements, or more simply via direct ionizationmeasurements - could provide complementary partial-waveinformation; the coupling of these two analyses could thusprovide τ w ( k, θ, φ ) . The cleanest measurement strategywould also make use of molecular alignment, in order tochoose only a single continuum symmetry. This would in-crease the complexity of the experiment, but allow for adecrease in the complexity of the analysis.Very similar considerations have been explored in thecontext of high-harmonic generation (HHG). In particular,angle and energy resolved phase measurements of Br wereperformed with the LAPIN technique [51]. In this tech-nique, a two part measurement strategy (similar to thatoutlined above) is used in order to provide data whichallows for reconstruction of the energy and angular de-pendence of the phase of the emitted high-harmonic ra-diation. In this case the measured emission phase in-cludes contributions from strong-field ionization, propaga-tion in the continuum and photo-recombination (this isthe three-step model of HHG); the final step here is ef-fectively equivalent to single-photon ionization. The ex-periments are based on two-source interferometry tech-niques: sensitivity to the angle-dependence of the phaseis obtained in the case of two spatially distinct harmonicsources, both with Br molecules, but with one sourcealigned and the other unaligned; sensitivity to the energydependence of the phase is obtained in the case of two dis-tinct species of emitter, with harmonics generated from amixed gas containing Br and a reference atom ( Xe ). Thecombination of the measurements, combined with a self-consistent phase-reconstruction procedure, provided angleand energy-dependent phase information. The reconstruc-ted phases agreed reasonably well with theoretical results,which were based on ePolyScat calculations similar to thoseemployed herein. Although a relatively involved procedure,the complete phase information obtained with the LAPINtechnique will contain τ gw ( k, θ ) from the recombination pro-cess, however other sources of delay will be present.Another related study from the field of HHG is that ofmeasurements on oriented CO , which was combined with atheoretical treatment (again within the standard three-stepmodel) in order to understand the various phase contribu-tions to the emitted harmonics [52]. In this case the predic-tion of even harmonics relied on both the difference in phasebetween the ends of the molecule, and the phase accruedduring the tunnel ionization and propagation steps (alsodirectionally dependent due to the shape of the molecularpotential). Although these measurements are made in thefrequency domain, the process can be understood in thetime domain as attosecond bursts of harmonics occurringon each half-cycle of the driving laser field. The spectral in-terference of these bursts at the detector (hence integratedover the driving pulse duration and the generation volume)then determines the magnitude of the harmonics. Althoughthis mechanism is responsible for all harmonic generation,in the CO experiments it is especially pertinent for un-derstanding the effect of the asymmetry of the molecularpotential, which results in different timings of the ioniza-tion and recollision leading to a phase difference which ismapped to the generated XUV bursts. In such measure-ments the global phase structure can be ascertained due tothe bandwidth, or energy multiplexing, present. Althoughthe spectral phase in this case was not measured directly,calculations based on a modified 3-step model using time-dependent ionization and propagation calculations, com-bined with accurate recombination matrix elements (hencescattering phases) were able to recreate the intensity en-velope of the harmonic spectrum and spectral phase differ-ences between opposites end of the molecule. While notproviding the full mapping of τ gw ( k, θ, φ ) discussed above,these types of measurements are very sensitive to phasedifferences in specific directions ( θ = 0 ◦ vs. θ = 180 ◦ inthis case), and could therefore provide an interesting steptowards full angle-resolved time-delay measurements, withthe benefit of significantly reduced experimental complex-ity. One might consider that this frequency domain meas-urement of coherent attosecond processes is a techniquesensitive to dynamics on the time-scale of τ w , so could ad-ditionally be a sensitive probe of electronic dynamics.A final point of note with regard to measurement of mo-lecular vs. atomic Wigner delays is the increased density ofstates in the molecular case. This suggests that the maindifficulty in application of the measurement schemes dis-cussed above, particularly to polyatomics, will likely bespectral congestion due to overlapping vibronic bands inthe photoelectron spectrum. There is no general solutionto this problem, since it is somewhat inherent to molecularionization, but in many cases the issue may be side-steppedby judicious choice of spectral window(s) for RABBIT orsimilar type measurement schemes, with the obvious costof reducing the energy range which can be investigated. Al-ternatively, it may be possible to make use of this additionalstructure by, for instance, probing the effects on the Wignerdelay of ionizing from different states, or via different inter-mediate states by making use of degenerate ionization pro-cesses of different photon orders. In both cases, degeneratephotoelectrons will interfere (providing ensemble coherenceis maintained, and the process is symmetry-allowed), andinformation on the Wigner delays associated with the dif-ferent ionizing transitions will be contained in the measure-ment. Conceptually this is similar to the measurements ofref. [10], which ascertained the difference in Wigner delaybetween photoelectron wavepackets originating from s and p ionizing states. However, in that case the measurementwas made via streaking of energetically separated photo- electron bands, rather than via direct interference betweenthe bands. Older frequency-domain work has investigatedexactly the case of degenerate photoelectron band inter-ferences suggested here, examples include coherent controland complete experiments [53, 54], and as a method sens-itive to the Breit-Wigner phase shift of an intermediatebound-state. However, in these cases only narrow energyranges were considered, so these older works did not con-sider the energy-dependence of the photoionization phaseand the associated Wigner delays. VI. CONCLUSIONS
Molecular ionization is a complex phenomenon, with theoutgoing photoelectron wavepacket experiencing a highlyanisotropic scattering potential. In the time-domain, thisresults in a highly-structured Wigner delay, as a functionof energy and angle in the molecular frame. With theuse of scattering calculations, the angle-dependent Wignerdelay τ gw ( k, θ, φ ) was examined for two simple diatomics,and these results illustrate the magnitudes of the delays,and types of structures, which might generally be expectedin molecular photoionization. The deep link between theWigner delay and the photoionization matrix elements isalso revealed in the correlation of energy-domain photoion-ization phenomena - in this case the shape resonance in N - with features in the Wigner delay. Physically, this cor-respondence arises from the mildly attractive and repuls-ive regions in the long-range part of the scattering poten-tial, which largely determine the continuum photoelectronwavefunction at the energy ranges investigated. In a wave-packet picture, the same considerations are manifested aslarge changes in the photoelectron wavepacket dwell-timesin these spatial regions, both as a function of energy andangle in the molecular frame. Finally, some concepts for theexperimental measurement of angle-resolved Wigner delayswere discussed, suggesting the possibility of experimentalmethodologies based on existing RABBIT measurements(and conceptually similar HHG studies) for the measure-ment of angle-resolved Wigner delays. While the outlookhere is promising, given the highly-structured nature of theWigner delay and molecular ionization continuum, such ex-periments will be very challenging. Acknowledgements
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