Dissociative ionization of H + 2 : Few-cycle effect in the joint electron-ion energy spectrum
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Dissociative ionization of H +2 : Few-cycle effect in the joint electron-ion energy spectrum V. Mosert and D. Bauer
Institut f¨ur Physik, Universit¨at Rostock, 18051 Rostock, Germany (Dated: October 15, 2015)Joint electron-ion energy spectra for the dissociative ionization of a model H +2 in few-cycle, infrared laserpulses are calculated via the numerical ab initio solution of the time-dependent Schr¨odinger equation. A strong,pulse-dependent modulation of the ionization probability for certain values of the protons’ kinetic energy (but al-most independent of the electron’s energy) is observed. With the help of models with frozen ions, this feature—which mistakenly might be attributed to vibrational excitations—is traced back to the transient population ofelectronically excited states, followed by ionization. This assertion is further corroborated employing a two-level model incorporating strong-field ionization from the excited state. PACS numbers: 33.80.Rv,33.20.Xx,33.60.+q,31.15.A-
I. INTRODUCTION
The hydrogen molecular ion H +2 is one of the few systemsfor which the interaction with intense, short laser pulses canbe simulated truly ab initio , i.e., based on the solution of thetime-dependent Schr¨odinger (TDSE) equation [1, 2] withoutfurther approximations such as, e.g., Born-Oppenheimer orEhrenfest dynamics. Only rotations are usually neglected,which is justified for short laser pulses. Despite the simplic-ity of H +2 , its joint electron spectra (JES) for electrons andnuclei are intriguingly complex [3–6]. In fact, on top of thealready complex features in photoelectron spectra from atoms[7] there is a nuclear degree of freedom added in H +2 (or itsisotopic sisters). Hence, for any feature observed in a strong-field JES at least one question arises: are there vibronic exci-tations involved?Experimental photoelectron spectra for H +2 and JES for H have been reported in Refs. [8, 9], simulated ones in Refs. [3–6]. In the multiphoton regime, energy sharing according to E + n ¯ hω = E e + E p is observed. Here, E is the initialenergy, n ¯ hω the absorbed photon energy, and E e , E p the en-ergy of the emitted electron and the nuclear kinetic energyrelease (KER), respectively. As E e + E p = const., this corre-lated energy sharing leads to diagonal, straight-line featuresin the E e , E p -plane of the JES. At longer wavelengths theJES are less simple, especially at low electron energy whereCoulomb effects are very important, as is well known fromatomic strong-field ionization [10]. The diagonal, correlatedfeatures tend to fade while pronounced oscillations in the ion-ization probability as function of the electron energy emerge.However, also oscillations of the probability for dissociativeionization (DI) as function of the KER are observed, whichhave been shown to depend on the initial vibrational state [3].One might be tempted to always attribute such variations inthe DI probability to vibrational excitations. In fact, an inter-esting application of DI is Coulomb explosion imaging [11]where one strives for reconstructing the initial configurationof the nuclei from the KER spectrum after rapid ionization bya strong laser field. In this way, e.g., interference structures inthe KER spectra due to a two-surface population dynamics inH +2 were observed experimentally [12, 13]. We will discussin this paper another mechanism that introduces a modulationin the KER. It is based on the oscillatory behavior of the ion- ization probability as function of the internuclear distance andthe few-cycle laser pulse duration.The paper is organized as follows. In Sec. II we start withthe full quantum H +2 model and introduce the effect we dis-cuss in the remainder of this work: the “vertical fringes” (VF)in JES, indicating strong variations of the DI yield as func-tion of the KER but almost independent of the electron en-ergy. The subsequent sections serve to prove that the VF effectis not due to vibrational excitations (Sec. III), not due to theone-dimensionality of our model (Sec. IV), and not due to thetwo-center nature of diatomic molecular potentials (Sec. V).In Sec. VI, a two-level model, combined with the strong-fieldapproximation, is introduced that is capable of qualitativelyreproducing the VF effect. We conclude in Sec. VII and giveall the numerical details in the Appendix, in particular on thet-SURFF approach for calculating the JES in the various ge-ometries.Atomic units ¯ h = m e = | e | = 4 πǫ = 1 are used unlessotherwise indicated. II. FULL QUANTUM H +2 MODEL
