Semiclassical two-step model for ionization by a strong laser pulse: Further developments and applications
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Semiclassical two-step model for ionization by a strong laserpulse: Further developments and applications.
N. I. Shvetsov-Shilovski
February 2, 2021
Abstract
We review the semiclassical two-step modelfor strong-field ionization. The semiclassical two-stepmodel describes quantum interference and accounts forthe ionic potential beyond the semiclassical perturba-tion theory. We discuss formulation and implementa-tion of this model, its further developments, as well assome of the applications. The reviewed applications ofthe model include strong-field holography with photo-electrons, multielectron polarization effects in ioniza-tion by an intense laser pulse, and strong-field ioniza-tion of the hydrogen molecule.
Keywords
Strong-field ionization · semiclassicaltwo-step model · quantum interference · Coulombpotential
Strong-field physics studies phenomena arising from theinteraction of strong laser pulses with atoms and mo-lecules. The most well-known examples of these highlynonlinear phenomena are above-threshold ionization(ATI), formation of the high-energy plateau in the elec-tron energy spectrum (High-order ATI), generation ofhigh-order harmonics (HHG) and nonsequential dou-ble ionization (NSDI), see Refs. [1,2,3,4,5] for reviews.Both experimental and theoretical approaches used toanalyze these processes are constantly being improved.The vast majority of the modern theoretical methodsused in strong-field physics are based on the strong-field approximation SFA [6,7,8], the direct numericalsolution of the time-dependent Schr¨odinger equation(see Refs. [9,10,11,12] and references therein), and the
Institut f¨ur Theoretische Physik, Leibniz Universit¨at Han-nover, D-30167, Hannover, Germany semiclassical models applying classical mechanics to de-scribe the electron motion in the continuum. The widelyknown examples of the semiclassical models are the two-step model [13,14,15] and the three-step model [16,17].In the SFA ionization is described as a transitionfrom an initial state unaffected by the laser field to aVolkov state, i.e., the wave function of an electron inan electromagnetic field. Therefore, the SFA neglectsthe intermediate bound states and the Coulomb inter-action in the final state. The SFA provides the illustra-tive physical picture of many strong-field phenomenaand often allows for the analytic solutions. Nevertheless,the approximations used in the SFA are strong enoughand may sometimes lead to wrong results. The widelyknown example is the fourfold symmetry of the photo-electron angular distributions in the elliptically polar-ized field predicted by the SFA [18]. In contrast to this,the experimental angular distributions show only theinversion symmetry: They are asymmetric in any half ofthe polarization plane [19]. The theoretical studies [20,21,22,23,24,25] have shown that the fourfold symme-try of the angular distributions is a direct consequenceof neglecting the effect of the Coulomb potential on theelectron motion in the continuum.In most cases the direct numerical solution of theTDSE provides a good agreement with the experimen-tal results. However, it is often difficult to understandthe physical mechanism of the phenomena under studywith only the numerical wave function. What is also im-portant, the capabilities of modern computers are notunlimited. One of the most prominent examples is thestrong-field ionization of molecules. The solution of theTDSE in three spatial dimensions is possible only forthe simplest molecules and with selection of the mostrelevant degrees of freedom [26,27]. Indeed, ionizationof a molecule by an intense laser pulse is much more
N. I. Shvetsov-Shilovski complicated than ionization of an atom. This is becauseof the existence of additional degrees of freedom (nu-clear motion), the associated time scales, and the com-plex shape of the electronic orbitals. For typical laserparemeters used in experiments nuclear motion shouldbe treated on an equal footing with the processes in-duced by a strong laser field. Simultaneously, the richnuclear structure of molecules results in orbitals of di-verse symmetries.Although the first semiclassical model (i.e., the two-step model) was formulated in 1988-1989 [13,14,15], thetrajectory-based models are still widely used for de-scription of various strong-field phenomena. This is dueto a number of important advantages characteristic tothe semiclassical approaches. The semiclassical modelsprovide a great insight into strong-field processes. Theyallow to reveal the specific mechanism responsible forthe process under investigation, as well as visualize itusing classical trajectories. This point needs to be dis-cussed in more details.In the ATI an electron absorbs more photons thannecessary for ionization. The studies of the ATI haverevealed that the majority of the ionized electrons donot experience hard recollisions with their parent ions.These electrons are referred to as direct electrons. Theycontribute to the low energy part of the ATI energyspectrum
E < U p , where U p = F / ω is the pomdero-motive energy. Here, in turn, F and ω are the ampli-tude and the frequency of the laser field (atomic unitsare used throughout the paper). The two-step modelallows to describe the spectrum of the direct electrons.In the first step of the this model an electron tunnelsout of an atom. In the second step it moves along aclassical trajectory in the laser field towards a detector.There are also rescattered electrons that are drivenback by the laser field to their parent ions. Upon theirreturns the rescattered electrons scatter from the par-ent ions by large angles close to 180 ◦ . These electronsform the high-energy plateau of the ATI spectrum. Therescattering scenario provides the basis for an under-standing of the HHG and NSDI. Indeed, the returningelectron can recombine to the parent ion and as the re-sult of the recombination a high-frequency photon (har-monic) radiation is emitted. Alternatively, if the energyof the scattered electron is sufficient enough, it can re-lease the second electron from the ion, e.g., by impactionization. The three-step model comprises the inter-action of the rescattered electron with the parent ionas the third step. As the result, the three-step modelprovides the qualitative description of the rescattering-induced processes.The three-step model explained a number of fea-tures revealed in the studies of the high-order ATI, HHG, and NSDI: the cutoffs in high-order ATI spec-trum [28] and HHG [16,29], the maximum angles ofthe angular distributions of ionized electrons [30], thecharacteristic recoil ion momenta in NSDI [31,32], etc.Originally the two-step and the three-step models didnot account for the effect of the ionic potential on theelectron motion in the continuum. The inclusion of theionic force in the Newton’s equation of motions allowedto uncover the Coulomb focusing effect [33], study theCoulomb cusp in the angular distributions of the photo-electrons [34], investigate the low-energy structures inionization by the strong midinfrared pulses [35,36,37,38,39,40,41,42,43] (the so-called ”ionization surprise”observed for the first time in experiment [44]), explorethe nonadiabatic effects in ionization by intense laserpulses (see, e.g., Refs. [45,46,47]), etc.The trajectory-based simulations are often (althoughnot always) computationally less expensive than the so-lution of the TDSE. Furthermore, for some strong-fieldprocesses the semiclassical simulations are presently theonly feasible approach. The most well-known exampleof such process is the NSDI of atoms by circularly [48]or elliptically polarized pulses [49,50,51], as well as theNSDI in molecules [52]. Therefore, further developmentof the semiclassical approaches to strong-field phenom-ena is an important objective.Until recently the trajectory-based models were notable to describe quantum interference effects. However,a significant progress along these lines has been madein the last decade. The trajectory-based Coulomb SFA(TCSFA) [37,53], the quantum trajectory Monte Carlomodel (QTMC) [54], the semiclassical two-step model(SCTS) [55], and the Coulomb quantum orbit strong-field approximation (CQSFA) [56,57,58,59,60] (see Ref.[61] for the foundations of the CQSFA approach) are re-cent trajectory-based models that are capable to repro-duce interference structures in photoelectron momen-tum distributions of the ATI process. These models as-sign certain phases to classical trajectories, and the con-tributions of different trajectories leading to the samefinal momentum are added coherently.The TCSFA is an extension of the CCSFA [62,63]that on an equal footing accounts the laser field andthe Coulomb force in the Newton’s equation for elec-tron motion in the continuum. The TCSFA applies thefirst-order semiclassical perturbation theory [64] to ac-count the Coulomb potential in the phase associatedwith every trajectory. The same first-order semiclassi-cal perturbation theory was used in the phase of theQTMC model. In contrast to this, the SCTS and theCQSFA approaches go beyond the perturbation theory.The SCTS model operates with large ensembles ofclassical trajectories that are propagated in the con- emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 3 tinuum to find the final asymptotic momenta and binthem (and, therefore, the corresponding contributionsassigned to these trajectories) in bins in momentumspace. This approach is often referred to as “shoot-ing method” (see, e.g., Ref. [37]). Instead, the CQSFAmodel solves the so-called “inverse problem”, i.e., findsall the trajectories leading to a given final momentum.This allows to avoid large ensembles of trajectories andestablish a better control over cusps and caustics thatare inevitable in trajectory-based simulations. The pricethat is to be paid is that the solution of the inverse prob-lem is a difficult task. In addition to this, the approachwith the inverse problem can often be less versatile.In this paper we review the SCTS model, as well astwo recent implementations of this model. We also dis-cuss some of the applications of the SCTS. The SCTSmodel has been applied to the investigation of the intra-half-cycle interference of photoelectrons with low ener-gies [65], to the studies of the interference patterns aris-ing in the strong-field photoelectron holography [66,67,68], to the analysis of the sub-cycle interference in ion-ization by counter-rotating two-color fields [69], to theinvestigation of sideband modulation by subcycle inter-ference in ionization by circularly polarized two-colorlaser fields [70], etc. Here we focus on the applicationsof the SCTS to the strong-field photoelectron hologra-phy, study of the multielectron polarization effects, andthe ionization of the H molecule.The paper is organized as follows. In Sec. II we re-view the SCTS and discuss different approaches used toimplement this model numerically. In Sec. III we discussthe further modifications of the SCTS model: the semi-classical two-step model with quantum input and theSCTS model accounting for the preexponential factorof the semiclassical propagator. In Sec. IV we briefly re-view applications of the SCTS model to the strong-fieldphotoelectron holography. The application of the SCTSto the study of the multielectron polarization effects inthe ATI are discussed in Sec. V. In Sec. VI we reviewthe usage of the SCTS model to describe the strong-field ionization of the H molecule. The conclusions ofthis colloquia paper are given in Sec. VII. d ~rdt = − ~F ( t ) − ∇ V ( ~r, t ) , (1) where ~F ( t ) is the laser field and V ( ~r, t ) is the ionic po-tential. In order to find the trajectory from Eq. (1), weneed to specify the initial conditions, i.e., the initial ve-locity of the departing electron and the starting point.In the original version of the SCTS model it is assumedthat the electron starts with zero initial velocity alongthe laser field v ,z = 0, but it can have a nonzero initialvelocity v , ⊥ in the perpendicular direction. We notethat the application of the SFA to describe the electronmotion under the potential barrier leads to a nonzeroinitial longitudinal velocity v ,z = 0. The effect of thenonzero v ,z will be discussed later. Let us first assumethat the interaction of the ionized electron with the ionis modelled by the Coulomb potential. Then the start-ing point of the trajectory, i.e., the tunnel exit point,can be obtained using the separation of the static tun-neling problem in parabolic coordinates. For the staticfield polarized along the z axis we define the paraboliccoordinates as ξ = r + z , η = r − z , and ϕ = arctan ( y/x )and find the tunnel exit coordinate η e from the follow-ing equation: − β ( F )2 η + m − η − F η − I p ( F )4 . (2)Here m is the magnetic quantum number of the initialstate, I p ( F ) is the Stark-shifted ionization potential,and β ( F ) = Z − (1 + | m | ) p I p ( F )2 . (3)The tunnel exit point is given by z e = − η e /
2. In thegeneral case, the ionization potential I p ( F ) in Eq. (2)is given by I p ( F ) = I p (0) + ( ~µ N − ~µ I ) · ~F + 12 ( α N − α I ) ~F . (4)Here I p (0) is the ionization potential in the absence ofthe field, and ~µ N,I and α N,I are the dipole momentsand static polarizabilities, respectively. The index N refers to the neutral atom (molecule), and the index I stands for its ion. We note that for atom the termlinear with respect to F is absent in Eq. (4). The staticfield F in Eqs. (2), (3), and (4) should be replaced bythe instantaneous value of the laser field at the time ofionization t .The instants of ionization and the initial transversevelocities are distributed in accord with the static ion-ization rate [71]: w ( t , v , ⊥ ) ∼ exp (cid:18) − κ F ( t ) (cid:19) exp − κv , ⊥ F ( t ) ! , (5)where κ = p I p . Following the original formulation ofthe SCTS model we omit the preexponential factor in N. I. Shvetsov-Shilovski
Eq. (5). For atoms it only slightly affects the shape ofthe electron momentum distributions.After the laser pulse terminates an electron movesin the Coulomb field only. If the electron energy at thetime t = t f at which the laser pulse terminates is neg-ative E <
0, the electron moves along the ellipticalorbit, and it should be treated as captured into a Ry-dberg state [72,73]. The corresponding process is of-ten referred to as frustrated tunnel ionization, see, e.g.,Refs. [74,75,76,77]. It is clear that the trajectories with
E <
E > ~k of the electron is determinedby its position ~r ( t f ) and momentum ~p ( t f ) at the time t = t f : ~k = k k (cid:16) ~L × ~a (cid:17) − ~a k L , (6)see Refs. [78,73]. In Eq. (6) ~L = ~r ( t f ) × ~p ( t f ) and ~a = ~p ( t f ) × ~L − Z~r ( t f ) /r ( t f ) are the angular mo-mentum and the Runge-Lenz vector, respectively. Themagnitude of the momentum k is determined by theenergy conservation: k ~p ( t f )2 − Zr ( t f ) . (7)The key ingredient of the SCTS model is the expres-sion for the phase associated with every trajectory. Thisphase corresponds to the phase of the matrix elementof the semiclassical propagator U SC ( t , t ) between theinitial state at time t and the final state at time t [79,80,81] (for a text-book treatment see Refs. [82,83]). De-pending on the variables used to describe the initial andfinal states there exist four equivalent forms of the semi-classical propagator U SC : h ~r | U SC ( t , t ) | ~r i = " − det (cid:0) ∂ φ ( ~r , ~r ) /∂~r ∂~r (cid:1) (2 πi ) / × exp [ iφ ( ~r , ~r )] , (8a) h ~r | U SC ( t , t ) | ~p i = " − det (cid:0) ∂ φ ( ~p , ~r ) /∂~p ∂~r (cid:1) (2 πi ) / × exp [ iφ ( ~p , ~r )] , (8b) h ~p | U SC ( t , t ) | ~r i = " − det (cid:0) ∂ φ ( ~r , ~p ) /∂~r ∂~p (cid:1) (2 πi ) / × exp [ iφ ( ~r , ~p )] , (8c) h ~p | U SC ( t , t ) | ~p i = " − det (cid:0) ∂ φ ( ~p , ~p ) /∂~p ∂~p (cid:1) (2 πi ) / × exp [ iφ ( ~p , ~p )] . (8d) Here ~r ( ~r ) and ~p ( ~p ) are the initial (final) coordi-nates and momenta, respectively. The phase φ thatcorresponds to the transition from the initial state tothe final state, which are both described by the posi-tion, is determined by the classical action: φ ( ~r , ~r ) = Z t t n ~p ( t ) ˙ ~r ( t ) − H [ ~r ( t ) , ~p ( t )] o dt, (9)where H [ ~r ( t ) , ~p ( t )] is the classical Hamiltonian func-tion that depends on the canonical coordinates ~r ( t ) andmomenta ~p ( t ). The other three phases φ , φ , and φ are related to φ by the canonical transformations: φ ( ~p , ~r ) = φ ( ~r , ~r ) + ~p · ~r , (10a) φ ( ~r , ~p ) = φ ( ~r , ~r ) − ~p · ~r , (10b) φ ( ~p , ~p ) = φ ( ~r , ~r ) + ~p · ~r − ~p · ~r , (10c)Then the question arises: Which of these phases shouldbe chosen for description of the strong-field ionizationprocess? On the assumption that for a given ionizationtime the starting-point of the electron trajectory is lo-calized in space [see Eq. (2)] and the final state is char-acterized by the asymptotic momentum ~k , the phase φ is used in the SCTS model. Indeed, the strong-fieldionization can be viewed as a half-scattering processof an electron that is initially localized near the atom(molecule) and detected with the final momentum ~k .We note that if the initial longitudinal velocity is equalto zero, the initial electron momentum ~p is orthogo-nal to the initial position vector ~r (i.e., ~p · ~r = 0),and therefore, the phases φ and φ coincide with eachother. For nonzero v ,z the term ~p · ~r is to be ac-counted in the phase. However, in most cases this termalmost does not affect the resulting electron momentumdistributions.As the result, the phase corresponding to a giventrajectory in the SCTS model is given by: Φ SCT S ( t , ~v ) = − ~v · ~r ( t ) + I p t − Z ∞ t dt n ˙ ~p ( t ) · ~r ( t ) + H [ ~r ( t ) , ~p ( t )] o , (11)where it is assumed that the trajectory has also the ini-tial phase exp ( iI p t ) that describes the time evolutionof the ground state. The expression (11) can be alsowritten as follows: Φ SCT S ( t , ~v ) = − ~v · ~r ( t ) + I p t − Z ∞ t dt (cid:26) p ( t )2 + V [ ~r ( t )] − ~r ( t ) · ~ ∇ V [ ~r ( t )] (cid:27) . (12)To arrive at the expression (12), we use the explicit formof the Hamiltonian for an arbitrary effective potential H [ ~r ( t ) , ~p ( t )] = ~p ( t )2 + ~F ( t ) · ~r ( t ) + V ( ~r ) (13) emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 5 and employed Newton’s equation of motion (1). Thisformula is applicable for any single-active-electron po-tential used to describe the multielectron system (atomor molecule), including pseudopotentials (see, e.g., Ref. [84]and references therein). For the specific case of theCoulomb potential, the phase (12) reads as Φ ST CS ( t , ~v ) = − ~v · ~r ( t ) + I p t − Z ∞ t dt (cid:26) p ( t )2 + 2 Zr ( t ) (cid:27) . (14)This formula should be compared with the phase usedin the QTMC model: Φ QT MC ( t , ~v ) = − ~v · ~r ( t ) + I p t − Z ∞ t dt (cid:26) p ( t )2 + Zr ( t ) (cid:27) . (15)It is seen that the QTMC phase can be obtained fromEq. (14) by neglecting the term ~r ( t ) · ~ ∇ V [ ~r ( t )] in theintegrand. This term leads to the double weight of theCoulomb term in the SCTS compared to the QTMC.Therefore, the QTMC phase can be considered as anapproximation to the SCTS one. The double weight ofthe Coulomb contribution leads to a better agreementwith the TDSE results [55].The SCTS phase (14) is divergent at t → ∞ , andtherefore, it is to be regularized. The regularization(see Ref. [55]) can be accomplished by decomposing theSCTS phase as Φ SCT S ( t , ~v ) = − ~v · ~r ( t ) + I p t − Z t f t (cid:26) p ( t )2 − Zr ( t ) (cid:27) − Z ∞ t f dt (cid:26) E − Zr ( t ) (cid:27) (16)and separating the time-independent part of the inte-grand in the term R ∞ t f dt { E − Z/r ( t ) } . Although thistime-independent part leads to the contributionlim t →∞ E ( t − t f ) that diverges linearly when t → ∞ , itdoes not produce a phase difference for electron tra-jectories ending up in the same bin. Indeed, the finalmomenta of such trajectories (and, therefore, their en-ergies) should be considered as equal. Using the solutionof the Kepler problem (see, e.g., Ref. [85]) we calculatethe divergent integral Φ Cf ( t f ) = Z Z ∞ t f dtr ( t ) (17)analytically: Φ Cf ( τ f ) = Z √ b [ ξ ( ∞ ) − ξ ( τ f )]. The pa-rameter ξ is used to parametrize the time t and thedistance r from the Coulomb center: r ( t ) = b ( g cosh ξ − ,t = √ b ( g sinh ξ − ξ ) + C (18) Here, in turn, b = 1 / (2 E ) and g = √ EL . Theconstant C in Eq. (18) is to be found using the initialconditions, i.e, ~r ( t f ) and ~p ( t f ). It is easy to verify that ξ ( t f → ∞ ) = ln (cid:18) tg √ b (cid:19) , (19)see Ref. [55]. Therefore, for trajectories arriving at thesame bin, we can discriminate between the commondivergent part ln (cid:16) t/ √ b (cid:17) and the finite contributionsdetermined by − ln ( g ). We note that the latter dependsnot only on the energy, but also on the angular