Forward-backward asymmetry of photoemission in C_{60} excited by few-cycle laser pulses
FForward-backward asymmetry of photoemission in C excited by few-cycle laserpulses C.-Z. Gao, P. M. Dinh ∗ , P.-G. Reinhard, E. Suraud, and C. Meier Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France Institut für Theoretische Physik, Universität Erlangen, Staudtstraße 7, D-91058 Erlangen, Germany Laboratoire Collisions-Agrégats-Réactivité, Université de Toulouse, CNRS, UPS, France (Dated: November 11, 2018)We theoretically analyze angle-resolved photo-electron spectra (ARPES) generated by the in-teraction of C with intense, short laser pulses. In particular, we focus on the impact of thecarrier-envelope phase (CEP) onto the angular distribution. The electronic dynamics is describedby time-dependent density functional theory, and the ionic background of C is approximated bya particularly designed jellium model. Our results show a clear dependence of the angular distribu-tions onto the CEP for very short pulses covering only very few laser cycles, which disappears forlonger pulses. For the specific laser parameters used in a recent experiments, a very good agreementis obtained. Furthermore, the asymmetry is found to depend on the energy of the emitted photo-electrons. The strong influence of the angular asymmetry of electron emission onto the CEP andpulse duration suggests to use this sensitivity as a means to analyze the structure of few-cycle laserpulses. PACS numbers:
I. INTRODUCTION
With the advance in laser technology, it has becomepossible to generate femtosecond laser pulses which coveronly few optical cycles [1, 2], and this has rapidly found abroad range of applications in many disciplines, such asthe generation of attosecond pulses and precision controlof chemical process. In this context, one of the most in-triguing aspects is that these extremely short pulses maygive access to time-resolved electronic dynamics in atomsand molecules. This paves the way to a multiplicity ofinteresting phenomena, such as high-order harmonic gen-eration (HOHG), above-threshold ionization (ATI), andlaser-induced molecular fragmentation, as has been seenalready in several earlier experimental and theoreticalstudies [2–9], for reviews see Refs. [10–13]. It has alsobeen shown that structural information of the target canbe retrieved using short light pulses [14, 15].The carrier-envelope phase (CEP) is the phase of thefast oscillations of the laser field relative to its envelope.For few-cycle lasers, this CEP becomes a decisive laserparameter because the CEP offsets modify the pattern ofthe pulse dramatically, which, in turn, can have a strongimpact on laser-induced electron dynamics. For example,photoelectron emission induced by few-cycle laser fieldscan be controlled by the CEP, leading to a pronouncedforward-backward (also called “right-left”) asymmetry inthe photoelectron spectra (PES). This has been exper-imentally reported in [16] where it was found that theoutcome depends on the photoelectrons’ kinetic energy.The energy dependence has been explained within a semi-classical model by different electron emission processes in ∗ corresponding author : [email protected] the low- and high-energy regimes [17]. In the low-energyregime, electrons are directly emitted with a kinetic en-ergy of up to the U p , where U p = I las / ω las (in atomicunits) is the ponderomotive energy of the laser field.In the high-energy regime, electron recollision with thetarget system dominates, forming a plateau-like structurein the PES delimited by a well-defined cutoff. The rescat-tered electrons can be accelerated to energies of roughlyup to the U p [18] provided that the tail of the laserfield is still sufficiently high which is, however, rathercritical for few-cycle laser fields. Interestingly, a cou-ple of theoretical calculations have shown that the partsof PES related to electron recollisions are more sensi-tive to CEP than those parts related to electrons emit-ted directly [5, 9, 19–21]. Experimentally, the depen-dence of high-energy PES on the CEP has been exploredfor atoms, e.g., xenon [16, 22], argon [23], krypton [22],as well as for small-sized dimer molecules, e.g., N andO [24]. Recently, it has been studied in solids, such astungsten [25] and gold nanotips [26] which were found tobe efficient and controllable nanoemitters of extreme ul-traviolet (XUV) electrons, thus allowing to investigate ul-trafast