The Hamiltonian for the two-dimensional H +2 model reads H = − µ ∂ x − M ∂ R − iβA ( t ) ∂ x + V pe − + 1 | R | . (1)Both the electronic degrees and the nuclear degrees of free-dom, i.e., electronic coordinate x ∈ ( −∞ , ∞ ) and internu-clear distance R ∈ (0 , ∞ ) , are restricted to the laser polariza-tion axis, M = 1836 is the proton mass, µ = 2 M/ (2 M + 1) ,and β = ( M + 1) /M . As we are dealing with a homonucleardiatomic molecule the laser field only couples to the electronicdegree of freedom [14]. The velocity-gauge coupling to thelaser field of vector potential A ( t ) (in dipole approximation)was chosen, with the purely time-dependent A -term trans-formed away.For the interaction between electron and protons we choose V pe − = − p ( x − R/ + ǫ − p ( x + R/ + ǫ (2)with the smoothing parameter ǫ = 1 . The JES for DI is calcu-lated via the time-dependent surface flux method (t-SURFF) x Rx = − R/ V pe − ≈ X DI − X DI x = R/ FIG. 1. (Color online) Computational grid for the full quantum H +2 model. Coordinates x and R are electronic and internuclear distance,respectively. In the blue areas around x = R/ and x = − R/ the electron is close to one of the protons. The relevant t-SURFFboundary for DI (with electrons escaping in positive x direction) isgiven by x = X DI . The red area indicates the region in which a maskfunction absorbs probability density. [15] (see Appendix B for details). Figure 1 depicts the ge-ometry of the system. Upon time-propagation, probabilitydensity will pass the surfaces defined by a sufficiently large | x | = X DI , “recorded” there for the calculation of the JES us-ing t-SURFF, and be absorbed by a mask function thereafter.For the t-SURFF approximation we have to assume V pe − ≈ which makes sense if X DI ≫ R/ .Figure 2 shows correlated spectra for the process of DI infew-cycle laser pulses of vector potential A ( t ) = A sin (cid:18) ωt n c (cid:19) sin( ωt ) (3)for < t < T p = 2 πn c /ω , calculated by the absolute squareof (B6). T p is the pulse duration and n c the number of lasercycles. The initial wave function for the TDSE simulationswas always the ground state of the Hamiltonian (1), whichhas the energy E = − . . The laser parameters are givenin the figure caption.The main features in the spectra in Fig. 2 are nearly verticaland horizontal fringe patterns. Diagonal features indicatingenergy sharing according to E + n ¯ hω = E e + E p are notobserved for the laser parameters and the direction of escap-ing electrons chosen. The modulation of the yield for fixednuclear kinetic energy E p as function of the electronic kineticenergy E e is well known from laser atom interaction. It can beattributed to the interference of several electron paths with dif-ferent ionization times that lead to the same final electron mo-mentum (see, e.g., [7, 16, 17]). Moreover, “direct electrons” . . . .
54 10 − − − − − − − − − e l ec tr o n i c k i n . e n e r g y ( H a rtr ee ) . .
52 10 − − − − − − − − − e l ec tr o n i c k i n . e n e r g y ( H a rtr ee ) FIG. 2. (Color online) Upper panel: JES for DI of H +2 in a sin -shaped laser pulse with parameters: n c = 3 , λ = 800 nm, I peak =2 . · W/cm , U p = 0 . . The horizontal line marks the cut-off for direct ionization ( U p ). The upper JES was calculated at theupper t-SURFF boundary + X DI . Lower panel: n c = 4 , I peak =1 . · W/cm . The number of VF per nuclear kinetic energyinterval increases with n c . The lower JES was calculated at the lowert-SURFF boundary − X DI . and rescattered electrons can be clearly distinguished. The“simple man’s” cut-off U p is indicated in both panels by ahorizontal line. The yield due to direct electrons stretches wellbeyond U p before it drops down to the level of rescatteredelectrons (approximately four orders of magnitude smaller,visible for . < E p < . ).The objective of this paper is to reveal the origin of the mod-ulation of the yield as function of the nuclear kinetic energy E p . In other words, why is the DI yield strongly suppressedfor certain proton energies? And why is this suppression al-most independent of the electronic energy (i.e., why are thecorresponding fringe patterns almost vertical in Fig. 2) butdependent on the pulse duration? Similar modulations havebeen reported in Ref. [3] for simulations starting from a vibra-tionally excited H +2 molecule. The number of VF was foundto increase with increasing vibrational quantum number ν ofthe initial state. However, in our simulations we started from ν = 0 so that the vertical pattern in Fig. 2 does not just reflectthe probability density of the initial vibrational wave packet.On one hand, experience shows that commonly all spectralfeatures in strong field ionization can be explained in termsof interfering quantum trajectories. On the other hand, the in-terference of the usual long and short trajectories starting atthe two nuclear sites [18] (including potential rescattering andthe generation of a double-slit type interference pattern [19]),should depend not only on the internuclear distance but alsoon the electronic energy E e . Hence the strong suppression − − − . − − . − . .