momen-tum L , and thus is different for different trajectoriesinterfering in a given bin. Since ξ ( t f ) = arsinh (cid:26) ~r ( t f ) · ~p ( t f ) g √ b (cid:27) , (20)we obtain the following contribution to the phase accu-mulated in the time interval [ t f , ∞ ] due to the Coulombpotential (see Ref. [55]):˜ Φ Cf ( t f ) = − Z √ b (cid:20) ln g + arsinh (cid:26) ~r ( t f ) · ~p ( t f ) g √ b (cid:27)(cid:21) . (21)This asymptotic correction of the phase which we callpost-pulse phase is missing in the QTMC model.2.2 Implementation of the semiclassical two-stepmodelThe expression for the phase can be conveniently treatedas an additional equation in the system of the first or-der ordinary differential equations for electron coordi-nates and velocity components following from (1). Thissystem can be solved using the fourth-order Runge-Kutta method with adaptive step size [86]. The abilityof the numerical method to change the integration stepis particularly important at small distances from theCoulomb center.It is clear that the convergence of the results must becontrolled with respect to both the size of the bin in themomentum space and the number of trajectories. It isparticularly convenient to control convergence by usingthe energy spectra. In contrast to the three-dimensional(3D) differential momentum distributions or their two-dimensional (2D) cuts, the spectra are functions of onlyone variable. They can be easily compared to each otherin, e.g., logarithmic scale.Already the first practical application of the SCTSmodel has shown that a large number of trajectories isneeded for convergence (see Ref. [55] for details). Typ-ically, for the same laser parameters a thousand timesmore trajectories are needed for the simulations with N. I. Shvetsov-Shilovski the phase compared to a semiclassical model disregard-ing the interference effect. This can be expected tak-ing into account the fine interference details of elec-tron momentum distributions generated in ionizationby strong laser pulses. For this reason, it is importantto consider optimization of the codes implementing theSCTS model. The most obvious way to speed up theSCTS calculations is to use parallelization. Indeed, anytrajectory-based simulation can be very easily and effi-ciently implemented on a computer cluster by parallel-ing the loop over the number of trajectories.Another approach consists in an efficient samplingof the initial conditions, i.e., times of ionization t j andinitial velocities v j , where index j enumerates the tra-jectories of an ensemble. In a standard trajectory-basedapproach the initial conditions are chosen either ran-domly or from a certain uniform grid. Neglecting inter-ference effect the ionization probability R (cid:16) ~k (cid:17) for the fi-nal momentum ~k that corresponds to the bin [ k i , k i + ∆k i ]( i = x, y, z ) is calculated as R (cid:16) ~k (cid:17) = n p X j =1 w (cid:16) t j , v j (cid:17) , (22)while the similar formula for the SCTS model reads as R (cid:16) ~k (cid:17) = n p X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r w (cid:16) t j , v j (cid:17) exp h iΦ SCT S (cid:16) t j , v j (cid:17)i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (23)The sums in Eqs. (22) and (23) are calculated over all n p trajectories arriving at the given bin. However, theapproach sketched here is not the only possible one.Importance sampling widely used in Monte-Carlo inte-gration (see, e.g., Ref. [86]) can be used to implementthe SCTS model.We turn first to the semiclassical simulations dis-regarding interference. In the important sampling ap-proach the weights (importance) of classical trajecto-ries are accounted already at the sampling stage. Morespecifically, the sets of initial conditions (cid:16) t j , v j (cid:17) aredistributed in accord with the tunneling rate w (cid:16) t j , v j (cid:17) and the ionization probability R (cid:16) ~k (cid:17) is given by a num-ber of trajectories reaching the bin corresponding to thefinal momentum ~k . It is easy to see that the ionizationprobability in the SCTS model based on the importancesampling reads as R (cid:16) ~k (cid:17) = n p X j =1 (cid:12)(cid:12)(cid:12) exp h iΦ SCT S (cid:16) t j , v j (cid:17)i(cid:12)(cid:12)(cid:12) . (24) with the initial conditions distributed in accord to thesquare root of the ionization probability. In many situa-tions the important sampling technique provides fasterconvergence compared to the standard approach ofEqs. (22)-(23). Its performance, however, depends onthe laser-atom parameters and the specific part of pho-toelectron momentum distribution under study.2.3 Benchmark case: Ionization of the H atomThe SCTS model was compared with the QTMC ap-proach and direct numerical solution of the TDSE forionization of the hydrogen atom (see Ref. [55]). The 2Delectron momentum distributions calculated in accordto the all three approaches are shown in Fig. 2 (a)-(c).The simulations are done for ionization by a few-cyclelaser pulse linearly polarized along the z -axis and de-fined through the vector-potential: ~A ( t ) = ( − n F ω sin (cid:18) ωt n (cid:19) sin ( ωt + ϕ ) ~e z . (25)Here n is the number of optical cycle within the pulse,and ~e z is the unit vector in the polarization direction.The pulse (25) is present between t = 0 and t = t f =(2 π/ω ) · n . The factor ( − n in (25) ensures that thefield has its maximum at the center of the pulse ( ωt = πn ) for even and odd n . The laser field is to be calcu-lated from Eq. (25) as ~F ( t ) = − d ~A/dt .It is seen that the most important features of theTDSE result are reproduced by the semiclassical mod-els [see Figs. 1 (a),(c),and (f)]. Indeed, the electron mo-mentum distributions are stretched along the z -axis andshow clear ATI rings as well as the pronounced inter-ference structure in their low-energy parts. The widthof the momentum distributions along the polarizationdirection is obviously underestimated by both semiclas-sical models. This is due to the initial condition v ,z = 0(see Ref. [55] for details).However, a closer examination of the low-energyparts of the distributions reveals remarkable deviationsIndeed, for | k | < . dR/dE showsthat the QTMC and the SCTS qualitatively reproduce emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 7 Fig. 1
Two-dimensional electron momentum distributionsfor ionization of the H atom by a laser pulse with a dura-tion of n = 8 cycles, peak intensity of 0 . × W/cm ,and wavelength of 800 nm calculated from the QTMC model[(a),(b)], numerical solution of the TDSE [(c),(d)], and theSCTS [(e),(f)]. Panels (b), (d), and (f) display the magnifica-tions for | k z | , | k ⊥ | < . z axis. The distributions are normalized to the totalionization yield. A logarithmic color scape in arbitrary unitsis used. the ATI peaks, see Figs. 2 (a)-(c). However, both semi-classical approaches can quantitatively reproduce theamplitude of interference oscillations only for a few low-order peaks. This is related to the fact that due to theinitial conditions [Eq. (5)] used in both semiclassicalmodels too few trajectories with large initial momentain the polarization direction are launched. This also ex-plains why the semiclassical energy spectra fall off toorapidly with the increase of energy. In order to testthis hypothesis, the initial longitudinal velocity for ev-ery ionization time is set to the value predicted by theSFA, see, e.g. Ref. [37]. This change in initial conditions leads to a better agreement between the SCTS modeland the TDSE, see Fig. 3 and Ref. [55]. Therefore, themain reason of deviations of the SCTS results from theTDSE solutions is not the semiclassical treating of theelectron motion in the continuum, but the fact that theSCTS model does not describe the tunneling step ac-curately enough. d R / d E d R / d E E (a.u.) d R / d E QTMC SCTS TDSE10 −4 −3 (a)(b)(c) Fig. 2
Photoelectron energy spectra for ionization of the Hatom by a laser pulse with a duration of n = 8 cycles and peakintensity of 0 . × W/cm calculated using the TDSE(thick light blue curve), the QTMC (dashed blue curve) andthe SCTS (solid red curve). Panels (a), (b), and (c) corre-spond to the wavelengths of 800 nm, 1200 nm, and 1600 nm,respectively. The spectra are normalized to the peak value. Substantial efforts have been recently made to mod-ify the SCTS model. These modifications are aimed atproviding not only a qualitative, but also a quantita-tive agreement with the TDSE. To achieve this goal, itis necessary to overcome the deficiencies of the SCTSmodel (as well as of any other semiclassical model) indescription of the ionization step. The simplest way is touse the SFA formulas to distribute the initial conditionsof classical trajectories. This approach dates back to thestudies of Refs. [91,92]. It is used in the various semi-classical models (see, e.g., Refs. [37,45,46,47]), as well
N. I. Shvetsov-Shilovski d R / d E QTMC SCTS TDSE −3 Fig. 3
Photoelectron energy spectra calculated from theTDSE (thick light blue curve), the QTMC model (dashedblue curve) and the SCTS (solid red curve). A nonzero ini-tial parallel velocity predicted by the SFA is used in both theQTMC and SCTS simulations. The pulse parameters are asin Fig 2(a). as in the implementations of the SCTS model developedin Refs. [93,70]. We note, however, that the validity ofthe SFA formulas used as initial conditions for classicaltrajectories requires a systematical study. To the best ofour knowledge, such a study has not been accomplishedso far. Here we discuss two modifications of the SCTSmodel: The semiclassical two-stem model with quan-tum input (SCTSQI) [94] and the SCTS model withthe prefactor [93].3.1 Semiclassical two-step model with quantum inputThe SCTSQI model combines the SCTS with initialconditions obtained from the solution of the TDSE.Such a combination leads to a novel quantum-classicalapproach. The SCTSQI model is formulated for ioniza-tion of a one-dimensional (1D) model atom. Therefore,before reviewing the SCTSQI, we briefly discuss the so-lution of the 1D TDSE, as well as the application of theSCTS model in 1D case.For the 1D model, the TDSE in the velocity gaugeis given by i ∂∂t Ψ ( x, t )= ( (cid:18) − i ∂∂x + A x ( t ) (cid:19) + V ( x ) ) Ψ ( x, t ) , (26)where Ψ ( x, t ) is the wave function in coordinate space.The 1D soft-core Coulomb potential V = − √ x + a (27)with a = 1 . (cid:26) − d dx + V ( x ) (cid:27) Ψ ( x ) = EΨ ( x ) . (28)Equation (28) can be easily solved on a grid using, e.g.,the well-known three-step