Impact of two-electron dynamics and correlations on high-order harmonic generation in He
aa r X i v : . [ phy s i c s . a t m - c l u s ] F e b Impact of two-electron dynamics and correlationson high-order harmonic generation in He
Anton N. Artemyev, Lorenz S. Cederbaum, and Philipp V. Demekhin
1, 3, ∗ Institut f¨ur Physik und CINSaT, Universit¨at Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany Theoretische Chemie, Physikalisch-Chemisches Institut,Universit¨at Heidelberg, Im Neuenheimer Feld 229, 69120 Heidelberg, Germany Research Institute of Physics, Southern Federal University, Stachki av. 194, 344090 Rostov-on-Don, Russia (Dated: August 3, 2018)The interaction of a helium atom with intense short 800 nm laser pulse is studied theoreticallybeyond the single-active-electron approximation. For this purpose, the time-dependent Schr¨odingerequation for the two-electron wave packet driven by a linearly-polarized infrared pulse is solvedby the time-dependent restricted-active-space configuration-interaction method (TD-RASCI) in thedipole velocity gauge. By systematically extending the space of active configurations, we investigatethe role of the collective two-electron dynamics in the strong field ionization and high-order harmonicgeneration (HHG) processes. Our numerical results demonstrate that allowing both electrons in Heto be dynamically active results in a considerable extension of the computed HHG spectrum.
PACS numbers: 32.80.Rm, 33.20.Xx, 42.65.Ky
I. INTRODUCTION
An electron released from an atom or molecule and fur-ther on steered in the field of the parent ion by intenselaser fields gives rise to many fundamental phenomena[1]. For instance, the field driven electron-ion recombi-nation results in the emission of coherent radiation withfrequencies which are odd integer multiples of the carrierfrequency of exciting pulse, a phenomenon known as thehigh-order harmonic generation (HHG) process. Overthe last two decades, HHG has been intensively studiedexperimentally and theoretically (see, e.g., review articles[2–4] and references therein), owing to its powerful ap-plication to the generation of coherent XUV laser pulsesdown to the attosecond regime [5–8].In principle, high harmonics can be generated by tran-sitions among excited bound electronic states as wellas by the involvement of unbound continuum electronicstates. Detailed calculations on an array of quantum dotswith only bound discrete electronic states have demon-strated an efficient HHG [9]. However, as the current ex-periments on atoms and molecules are performed in thestrong-field ionization regime, below we concentrate onthis regime. Here, the essential mechanism behind theHHG process is explained within the simplified three-step model [1, 10, 11], in which an electron: (i) es-capes from the potential formed by the superpositionof the ionic core and linearly-polarized laser field po-tentials; (ii) is accelerated back to the parent ion dur-ing the next half-cycle of the driving pulse; and (iii) re-combines with the ion emitting thereby high-energy pho-tons. The validity of this illustrative model has beenconfirmed by numerous theoretical studies of HHG pro-cess in atoms and molecules performed within the single- ∗ Electronic address: [email protected] active-electron (SAE) approximation (for recent resultssee, e.g., Refs. [12–20] and references therein).The role of inactive electrons, which are kept frozenwithin the SAE approximation, is not negligible for thestrong-field processes. However, exact numerical solutionof the time-dependent Schr¨odinger equation (TDSE) formany-electron systems in laser fields is very formidabletask. At present, it has been realized only for He atom[21–27] to calculate single- and double-electron multi-photon ionization rates. The method of full dimensionalnumerical integration of TDSE for two electrons, devel-oped in these works, utilizes a basis set of coupled spheri-cal harmonics [21], and it is particularly efficient to studystrong-field problems which can be described by implyingvery small radial grids of