95 1 1 .
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95 1 1 .
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25 1 . . . . ionization probability − |h ψ | GS i| |h ψ | i| e n e r g y ( H a rtr ee ) ionization potential (Hartree) d i p o l e m o m e n t( a . u . ) ionization potential (Hartree) E E d /R (Hartree)0.3 0.4 0.5 . . . . e l ec tr o n i ce n e r g y ( H a rtr ee ) − − − − − − − − − . . . FIG. 3. (Color online) Top panel: electronic spectra (each for fixed R ) for H +2 (same laser parameters as in Fig. 2, upper panel). Inlay intop panel: spectrum of electrons escaping in the opposite direction.Middle panel: occupation of the first excited state, − occupation ofthe ground state, and the total ionization probability. Bottom panel:energies of ground and first excited state, and their transition dipolemoment. of the DI yield for certain values of E p but almost indepen-dent of E e cannot be explained by such interfering quantumtrajectories. III. FIXED INTERNUCLEAR DISTANCE
To rule out vibrations as the origin of the VF in Fig. 2 resultsfor H +2 with fixed internuclear distances are discussed now.In order to calculate electronic spectra for the H +2 modelwith fixed inter nuclear distance R the electronic TDSE i∂ t Ψ( x, t ) = H ( t )Ψ( x, t ) for the Hamiltonian H ( t ) = − ∂ x − iA ( t ) ∂ x + V ( x ) (4)and binding potential V ( x ) = − p ( x − R/ + ǫ − p ( x + R/ + ǫ (5)was solved for many R in the range where /R ≃ E p coversthe relevant KER E p in Fig. 2.In the top panel of Fig. 3 all these electronic spectra arecollected for comparison with Fig. 2, upper panel. The over-all trend is an increasing ionization yield with increasing R because of the decreasing ionization potential I p = | E | (seebottom panel). The most important insight gained from thesefixed- R simulations is that pronounced suppressions of the ionization yield are observed for certain internuclear distances R ≃ /E p as well. This proves that vibrational excitationcannot be the origin of the VF visible in both Fig. 3 and Fig. 2.The bottom panel of Fig. 3 shows the energies of the twolowest bound electronic states in the H +2 potential (5) vs theionization potential I p = | E | . For large internuclear dis-tances R the two levels are almost degenerate with the groundstate energy rising asymtotically towards the ground state en-ergy value for the potential V ( x ) = − / √ x + ǫ . Because ofthis asymptotic degeneracy and the related diverging transi-tion dipole moment d these two states were dubbed “chargeresonance states”. However, we should stress that the VF inthe DI yield as discussed in this work occur at smaller dis-tances than “charge resonance enhanced ionization” (CREI)[20].The middle panel of Fig. 3 shows the total ionization prob-ability P ion = 1 − Z X I − X I d x | Ψ( x, T ) | , (6)and the occupations of the ground and first excited states afterthe interaction with the laser pulse. The modulations in thetotal ionization probability is less pronounced than in the en-ergy resolved spectrum, which can be explained by the “left-right asymmetry” [7] of the spectra for electrons escaping inpolarization direction (top panel) and opposite to it (inset intop panel). Moreover, the modulations in the energy resolvedspectrum are not strictly independent of the electronic energy E e , i.e., not perfectly vertical but slightly tilted.The fact that ionization probability and bound state occupa-tions at the end of the pulse oscillate similarly as function of I p (or /R ≃ E p ) suggests that the electronically excited stateplays an important role in the DI process in few-cycle laserpulses. However, the minima in the ionization yield do notperfectly coincide with the minima in the excited-state popu-lation. For ionization probabilities smaller than − the oc-cupation of the excited state rather oscillates with twice thefrequency of P ion as function of I p . Hence, the ionization stepintroduces an additional, nontrivial I p -dependence. Note thatthe excited state energy | E | ≃ . varies little with R so thatin any case at least about photons are required for ioniza-tion. Additionally, 2 up to 10 photons are needed to couplethe initial electronic ground state of energy E with the ex-cited state of energy E . Below, in Sec. VI, we reproduce theVF qualitatively, using a two-level approximation in combi-nation with the strong-field approximation (SFA). IV. 3D H +2 WITH FIXED INTERNUCLEAR DISTANCE
In order to show that the modulation in the DI yield is notan artifact of the low dimensionality of our models the 3Dmolecular ion with fixed ions and aligned along the laser po- .