formula for approximation ofthe second derivative and subsequent diagonalization.In Ref. [94] the TDSE (26) is solved using slit-operatormethod [96]. In the regions x b ≤ | x | ≤ x max the wavefunction is multiplied by a mask M ( x ) = cos / (cid:20) π ( | x | − x b )2 ( x max − x b ) (cid:21) , (29)where x = ± x b correspond to the internal boundariesof the absorbing regions, and x max is the size of thecomputational box. The mask prevents unphysical re-flections of the propagating wave function from the gridboundary and allows to calculate the electron momen-tum distributions using the mask method [97].In the 1D case the Newton’s equation for an electronmoving in the laser field and the field of the potential(27) reads as d xdt = − F x ( t ) − x ( x + a ) / . (30)The corresponding SCTS phase is given by (see Ref. [94]): Φ SCT S ( t , ~v ) = I p t − Z ∞ t dt ( v x ( t )2 − x ( x + a ) / − √ x + a ) . (31)We note that the ionization rate (5) in the 1D case isto be replaced by w ( t ) ∼ exp − | E | ) / F ( t ) ! , (32)where E = − . t = t f .The asymptotic momentum of the photoelectron can befound from x ( t f ) and p x ( t f ) using the energy conser-vation law. We note that after the end of the pulse theunbound electron cannot change its direction of motion,and, therefore, k x has the same sign as that of p x ( t f ).In order to correctly apply the SCTS model in the1D case, the post-pulse phase is to be calculated. Thiscalculation can be performes as follows (see Ref. [94]).At first, we decompose the phase as: Φ SCT S ( t , ~v ) = I p t − Z t f t dt ( v x ( t )2 − x ( x + a ) / − √ x + a ) + Φ Vf , (33) emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 9 As in the 3D case, we separate the post-pulse phaseinto parts with time-dependent and time-independentintegrands and disregard the linearly divergent contri-bution from the first part. As the result, the post-pulsephase is determined by:˜ Φ Vf = Z ∞ t f x ( t )[ x ( t ) + a ] / dt. (34)The divergent part of this integral can be efficientlyisolated. Indeed, Eq. (34) can be equivalently rewrittenas follows:˜ Φ Vf = Z ∞ t f " x ( x + a ) / − Et (2 Et + a ) / dt + Z ∞ t f Et (2 Et + a ) / dt. (35)Since the second divergent term in Eq. (35) depends onthe electron energy E and the parameter a , it is thesame for every trajectory that arrives at a given bin[ k x − ∆k x , k x + ∆k x ]. Therefore, it does not affect theresulting interference pattern and can be omitted [94].The post-pulse phase is determined by the first term inEq. (35). This converging integral is easily calculatednumerically. It depends on the position x ( t f ) and ve-locity p x ( t f ) at the end of the laser pulse what suggestsan efficient way to calculate it by interpolation [94].This is not a simple task to unify the direct solu-tion of the TDSE and the trajectory-based approach inone single model. The main problem of such combina-tion has a fundamental origin. Indeed, both the startingpoint and the initial velocity are needed to uniquely de-termine the classical trajectory. On the other hand, theHeisenberg’s uncertainty principle imposes a limit tothe precision with which position and momentum (asother canonically conjugated variables) can be simul-taneously known. The application of quasiprobabilitydistribution allows to extract the information from thewave function about both the coordinate and momen-tum.The most widely-known examples of the quasiprob-ability distributions are the Wigner function and Husimidistribution [98] (see Ref. [99] for a textbook treat-ment). The latter can be obtained by smoothing of theWigner function with a Gaussian weight. The Gabortransformation [100] was used in Ref. [94]. The Gabortransformation is presently widely used in studies of theATI (see, e.g., Ref. [101]) and, especially, the HHG (see,e.g., Refs. [102,103,104]). The Gabor transform of thewave function ˜ Ψ ( x, t ) near the point x is given by: G ( x , p x , t ) = 1 √ π Z ∞−∞ ˜ Ψ ( x ′ , t ) exp " − ( x ′ − x ) δ × exp ( − ip x x ′ ) dx ′ , (36) where δ is the width of the Gaussian window. Thesquare modulus | G ( x , p x , t ) | corresponds to the mo-mentum distribution of the particle in the vicinity of x = x at time t and is just the Husimi distribution[98]. The Hisimi distribution is a positive semidefinitefunction, which helps to interpret it as a quasiprobabil-ity distribution.The SCTSQI model employs the solution of theTDSE in the length gauge: i ∂∂t Ψ ( x, t )= (cid:26) − ∂ ∂x + V ( x ) + F x ( t ) x (cid:27) Ψ ( x, t ) . (37)Two additional spatial grids containing N points areintroduced the absorbing regions | x | ≥ x b : x j , ± = ∓ ( x b + ∆x · j ) , (38)Here j = 0 , ..., N and ∆x = ( x max − x b ) /N . In theSCTSQI the Gabor transforms of the absorbed partof the wave function ˜ Ψ ( x, t ) = [1 − M ( x )] Ψ ( x, t ) arecalculated at every time at the points x j , − and x j , − of the grids (38). The value of the Gabor transfor-mation at an arbitrary point belonging to D or D can be obtained by interpolation (see Ref. [94] for adetails of the implementation of the SCTSQI model).Hence, at every time t the Gabor transform G ( x, p x , t )is known on the grids in the phase-space domains D =[ − x max , − x b ] × [ − p x, max , p x, max ] and D = [ x b , x max ] × [ − p x, max , p x, max ]. An example of the Husimi distribu-tion obtained in the domains D and D at t = 3 t f / x max used in the SCTSQIcan be much smaller than the one required to obtainaccurate momentum distributions by using the maskmethod.At every time t an ensemble of n p classical trajec-tories with random initial positions x j and momenta p jx, ( j = 1 , ..., n p ) is launched in the SCTSQI model.Every trajectory of the ensemble is assigned with theamplitude G (cid:16) t , x j , p jx, (cid:17) and the SCTSQI phase Φ SCT SQI (cid:16) t , x j , p jx, (cid:17) = − Z ∞ t dt ( v x ( t )2 − x ( x + a ) / − √ x + a ) . (39)This phase coincides with the phase of the semiclassi-cal propagator describing a transition from an initialstate characterized by the momentum to a final state,which is also described by the momentum value. The x (a.u.) k x ( a . u . ) −400 −200 0 200 400−2−1012 −10−8−6−4−20 D D S S S Fig. 4
The Husimi quasiprobability distribution | G ( x, p x , t ) | at t = 3 t f / n = 4 cycles, wavelength of 800 nm, and peak intensity of2 . × W/cm . The distribution is calculated in thephase space domains D and D (see text). The points S , S , and S depicted by a green circle, magenta square, andcyan rectangle, respectively, show the three main maxima ofthe Husimi distribution. A logarithmic color scale is used. ionization probability R ( k x ) is calculated as: R ( k x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N T X m =1 n kx X j =1 G (cid:16) t m , x j , p jx, (cid:17) × exp h iΦ SCT SQI (cid:16) t m , x j , p jx, (cid:17)i (cid:12)(cid:12)(cid:12)(cid:12) , (40)where N T is the number of steps that is used in theTDSE propagation and n k is the number of trajecto-ries arriving at the same bin centered at k x . It is impor-tant to stress that G (cid:16) t m , x j , p jx, (cid:17) is a complex functionhaving both modulus and the phase.The SCTSQI model was tested by comparing itspredictions with the numerical solution of the TDSEand the SCTS model, see Figs. 5 (a) and (b). It isseen that the SCTSQI provides not only qualitative,but also quantitative agreement with the TDSE result.This is true for both the width of the electron mo-mentum distributions and the positions of the inter-ference maxima and minima. The small discrepancy inthe heights of some interference peaks [see in Fig. 5 (a)]is attributed to the fact that the SCTSQI model doesnot account for the preexponential factor of the semi-classical matrix element. We note that as in the 3Dcase (see Sec. 2.3) the 1D SCTS model shows only aqualitative agreement with the fully quantum results,see Fig. 5 (b). Specifically, the SCTS model underesti-mates the width of the momentum distributions. Theelectron energy spectra calculated within the SCTSQImodel and from the solution of the TDSE are in almostperfect agreement. Simultaneously, the spectrum calcu-lated using the SCTS model falls off too rapidly withthe increase of the energy. This is caused by the un-derestimation of the width of the electron momentumdistributions in the SCTS model. It was shown that the phase of the Gabor transform is very important in theSCTSQI [94]. Without this phase the SCTSQI modeldoes not provide even a qualitative agreement with theTDSE result. This could be expected, since the ampli-tude G ( t, x, p x ) contains all the information about thequantum dynamics of the absorbed part of the wavefunction before it was transformed in an ensemble oftrajectories. In a way the term I p t in the phase (11)of the SCTS model plays the same role as the phase of G ( t, x, p x ) in the SCTQI approach. −2 −1.5 −1 −0.5 0 0.5 1 1.5 200.51 k x (a.u.) y i e l d ( a r b . un it s ) E (a.u.) d R / d E TDSESCTSQISCTSTDSESCTSQI10 −4 −6 (b)(a) Fig. 5 (a) The photoelectron momentum distributions forionization of a 1D atom by a laser pulse with a duration of n = 4 cycles, wavelength of 800 nm, and peak intensity of2 . × W/cm calculated from the solution of the TDSE(thick light blue curve) and the SCTSQI model (dashed greencurve). (b) Electron energy spectra obtained from the TDSE(thick light blue curve), SCTSQI (dashed green curve), andthe SCTS (red curve). The distributions and spectra are nor-malized to the peak values. As any semiclassical approach, the SCTSQI modelcan visualize the physical mechanism responsible for thestrong-field process under study using classical trajecto-ries (see Ref. [94] for details). Since the initial conditionsin the SCTSQI model are determined from the directsolution of the TDSE, we expect that this model will beable to provide more accurate trajectory-based picturesof strong-field phenomena compared to the standardsemiclassical approaches. This advantage of the SCT-SQI model should be used in studies of complicatedstrong-field processes. In addition to this, after somemodification the SCTSQI model can be applied to stud-ies of the rescattering-induced phenomena, especiallythe high-order ATI and the HHG. The ways of thismodification are suggested in Ref. [94]. Finally, the ex-tension of the SCTSQI model to the three-dimensional emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 11 (3D) case is straightforward and developments in thisdirection are on the way. Most importantly, the modelsolves