electron dynamics in solids at an attosecond timescale, for a recent review see Ref. [27].Compared to atoms and dimers, the C fullerene isa typical example for a large system on the way frommolecules to solids. Thus the study of C might helptoward understanding dynamical properties of nanosys-tems. The advantage of C is its stability and accessibil-ity which renders it a useful laboratory for the study, e.g.,of thermal electron emission, charge migration, fragmen-tation channels, HHG, and ATI, see Refs. [28–31]. Froma geometrical point of view, C is similar to the outer-most part of capped-carbon nanotips which are promisingXUV electron nanoemitters. In a previous work [32], wehave theoretically investigated the PES of C in strong a r X i v : . [ phy s i c s . a t m - c l u s ] J a n fields using a near-infrared laser pulse ( λ las =912 nm). Inthat study, CEP effects were neglected since we consid-ered comparatively long pulses comprising about 8 opti-cal cycles. Recently, experiments on C using intensefew-cycle infrared laser pulses (720 nm and 4 fs) [33]were reported, in which a dramatic dependence of thePES on CEP was observed and qualitatively reproducedby Monte Carlo (MC) and Quantum Dynamical (QD)simulations.The aim of this article is to study the dependence of thePES on the CEP for C illuminated by intense, infrared,few-cycle laser pulses in a fully quantum-mechanicalframework. Our modeling is based on Time-DependentDensity-Functional Theory [34] with the time-dependentlocal-density approximation using the jellium approxima-tion for the ionic background [32]. We will focus on thedependence of the CEP effect on pulse length and onthe forward-backward asymmetry of photoemission dueto electron rescattering.The paper is outlined as follows, Section II briefly de-scribes the theoretical approach and the numerical anal-ysis. Results are presented and analyzed in Section III.Finally, conclusions are summarized in Section IV. II. FORMAL FRAMEWORK
We describe the electronic dynamics of C by time-dependent density functional theory (TDDFT) at thelevel of the time-dependent local density approximation(TDLDA) [35] using the exchange-functional from [36].For an appropriate modeling of electron emission, we aug-ment TDLDA by a self-energy correction (SIC) [37]. Asa full SIC treatment is computationally cumbersome [38],we use it in a simplified, but reliable and efficient versionas an average density SIC (ADSIC) [39]. The ADSIC suf-fices to put the single-particle energies into right relationto continuum threshold such that the ionization poten-tial (IP) is correctly reproduced in a great variety of sys-tems [40] from simple atoms to large organic molecules.In this context, a correct description of IP is crucial forphotoemission excited by external fields, in particular bystrong fields as it is assumed to be dominated by electronsin the highest occupied molecular orbitals (HOMO) [41].The ionic background (here carbon ions) is modeledwithin the jellium approximation by a sphere of positivecharge with a void at the center [42–44]. The jelliumpotential reads (in atomic units) : υ jel ( r ) = − (cid:90) d r (cid:48) ρ jel ( | r (cid:48) | ) | r − r (cid:48) | + υ ps ( | r | ) , (1a) ρ jel ( r ) = ρ g ( r ) , (1b) υ ps ( r ) = υ g ( r ) , (1c) g ( r ) = 11 + e ( r − R − ) /σ
11 + e ( R + − r ) /σ , (1d) R ± = R ± ∆ R . (1e) Here g denotes the Woods-Saxon profile, providing a softtransition from bulk shell to the vacuum. The jellium po-tential ( υ jel ) is augmented by an additional potential υ ps which is tuned to obtain reasonable values of the single-particle energies [45]. The shell radius R is taken fromexperimental data as R = 6 . [46]. The other parame-ters are the same as those in Ref. [32]. With the presentscheme, we reproduce rather well the electronic proper-ties of C : an IP at E IP = 7 . eV, a HOMO-LUMOgap of 1.77 eV, and a reasonable description of the photo-absorption spectrum [45]. This is in nice agreement withexperimental values [47, 48].It should be noted that the bulk density ρ is deter-mined such that (cid:82) d r ρ jel ( r ) = N el = 238 . Note thatthis number of electrons is different from 240 for a realC . This is because no jellium model so far manages toplace the electronic shell at N el = 240 as it should be.Most have the closure at N el = 250 [42–44]. The presentmodel with soft surfaces comes to N el = 238 which ismuch closer to the reality. Nevertheless, we have to