about 100 Bohr.There are several theoretical studies of HHG processperformed beyond the SAE approximation in atoms [28–31] and molecules [32–36]. Ref. [28], for instance, pro-poses generalization of the three-step model for many-electron systems by introducing perturbative correctionsdue to exchange and electron correlations. A more con-sistent tracking of electron correlations and dynamicsis provided by the multi-configuration time-dependentHartree-Fock (MCTDHF) method [37–42]. Its straight-forward implementation to the solution of the strong-fieldproblems is, however, a challenging computational task[43, 44]. In order to be able to study HHG by MCTDHFmethod in realistic systems, one can either lower dimen-sionality of the problem [35] (i.e., consider only one ortwo dimensions for each electron), or/and neglect the ex-change interaction [29] (i.e., utilize the MCTDH method[45]).Alternatively, relaxing the full configuration interac-tion character of MCTDHF by cleverly selecting and in-corporating only the important electron correlation anddynamical effects into the ansatz for the wavepacket, onecan fully catch the essential physics of the process athand, and at the same time reduce considerably the nu-merical effort and thus make the computation tractable.This can be realized by limiting the space of active elec-tron configurations utilizing, e.g., the time-dependentrestricted-active-space configuration-interaction method(TD-RASCI, [46, 47]), the time-dependent generalized-active-space configuration-interaction (TD-GASCI, [36,48]), or the time-dependent restricted-active-space self-consistent-field theory (TD-RASSCF, [30]). The latterapproach has already been applied to calculate HHGspectra of the 1D beryllium atom [30].Recently [47], we have applied the TD-RASCI methoddeveloped in Ref. [46] to investigate the photoioniza-tion of He by intense high-frequency laser pulses. Inthe present work, we utilize this method to investigatethe influence of the correlative electron dynamics on theHHG spectra of this simplest system with two electrons,treated each in three dimensions (i.e., we treat here a sixdimensional problem). The paper is organized as follows.Sec. II outlines our theoretical approach and justifies thepresent choice of the active space of electron configura-tions. In the numerical calculations, we systematicallyincreased the space of active configurations and proceedas far as we could. These numerical results are presentedand discussed in Sec. III. We conclude with a brief sum-mary.
II. THEORY
The present theoretical approach is fully described inour previous work [47]. More details on its numericalimplementation can be found in Refs. [49–53]. Therefore,only essential relevant points are discussed below.We describe the light-matter interaction in the velocitygauge, which is the most suitable gauge for strong fieldproblems [54], since it ensures rapid convergence of thenumerical solution over angular momentum of releasedphotoelectrons [17]. In the electric dipole approximation,the total Hamiltonian governing dynamics of two elec-trons of He exposed to intense coherent linearly-polarizedlaser pulse reads (atomic units are used throughout)ˆ H ( t ) = − ~ ∇ − ~ ∇ − r − r + 1 | ~r − ~r |− i ( ∇ z + ∇ z ) A g ( t ) sin( ωt ) . (1)Here, g ( t ) is the time-envelope of the pulse, ω is its car-rier frequency, A is the peak amplitude of the vectorpotential (the vector potential and the electric field vec-tor are related via E = − ∂ t A ), and the peak intensityof the pulse is I = ω πα A ( α ≃ / .
036 is the finestructure constant, and 1 a.u. of intensity is equal to6.43641 × W/cm ).The present calculations were performed for laserpulses with carrier frequency of ω = 0 . λ = 800 nm) and peakintensity of 5 × W/cm . For this photon energy,ionization of He requires the absorption of at least 16 photons (the ionization potential is 24.587 eV [55]). Thecorresponding Keldysh [56] parameter γ = 0 .
64 indicatesthat ionization takes place in the intermediate regime be-tween the strong-field tunnel-ionization ( γ ≪