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55 0 . . /R (Hartree) . . . . e l ec tr o n i c k i n . e n e r g y ( H a rtr ee ) − − − − − − − − − FIG. 4. (Color online) Same as Fig. 3 but for 3D H +2 aligned inpolarization direction of the laser. larization axis was considered. The Hamiltonian H = 12 ( − i ∇∇∇ + A ( t ) eee z ) − p ( z − R/ + ρ − p ( z + R/ + ρ (7)is cylindrically symmetric so that the natural choice for the t-SURFF boundary is the surface of a cylinder with radius R I and height Z I (see Appendix D for details).Figure 4 shows spectra for various internuclear distance R and electrons escaping in polarization direction. The mod-ulation of the ionization yield as function of /R is clearlyvisible although at low electron energies the fringes are moretilted than in the 1D results, making the suppression of theyield for certain internuclear distances less electron energy-independent. The fringe pattern for the rescattered electronsinstead is as vertical as in the 1D results. Revealing the originof this difference between 1D and 3D results requires furthersystematic investigations. In this work we are content with thefact that the modulation in the ionization yield as function of /R exists in 3D as well. V. SINGLE-CENTER POTENTIAL
Next we show that the two-center nature of the binding po-tential is not essential for the observed modulations of the ion-ization yield, while the existence of an excited state is. To thatend we consider a P¨oschl-Teller potential of the form V ( x ) = − b ( b − a cosh [ x/ (2 a )] (8)for which the finite number of energy levels of energy E n = − ( b − n − / (8 a ) , n = 0 , , , . . . < b − can be adjustedvia the parameters a > and b > .First, we aim at mimicking the behavior of ground and ex-cited state in the molecular model, i.e., E = − . is keptconstant, and E covers the range − . < E < − . Figure5 shows the electron spectra collected such that they can bedirectly compared to Fig. 3. The VF are there, proving thatthey are not due to a two-center interference. − − .
05 1 . .
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25 1 . . ionization potential (Hartree)ionization probability − |h ψ | GS i| |h ψ | i| . . . . e l ec tr o n i c k i n . e n e r g y ( H a rtr ee ) − − − − − − − − − FIG. 5. (Color online) Same as Fig. 3 but for a P¨oschl-Teller potential(8) with ground and excited state energy tuned close to the molecularcase. The laser intensity was increased to I peak = 3 . · W/cm in order to have an ionization yield similar to the molecular models. − − .
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25 1 . . ionization potential (Hartree)ionization probability − |h ψ | GS i| . . . . e l ec tr o n i c k i n . e n e r g y ( H a rtr ee ) − − − − − − − − − FIG. 6. (Color online) Same as Fig. 5 but for a P¨oschl-Teller potential(8) supporting only a single bound state. The intensity was increasedfurther to I peak = 6 . · W/cm . Second, Fig. 6 shows the case of a P¨oschl-Teller potentialwith a single bound state only. The intensity was increasedto I peak = 6 . · W/cm to compensate for the decreasingionization in the narrower and deeper potential. Each individ-ual photoelectron spectrum looks standard “SFA-like”. Boththe VF and oscillations in the occupation of the groundstateat the end of the laser pulse are absent. This substantiates ourassertion that the occupation of an excited state is crucial forthe modulation of the (dissociative) ionization yield. VI. TWO LEVELS + SFA
As long as the ionization probability is small we maymodel the occupation of the ground and first excited states − − .