the non-trivial problem how to choose initial con-ditions for classical trajectories. In the SCTSQI modelthese initial conditions are determined by the exactquantum dynamics.3.2 SCTS model with preexponential factorAn efficient modification and extension of the SCTSmodel was proposed recently in Ref. [93]. This studyfor the first time investigates systematically the influ-ence of the preexponential factor of the semiclassicalmatrix element (8c) (see Refs. [105,106]) that was notexplicitly considered in all other versions of the SCTS.The modulus of this prefactor that corresponds to themapping from initial conditions to the final momentumcomponents influences the weights of the classical tra-jectories. Its phase known as the Maslov phase can beidentified as a case of Gouy’s phase anomaly and mod-ifies the interference structures [93]. In addition, theauthors propose a novel way of solving the so-calledinverse problem based on a clustering algorithm.Since the SCTS implementation of Ref. [93] employsthe SFA and the saddle-point approximation to calcu-late the ionization weight of the classical trajectoriesand their initial positions, the ionization time t foreach initial electron momentum ~k ′ is determined bythe real part of the corresponding saddle-point time t s = t + it . The saddle point t s satisfies the equation:12 h ~k ′ + ~A ( t s ) i + I p = 0 . (41)The ionization probability is calculated as: R (cid:16) ~k (cid:17) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X DC Coul p | J ( t → ∞ ) | exp h i (cid:16) S ↓ + S → − νπ (cid:17)i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (42)Here, the summation is over all the initial momenta ~k ′ leading to the final momentum ~k . D is the matrix ele-ment emerging when the saddle-point method is appliedto calculate the SFA ionization amplitude and C Coul isthe Coulomb correction of the ionization rate [64]. Thephase associated with every trajectory is decomposedin Eq. (42) as S ↓ + S → , where S ↓ = I p t s − Z t t s dt h ~k ′ + ~A ( t ) i (43)corresponds to the ionization step (motion under thepotential barrier), and S → = − Z ∞ t dt (cid:26) p ( t )2 + V [ ~r ( t )] − ~r ( t ) · ~ ∇ V [ ~r ( t )] (cid:27) (44) accounts for the electron motion in the continuum. Wenote that coincides with the third term of Eq. (12). TheJacobian J is calculated as J ( t ) = det ∂~k ( t ) ∂~k ′ ! . (45)The Maslov index ν changes at focal points, i.e, at times T when the Jacobian is zero J ( T ) = 0. The change(jump) of the Maslov index when the trajectory passesthrough a focal point is calculated as: ∆ν ( T ) = m − g ) , (46)where the m × m matrix g is given by g i,j = δ~r ( i ) · Hesse ~r,~r ( H ) δ~r ( j ) (47)Here, in turn, m is the number of linearly independentdirections ~d ( i ) ( i = 1 , ..., m ), which can be found at thefocal points, such that infinitesimal changes of the ini-tial momenta in these directions ~k ′ → ~k ′ + ǫ ~d ( i ) do notaffect ~k ( T ) in the first order of ǫ . These changes of theinitial momenta correspond to the changes of the posi-tion δ~r ( i ) = ǫ X j ∂~r ( T ) ∂k ′ j d ( i ) j , (48)see Eq. (47). The Hessian Hesse ~r,~r ′ ( H ) of the Hamilto-nian function H = 12 h ~k + ~A ( t ) i + V ( ~r ) (49)is calculated with respect to the position vector ~r .The inverse problem is solved in Ref. [93] by usingclustering algorithms. More specifically, density-basedspatial clustering of applications with noise algorithmswas applied. The solution of inverse problem with clus-tering shows an example of the application of machinelearning (see Ref. [107] for a text-book treatment) tostrong-field phenomena. Other recent applications ofthe machine learning in strong-field physics are dis-cussed in, e.g., Refs. [108,109].The fact that the Jacobian is explicitly taken intoaccount in Eq. (42) along with the solution of the in-verse problem ensures the correct preexponential weightof every trajectory, namely, 1 / p | J | . It should be em-phasized that this weight cannot be reproduced in “shoot-ing method”, since the distribution of the trajectoriesover the cells in accord with their final momenta auto-matically creates a factor of 1 / | J | instead of the 1 / p | J | .This problem was ignored in the implementation of theSCTS [55], since the implementation of Ref. [55] ac-counts only for the exponential factors in the trajecto-ries weights. The simple relation between the Jacobian in the 3Dcase and the corresponding Jacobian for two spatial di-mensions was derived in [93] for systems (ionic potentialand the laser field) with cylindrical symmetry: | J D | = k ⊥ k ′⊥ | J D | , (50)where k ⊥ = q k x + k y (the field is polarized along the z -axis). This correction weight allows to obtain the re-sults for the 3D system performing only the 2D sim-ulations, and, by doing so, reduce the computationalcosts of the SCTS model significantly. We note thatEq. (50) has been already used in the SCTS simula-tions of Ref. [55].The modified version of the SCTS is in excellentagreement with solution of the TDSE. This applies forboth electron momentum distributions and energy spec-tra [93]. It is shown that the inclusion of the preex-ponential factors is crucial for quantitative agreementwith the TDSE results. The extended version of theSCTS can be applied not only to the linearly polarizedpulses, but also to non-cylindrically-symmetric laserfields, e.g., bicircular ones, see Ref. [93]. Undoubtedlythe version of the SCTS developed in [93] is a valuabletool that is extremely useful in studies of strong-fieldionization. Development of the techniques capable to image theatomic positions that change in time in a chemical re-action will lead to a revolution in chemistry, biology,nanoscience, etc. At present there are many methods fortime-resolved molecular imaging (see Ref. [110] for a re-view). These methods have been developed due to theprominent progress in laser technologies. This appliesabove all to the development of the technology for pulsecompression and the emergence of free-electron lasers.Moreover, the availability of table-top intense femtosec-ond lasers, which led to the emergence of strong-field,ultrafast, and attosecond physics, gave a strong impulseto the development of new techniques for time-resolvedmolecular imaging. Among these techniques are: laser-induced Coulomb-explosion imaging [111,112,113,114],laser-assisted electron diffraction [115,116], high-orderharmonic orbital tomography [117,118], laser-inducedelectron diffraction (see, e.g., Refs. [119,120,121]), andstrong-field photoelectron holography (SFPH) [122].The SFPH method implements the widely-knownidea of holography (1971 Nobel Prize in Physics awardedto Dennis Gabor, see Ref. [123]) in strong-field physics. It was for the first time shown in 2011 by Y. Huis-mans et al. [122] that a holographic pattern can beclearly recorded in experiment. This pattern in the elec-tron momentum distributions is created by the signal(rescattered) and reference (direct) electrons. The SFPHcan be implemented in a table-top experiment. It wasshown that the holographic patterns encode a lot ofspatio-temporal information about both the parent ionand the recolliding electron [122]. Last but not least, theelectron dynamics can be imaged with subcycle (i.e., at-tosecond) time resolution. These advantages have trig-gered extensive studies of the SFPH, both experimental[124,125,126,127,66] and theoretical [122,124,125,128,129,130,131,132,133,56,57,58,59,60].However, the first SFPH experiments [122,124,125,128,126] investigated the ionization process and the dy-namics of the electron wave packet rather than molec-ular structure or dynamics. This is because of the factthat for diatomic and small molecules the holographicstructures are mostly determined by the long-range andthe alignment-independent Coulomb potential. As theresult, the short-range effect reflecting the molecularstructure cannot be observed on the background of themore intense Coulomb contribution. This problem waselegantly solved in experiment of Ref. [127] by consid-ering the difference between the normalized photoelec-tron holograms for aligned and antialigned molecules.This approach is based on the fact that for large scat-tering angles the differential cross section deviates fromthe Coulomb one and depends on the alignment of themolecule at the ionization instant. A similar methodwas also used in Ref. [66]. Various approaches wereused for theoretical analysis of the SFPH: the three-stepmodel [129,130,131,133], the SFA version that accountsfor rescattering [122,128], the Coulomb-corrected strong-field approximation [122,128], the CQSFA [56,57,59,60], etc. (see Ref. [134] for recent review).4.1 SCTS model and experimental holographicpatternsThe SCTS model was applied to the simulations of theholographic interference patterns observed in the ex-periment [66]. In the study [66] the electron momentumdistribution produced in ionization of the NO moleculewere calculated for two different cases. In the first casethe electron density of the highest occupied molecularorbital (HOMO) is aligned along the polarization direc-tion, whereas in the second case this density is orthog-onal to it. These distributions, as well as their normal-ized difference are shown in Fig. 6. To apply the SCTSmodel, the distributions over the initial transverse ve-locities are needed for both these cases. These distribu- emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 13 tions were determined using the approach based on par-tial Fourier transform generalized to molecules (MO-PFT) [135,136,137]. The MO-PFT approach works withthe electron wave function in mixed (coordinate-momentum)representation and uses the Wentzel-Kramers-Brillouin(WKB) approximation. The MO-PFT requires the cor-responding HOMO’s that were obtained using the GAMESSpackage [138]. The semiclassical simulations are in aperfect agreement with the experimental results [66].The simulations within the SCTS model reproduce allcharacteristic features of the holographic patterns. Theregions of constructive and destructive interference pre-dicted by the model of Ref. [129] that neglects theCoulomb potential are shown in Fig. 6 with white andblack color, respectively. It is seen that the there-stepmodel overestimates the spacing between the holographicfringes in the direction perpendicular to laser polariza-tion. Therefore, the account of the Coulomb potentialleads to the improved agreement between the experi-ment and the semiclassical simulations.