1) and mul-tiphoton ionization ( γ ≫
1) extremes. At the chosen fieldstrength, the rates for double ionization of He are verysmall compared to its single ionization rates [23, 26, 57].We may, therefore, neglect in the present study of theHHG process the double ionization and permit only oneof the electrons of He to be ionized by the pulse. Nev-ertheless, we allow the bound electron to interact withthe laser field as well as with the photoelectron, i.e., it isfully active but kept bound.To accomplish the above description, we utilized thefollowing symmetrized ansatz for the spatial part of thetotal two-electron wave function Ψ( ~r , ~r , t ) in the singletspin state:Ψ( ~r , ~r , t ) = X α a α ( t ) φ α ( ~r ) φ α ( ~r )+ X α>α ′ b αα ′ ( t ) 1 √ φ α ( ~r ) φ α ′ ( ~r ) + φ α ′ ( ~r ) φ α ( ~r )]+ X αβ √ φ α ( ~r ) ψ β ( ~r , t ) + ψ β ( ~r , t ) φ α ( ~r )] . (2)As justified above, the wave function (2) is constructedby using two different mutually-orthogonal one-electronspatial basis sets { φ α ( ~r ) } and { ψ β ( ~r, t ) } . The for-mer describes dynamics of the electron which remainsbound to the nucleus, and it includes selected discreteorbitals { φ α ( ~r ) ≡ φ nℓm ( ~r ) } . The latter is formed bythe time-dependent wave packets of a photoelectron { ψ β ( ~r, t ) ≡ ψ αℓm ( ~r, t ) } , which are built to be orthogonalto all incorporated discrete orbitals, i.e., h φ α | ψ β ( t ) i = 0.In order to describe these one-electron basis sets,we applied the finite-element discrete-variable repre-sentation (FEDVR) scheme and introduced the three-dimensional basis element ξ λ ( ~r ) as: ξ λ ( ~r ) ≡ ξ ik,ℓm ( ~r ) = χ ik ( r ) r Y ℓm ( θ, ϕ ) . (3)Here, the radial coordinate is represented by the basis setof the normalized Lagrange polynomials χ ik ( r ) [49–53]constructed over a Gauss-Lobatto grid { r ik } (index i runsover the finite intervals [ r i , r i +1 ] and index k counts thebasis functions in each interval). Using this FEDVR, thenormalized stationary orbitals and the time-dependentwave packets can be expanded as follows (note that λ ≡{ ik, ℓm } is four-dimensional index): φ α ( ~r ) = X λ d αλ ξ λ ( ~r ) , (4a) ψ β ( ~r, t ) = X λ c βλ ( t ) ξ λ ( ~r ) . (4b)In the used basis set of the three-dimensional elementsEq. (3), all matrix elements of the Hamiltonian (1) canbe evaluated analytically. The corresponding explicit ex-pressions can be found in our previous work [47], apartfrom the light-matter interaction term which was treated there in the dipole length gauge. In the velocity gaugeused here, the dipole transition matrix element reads h ξ λ |∇ z | ξ λ ′ i = (cid:28) χ ik r Y ℓm (cid:12)(cid:12)(cid:12)(cid:12) cos θ ∂∂r − sin θr ∂∂θ (cid:12)(cid:12)(cid:12)(cid:12) χ i ′ k ′ r Y ℓ ′ m ′ (cid:29) = "r (2 ℓ + 3)( ℓ − m + 1)( ℓ + m + 1)2 ℓ + 1 δ ℓ ′ ,ℓ +1 δ i,i ′ δ k,k ′ r ik + s ( ℓ − m + 1)( ℓ + m + 1)(2 ℓ + 3)(2 ℓ + 1) δ ℓ ′ ,ℓ +1 + s ( ℓ − m )( ℓ + m )(2 ℓ + 1)(2 ℓ − δ ℓ ′ ,ℓ − ! ∞ Z χ ik ddr χ i ′ k ′ dr − ( ℓ ′ + 1) δ i,i ′ δ k,k ′ r ik δ m,m ′ , (5)where the first derivative dχ ik /dr can be evaluated ana-lytically via Eqs. (10) from Ref. [47].Evolution of the total wave function (2) in time is givenby the vector of the time-dependent expansion coeffi-cients ~A ( t ) = n a α ( t ); b αα ′ ( t ); c βλ ( t ) o , which was propa-gated according to the Hamiltonian (1): ~A ( t + ∆ t ) = exp n − iP ˆ H ( t ) P ∆ t o ~A ( t ) . (6)The one-particle projector P = 1 − P α | φ α ih φ α | in Eq. (6)acts on the n c βλ ( t ) o subspace and ensures the orthogonal-ity condition h φ α | ψ β ( t ) i = 0. The propagation was car-ried out by the short-iterative Lanczos method [58]. Theinitial ground state ~A ( t = 0) was obtained by the propa-gation in imaginary time (by relaxation) in the absenceof the field. The total three-dimensional photoemissionprobability was computed as the Fourier transformationof the final electron wave packets at the end of laser pulse: W ( ~k ) = 1(2 π ) / X β (cid:12)(cid:12)(cid:12)(cid:12)Z ψ β ( ~r ) e − i~k · ~r d ~r (cid:12)(cid:12)(cid:12)(cid:12) . (7)Finally, the HHG spectrum I ( ω ) was computed as thesquared modulus of the Fourier transformed accelerationof the total electric dipole moment: I ( ω ) = 1(2 π ) / (cid:12)(cid:12)(cid:12)(cid:12)Z d D ( t ) dt e − iωt dt (cid:12)(cid:12)(cid:12)(cid:12) , (8)with D ( t ) given by D ( t ) = h Ψ( ~r , ~r , t ) | ~r + ~r | Ψ( ~r , ~r , t ) i . (9)The present study was conducted for two differentpulse envelopes g ( t ). The main set of calculations wasperformed for a trapezoidal pulse with a linearly-growingfront-edge, a constant plateau with unit height, and alinearly-falling back-edge, each supporting five optical cy-cles. The propagation was thus performed in the timeinterval of [0 , T f ] with T f = 15 πω ≃
40 fs. In addition, we performed a few key calculations for a sine-squaredpulse of the same length g ( t ) = sin ( π tT f ). The size ofthe radial box was chosen to be R max = 3500 a.u. Theinterval [0 , R max ] was represented by 1750 equidistantfinite elements of the 2 a.u. size, each covered by 10Gauss-Lobatto points. The photoelectron wave packetswere described by the partial harmonics with ℓ ≤ R max , they were multiplied at eachtime-step by the following mask-function [13, 59] g ( r ) = ( , r < R (cid:16) cos h π r − R R max − R i(cid:17) , R < r < R max , (10)The mask-function (10) was set-in at R = 3400 a.u.Simultaneously, we paid attention that the total electrondensity in the interval of [0 , R ] does not deviate from itsinitial value of 2 by more than 10 − . III. RESULTS AND DISCUSSION
The present calculations were performed on differ-ent levels of approximation which are systematically im-proved by extending in the total wave function (2) theset of electronic configurations describing the dynamicsof the electron which remains bound. The photoelectronis described fully. For this purpose, in each improve-ment step we expanded the basis set { φ α ( ~r ) } describ-ing the bound electron by one additional hydrogen-like nℓ + function of the He + ion with large quantum num-bers n and ℓ . The smallest basis set used includes onlyone 1 s + orbital of He + . Hence, in this simplest level ofapproximation the bound electron is frozen and exhibitsno dynamics. This approximation is an analogue of theSAE approximation. One should stress, however, that itdiffers from the usual one-electron SAE approximationin hydrogen, since the Hamiltonian (1) includes directand exchange Coulomb interactions between bound andcontinuum electrons. The largest basis set of active or- -15 -13 -11 -9 -7 -5 -3 -1 -5 -3 -1 Radial coordinate (a.u.) E l e c t r on den s i t y ( a . u . ) n = = = = E l e c t r on s pe c t r u m ( a . u . ) Electron energy (eV) n = = = = FIG. 1: (Color online) The final photoelectron radial densi-ties ( upper panel ) and the final photoionization spectra ( lowerpanel ) after the linearly-polarized 800 nm trapezoidal pulsehas expired. The time-shape, duration and intensity of thepulse are indicated in the text. Note the logarithmic scaleson the vertical axes. The calculations are performed in a sys-tematic series of improving approximations by sequentiallyextending the basis set of discrete orbitals { nℓ + } , used to de-scribe dynamics of the bound electron. In each step, this basiswas extended by the layer of all nℓ + orbitals with larger prin-cipal quantum number n , as indicated at the right-hand sideof each spectrum. The results labeled by n = 1 correspond tothe SAE approximation. To enable for a better comparison,the different spectra in each panel are vertically shifted up-wards by multiplying successively with 10 starting with thespectrum of the SAE ( n = 1 layer) approximation. bitals used in the calculations consists of all the discreteone-electron functions { nℓ + } with n ≤ ℓ ≤ n = 1) falls exponentially as a function of distance to thenucleus (note the logarithmic scale on the vertical axis).This density is modulated by weak sharp features whichrepresent bunches of fast electrons released by the strong-field multiphoton above-threshold ionization. One can = = = = T o t a l d i po l e m o m en t ( a . u . ) Time (a.u.)