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32 1 . ionization potential (Hartree) | a | TDSE ρ ρ analytical . e l ec tr o n i c k i n . e n e r g y ( H a rtr ee ) − − − − − FIG. 7. (Color online) Upper panel: photoelectron spectra for themodel of Fig. 3 calculated using the two-level SFA. Lower panel:occupation of the first excited state as calculated from the TDSE, thenumerical solution of Eqs. (10) and (11), and the analytical result(14). by a simple two-level model. Plugging the ansatz | ψ ( t ) i = a ( t ) | ψ i + a ( t ) | ψ i into the TDSE in length gauge i∂ t ψ ( x, t ) = (cid:18) − ∂ x + E ( t ) x + V ( x ) (cid:19) ψ ( x, t ) (9) one finds the well-known equations of motion for the densitymatrix elements ρ ij = a ∗ i a j , i = 0 , , j = 0 , , ˙ ρ = id E ( t )( ρ − ρ ∗ ) (10) ˙ ρ = id E ( t )( ρ − ρ ) + i ∆ Eρ (11)where ∆ E = E − E , d = h ψ | x | ψ i (assumed real), ρ = ρ ∗ , ρ = 1 − ρ . As we are interested in few-cyclepulses and the transient dynamics induced by them we can-not apply the rotating wave approximation, and a dressed orFloquet state approach does not make sense either. Instead, inthe bottom panel of Fig. 7 the density matrix element ρ atthe end of the laser pulse t = T p from the numerical solutionof the two-level model Eqs. (10) and (11) [initial conditions ρ ( t = 0) = 1 and ρ ( t = 0) = 0 ] is compared to the occu-pation of the first excited state from the numerical solution ofthe full TDSE. The agreement is very good apart from a shiftalong the ionization potential axis. This shift is caused by ne-glecting higher excited states and the coupling to the contin-uum in the two-level model. We checked that for lower fieldstrengths (where even less ionization occurs and other excitedstates are negligibly populated) the agreement improves.The results of the two-level system can be used to modelionization as well. In “standard” SFA only a single boundstate (besides the continuum states of momentum k ) is con-sidered, and depletion of its population is neglected (see, e.g.,Ref. [7]). Instead, we plug the modulus of the occupation | a ( t ) | = p ρ ( t ) of the first excited state into the SFA am-plitude for direct ionization, | a I, SFA ( k, t ) | ≃ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t d t ′ p ρ ( t ) d k ( A ( t ′ ) + k ) E ( t ′ ) e i R t ′ d t ′′ [ A ( t ′′ )+ k ]22 − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (12)Here, d k ( k ) = h ψ | x | k i , and we neglect the amplitude for ionization from the ground state because its contribtion is severalorders of magnitude smaller than the contribution of the first excited state. This is fortunate, as otherwise the phases of thecomplex a ( t ) and a ( t ) matter and should be calculated from a full SFA with two bound states and all relevant bound-boundand bound-continuum couplings. The top panel in Fig. 7 shows the collected electronic spectra for the molecular potential withfixed protons of Sec. III, calculated using our simplified two-level SFA. The dipole moments and the eigenenergies E and E were calculated numerically from the TDSE data of Sec. III (see the bottom panel of Fig. 3 for the R -dependent energiesand transition dipole moment d ). Figure 7 shows that the simple two-level SFA reproduces the I p -dependent features in theionization probability qualitatively. In