A semi-classical approach for solving the time-dependent Schrödinger equation in spatially inhomogeneous electromagnetic pulses
AA semi-classical approach for solving the time-dependent Schrödinger equation inspatially inhomogeneous electromagnetic pulses
Jianxiong Li and Uwe Thumm
Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA (Dated: December 19, 2019)To solve the time-dependent Schrödinger equation in spatially inhomogeneous pulses of electro-magnetic radiation, we propose an iterative semi-classical complex trajectory approach. In numer-ical applications, we validate this method against ab initio numerical solutions by scrutinizing (a)electronic states in combined Coulomb and spatially homogeneous laser fields and (b) streaked pho-toemission from hydrogen atoms and plasmonic gold nanospheres. In comparison with streakedphotoemission calculations performed in strong-field approximation, we demonstrate the improvedreconstruction of the spatially inhomogeneous induced plasmonic infrared field near a nanoparticlesurface from streaked photoemission spectra.
I. INTRODUCTION
The exposure of gaseous atomic, mesoscopic, and solidtargets to incident pulses of electromagnetic radiation ofsufficiently high photon energy or intensity leads to theemission of photoelectrons [1]. For more than a cen-tury, photoelectron spectroscopy has very successfullyexploited this phenomenon and has long become estab-lished as one of the most prolific techniques for unravelingthe static electronic structure of matter by examining thekinetic-energy or momentum distribution of emitted pho-toelectrons. More recently, starting in the 21st century,advances in ultrafast laser technology started to extendphotoemission spectroscopy into the time domain [2–4]. Importantly, the development of attosecond streak-ing [5, 6] and interferometric [7–9] photoelectron spec-troscopy enabled the observation of electron dynamics atthe natural time scale of the electron motion in matter(attoseconds, as = 10 − s). This was demonstrated inproof-of-principle experiments for gaseous atomic [10–16]and molecular [17–19] targets. Attosecond time-resolvedphotoemission spectroscopy is currently being extendedto complex targets [6, 20], such as nanostructures andnanoparticles [21–28], and solid surfaces [9, 29–36], mak-ing it possible to examine, for example, the dynamicsof photoemission from a surface on an absolute timescale [37] and suggesting, for example, the time-resolvedobservation of the collective motion of electrons (plas-mons) in condensed-matter systems [38–40].In combination with advances in nanotechnology, al-lowing the production of plasmonic nanostructures withincreasing efficiency at the nm length scale, attosecondphotoemission spectroscopy has started to progress to-wards the spatiotemporal imaging of electron dynam-ics in complex targets, approaching the atomic lengthand time scales (nm and attoseconds) [20, 21, 25–28, 31, 41, 42]. Photoemission spectroscopy thereforeholds promise to become a powerful tool for examin-ing nm-attosecond scale processes that are operative inplasmonically enhanced photocatalysis [43], light har-vesting [44], surface-enhanced Raman spectroscopy [45],biomedical and chemical sensing [46], tumor detection and treatment [47], and ultrafast electro-optical switch-ing [48]. The concurrent development and provision oflarge-scale light sources, capable of producing intense ul-trashort pulses in the extreme ultraviolet (XUV) to X-ray spectral range at several leading laboratories in Eu-rope, the United Stated, and Japan [49], promises to fur-ther boost the value of spatiotemporally resolved elec-tron spectroscopy as a tool for imaging electronic dy-namics within a wide array of basic and applied researchprojects.Being able to take advantage of the full potential of-fered by current and emerging atomic scale photoelectronimaging techniques relies on theoretical and numericalmodeling. This is true for comparatively simple atomsin the gas phase, and for complex nanostructured tar-gets