keepin mind that a jellium model is a rough approximation toa detailed ionic structure. But it is a powerful approxima-tion as it allows to appropriately describe many featuresof electronic structure and dynamics in solids [49, 50] andcluster physics, see Refs. [51, 52]. Recently, the presentmodel has been validated as one of efficient and reliabletools to describe electron recollisions in strong fields inC , see Ref. [32]. The jellium model stands naturally fora frozen ionic background. It is justified for the presentstudy where the laser pulses considered are so short thatthe nuclear dynamics can be neglected.Within the dipole approximation, and assuming a lin-early polarized laser pulse with the polarization vectoralong the z -axis e z , the interaction with the laser field(in atomic units) is given by v las ( r, t ) = E ( t ) r · e z (2)with the electric field chosen to be E ( t ) = E cos (cid:18) πtT las (cid:19) cos( ω las t + φ CEP ) (3)for − T las / ≤ t ≤ T las / . Here, E denotes the peakelectric field, ω las the carrier frequency, and T las the totalpulse duration. The CEP is comprised in the parameter φ CEP which defines the phase between oscillations at fre-quency ω las and the maximum of the cos envelope. Inwhat follows, we use laser parameters close to those inrecent experiments [33]: laser frequency ω las = 1 . eV(a wavelength of 720 nm), intensity I = 6 × W / cm ,corresponding to field amplitude E = 1 . eV/a , andtotal duration T las =4 fs, 6 fs, and 8 fs, corresponding to1.7, 2.5, and 3.3 optical cycles (1 optical cycle = 2.4 fs).Note that these laser parameters are associated with aponderomotive energy U p = 2 . eV. Figure 1 illustratesthe temporal part of the laser field for the three T las underconsideration, each one for the CEP at ◦ (black curves)and ◦ (red curves). For φ CEP = 0 ◦ , the center of the -0.04-0.0200.020.04 -4 -3 -2 -1 0 1 2 3 4Time (fs)(c) T las = 8 fs-0.04-0.0200.020.04 -4 -3 -2 -1 0 1 2 3 4 E l ec tr i c fi e l d E ( t )( a . u . ) (b) T las = 6 fs-0.04-0.0200.020.04 -4 -3 -2 -1 0 1 2 3 4(a) T las = 4 fs φ = 0 ◦ φ = 90 ◦ Envelope
Figure 1: (Color online) The temporal part of the laserfield (3) for T las =4 fs (a), 6 fs (b), 8 fs (c). Each panel isplotted at φ CEP = 0 ◦ (black lines) and ◦ (red lines). Theenvelope is shown in dashed blue lines. Horizontal ( E =0) andvertical ( t =0) solid lines are depicted to facilitate the CEPcomparison. Other laser parameters are given in the text. envelope coincides with a maximum of the oscillations,while for φ CEP = 90 ◦ , it is shifted to match with thenodal points of the electric field. Clearly, we see a sub-stantial change of the electric field due to the differentCEP in each case. A. Numerical details
The TDLDA equations are solved numerically on acylindrical grid in coordinate space [53]. The static iter-ations towards the electronic ground state are done withthe damped gradient method [54] and time evolution em-ploys the time-splitting technique [55]. For details of thenumerical methods, see [56–58]. We use a numerical boxwhich extends
500 a in z direction (along the laser polar- ization) and
250 a orthogonal to it (radial r coordinate),with a grid spacing of . in both directions. Timepropagation is followed up to after 44 fs with a smalltime step of − fs. Box size and time span are suffi-ciently large to track completely the rescattering of elec-trons in the laser field (ponderomotive motion). To ac-count for ionization, absorbing boundary conditions areimplemented using a mask function [59]. The absorbingmargin extends over 35 a (70 grid points) at each side.The central observable of electron emission in our anal-ysis are angle-resolved photoelectron spectra (ARPES),i.e., the yield of emitted electrons [ Y ( E kin , θ ) ] as functionof kinetic energy E kin and emission angle θ . We calcu-late ARPES by recording at each time step the single-electron wave functions { ψ j ( t, r M ) , j = 1 , . . . , N el } atselected measuring points r M near the absorbing layerand finally transforming this information from time- tofrequency-domain, see [60–63]. Finally, the PES is writ-ten as Y ( E kin , θ ) ∝ N el (cid:88) j =1 | (cid:102) ψ j ( E kin , r M ) | (4)where (cid:102) ψ j are the transformed wave functions in energydomain. In case of strong fields, as we encounter here,the (cid:102) ψ j are to be augmented by a phase factor accountingfor the ponderomotive motion, for technical details see[62]. The angle θ is defined with respect to e z , i.e. θ =0 ◦ means electronic emission in the direction of e z . Adetailed ARPES analysis requires a fine resolution. Tothat end, we use an increment of 0.04 eV in energy and ◦ for the angular bins. III. RESULTS AND DISCUSSIONSA. CEP-averaged PES