FIG. 2: (Color online) The total time-dependent electricdipole moment Eq. (9), induced by the linearly-polarized800 nm trapezoidal pulse described in the text. The calcu-lations are performed in a systematic series of improving ap-proximations as indicated in the caption of Fig. 1. see from this figure that enabling the bound electron inHe to be dynamically active causes considerable changesin the computed radial density. Indeed, systematicallyextending the { nℓ + } basis set by the layers of orbitalswith n =2, 3, and 4 results in a significant lowering ofthe computed electron density around the nucleus and toits slight enhancement in the outer region. This alreadyindicates the loss of low-kinetic-energy and gain of high-kinetic-energy electrons in the spectrum.The final photoelectron spectra obtained as a Fouriertransformation from the final wave packets via Eq. (7)are compared in the lower panel of Fig. 1. Because of thetrapezoidal pulse envelope, the computed spectra consistof a comb of sharp peaks. These peaks with exponen-tially falling intensity are separated by the photon energy ω (note also the logarithmic scale on the vertical axis).Each peak represents photoelectrons with kinetic energyof ε nj = E n − E − jω released by the above-thresholdmultiphoton ionization of He (here j is the number of ab-sorbed photons, E is the ground state energy of He, and E n stands for the energy of the nℓ + ionic state of He + ).By comparing slopes of the electron spectra obtained indifferent approximation (see lower part of Fig. 1), one cannow directly observe the already announced diminutionof the low-energy and enhancement of the high-energyparts of the computed spectra, caused by the dynamicsof the bound electron.Figure 2 compares the total time-dependent electricdipole moments (9) computed in different approxima-tions. As is evident from this figure, the impact of thedynamics of the bound electron on this quantity is dra-matic: The maximal value of D ( t ), computed for thelargest basis set used here (labeled as n = 4), drops byalmost a factor of 7 as compared to the SAE approxima-tion (labeled as n = 1).We now turn to Fig. 3 which collects all HHG spec-tra computed for the trapezoidal pulse by sequentiallyextending the basis set of discrete orbitals { nℓ + } by oneadditional function. For a better eye view, the spectraobtained in each step are shifted vertically: The lower-most spectrum corresponds to the SAE approximation,whereas the uppermost one corresponds to the largestbasis set used here. The HHG spectrum computed in theSAE approximation (the lowermost spectrum in Fig. 3 la-beled as 1 s + ) exhibits a set of sharp harmonics kω withodd numbers k ≤ s + state to the basis set of active orbitals for the boundelectron extends the number of generated harmonics. Al-lowing the bound electron to occupy the 2 s + and 2 p + or-bitals, further extends the number of generated harmon-ics. One can speak of the formation of a second relativelyweak plateau in the HHG spectrum, which starts at thecutoff of the main plateau and exhibits its own cutoff atmuch higher kω . Including sequentially the additional3 s + , 3 p + , 3 d + , and further on the 4 s + , 4 p + , 4 d + , and4 f + orbitals in the { nℓ + } basis set results in a systemati-cal broadening of the second plateau shifting its cutoff toharmonics kω of higher and higher order k (see Fig. 3).The HHG spectra computed by sequentially adding alayer of { nℓ + } states with fixed principal quantum num-ber n = 1, 2, 3, and 4 are summarized again in the upperpanel of Fig. 4. One can see that the extension of thefirst plateau of the HHG spectrum does not change asthe basis set is enlarged. The extension of the secondplateau, which is due to the presence of a second activeelectron, appears once the basis set contains quantumnumbers n larger than 1, and grows fast with additionalbasis functions n = 2 and n = 3. The extension of thisplateau computed for the largest basis set with n = 4does, however, not differ much from that obtained with n = 3 indicating a noticeable trend in the convergenceof the present computational results with respect to thebasis set of discrete orbitals. We stress that the calcula-tions performed here for the largest basis with n = 4 werealready at the limit of our computational capabilities.As a final point of our study, we ensure that the effectobserved here is independent of the time-envelope of thelaser pulse employed. For this purpose, we performed ananalogous set of calculations using a sine-squared pulseof the same length T f and intensity I (for details seethe last paragraph of the preceding section). The resultsof these calculations are collected in the lower panel ofFig. 4. The HHG spectrum computed in the SAE ap-proximation (labeled as n = 1) exhibits a main plateauand cutoff which are very similar to those obtained forthe trapezoidal pulse in the same approximation (com-pare with the lowermost spectrum in the upper panel ofthis figure). From the lower panel of Fig. 4 one can also -17 -15 -13 -11 -9 -7 -5 -3 -1 + + + + + + + + HH G s pe c t r u m ( a r b . un i t s ) Harmonic order + + FIG. 3: (Color online) The HHG spectra of He computed forthe trapezoidal laser pulse. The calculations are performed ina systematic series of improving approximations by sequen-tially extending the basis set of discrete orbitals { nℓ + } , usedto describe dynamics of the bound electron. In each step,this basis was extended by one orbital, which is indicated atthe right-hand side of each spectrum. The results labeled by1 s + correspond to the SAE approximation. To enable for abetter comparison, the spectra for each nℓ -state are verticallyshifted upwards by multiplying successively with 10 startingwith the 1 s + spectrum. -17 -15 -13 -11 -9 -7 -5 -3 -1 -16 -14 -12 -10 -8 -6 -4 -2 n = HH G s pe c t r u m ( a r b . un i t s ) n = = = = = = Harmonic order