particular, the correlation between the oscillations of the excited-state population at theend of the pulse as function of I p and the oscillations in the ionization probability with only half the frequency is as observed inthe TDSE results of Sec. III.Employing ρ ≪ ρ = 1 − ρ ≃ in (11), Eqs. (10) and (11) can be solved analytically, leading to ρ ( τ ) = (cid:12)(cid:12)(cid:12)(cid:12) d ω Z τ d τ ′ E ( τ ′ ) e − inτ ′ (cid:12)(cid:12)(cid:12)(cid:12) , τ = ωt, n = ∆ E/ω. (13)Using E ( t ) = − ∂ t A ( t ) with the vector potential (3) yields the occupation of the first excited state at the end of the sin pulse ρ = " A d ω ( n c ω − ω + 3∆ E n c )∆ E sin (cid:0) π ∆ Eω n c (cid:1) ω − ω + ( n c ω − − ω )( n c ω + − ω )( n c ω − + ω )( n c ω + + ω ) (14)where ω + = ω + ∆ E and ω + = ω − ∆ E . Expanding this expression in the small parameter η = ω/ ∆ E = 1 /n gives ρ ≃ A d sin ( π ∆ En c /ω ) η n c . (15)The occupation of the first excited state after the pulse thusdecreases as n c increases. Hence the observed VF in the (dis-sociative) ionization yield are a few-cycle effect. Moreover,inspection of the sine’s argument in (14) shows that the fre-quency of the oscillation depends on the number of cycles n c .The higher n c the more oscillations within a given ∆ E/ω in-terval. This is in agreement with the TDSE results in Fig. 2where the 4-cycle laser pulse was found to generate more VFthan the -cycle pulse. As E is almost constant in the H +2 model the oscillations in Fig. 7 are of almost constant periodwhen plotted vs I p = | E | .Note that ρ is very sensitive to the pulse shape. In fact,for a Gaussian pulse ρ according (13)—with the integra-tion limits stretched to ±∞ —becomes the (modulus squared)Fourier transform of a Gaussian, which is a Gaussian and thusdoes not oscillate with ∆ E and n c (full-width half maximum)at all. VII. CONCLUSION
Numerical simulations of the dissociative ionization pro-cess in H +2 for short laser pulses reveal patterns of verticalfringes in the joint energy spectra, i.e., strong variations of theyield as function of the ion energy that are almost independentof the electron energy. Identifying the kinetic energy releasewith the inverse internuclear distance, the effect is also foundin calculations with fixed ions, ruling out vibrational excita-tions as its origin. Instead, ionization proceeds via the firstexcited electronic state. In few-cycle pulses the population ofthe first excited state depends strongly on the number of cy-cles and the pulse shape in general. The vertical fringes in thecontinuous dissociative ionization spectra are clearly corre-lated with the population of the first excited state at the end of the pulse, as qualitatively reproduced using a simple two-statemodel combined with the strong-field approximation.The observed effect relies on the ultrashort, transient dy-namics in few-cycle laser pulses and not on resonances, spe-cially chosen detunings, or interference. In fact, in the limitof long laser pulses the vertical fringes disappear and oneapproaches—depending on the laser frequency—either or-dinary non-resonant multiphoton or tunneling ionization, orwell-known resonance-enhanced multiphoton ionization. ACKNOWLEDGMENTS
This work was supported by the SFB 652 of the GermanScience Foundation (DFG).
Appendix A: Numerical details