We first look at CEP-averaged photoelectron spectra,simply denoted by PES, of C in a forward emissioncone as measured in the experiments of [33]. The com-puted PES are thus averaged over CEP in a range of ◦ - ◦ with ∆ φ CEP = 15 ◦ and collected in a forwardcone with opening angle of ◦ . Figure 2 shows thecalculated results for the three pulse lengths togetherwith the experimental results (black solid circles) [33].The patterns of the CEP-averaged PES are found to de-pend sensitively on the pulse length T las . At low energies( ≤ E kin ≤ eV), the photoelectron yield decreaseswith increasing T las . It is interesting to note that the PESfor T las =4 fs reaches a maximum around 10 eV. This co-incides approximately with 3.2 U p which is the maximalenergy upon the first return of rescattered electrons. Asimilar pattern has been obtained in C by quantumdynamical (QD) calculations [33] under the same laserconditions. For the longest pulse considered, T las =8 fs,we find pronounced peaks which are the ATI peaks sep-arated by the photon energy. For shortest pulse lengths, − − −
10 15 20 25 30 35 Y ( E k i n )( a r b . un i t s ) E kin (eV) 10 U p U p Experiment T las =8 fs T las =6 fs T las =4 fs Figure 2: (Color online) CEP-averaged photoelectron spec-tra (PES) of C excited by intense, linearly polarized, few-cycle laser pulses. Experimental data [33] are shown as blackcircles and calculated results as full curves for different pulselengths as indicated. The PES are collected in a cone withopening angle of ◦ at a fixed CEP and then averaged overa CEP range of ◦ - ◦ at a step of ∆ φ CEP = ◦ . The high-energy cutoff position at U p is indicated by a black arrow. these structures vanish because the pulse does not havesufficient energy resolution any more. In the high-energyregime ( ≤ E kin ≤ eV), we see a reverse depen-dence on T las , where the longest pulse ( T las =8 fs) leadsto the highest yield because there is more time to ac-celerate emitted electrons in the still ongoing laser field.The most satisfactory agreement between TDLDA re-sults and experimental data is found for T las =6 fs, whichcan nearly reproduce the measured PES data in the fullenergy range. This strong dependence of PES on pulselength may provide an opportunity to characterize theexperimental pulse duration by comparing the pattern ofPES to calculated results. B. Angle-resolved PES (ARPES)
In a next step, we analyze the full ARPES at fixedCEP values for the pulse length T las =6 fs where com-puted CEP-averaged PES agree best with experiments,as seen in the previous section. Figure 3 shows ARPESfor two typical CEP values, φ CEP = 0 ◦ in panel (a) and ◦ in (b). The most prominent feature in both casesis the remarkable θ = 0 ◦ ↔ ◦ asymmetry of thePES, particularly for electrons in the high-energy regime( E kin ≥ eV). At φ CEP = 0 ◦ in Fig. 3(a), high-energyphotoelectrons are emitted favorably in the direction to-wards θ = 180 ◦ , which is characterized by a rather broadramp extending to 35 eV. In contrast, low-energy elec-trons ( ≤ E kin ≤ eV) are emitted preferentially to-wards θ = 0 ◦ . The observed ARPES pattern is consis-tent with angular-integrated asymmetry maps in experi-ments [33] for the CEP analyzed here. The preferences of Figure 3: (Color online) ARPES of C excited by a pulsewith duration T las =6 fs, frequency ω las = 1 . eV, and in-tensity I las = 6 × W / cm for φ CEP = 0 ◦ (a) and ◦ (b). electron emission change for φ CEP = 90 ◦ in Fig. 3(b), inwhich the low-energy electrons have more weight towards θ = 180 ◦ while the high-energy electrons prefer the otherway towards θ = 0 ◦ . A similar asymmetry of the PES isalso observed for T las = ¨o dinger equation. A semiclassical explanation is thatthe generation of high-energy electrons originate from theelectron recolliding with the target, thus depending onthe two time instants at which electrons were releasedand scattered off, respectively. It is difficult for few-cyclelaser pulses to fulfill this condition simultaneously in θ =0 ◦ and ◦ directions. However, it is possible to realizeit in one of the two directions by tuning the CEP offsetof the laser field, as shown in Fig. 3. This has beenfirst suggested in [2] as a phase-meter to determine theabsolute phase of an ultrashort laser pulse. We shallshow in the next step that such a phase-meter stronglydepends on pulse duration and that it becomes invalidwith increasing pulse length. C. Asymmetry versus CEP