Fig. 6
Photoelectron momentum distributions for ionizationof the NO molecule by a laser pulse with a duration of 35 fs,intensity of 2 . × W/cm , and wavelength of 800 nmcalculated using the SCTS model. The panels (a) and (b)show the distributions obtained in the cases where the elec-tron density of the HOMO is aligned along the laser polariza-tion direction and perpendicular to it, respectively. Panel (c)presents the normalized difference of the distributions shownin (a) and (b). The figure is reprinted from Ref. [66]. | z e ( t ) | = − I p F ( t ) , (51)where the sign of z e ( t ) is to be chosen to ensure theelectron tunnels in the direction opposite to the instan-taneous field ~F ( t ). This makes it possible to directlycompare the resulting interference patterns with thepatterns of the three-step model. Third, the weights(5) of classical trajectories were not taken into account,and the trajectories were distributed uniformly, whichis justified by the fact that holographic patterns andnot electron momentum distributions were calculated inRef. [67]. Finally, a special approach instead of Eq. (23)has to be used in the semiclassical model to obtainthe phase difference between the signal and referenceelectrons. Indeed, to calculate the phase difference weneed to isolate only one kind of rescattered trajectoriesand only one kind of the direct ones. This is a com-plicated task if the Newton’s equation of motion (1) issolved treating the laser field and the Coulomb forceon equal footing. First of all, it is necessary to answerthe question: How to distinguish between the direct andrescattered electron trajectories in the presence of theCoulomb field? Indeed, all the trajectories are, to someextent, affected by the Coulomb potential.The following simple recipe is used in Ref. [67]. Thereference trajectories were defined as those passing theionic core at large distances and thus experiencing small-angle scattering only. More precisely, the reference elec-trons obey the condition v , ⊥ k y ≥
0. In contrast tothem, the signal trajectories come close to the parention and undergo large-angle scattering that changes di-rection of the k y component compared to the initialone. Therefore, the signal trajectories can be defined as obeying the condition v , ⊥ k y ≤
0. However, these con-ditions are not sufficient to calculate the holographicstructures correctly. The fact is that in the presence ofthe Coulomb field the mapping from the plane of ini-tial conditions ( t , v , ⊥ ) to the ( k x , k y ) plane is a com-plicated function. For example, in the domain wherethe condition v , ⊥ k y ≤ ~k , see Ref. [67] for details. The separation of trajecto-ries of different kinds can be efficiently done by usingthe clusterization algorithms. In Ref. [67] this trajec-tory separation was accomplished manually by carefulinspection of the mapping ( t , v , ⊥ ) → ( k x , k y ).It was found that the Coulomb potential changesinterference patterns significantly. Three main effectsof the Coulomb field in the holographic patterns wereidentified in Ref. [67]. These are: shift of the interfer-ence pattern as a whole, filling of the parts of the pat-tern that are unfilled when the Coulomb potential isdisregarded, and the characteristic kink of the interfer-ence pattern in the vicinity of k y = 0 [cf. Figs. 7 (a)and (b)]. This kink at zero transverse momenta was at-tributed to the Coulomb focusing effect [139]. However,the question remains, how sensitive are the predictedCoulomb effects to focal averaging. Therefore, furtherstudies are required to understand which of these effectscan be observed in experiment. p x (a.u.) p y ( a . u . ) −2 −1.5 −1 −0.5 0−2−1012 p x (a.u.) p y ( a . u . ) −2 −1.5 −1 −0.5 0−2−1012 −1−0.500.51(a) (b) Fig. 7
Holographic patterns emerging due to interference ofa direct electron with a rescattered one that has the shortesttravel time (see Ref. [67]) calculated (a) using the three-stepmodel with time-dependent exit point, and (b) accounting forthe Coulomb potential of the ion. The interference patternsare calculated for ionization of the H atom at a wavelengthof 800 nm and intensity of 6 . × W/cm . The theoretical methods used in strong-field physicsusually employ the single-active electron approxima-tion (SAE). In the SAE an atom or molecule interact- ing with the laser pulse is replaced by a single elec-tron. This single electron moves in the laser field andin the field of an effective potential. Therefore, the ion-ization is treated as a one-electron process. The SAEis a basis for understanding of many strong-field pro-cesses, including ATI and HHG [2,83]. Nevertheless,the role of the multielectron effects (ME) in strong-field and ultrafast physics has been attracting partic-ular attention (see, e.g., Refs. [140,141] and referencestherein). By now many theoretical approaches aimed atthe description of the ME effects have been developed.The most well-known and widely used of them are:the time-dependent density-functional theory [142] (seeRefs. [143,144] for a text-book treatment), multiconfig-uration time-dependent Hartree-Fock theory [145,146],time-dependent restricted-active-space [147] and time-dependent complete-active-space self-consistent field the-ory [148], time-dependent R-matrix theory [149,150]and R-matrix theory with time-dependence [151,152],time-dependent analytical R-matrix theory [153], etc.(see Ref. [154]). There are also many semiclassical ap-proaches capable to account for the ME effects, see, e.g.Refs. [155,78,156,43,157,140]. The advantages of thetrajectory-based models discussed in Sec. 1 are partic-ularly valuable in studies of complex ME effects.One of the most well-known ME effects in strong-field ionization is laser-induced polarization of the par-ent ion. Recently the polarization effects in the ATIhave been actively studied, see, e.g., Refs. [158,155,78,156,43,157,140]. In Refs. [159,160] and [158] the ef-fective potential for the outer electron that accountsfor the external laser field, the Coulomb interaction,and the polarization effects of the ionic core, is de-rived in the adiabatic approximation. It was for thefirst time found in Ref. [155] that the time-independentSchrodinger equation with this effective potential andaccounting for the Stark-shift of the ionization poten-tial can be approximately separated in parabolic coor-dinates. This separation determines a certain tunnelinggeometry. The emerging physical picture of the flow ofthe electron charge associated with the tunneling elec-tron is referred to as tunnel ionization in parabolic co-ordinates with induced dipole and Stark shift (TIPIS).The semiclassical model based on the TIPIS approachand disregarding the interference effect has shown agood agreement with experimental data (see Refs. [155,156,157]) and the TDSE results [155,78].The electron momentum distributions generated inionization of different atoms and molecules, includingAr, Mg, CO, naphthalene, etc., are very sensitive tothe ME effects accounted by the induced dipole of theionic core [155,78,156,43,157]. These studies considerionization by circularly or elliptically polarized laser emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 15 pulses. This is due to the fact that the effective poten-tial derived in Refs. [159,160] and [158] is valid only atlarge and intermediate distances from the ionic core. Inclose to circularly polarized laser fields the rescattering-induced processes are suppressed (see Ref. [161]), and,therefore, the vast majority of the ionized electrons donot return to the parent ion. However, this is not truefor linearly polarized field, and the applicability of theTIPIS approach in semiclassical simulations in the caseof linear polarization raised questions. This problem isaddressed in Ref. [154]. Furthermore, the study [154]combines the TIPIS approach with the SCTS model.The resulting two-step semiclassical model for strong-field ionization is capable to describe quantum inter-ference and accounts for the Stark-shift, the Coulombpotential, and the polarization induced dipole poten-tial.5.1 Combination of the TIPIS model and the STCSThe ionic potential derived in Refs. [159,160,158] readsas: V ( ~r, t ) = − Zr − α I ~F ( t ) · ~rr , (52)where ME effect is accounted through the induced dipolepotential h α I ~F ( t ) · ~r/r i . For the potential of Eq. (52)the starting point of a classical trajectory can be ob-tained as the tunnel exit in the TIPIS model. Morespecifically, the tunnel exit point is given bt z e ≈ − η e / η e satisfies the equation: − β ( F )2 η + m − η − F η α I Fη = − I p ( F )4 , (53)It is seen that Eq. (53) has the additional ME term inthe left-hand side compared to the equation (2). Sincethe ME term in the potential (52) is proportional tothe laser field ~F ( t ), it is absent at t > t f . Therefore,after the laser pulse terminates, the electron moves inthe Coulomb field only. This makes it possible to useEq. (6) for calculation of the asymptotic momentum ofthe electron from its position and momentum at t = t f .The SCTS phase (12) with the potential V ( ~r, t ) definedby Eq. (52) reads as: Φ SCT S ( t , ~v ) = − ~v · ~r ( t ) + I p t − Z ∞ t dt ( p ( t )2 − Zr − α I ~F ( t ) · ~rr ) . (54)In order to implement the resulting semiclassical model,the importance sampling method was used in Ref. [154].We note that in addition to the inapplicability of the potential (52) at small distances, there exist other con-ditions that restrict the range of applicability of theTIPIS model (see Refs. [78,154] for details). The study[154] focuses on the cases of Mg ( I p = 0 .
28 a.u., α N =71 .
33 a.u., α I = 35 .
00 a.u.) and Ca ( I p = 0 .
22 a.u., α N = 169 . α I = 74 .