The TDSE was solved numerically by propagating thewavefunction with the Crank-Nicolson time propagator. Thewavefunction and the potentials were discretized on a Carte-sian grid with the spatial derivatives in the Hamiltonian ap-proximated by finite differences. An iterative block Gauss-Seidel method and the Thomas algorithm were applied forthe solution of the linear system of equations of the Crank-Nicolson method in the two and one dimensional case, re-spectively. The initial ground-state wavefunctions for the timepropagation and the first excited state were obtained by theshift-invert method [21]. For the numerical solution of thecylindrically symmetric Hamiltonian (7) the coordinate trans-formation ξ = ρ / was used [22]. Numerical parameters forthe TDSE simulations are summarized in Table I. Appendix B: t-SURFF for the H +2 model Assuming that V pe − in (2) can be neglected for x > X DI the wavefunction there separates in the form ψ k ( x, t ) φ p ( R ) e − itE p where φ p ( R ) are the solutions of the Coulomb scattering problem (cid:18) T N + 1 R (cid:19) φ p ( R ) = E p φ p ( R ) , T N = − M ∂ R , (B1)and ψ k ( x, t ) = (2 π ) − / e − iα ( t )+ ikx , α ( t ) = 12 Z t dt ′ [ k + 2 kA ( t ′ )] (B2)are Volkov wavefunctions.The DI amplitude (restricted to the electrons escaping in positive direction) is approximated by the integral h Ψ( T p ) | DI i ≃ a DI ( k, p ) ≡ h Ψ( T ) | Θ( x − X DI ) | p ( T ) i| k ( T ) i = Z dR Z x>X DI dx Ψ ∗ ( x, R, T ) ψ k ( x, T ) φ p ( R ) e − iT E p . (B3)This expression is not yet useful for practical purposes because T needs to be large enough to allow the slow electrons arrivingin the region x > X DI . On the other hand, the fast electrons need to be kept on the grid as well, necessitating a huge grid size. Inorder to avoid large grids the t-SURFF method [5, 15] was adapted to the problem at hand. Writing the right hand side of (B3)as a time integral we obtain a DI ( k, p ) = h Ψ(0) | Θ( x − X DI ) | p (0) i| k (0) i + Z T dt ∂ t h Ψ( t ) | Θ( x − X DI ) | p ( t ) i| k ( t ) i . (B4)For sufficiently large X DI and a bound initial state | Ψ i we have h Ψ(0) | Θ( x − X DI ) | p (0) i| k (0) i ≃ . Employing the TDSE with V pe − ≃ yields a DI ( k, p ) ≃ a DI,t-SURFF ( k, p ) ≡ i Z T dt h Ψ( t ) | h − µ ∂ x − M ∂ R − iβA ( t ) ∂ x + 1 R , Θ( x − X DI ) i | p ( t ) i| k ( t ) i . (B5)Only terms of the Hamiltonian containing derivatives with respect to x contribute in the commutator, leading to a DI,t-SURFF ( k, p ) = Z T dt Z ∞ dR h βA ( t )Ψ ∗ ( X DI , R, t ) ψ k ( X DI , t ) − i µ (cid:0) Ψ ∗ ( X DI , R, t ) ∂ x ψ k ( x, t ) | x = X DI − ψ k ( X DI , t ) ∂ x Ψ ∗ ( x, R, t ) | x = X DI (cid:1)i φ p ( R, t ) . (B6)The scattering states φ p ( R ) were used as implemented in the GNU Scientific Library (GSL) [23]. Appendix C: t-SURFF for 1D calculations
For the one dimensional systems the probability amplitude for ionization with final electron momentum k is approximated as h Ψ( T ) | Θ( x − X I ) | k ( T ) i ≃ Z T dt (cid:16) A ( t )Ψ ∗ ( X I , t ) ψ k ( X I , t ) − i ∗ ( X I , t ) ∂ x ψ k ( x, t ) | x = X I − ψ k ( X I , t ) ∂ x Ψ ∗ ( x, t ) | x = X I ) (cid:17) . (C1)Again, only electrons escaping in positive x direction, passing the t-SURFF boundary X I , are considered, and the bindingpotential is neglected for distances x > X I . Appendix D: t-SURFF for cylindrically symmetric system