The dominant feature of the ARPES in Fig. 3 is thestrong influence of CEP on the energy resolved forward-backward asymmetry. This was also found in the pre-vious work where the asymmetry often produces regularoscillations for the CEP as function of kinetic energy [12].To investigate such oscillations for the present example,we define the asymmetry η as η ( φ CEP ) = Y ( E − , θ + ) − Y ( E − , θ − ) Y ( E − , θ + ) + Y ( E − , θ − ) (5)where E − denotes in brief the integration in the energyinterval [ E : E ] and θ + and θ − stand for cones of emis- -1-0.500.51 0 45 90 135 180 225 270 315 360 A s y mm e tr y p a r a m e t e r CEP offset (deg) experiment4 fs6 fs8 fs12 fs20.4 fs
Figure 4: (Color online) Comparison of the asymmetry pa-rameter η , see Eq. (5), from experiments [33] (black solidcircles) with theoretical results (red solid curves for T las = [ E : E ] = [23 eV : 30 . as done in experiments. sion angles. In most experiments, photoelectron yieldsare collected in a cone angle of ◦ . This means that θ + corresponds to | θ + | ≤ ◦ and θ − to | − θ − | ≤ ◦ .We use the same convention for our theoretical analysis.Figure 4 compares the asymmetry parameter η be-tween experiments (black solid circles) [33] and calcu-lated results for various pulse lengths T las . We find agood agreement between experimental data and presentresults for T las = η ( φ CEP ) . Yet, slight differences remain for the shapeof η ( φ CEP ) . The experimental curve shows softer transi-tions than the two theoretical curves. The cos envelopeof the theoretical pulse, see Eq. (3), is surely more con-fined than the experimental pulse which is often assumedto have Gaussian envelope, but may easily be plagued byprepulses [64].The pattern of η ( φ CEP ) differ substantially from theexperiment for other T las (smaller than 6-8 fs and largerones) concerning position of maxima/minima, amplitudeof oscillations, and softness on T las . Clear trends areseen for the amplitude which decreases with increasing T las and the softness which increases with T las , both to-gether eventually wiping out the signal for long pulses.The trends are plausible. Very short pulse shrink basi-cally to one oscillation and so become extremely sensitiveto the CEP while more and more comparably high oscil-lations in longer pulses render the CEP less crucial. Thisstrong sensitivity of the signal η ( φ CEP ) to pulses param-eters raises the question how sensitive the result is todetails of the pulse profile. To check that, we have alsorun calculations with a Gaussian envelope for the laserpulse instead of the cos envelope used above, see Eq. (3).The results are practically the same if the same FWHM isused. Therefore, we conclude that pulse length is the de-cisive parameter and measuring η ( φ CEP ) can give accessto this parameter. However, η emerges from combined action of laser pulses and responding system. This as-pect, i.e., the influence of the system, will be addressedin future work.Although it is plausible that the impact of CEP fadesaway for longer pulses, CEP-dependent asymmetry of thePES can be recovered also for longer pulses by a collinear,two-color pump-probe scheme, namely, a combination ofa fundamental laser ( ω ) and its n -th order harmonic ( nω )typically represented by a ω - ω laser setup. The presenceof a second harmonic is used to twist the field strengthof the fundamental mode by varying the delay phase,resulting in the asymmetry in the field amplitude influ-encing ionization as well as the rescattering. The ω - nω scheme has previously been proposed in [65], and haslater been used to explore the PES asymmetry in sodiumclusters (Na and Na +4 ) excited by intense 7-cycle laserfields [66]. A more recent experimental application of ω - ω combined laser pulses on rare gas atoms and CO molecule is found in Ref. [67]. On this basis, the com-parison of controlling efficiency of CEP-dependent asym-metry between one-color few-cycle fields and two-colormultiple-cycle fields is an intriguing topic, yet this is be-yond the scope of present study, thus we postpone it tothe next exploration.Since the asymmetry parameter (5) depends on the ki-netic energy of the photoelectrons, it is also interestingto study its dependence on the energy. Figure 5 shows η as a function of the kinetic energy and the φ CEP , us-ing T las = ∼ and ∼