11 a.u.) atoms, which havesimilar ionization potentials. But their static ionic po-larizabilities are different by approximately two times.In order to avoid the application of the potential(52) at small distances, a special cutoff radius r C wasintroduced in Ref. [154], and all the trajectories enter-ing the sphere r < r C were ignored. The remainingtrajectories do not reach the vicinity of the ion. It isclear that the elimination of the whole class of the tra-jectories (the returning ones) depletes some parts ofelectron momentum distributions. However, these de-pleted parts usually correspond to the boundary of thedirect ionization spectrum. Therefore, they do not af-fect the main part of the momentum distributions thatprovides major contribution to the ionization yield, seeRef. [154] for details.5.2 Application of the combined semiclassical modelThe 2D photoelectron momentum distributions calcu-lated in accord with the resulting semiclassical modelare shown in Figs. 8 (a)-(d). Figures 8 (a) and (c) cor-respond to the distributions calculated accounting forthe laser and Coulomb fields. Figures 8 (b) and (d) dis-play the results of the combined TIPIS + SCTS model,i.e., with the account of the ME potential. The pan-els [(a), (b)] and [(c), (d)] correspond to ionization ofMg and Ca, respectively. It is seen that the presence ofthe ME term in the potential of Eq. (52) results in anarrowing of the longitudinal momentum distributionsand modification of the interference structures.We first discuss the narrowing of the longitudinaldistributions. This effect is further illustrated in Figs. 9(a) and (c) that show the longitudinal momentum dis-tributions obtained with and without the ME term forMg and Ca, respectively. Since the widths of the distri-butions do not change due to the interference effects,the phase is disregarded in the calculations of Figs. 9 (a)and (c). The corresponding electron energy spectra areshown in Figs. 9 (b) and (d). It is seen that the spec-tra calculated accounting for the ME term fall off morerapidly with increase of the energy than the ones ob-tained neglecting the ME effects. This is a direct conse-quence of the narrowing of the corresponding 2D elec-tron momentum distributions.The mechanism underlying the narrowing effect hasa kinematic origin [154]. The analysis of classical trajec-tories has shown that there is a certain class of trajecto- k z (a.u.) k ⊥ ( a . u . ) −1 −0.5 0 0.5 1−1−0.500.51 k z (a.u.) −1 −0.5 0 0.5 1−1−0.500.51 k z (a.u.) k ⊥ ( a . u . ) −1 −0.5 0 0.5 1−1−0.500.51 k z (a.u.) −1 −0.5 0 0.5 1−1−0.500.51 −10 1 2 3 −10 1 2 3 MEno MEno ME ME Ca(d)(c) Ca(a) Mg Mg(b)
Fig. 8
Two-dimensional electron momentum distributionsfor the Mg [(a),(b)] and Ca [(c),(d)] atoms calculated by com-bining the TIPIS approach with the SCTS model. The wave-length is 1600 nm and the pulse duration is n = 8 cycles. Pan-els [(a),(b)] and [(c),(d)] show the distributions calculated atthe intensities of 3 . × W/cm and 1 . × W/cm , re-spectively. The distributions [(a), (c)] are obtained neglectingthe ME terms in Eqs. (52), (53), and (54), whereas the distri-butions [(b),(d)] are calculated accounting the ME terms inall these equations. The momentum disributions are normal-ized to the total ionization yield. A logarithmic color scale inarbitrary units is used. ries strongly affected by the induced polarization of theionic core. The trajectories of this class start closer tothe parent ion that other trajectories and their initialtransverse velocities are not too large (see Ref. [154] fordetails). Indeed, the force acting on the electron dueto the ME polarization effect (the ME force) decays as1 /r with increasing r . Therefore, this force can changethe electron motion only at the initial part of the tra-jectory adjacent to the tunnel exit. The ME force re-duces both longitudinal and transverse components ofthe electron final momentum, and, as the result, thetrajectories belonging to this class lead to the bins withsmaller ~k . We note that for close to circularly polarizedlaser pulses the ME effects result in the rotation of the2D electron momentum distributions towards the smallaxis of polarization ellipse [155].It is seen that the presence of the ME term in theequations of motion and the phase does not dramat-ically change the interference patterns. The interfer-ence structure is modified only in the first and the sec- d R / d E d R / d E E / U p d R / dk z −1.5 −1 −0.5 0 0.5 1 1.500.20.40.60.81 k z (a.u.) d R / dk z ME no ME (b)(d)(c)(a)
Fig. 9 (a),(c) Electron momentum distributions in the longi-tudinal direction, and (b),(d) energy spectra. Panels (a,b) and(c,d) correspond to the ionization of Mg and Ca, respectively.Thick green curve and thin blue curve show the semiclassicalresults obtained with and without ME terms, respectively.The wavelength and duration of the pulse are as in Fig. 8.The panels (a,b) and (c,d) are calculated for the intensitiesof 3 . × W/cm and 1 . × W/cm , respectively. Thelongitudinal distributions and energy spectra are normalizedto the maximum value. ond ATI peaks and also in the vicinity of the k z axis.The analysis of the mechanism behind the polarization-induced interference effect showed that the changes ininterference patterns are mostly caused by the ME termin the equation of motion, whereas the presence of theterm − α I ~F · ~r/r in the phase (54) does not play asubstantial role [154].It was found that the trajectories interfering in agiven bin often have similar ME contributions to thephase − Z ∞ t dt α I ~F · ~r ( t ) r ( t ) , (55)and therefore, the difference of these contributions issmall. This difference is the only important quantityfor the interference effect. The ME contributions to thephase are similar due to the combination of the follow-ing reasons: (i) the tunneling probability is a sharp func-tion of the laser field F ( t ) at time of ionization, (ii) thetunnel exit depends only on ~F ( t ) and the parametersof the atom (molecule), and (iii) only the initial part ofthe electron trajectory is relevant in the integral (55).Nevertheless, for atoms and molecules with large valuesof the ionic polarizability α I , the difference of the MEcontributions to the phase is essential. As the result, thechanges in the interference patters due to the ME effectcan be significant. This is illustrated in Figs. 10 (a)-(d).It is seen that the number of radial nodal lines in the emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 17 k ⊥ ( a . u . ) k z (a.u.) −0.8−0.4 0 0.4 0.8−0.8−0.400.40.8 k z (a.u.) −0.8−0.4 0 0.4 0.8−0.8−0.400.40.8 −10123 k z (a.u.) k ⊥ ( a . u . ) −0.2 0 0.2−0.200.2 k z (a.u.) −0.2 0 0.2−0.200.2 −10123 (c) (d)(a) (b)no ME MEMEno ME Fig. 10
Two-dimensional photoelectron momentum distri-butions for the Ba atom ionized by a laser pulse with a du-ration of n = 4 cycles, intensity of 3 . × W/cm anda wavelength of 1600 nm obtained by semiclassical simula-tions neglecting the ME term in the phase (54) [(a),(c)] andincluding this term [(b),(d)]. Panels (c) and (d) display themagnification for | k z | ≤ . | k ⊥ | ≤ .
25 a.u. of themomentum distributions shown in (a) and (b), respectively.In both cases the ME force is included in the Newton’s equa-tion of motion. The normalization to the total ionization yieldis used. The color scale is logarithmic with arbitrary units. fanlike interference pattern at low energies is differentwhen calculated with and without the ME term in thephase of Eq. (54). In Fig. 10 (c) there are six nodal linesfor positive k ⊥ , while in the presence of the ME termonly five such lines are visible [see Fig. 10 (d)]. molecule To the best of our knowledge, there are only a fewworks that apply semiclassical models accounting forthe quantum interference effect to describe strong-fieldionization of molecules, see Refs. [162,137,163]. Thestudies [162,137] extend the QTMC model to the molec-ular case. The SCTS model was applied to the hydro-gen molecule in Ref. [163]. Two-dimensional electronmomentum distributions, energy spectra, and angulardistributions were compared to the ones calculated forionization of the atomic hydrogen. The study [163] re-vealed substantial differences in electron momentumdistributions and energy spectra as compared to theatomic case. 6.1 SCTS model for hydrogen moleculeThe ionic potential experienced by a single-active-electronin the H molecule is given by V ( ~r ) = − Z (cid:12)(cid:12)(cid:12) ~r − ~R/ (cid:12)(cid:12)(cid:12) − Z (cid:12)(cid:12)(cid:12) ~r + ~R/ (cid:12)(cid:12)(cid:12) (56)Here ~R is the vector pointing from one nucleus to an-other. It is assumed that the origin of the coordinatesystem is located in the center of the molecule. Theeffective charges Z and Z are chosen to be equalto 0 . molecule is thebonding superposition of the two 1s atomic orbitals lo-cated at the centers of the atoms: Ψ H ( ~r ) = 1 p S OI ) h ψ atom (cid:16) ~r − ~R/ (cid:17) + ψ atom (cid:16) ~r + ~R/ (cid:17)i . (57)The corresponding partial Fourier transform is given by(see Ref. [136]): Π H ( p x , p y , z ) = exp (cid:18) − i R sin θ m [ p x cos ϕ m + p y sin ϕ m ] (cid:19) Π atom (cid:18) p x , p y , z − R θ m (cid:19) + exp (cid:18) i R sin θ m [ p x cos ϕ m + p y sin ϕ m ] (cid:19) Π atom (cid:18) p x , p y , z + R θ m (cid:19) . (58)Here θ m and ϕ m are the polar and azimuthal angles ofthe molecular axis, respectively, and Π atom ( p x , p y , z ) isthe partial Fourier transform of the 1 s orbital. Substi-tuting the expression for Π atom ( p x , p y , z ) (see Ref. [135])in Eq. (58) we obtain the following formula for themixed-representation wave function of the H molecule applicable just beyond the tunnel exit: Π ( p x , p y , z e ) ∼ (cid:26) exp (cid:18) − i R sin θ m [ p x cos ϕ m + p y sin ϕ m ] (cid:19) × exp (cid:18) − κR cos θ m (cid:19) + exp (cid:18) i R sin θ m [ p x cos ϕ m + p y sin ϕ m ] (cid:19) × exp (cid:18) − κR cos θ m (cid:19)(cid:27) × exp " − κ F − κ (cid:0) p x + p y (cid:1) F . (59)As in Ref. [137], this expression (without prefactor)was used in [163] as a complex amplitude describingionization at time t with initial transverse velocity v , ⊥ = p , ⊥ . In the simplest case analyzed in Ref. [163]the molecule is oriented along the laser polarization di-rection ( θ m = ϕ m = 0), and the factor in brackets inEq. (59) is constant for a fixed internuclear distance R . This allows to use only the exponential factor ofEq. (59).Different approaches can be used to find the tunnelexit point, i.e., the starting point of the trajectory, inthe