The probability amplitude for an electron escaping with a momentum kkk = k ρ eee x + k z eee z can be approximated by the integral h kkk ( T ) | Ψ( T ) i ≃ Z T dt Z dV ∂ t (cid:16) ψ kkk ( ρ, z, t ) ∗ (cid:0) θ ( R I − ρ ) θ ( z − Z I ) + θ ( R I − ρ ) θ ( − z − Z I ) + θ ( ρ − R I ) (cid:1) Ψ( ρ, z, t ) (cid:17) . (D1)The integral (D1) can be divided into three terms which are evaluated separately. Using the TDSE, the first term reads (droppingthe arguments of ψ kkk and Ψ ) s ( k ρ , k z ) = Z T dt Z dV ∂ t (cid:16) ψ ∗ kkk θ ( R I − ρ ) θ ( z − Z I )Ψ (cid:17) = − i Z T dt Z dV ψ ∗ kkk (cid:2) ρ − ∂ ρ ρ∂ ρ + ∂ z + 2 iA ( t ) ∂ z , θ ( R I − ρ ) θ ( z − Z I ) (cid:3) Ψ= i Z T dt Z dV ψ ∗ kkk (cid:16) θ ( z − Z I ) (cid:0) Ψ ∂ ρ δ ( R I − ρ ) + 2( ∂ ρ Ψ) δ ( R I − ρ ) + ρ − Ψ δ ( R I − ρ ) (cid:1) − θ ( R I − ρ ) (Ψ ∂ z δ ( z − Z I ) + 2( ∂ z Ψ) δ ( z − Z I ) + 2 iA ( t )Ψ δ ( z − Z I )) (cid:17) = i Z T dt Z π dϕ (cid:18) Z ∞ Z I dz ( − ∂ ρ ( ρψ ∗ kkk Ψ) + 2 ρψ ∗ kkk ( ∂ ρ Ψ) + ψ ∗ kkk Ψ) | ρ = R I − Z R I ρdρ ( − ∂ z ( ψ ∗ kkk Ψ) + 2 ψ ∗ kkk ( ∂ z Ψ) + 2 iA ( t ) ψ ∗ kkk Ψ) | z = Z I (cid:19) . (D2)The second term is, analogously, s ( k ρ , k z ) = i Z T dt Z π dϕ (cid:18) Z − Z I −∞ dz ( − ∂ ρ ( ρψ ∗ kkk Ψ) + 2 ρψ ∗ kkk ( ∂ ρ Ψ) + ψ ∗ kkk Ψ) | ρ = R I + Z R I ρ dρ ( − ∂ z ( ψ ∗ kkk Ψ) + 2 ψ ∗ kkk ( ∂ z Ψ) + 2 iA ( t ) ψ ∗ kkk Ψ) | z = − Z I (cid:19) , (D3)and s ( k ρ , k z ) = Z T dt Z dV ∂ t (cid:16) ψ ∗ kkk θ ( ρ − R I )Ψ (cid:17) = − i Z T dt Z π dϕ Z ∞−∞ dz ( − ∂ ρ ( ρ Ψ ψ ∗ kkk ) + 2 ρψ ∗ kkk ( ∂ ρ Ψ) + ψ ∗ kkk Ψ) | ρ = R . (D4)Inserting the Volkov wavefunction ψ kkk ( ρ, z, t ) = (2 π ) − / e − iα ( t )+ ik z z + ik ρ ρ cos ϕ , α ( t ) = 12 Z t dt ′ [ k + 2 k z A ( t ′ )] (D5)and collecting all integrals over z in s ′ such that s ′ + s ′ + s ′ = s + s + s yields s ′ ( k ρ , k z ) = − i π ) − / e − ik z Z I Z T dt e iα ( t ) Z R I ρ dρJ ( k ρ ρ ) ( ik z Ψ + ∂ z Ψ + 2 iA Ψ) | z = Z I , (D6) s ′ ( k ρ , k z ) = i π ) − / e ik z Z I Z T dt e iα ( t ) Z R I ρ dρJ ( k ρ ρ ) ( ik z Ψ + ∂ z Ψ + 2 iA Ψ) | z = − Z I , (D7)and s ′ ( k ρ , k z ) = − i π ) − / R I Z T dt e iα ( t ) Z Z I − Z I dz e − ik z z (Ψ k ρ J ( k ρ R I ) + J ( k ρ R I ) ∂ ρ Ψ) | ρ = R I (D8)where J and J are Bessel functions of the first kind. Denoting the Fourier transform Ψ( ρ, k z , t ) = Z Z I − Z I dz e − ik z z Ψ( ρ, z, t ) (D9)and the Hankel transform Ψ( k ρ , z, t ) = Z R I dρ ρJ ( k ρ ρ )Ψ( ρ, z, t ) (D10)the approximation of the probability amplitude reads s + s + s = − i √ π Z T dt e iα ( t ) (cid:0) e − ik z Z I ( ik z + ∂ z + 2 iA ) Ψ( k ρ , z, t ) | z = Z I − e ik z Z I ( ik z + ∂ z + 2 iA ) Ψ( k ρ , z, t ) | z = − Z I + R I ( k ρ J ( k ρ R I ) + J ( k ρ R I ) ∂ ρ ) Ψ( ρ, k z , t ) | ρ = R I (cid:1) . (D11)The GSL [23] was used for the Hankel transform and the Bessel functions.In order to suppress spurious effects introduced by the finite time T in the t-SURFF time integrals a Hanning window H ( t ) = ( if t < T / − cos(2 πt/T )] / if t ≥ T / (D12)was multiplied to the integrands (B6), (C1), and (D11). [1] S. Chelkowski, T. Zuo, O. Atabek, and A. D. Bandrauk,Phys. Rev. A , 2977 (1995). [2] S. X. Hu, L. A. Collins, and B. I. Schneider,Phys. Rev. A , 023426 (2009). TABLE I. 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