30 eV , wefind the strong CEP dependence already visible in Fig. 4.For low energy electrons, those less than ∼
15 eV , we findan opposite behavior. This reflects the complex electron o )51015202530 E k i n ( e V ) -1-0.8-0.6-0.4-0.200.20.40.60.81 a s y mm e tr y Figure 5: (Color online) Dependence of the asymmetry pa-rameter on the CEP and on the kinetic energy of the emittedelectrons (for T las = <
15 eV and high energyelectrons (between
23 eV and
31 eV ). dynamics taking place during the interaction with theultrashort laser pulse, which changes drastically for dif-ferent CEP’s. -0.500.51 e x t . fi e l d E ( t )( e V / a ) z ( a ) -1.5-1-0.500.511.5 l o c a l v e l o c i t y ( a / f s ) -200-150-100-50050100150200 z ( a ) − − − − − l og d e n s i t y ρ ( a − ) Figure 6: Time evolution of external field (upper panel),local density along z -axis (middle panel), and local velocityalong z -axis (lower panel) for C excited by a laser pulse witha duration of T las =6 fs, frequency ω las = 1 . eV, intensity I las = 6 × W/cm , and CEP φ CEP = 0 ◦ . The cyan linesin the density plot (middle panel) indicate a velocity ± v = 58 a /fs corresponding to a kinetic energy of 13.6 eV and theyellow lines of ± v = 41 a /fs corresponding to 27.2 eV. To explain these features in more detail, we have an-alyzed the electron dynamics in the case of φ CEP = 0 ◦ .These results are depicted in Fig. 6 It shows the time evo-lution of the density (middle panel) and the velocity dis-tribution (bottom panel), together with the electric fieldof the laser pulse (top panel). In analyzing the results,one has to concentrate on the low-density tail correspond-ing to the finally emitted flow. The velocities need to belooked together with the density because velocity alonedoes not indicate the importance of a contribution (whichis weighted, of course, with the density). As soon as theelectric field sets on, both the density and the velocityshow oscillations synchronous with the external field. Abunch of high positive velocities (cyan line with positive slope) develops followed by a density shift to positive z short after the first negative peak in E at 2 fs. Thisturns into opposite direction following the strong posi-tive peak in E after 3 fs and a last swap back withthe counter peak after 4 fs. A particularly interestingprocess takes places with the largest peak at 3 fs. Thenegative peak before has triggered a strong flow to posi-tive z . This is abruptly stopped and counter-weighted bythe subsequent large positive field (exerting a force in thenegative direction). The positive flow recovers only withthe second negative E peak and leaves the system with akinetic energy of about 13.6 eV (yellow line with positiveslope). The large positive peak, on the other hand, re-leases a bunch of fast electrons towards negative directionwhich escapes eventually with the higher kinetic energyof 27.2 eV (cyan line with negative slope). The detailedtime-resolved picture so nicely elucidates the peaks on ◦ and ◦ direction in the previous figure. IV. CONCLUSIONS
We have investigated the impact of the carrier-envelopephase (CEP) of very short laser pulses on the angle-resolved photo-electron spectra (ARPES) of the C clus-ter. To this end, we used as a tool time-dependentdensity-functional theory at the level of the time-dependent local-density approximation. This was aug-mented by self-interaction correction to achieve a correctionization potential which is crucial for an appropriatedescription of photoemission dynamics. The ionic back-ground is assumed to be frozen during the femtoseconddynamics. For C , it is approximated by a jellium modelwhich is particularly tuned to its special geometry. Ab-sorbing boundary conditions are used to describe elec-tron emission and ARPES are computed with samplingthe time evolution of the single-electron wave functionsat selected measuring points close to the absorbing mar-gins.Our results depend sensitively on the laser pulselength. This holds already for the global signal of CEP-averaged photo-electron spectra where we find a goodagreement with experimental results when using the ap-propriate pulse length (and huge deviations from data forother pulse lengths). Particular attention was paid to theangular asymmetry of ARPES. A