additional theoretical challenges arise [6, 20]. Whilefor atomic photoionization by visible and near UV light,the size of the target is small compared to the wave-length of the incident light pulse, this is no longer truefor X-ray ionization, leading to the well-know break-down of the dipole approximation [50]. Furthermore, fornanoparticles [22–28], (nanostructured) surfaces [36, 51–53], and layered structures [35, 36, 42, 54], not onlythe comparability of the wavelength and structure sizerequires careful quantum-mechanical modeling beyondthe dipole approximation, but also the target’s spatiallyinhomogeneous dielectric response to the incident lightpulse [52, 53]. Most numerical models for streaked andinterferometric photoemission from atoms are based onthe so-called ‘strong-field approximation (SFA)’ [6]. TheSFA builds on the assumption that photo-emitted elec-trons are solely exposed to spatially homogeneous exter-nal fields. It discards all other interactions photo-releasedelectrons may be subject to (e.g., with the residual parention) and cannot accommodate spatially inhomogeneousfinal-state interactions.While the SFA was shown to deteriorate for lower pho-toelectron energies [55], it completely looses its applica-bility for complex targets as screening and plasmonic ef-fects expose photoelectrons to inhomogeneous net electro-magnetic fields [6, 38, 52, 53]. The convenient use of an-alytically known so-called ‘Volkov wavefunctions’ for the a r X i v : . [ phy s i c s . a t m - c l u s ] N ov photoelectron’s motion in homogeneous electromagneticfields [56] is no longer acceptable, since dielectric responseeffects entail screening length and induced plasmonicfields at the nm length scale [22, 24–28, 36, 38]. Thus,the numerical modeling of photoemission from complextargets with morphologies or plasmonic response lengthsat the nm scale by intense short wavelength pulses (madeincreasingly available at new (X)FEL light sources [49]),necessitates photoemission models beyond the SFA.To this effect we previously employed heuristically gen-eralized Volkov states to model photoemission from bareand adsorbate-covered metal surfaces [35, 36, 52, 53]and plasmonic nanoparticles [22, 26, 28]. While thisallowed us to numerically model streaked [42, 52, 53,57] and interferometric photoemission spectra from sur-faces [35, 36], in fair to good agreement with experimen-tal data, and to reconstruct plasmonic fields near goldnanospheres [28], a systematic mathematical solution ofthe time-dependent Schrödinger (TDSE) for a single ac-tive electron exposed to inhomogeneous external fieldsremains to be explored. We here discuss a semiclassi-cal model for obtaining such solutions. While being ap-proximate, our complex-phase Wentzel-Kramer-Brillouin(WKB)-type approach lends itself to systematic itera-tive refinement. Our proposed method, termed ACC-TIVE (Action Calculation by Classical Trajectory Inte-gration in Varying Electromagnetic fields), employs com-plex classical trajectories to solve the TDSE in the pres-ence of spatially inhomogeneous electromagnetic pulsesthat are represented by time-dependent inhomogeneousscalar and vector potentials. Our approach is inspiredby the semiclassical complex-trajectory method for solv-ing the TDSE with time-independent scalar interactionsof Boiron and Lombardi [58] and its adaptation to time-dependent scalar interactions by Goldfarb, Schiff, andTannor [59].Following the mathematical formulation of ACCTIVEin Sec. II, we validate this method by discussing fiveexamples in Sec. III. We first compare ACCTIVE cal-culations with ab initio numerical solutions by scru-tinizing electronic states in a (i) homogeneous laserfield, (ii) Coulomb field, and (iii) combination of laserand Coulomb fields. Next, we apply ACCTIVE tostreaked photoemission from (iv) hydrogen atoms and(v) plasmonic nanoparticles. In the application to Aunanospheres, we examine final states for the simultaneousinteraction of the photoelectron with the spatially inho-mogeneous plasmonically enhanced field induced by thestreaking infrared (IR) laser pulse and demonstrate theimproved reconstruction of the induced nanoplasmonicIR field from streaked photoemission spectra. Section IVcontains our summary. In three appendices we adddetails of our calculations within ACCTIVE of Volkovwavefunctions (Appendix A) and Coulomb wavefunctions(Appendix B), and additional comments on streaked pho-toemission from Au nanospheres (Appendix C) withinACCTIVE. II. THEORY