FIG. 4: (Color online) Summary of the HHG spectra of He,computed for the trapezoidal ( upper panel ) and sine-squared( lower panel ) laser pulses in different approximations (see cap-tion of Fig. 1 and text for details). To enable for a bettercomparison, the spectra for each n -layer are vertically shiftedupwards by multiplying successively with 10 starting withthe spectrum of the SAE ( n = 1 layer) approximation. see that allowing the bound electron in He to occupy thenext layer of orbitals with n = 2 results in the forma-tion of a second plateau in the computed HHG spectrum.As demonstrated by the calculations which also includethe next layer of orbitals with n = 3, the second cutofffurther moves toward higher photon energies. IV. CONCLUSION
Generation of high-order harmonics in the He atomexposed to intense linearly-polarized 800 nm laser pulseis studied beyond the single-active-electron approxi-mation by the time-dependent restricted-active-spaceconfiguration-interaction method. During the propaga-tion of the two-electron wave packets in strong laserfields, we allowed only one of the electrons to be ion-ized and kept the other electron always bound to the nucleus, neglecting thereby the double ionization processwhich is very weak for the pulse applied. For this pur-pose, the present active space was restricted to configura-tions which permit only one of the electrons to populatecontinuum states. This photoelectron was described inthe time-dependent wave packets with angular momenta ℓ ≤
50. The dynamics of the bound electron was de-scribed by a set of selected discrete orbitals { nℓ + } of theHe + ion. In the numerical calculations, this discrete one-electron basis was systematically increased by includingstates with larger quantum numbers n and ℓ up to n ≤ ℓ ≤ kω up to k of about 100. For the largest basis set of discrete orbitalsused here, the second plateau, which is about three or-ders of magnitude weaker than the main one, consists ofadditional harmonics with 100 < k < { nℓ + } , it is rather difficult to exactly predict thefinal fate of the second plateau found at higher order har-monics. Nevertheless, the main theoretical conclusion ofthe present work – that going beyond the SAE approxi-mation and allowing more electrons to be active and tointeract is important and leads to the generation of con-siderably more harmonics – will remain unchanged. Afull understanding of how two or more active electronsimpact the HHG is a rather involved subject and goesmuch beyond the present work.Nevertheless, we can say already now that having morediscrete orbitals implies many more bound states of thetwo and in other systems possibly more electrons partici-pating in the process. These bound states alone can alsogive rise to some HHG [9]. In Ref. [9] a realistic modelfor an array of quantum dots with six active correlatedbound electrons has been solved numerically exactly andshown to unambiguously give rise to a second plateau inthe HHG spectrum. Although the situation in Ref. [9]differs from ours, this result supports our finding that al-lowing more electrons to be active generates higher har-monics and is likely to give rise to a second plateau. Inour present example of He, the ionization of an electronplays a crucial role, but the existence of the additionalbound states certainly leads to additional different path-ways of the important ionization channel and this willalso influence the HHG. Finally, after one electron is ion-ized, the He + ion can stay in several excited states. Therecombination step is then also different from that in theSAE approximation as there are many new pathways forit now. Acknowledgments
This work was partly supported by the State Hes-sen initiative LOEWE within the focus-project ELCH, by the F¨orderprogramm zur weiteren Profilbildung inder Universit¨at Kassel (F¨orderlinie
Große Br¨ucke ), bythe Deutsche Forschungsgemeinschaft the within DFGproject No. DE 2366/1-1, and by the U.S. ARL andthe U.S. ARO under Grant No. W911NF-14-1-0383.Ph.V.D. acknowledges Research Institute of Physics,Southern Federal University for the hospitality and sup-port during his research stay there. [1] P.B. Corkum, Phys. Rev. Lett. , 1994 (1993).[2] T. Brabec and F. Krausz, Rev. Mod. Phys. , 545(2000).[3] C. Winterfeldt, C. Spielmann, and G. Gerber, Rev. Mod.Phys. , 117 (2008).[4] F. Krausz and M. Ivanov, Rev. Mod. Phys. , 163(2009).[5] P. Agostini and L.F. DiMauro, Rep. Prog. Phys. , 813(2004).[6] G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L.Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci,R. Velotta, S. Stagira, S. De Silvestri, M. Nisoli, Science , 443 (2006).[7] J.J. Carrera, X.M. Tong, and S.-I. Chu, Phys. Rev. A , 023404 (2006).[8] E. Goulielmakis, V.S. Yakovlev, A.L. Cavalieri, M. Uib-eracker, V. Pervak, A. Apolonski, R. Kienberger, U.Kleineberg, and F. 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