short glance at the fullARPES distribution and a detailed evaluation of asym-metry as a function of energy show that the asymmetrybehaves different in low- and high-energy regime. Thisis explained in detail by analyzing the time-dependentdensity and velocity distributions of the accelerated elec-trons.Following experimental data, we have focused then onthe dependence of asymmetry on CEP in the regime ofhigh-energy emission. For very short pulses, we find astrongly varying function oscillating with steep slopesbetween the forward/backward extremes of asymmetry.These patterns change significantly with the pulse pa-rameters. The amplitude of oscillations shrinks with in-creasing pulse length while the pattern become softer.In particular, the signal practically disappears for longerpulses covering 8 laser cycles or more. We find againa good agreement with experimental data for the ap-propriate pulse length, the same which also allowed toreproduce the CEP-averaged photo-electron spectra.We have studied the sensitivity of the results to thedetailed profile of the laser pulse (Gaussian versus cos ).The differences are so small that pulse profiles cannot beidentified clearly from the asymmetry signal.This study emphasizes the amount of detailed infor-mation that can be gained from a systematic scan of theARPES as a function of the CEP. For example, the highsensitivity of asymmetry versus CEP to the laser pulsemay be used for an independent measurement of pulseparameters. This may constitute an interesting aspectfor ultrashort pulse characterization. However, in orderto develop the shown methodology in this direction, one needs to carefully disentangle pulse properties from sys-tem properties, as resonances. Research along these linesare currently being undertaken. Acknowledgments:
We thank Institut Universitaire de France, EuropeanITN network CORINF and French ANR contractLASCAR (ANR-13-BS04-0007) for support during therealization of this work. One of authors (C.-Z.G.) isgrateful for the financial support from China ScholarshipCouncil (CSC) (No. [2013]3009). It was also grantedaccess to the HPC resources of CalMiP (Calcul enMidi-Pyrénées) under the allocation P1238, and ofRRZE (Regionales Rechenzentrum Erlangen). [1] M. Nisoli, S. De Silvestri, O. Svelto, R. Szipöcs, K. Fer-encz, C. Spielmann, S. Sartania, and F. Krausz, Opt.Lett. , 522 (1997).[2] G. Paulus, F. Grasbon, H. Walther, P. Villoresi,M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, Na-ture , 182 (2001).[3] G. Tempea, M. Geissler, and T. Brabec, JOSA B , 669(1999).[4] A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel,E. Goulielmakis, C. Gohle, R. Holzwarth, V. Yakovlev,A. Scrinzi, T. Hänsch, et al., Nature , 611 (2003).[5] S. Chelkowski and A. D. Bandrauk, Phys. Rev. A ,053815 (2005).[6] M. Kling, C. Siedschlag, A. J. Verhoef, J. Khan,M. Schultze, T. Uphues, Y. Ni, M. Uiberacker,M. Drescher, F. Krausz, et al., Science , 246 (2006).[7] Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong,Phys. Rev. Lett. , 073004 (2011).[8] X. Xie, K. Doblhoff-Dier, S. Roither, M. S. Schöffler,D. Kartashov, H. Xu, T. Rathje, G. G. Paulus, A. Bal-tuška, S. Gräfe, et al., Phys. Rev. Lett. , 243001(2012).[9] N. Suárez, A. Chacón, M. F. Ciappina, J. Biegert, andM. Lewenstein, Phys. Rev. A , 063421 (2015).[10] T. Brabec and F. Krausz, Rev. Mod. Phys. , 545(2000).[11] W. Becker, F. Grasbon, R. Kopold, D. Milosevic,G. Paulus, and H. Walther, Adv. Atom Mol. Opt. Phys. , 35 (2002).[12] D. Milošević, G. Paulus, D. Bauer, and W. Becker, J.Phys. B: At. Mol. Opt. Phys. , R203 (2006).[13] S. Haessler, J. Caillat, and P. Salieres, J. Phys. B: At.Mol. Opt. Phys. , 203001 (2011).[14] T. Morishita, A.-T. Le, Z. Chen, and C. D. Lin, Phys.Rev. Lett. , 013903 (2008).[15] H. Kang, W. Quan, Y. Wang, Z. Lin, M. Wu, H. Liu,X. Liu, B. B. Wang, H. J. Liu, Y. Q. Gu, et al., Phys.Rev. Lett. , 203001 (2010).[16] G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev.Lett. , 253004 (2003).[17] P. B. Corkum, Phys. Rev. Lett. , 1994 (1993).[18] G. G. Paulus, W. Becker, W. Nicklich, and H. Walther,J. Phys. B: At. Mol. Opt. Phys. , L703 (1994).[19] D. Milošević, G. Paulus, and W. Becker, Opt. Express , 1418 (2003).[20] X. M. Tong, K. Hino, and N. Toshima, Phys. Rev. A ,031405 (2006).[21] Q. Liao, P. Lu, P. Lan, W. Cao, and Y. Li, Phys. Rev. A , 013408 (2008).[22] M. Kling, J. Rauschenberger, A. Verhoef, E. Hasović,T. Uphues, D. Milošević, H. Muller, and M. Vrakking,New J. Phys. , 025024 (2008).