We seek approximate solutions of the TDSE for a par-ticle of (effective) mass m and charge q in an inhomo-geneous time-dependent electro-magnetic field given bythe scalar and vector potentials φ ( r , t ) and A ( r , t ) andan additional scalar potential V ( r , t ) , i (cid:126) ∂∂t Ψ( r , t ) = (cid:40) m (cid:104) i (cid:126) ∇ + q A ( r , t ) (cid:105) + ϕ ( r , t ) (cid:41) Ψ( r , t ) , (1)where ϕ ( r , t ) = qφ ( r , t ) + V ( r , t ) and V ( r , t ) is any scalarpotential. Representing the wavefunction in eikonalform, Ψ( r , t ) = e iS ( r ,t ) / (cid:126) , Eq. (1) can be rewritten interms of the complex-valued quantum-mechanical action S ( r , t ) , ∂∂t S ( r , t ) + 12 m (cid:104) ∇ S ( r , t ) − q A ( r , t ) (cid:105) + ϕ ( r , t )= i (cid:126) m ∇ · (cid:104) ∇ S ( r , t ) − q A ( r , t ) (cid:105) . (2)Expanding the action in powers of (cid:126) [58, 59], S ( r , t ) = ∞ (cid:88) n =0 (cid:126) n S n ( r , t ) , (3)substituting Eq. (3) into Eq. (2), and comparing termsof equal order, results in the set of coupled partial differ-ential equations ∂∂t S ( r , t ) + (cid:2) ∇ S ( r , t ) − q A ( r , t ) (cid:3) m + ϕ ( r , t ) = 0 (4a) ∂∂t S ( r , t ) + (cid:20) ∇ S ( r , t ) − q A ( r , t ) m (cid:21) · ∇ S ( r , t )= i ∇ · (cid:20) ∇ S ( r , t ) − q A ( r , t ) m (cid:21) (4b) ∂∂t S n ( r , t ) + (cid:20) ∇ S ( r , t ) − q A ( r , t ) m (cid:21) · ∇ S n ( r , t )= − m n − (cid:88) j =1 ∇ S j ( r , t ) · ∇ S n − j ( r , t )+ i m ∇ S n − ( r , t ) ( n ≥ , (4c)where the lowest-order contribution S ( r , t ) is the classi-cal action of a charged particle moving in the electromag-netic field given by E ( r , t ) = −∇ ϕ ( r , t ) /q − ∂ A ( r , t ) /∂t and B ( r , t ) = ∇ × A ( r , t ) .Solving the classical Hamilton-Jacobi equation (HJE)Eq. (4a) leads to Newton’s Second Law, ddt v ( r , t ) = qm (cid:104) E ( r , t ) + v ( r , t ) × B ( r , t ) (cid:105) , (5)where the classical velocity field v ( r , t ) and kinetic mo-mentum, p ( r , t ) ≡ m v ( r , t ) ≡ ∇ S ( r , t ) − q A ( r , t ) , (6)are given in terms of the canonical momentum ∇ S ( r , t ) [60]. The combination of the HJE (4a) andEq. (6) provides the Lagrangian L (cid:2) r , v ( r , t ) , t (cid:3) as a totaltime differential of S ( r , t ) , ddt S ( r , t ) = L (cid:2) r , v ( r , t ) , t (cid:3) = 12 m v ( r , t ) + q v ( r , t ) · A ( r , t ) − ϕ ( r , t ) . (7)Similarly, by substituting Eq. (6) into Eqs. (4b) and(9), we find the total time derivatives of the first-ordercontribution to S ( r , t ) , ddt S ( r , t ) = i ∇ · v ( r , t ) , (8)and of all higher order terms, ddt S n ( r , t ) = − m n − (cid:88) j =1 ∇ S j ( r , t ) · ∇ S n − j ( r , t )+ i m ∇ S n − ( r , t ) ( n ≥ . (9)Approximate solutions to S ( r , t ) can be obtained by iter-ation of Eq. (9), after integrating the total time deriva-tives in Eqs. (7), (8), and (9) along classical trajectories ˜ r ( t ) that are defined by ddt ˜ r ( t ) ≡ v (cid:2) ˜ r ( t ) , t (cid:3) (10)with respect to a reference time (integration constant) t r .The wavefunction at t r , Ψ r ( r ) = Ψ( r , t r ) , provides initial( t r (cid:28) ) or asymptotic ( t r (cid:29) ) conditions in terms ofthe action S ( r , t r ) = − i (cid:126) ln[Ψ r ( r )] (11)and the velocity field v ( r , t r ) = − m ∇ S ( r , t r ) − qm A ( r , t r ) ≈ − m ∇ S ( r , t r ) − qm A ( r , t r )= − i (cid:126) ∇ Ψ r ( r ) m Ψ r ( r ) − qm A ( r , t r ) . (12)The semiclassical solution of Eqs. (7), (8), and (9) re-quires an appropriate classical trajectory ˜ r ( t (cid:48) ) - for anygiven ‘current’ event ( r , t ) - that connects the ‘current’coordinate