molecular case. The simplest one consists in neglect-ing the molecular potential, i.e., considering triangularpotential barrier (51). An alternative approach, the so-called field direction model (DFM) (see Ref. [78]), ac-counts for the molecular potential. The potential bar-rier in the FDM model is formed by the molecular po-tential and the laser field in a 1D cut along the fielddirection. Therefore, the tunnel exit point in the FDMmodel is defined by the equation: V ( ~r ) + F ( t ) z e = − I p . (60)To finalize the generalization of the SCTS model to thecase of the H molecule, we need to obtain the phase,which is assigned to a classical trajectory. This phase isderived by substituting the potential (56) in Eq. (12): Φ SCT SH ( t , ~v ) = − ~v · ~r ( t ) + I p t − Z ∞ t dt p ( t )2 − Z (cid:16) ~r − ~R/ (cid:17) · (cid:16) ~r − ~R/ (cid:17)(cid:12)(cid:12)(cid:12) ~r − ~R/ (cid:12)(cid:12)(cid:12) + Z (cid:16) ~r + ~R/ (cid:17) · (cid:16) ~r + ~R/ (cid:17)(cid:12)(cid:12)(cid:12) ~r + ~R/ (cid:12)(cid:12)(cid:12) , (61)see Ref. [163]. It is seen that for r ≫ R/ − Z/r with the effective charge Z = Z + Z . In contrast to this, the QTMC phase for the H moleculeis given by: Φ QT MCH ( t , ~v ) = − ~v · ~r ( t ) + I p t − Z ∞ t dt p ( t )2 − Z (cid:12)(cid:12)(cid:12) ~r − ~R/ (cid:12)(cid:12)(cid:12) − Z (cid:12)(cid:12)(cid:12) ~r + ~R/ (cid:12)(cid:12)(cid:12) . (62)The expression 61 can be simplified at large distancesand, as the result, the SCTS phase for the H atom isreproduced. Finally, it is assumed in Ref. [163] that atthe end of the laser pulse the ionized electron is farenough from both nuclei, i.e., r ( t f ) ≫ R . If this con-dition is met, after the end of the pulse the electronmoves in the Coulomb field with the effective charge Z .Therefore, its asymptotic momentum can be calculatedfrom Eq. (6), and the post-pulse phase is determinedby Eq. (21).6.2 Application of the SCTS model to H moleculeIn Fig. 11 we compare the photoelectron momentumdistributions calculated within the SCTS model for thehydrogen atom [Fig. 11(a)] and hydrogen molecule [Figs.11 (b) and 11 (c)], see Ref. [163]. The starting point ofthe trajectory for H is calculated using the triangularpotential barrier (51). The distribution of Fig. 11 (a) forH is also obtained for the exit point calculated fromEq (51). We note that the molecular potential is fullytaken into account in the classical equations of motion(1) and in the phase (12) when calculating Fig. 11 (a).The electron momentum distribution of Fig. 11 (a) cor-responds to the tunnel exit obtained by using the FDMmodel. The electron momentum distributions shown inFigs. 11 (a) and 11 (b) are similar to each other. There-fore, it can be concluded that if the molecular poten-tial is not accounted in calculating the starting point,the effects of the molecular structure are not visible inelectron momentum distributions. This result can beexpected bearing in mind that r = I p /F ≫ R/ emiclassical two-step model for ionization by a strong laser pulse: Further developments and applications. 19 case are more aligned along the polarization direction.Second, at the same parameters of the laser pulse theholographic interference fringes are more pronouncedfor H than for H (see Fig. 11). The comparison of the k z (a.u.) k ⊥ ( a . u . ) −1 −0.5 0 0.5 1−0.500.5 k z (a.u.) k ⊥ ( a . u . ) −1 −0.5 0 0.5 1−0.500.5 k z (a.u.) k ⊥ ( a . u . ) −1 −0.5 0 0.5 1−0.500.5 −2−10 1 2 −2−10 1 2 −2−10 1 2 (b)(a) H , FDM(c) H, triangularH , triangular Fig. 11
Two-dimensional electron momentum distributionsfor ionization of (a) the H atom and (b),(c) the H moleculeby a laser pulse with the duration of n = 4 cycles, intensityof 2 . × W/cm , and wavelength of 800 nm. The distri-butions shown in panels (a) and (b) correspond to the tunnelexit point calculated from Eq. (51). The distribution of panel(c) is obtained using the FDM expression for the tunnel exit.The H molecule is oriented along the polarization directionof the laser field ( z -axis). The holographic fringes are shownby white lines in panel (a). The normalization to the totalionization yield is used. The color scale is logarithmic witharbitrary units. distributions calculated using the SCTS and QTMCmodels for ionization of the H molecule is presentedin Figs. 12 (a)-(d). Two different pulse envelopes wereused in Figs. 12 (a)-(d). Figures 12 (a) and (b) corre-spond to the sine squared pulse, whereas Figures 12 (c)and (d) show the distributions obtained for the trape- zoidal pulse (see Ref. [163] for details). Figures 12(a)and (c) display the momentum distributions calculatedwithin the QTMC model, and Figs. 12 (b) and (d) showthe corresponding SCTS results. For the sine squaredpulse these distributions have a pronounced fan-likestructure in their low-energy part. For the trapezoidalenvelope the fans are substituted by the characteristicblobs [see Figs. 12(c) and (d)] lying on a circle withthe radius k = 0 .
30 a.u. Similar to the atomic case, theQTMC predicts fewer nodal lines in the interferencestructure at low energies than the SCTS model. Thisfact can be again attributed to the underestimation ofthe Coulomb potential in the QTMC phase [163]. k z (a.u.) k ⊥ ( a . u . ) −0.2 0 0.2−0.200.2 k z (a.u.) −0.2 0 0.2−0.200.2 k z (a.u.) k ⊥ ( a . u . ) −0.2 0 0.2−0.200.2 k z (a.u.) −0.2 0 0.2−0.200.2 −2−10 1 2 −2−10 1 2 QTMC SCTSQTMC SCTS(d)(b)(a)(c)
Fig. 12
The low-energy parts of the two-dimensional photo-electron momentum distributions for the H molecule ionizedby a laser pulse with a duration of n = 4 cycles, wavelengthof 800 nm, and peak intensity of 1 . × W/cm . The leftcolumn [panels (a) and (c)] show the results of the QTMCmodel. The right column [panels (b) and (d)] present the dis-tributions calculated within the SCTS model. Panels (a,b)and (c,d) are calculated for the sine squared and trapezoidalenvelopes of the laser pulse, respectively (see Ref. [163]). Themolecule is oriented along the laser polarization direction ( z -axis). A logarithmic color scale in arbitrary units is used. The semiclassical models using classical mechanics todescribe the electron motion after it has been releasedfrom an atom or molecule are one of the powerful meth-ods of strong-field, ultrafast, and attosecond physics.The standard formulation of the trajectory-based mod- els does not allow to describe the effects of quantum in-terference. Nevertheless, a substantial progress in simu-lations of the interference effects using the semiclassicalmodels has been achieved recently. By present severaltrajectory-based models capable to describe the inter-ference effects have been developed and successfully ap-plied to the studies of the ATI. Here we discuss one ofthese models, namely, the SCTS.The SCTS model allows to reproduce interferencepatterns of the ATI process and accounts for the ionicpotential beyond the semiclassical perturbation theory.In the SCTS the phase assigned to every classical tra-jectory is calculated using the semiclassical expressionfor the matrix element of the quantum mechanical prop-agator [79,80,81]. As the result, the SCTS model yieldsa good agreement with the direct numerical solution ofthe TDSE, better than, e.g., the QTMC model apply-ing the first order semiclassical perturbation theory toaccount for the Coulomb potential in the phase.Here we review further developments and applica-tions of the SCTS. At first, we review the formulationof the SCTS and its numerical implementation. The ap-plication of the model was illustrated in the case of theH atom. We next turn to the further developments ofthe SCTS: the SCTSQI model [94] and the SCTS modelwith the prefactor [93]. In the SCTSQI model the ini-tial conditions for classical trajectories are determinedfrom the exact quantum dynamics of the wavepacket.For ionization of the 1D atom the SCTSQI model yieldsnot only qualitative, but also quantitative agreementwith the numerical solution of the TDSE. Further workis needed to accomplish the generalization of the SCT-SQI on the 3D case. The developments in this direc-tion have already begun. The quantitative agreementwith the TDSE was also achieved by the extension ofthe SCTS model that accounts for the prefactor of thesemiclassical matrix element. Furthermore, the 3D im-plementation of the SCTS [93] has a number of otherimportant modifications.We discuss the application of the SCTS approachto the SFPH. The semiclassical simulations within theSCTS model are in perfect agreement with the resultsof the recent experiment [66]. The model is able to re-produce all characteristic features of the observed holo-graphic patterns. The SCTS model also allows to inves-tigate the effect of the Coulomb potential on the holo-graphic structures. Three main Coulomb effects in theinterference patterns were predicted [67]. However, itshould be investigated how sensitive are these Coulombeffects to focal averaging. This further work will allowto understand, which of the predicted effects can beobserved. We also present a quick review of the applicationof the SCTS to study of the multielectron polariza-tion effects. We discuss the modification of the SCTSmodel accounting for the multielectron polarization-induced dipole potential. The semiclassical simulationspredict narrowing of the electron momentum distribu-tions along the polarization direction. This narrowingarises due to the focusing of the ionized electrons by theinduced dipole potential. Furthermore, the polarizationof the ionic core can also modify the interference pat-terns in electron momentum distributions.Finally, we briefly reviewed the extension of theSCTS model to ionization of the hydrogen molecule.The SCTS model for the H can be generalized to an ar-bitrary laser polarization and orientation of the molecule,as well as to heteronuclear and polyatomic molecules.We believe that these generalizations being combinedwith the extended versions of the SCTS will result to anemergence of powerfool tools for studies of the strong-field processes. Acknowledgements
We are grateful to M. Lein, L. B. Madsen, J. Burgd¨orfer,H. J. W¨orner, C. Lemell, D. G. Arb´o, E. R¨as¨anen,and K. T˝ok´esi for fruitful collaboration that resultedin some of the works discussed in this colloquium pa-per. We would also like to thank S. Brennecke, N. Eicke,C. Faria, A. Landsman, H. Ni, F. Oppermann, J. Solanp¨a¨a,S. Yue, and B. Zhang for valuable discussions. Thiswork was supported by the Deutsche Forschungsgemein-schaft (Grant No. SH 1145/1-2).
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