[23] F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška,E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer,W. Becker, and G. G. Paulus, Phys. Rev. Lett. ,040401 (2005).[24] A. Gazibegović-Busuladžić, E. Hasović, M. Busuladžić,D. B. Milošević, F. Kelkensberg, W. K. Siu, M. J. J.Vrakking, F. Lépine, G. Sansone, M. Nisoli, et al., Phys.Rev. A , 043426 (2011).[25] M. Krüger, M. Schenk, and P. Hommelhoff, Nature ,78 (2011).[26] D. J. Park, B. Piglosiewicz, S. Schmidt, H. Kollmann,M. Mascheck, and C. Lienau, Phys. Rev. Lett. ,244803 (2012).[27] M. Krüger, M. Schenk, M. Förster, and P. Hommelhoff,J. Phys. B: At. Mol. Opt. Phys. , 074006 (2012).[28] I. Hertel, T. Laarmann, and C. Schulz, Adv. Atom Mol.Opt. Phys. , 219 (2005).[29] E. E. Campbell, K. Hansen, M. Hedén, M. Kjellberg,and A. V. Bulgakov, Photochem. Photobiol. Sci. , 1183(2006).[30] R. Ganeev, Laser Phys. , 25 (2011).[31] F. Lépine, J. Phys. B: At. Mol. Opt. Phys. , 122002(2015).[32] C.-Z. Gao, P. M. Dinh, P. Klüpfel, C. Meier, P.-G. Rein-hard, and E. Suraud, Phys. Rev. A , 022506 (2016). [33] H. Li, B. Mignolet, G. Wachter, S. Skruszewicz,S. Zherebtsov, F. Süßmann, A. Kessel, S. Trushin, N. G.Kling, M. Kübel, et al., Phys. Rev. Lett. , 123004(2015).[34] E. Runge and E. K. Gross, Phys. Rev. Lett. , 997(1984).[35] R. M. Dreizler and E. K. U. Gross, Density FunctionalTheory: An Approach to the Quantum Many-Body Prob-lem (Springer-Verlag, Berlin, 1990).[36] J. P. Perdew and Y. Wang, Phys. Rev. B , 13244(1992).[37] J. P. Perdew and A. Zunger, Phys. Rev. B , 5048(1981).[38] J. Messud, P. M. Dinh, P.-G. Reinhard, and E. Suraud,Ann. Phys. (N.Y.) , 955 (2008).[39] C. Legrand, E. Suraud, and P.-G. Reinhard, J. Phys. B:At. Mol. Opt. Phys. , 1115 (2002).[40] P. Klüpfel, P. M. Dinh, P.-G. Reinhard, and E. Suraud,Phys. Rev. A , 052501 (2013).[41] H.-G. Müller and M. Fedorov, Super-intense laser-atomphysics IV , vol. 13 (Springer Science & Business Media,1996).[42] M. Puska and R. M. Nieminen, Phys. Rev. A , 1181(1993).[43] D. Bauer, F. Ceccherini, A. Macchi, and F. Cornolti,Phys. Rev. A (2001).[44] E. Cormier, P.-A. Hervieux, R. Wiehle, B. Witzel, andH. Helm, Eur. Phys. J. D , 83 (2003).[45] P.-G. Reinhard, P. Wopperer, P. M. Dinh, and E. Suraud,in ICQNM 2013, The Seventh International Conferenceon Quantum, Nano and Micro Technologies (2013), pp.13–17.[46] K. Hedberg, L. Hedberg, D. S. Bethune, C. A. Brown,H. C. Dorn, R. D. Johnson, and M. de Vries, Science , 410 (1991).[47] D. L. Lichtenberger, M. E. Jatcko, K. W. Nebesny, C. D.Ray, D. R. Huffman, and L. D. Lamb, Mater. Res. Soc.Symp. Proc. , 673 (1990).[48] K. Sattler,
Handbook of Nanophysics: Clusters andFullerenes , Handbook of Nanophysics (CRC Press, 2010).[49] N. W. Ashcroft and N. D. Mermin,
Solid State Physics (Saunders College, Philadelphia, 1976). [50] C. Lemell, X.-M. Tong, F. Krausz, and J. Burgdörfer,Phys. Rev. Lett. , 076403 (2003).[51] U. Kreibig and M. Vollmer, Optical properties of metalclusters , vol. 25 (Springer Science & Business Media,2013).[52] M. Brack, Rev. Mod. Phys. , 677 (1993).[53] B. Montag and P.-G. Reinhard, Z. Phys. D: At., Mol.Clusters , 265 (1995).[54] P.-G. Reinhard and R. Cusson, Nucl. Phys. A , 418(1982).[55] M. Feit, J. Fleck, and A. Steiger, J. Comput. Phys. ,412 (1982).[56] F. Calvayrac, P.-G. Reinhard, E. Suraud, and C. A. Ull-rich, Phys. Rep. , 493 (2000).[57] P.-G. Reinhard and E. Suraud, Introduction to ClusterDynamics (Wiley, New York, 2003).[58] P. Wopperer, P. M. Dinh, P.-G. Reinhard, and E. Suraud,Phys. Rep. , 1 (2015).[59] P.-G. Reinhard, P. D. Stevenson, D. Almehed, J. A.Maruhn, and M. R. Strayer, Phys. Rev. E , 036709(2006).[60] A. Pohl, P.-G. Reinhard, and E. Suraud, J. Phys. B ,4969 (2001).[61] U. De Giovannini, D. Varsano, M. A. L. Marques, H. Ap-pel, E. K. U. Gross, and A. Rubio, Phys. Rev. A ,062515 (2012).[62] P. M. Dinh, P. Romaniello, P.-G. Reinhard, andE. Suraud, Phys. Rev. A , 032514 (2013).[63] M. Dauth and S. Kümmel, Phys. Rev. A , 022502(2016).[64] A. Giulietti, P. Tomassini, M. Galimberti, D. G. L. A.Gizzi, P. Koester, L. Labate, T. Ceccotti, P. DâĂŹO-liveira, T. Auguste, P. Monot, et al., Phys. Plasmas ,093103 (2006).[65] G. Paulus, W. Becker, and H. Walther, Phys. Rev. A ,4043 (1995).[66] H. Nguyen, A. Bandrauk, and C. A. Ullrich, Phys. Rev.A , 063415 (2004).[67] S. Skruszewicz, J. Tiggesbäumker, K.-H. Meiwes-Broer,M. Arbeiter, T. Fennel, and D. Bauer, Phys. Rev. Lett.115