and velocity, r = ˜ r ( t ) , v = d ˜ r ( t (cid:48) ) dt (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) t , (13)to the proper coordinate and velocity at t r , r r = ˜ r ( t r ) , (14a) v r = d ˜ r ( t (cid:48) ) dt (cid:48) t r = − i (cid:126) ∇ Ψ r ( r r ) m Ψ r ( r r ) − qm A ( r r , t r ) . (14b) Figure 1. (Color online) Illustration of the shooting methodused for determining classical trajectories. For any givenevent ( r , t ) and a predetermined reference time t r , trajecto-ries are classically propagated from trial points in phase space, ( r , v trial ) , at time t along trial trajectories ˜ r trial ( t (cid:48) ) . The ve-locity field v and appropriate trajectory ˜ r ( t (cid:48) ) are determinedby iterating the trial velocity v trial , in order to find the rootsof f ( v trial ) in Eq. (15). The known quantities in Eqs. (13) and (14) are r , t , and t r , while v , r r , and v r are to be determined. To numer-ically calculate the undetermined quantities, we employa shooting method, starting with a ‘trial’ velocity v trial at position r and time t . Propagating r to the referencetime according to Eq. (5) results in r trialr = ˜ r trial ( t r ) and v trialr = d ˜ r trial ( t (cid:48) ) /dt (cid:48) | t r (Fig. 1).The velocity field v that satisfies Eq. (5) can now befound numerically by minimizing the function f ( v trial ) = | v trialr + i (cid:126) ∇ Ψ r ( r trialr ) m Ψ r ( r trialr ) + qm A ( r trialr , t r ) | (15)for an appropriate range of start trial velocities. In ournumerical applications this is accomplished by an efficientmulti-dimensional quasi-Newton root-finding algorithm(Broyden’s method) [61, 62]. Once the correct trajecto-ries ˜ r ( t (cid:48) ) are determined by finding the roots of Eq. (15),the actions in Eqs. (7), (8), and (9) are integrated alongthese trajectories and composed - by truncating Eq. (3)- into an approximate solution of Eq. (1).Since each term S n ( r , t ) in Eq. (3) depends only onterms of lower orders, ACCTIVE enables, in principle,the systematic iterative refinement of approximate solu-tions of Eq. (1) by including successively higher orders n .The iteration is started with S ( r , t ) , which is determinedby the velocity field v ( r , t ) , and continued by integratingEqs. (8) and (9).In the numerical examples discussed in Sec. III below,we find that retaining only the zero’th and first-orderterms, S ( r , t ) and S ( r , t ) , provides sufficiently accurateand physically meaningful solutions at modest numericalexpense. Thus, according to Eqs. (7) and (8), we apply Ψ( r , t ) ≈ exp (cid:8) iS ( r , t ) / (cid:126) + iS ( r , t ) (cid:9) = e iS ( r r ,t r ) / (cid:126) exp (cid:40) − (cid:90) tt r ∇ · v (cid:16) ˜ r ( t (cid:48) ) , t (cid:48) (cid:17) dt (cid:48) + i (cid:126) (cid:90) tt r L (cid:20) ˜ r ( t (cid:48) ) , v (cid:16) ˜ r ( t (cid:48) ) , t (cid:48) (cid:17) , t (cid:48) (cid:21) dt (cid:48) (cid:41) . (16)For real classical trajectories and potentials, the integralof S ( r , t ) is real, representing a local phase factor, while S ( r , t ) is purely imaginary and defines the wavefunctionamplitude, as in the standard WKB approach [50]. Thequantum-mechanical probability density ρ ( r , t ) then sat-isfies the continuity equation, dρ ( r , t ) dt = ddt | Ψ( r , t ) | = − ρ ( r , t ) ∇ · v ( r , t ) , (17)for the classical probability flux ρ ( r , t ) v ( r , t ) [63]. III. EXAMPLES
We validate the ACCTIVE method by discussing fiveapplications to electron wavefunctions in Coulomb andlaser fields.
A. Volkov wavefunction
For the simple example of an electron in a time-dependent, spatially homogeneous laser field, the poten-tials in Eq. (1) and reference wavefunction are (in theCoulomb electromagnetic gauge [50]) A ( r , t ) = A ( t ) , ϕ ( r , t ) = 0 , Ψ r ( r ) = e i p · r / (cid:126) , (18)and the first-order wavefunction in Eq. (16) reproducesthe well-known analytical Volkov solution [56], Ψ V ( r , t ) = exp (cid:26) i p · r (cid:126) − i m (cid:126) (cid:90) tt r (cid:2) p − q A ( t (cid:48) ) (cid:3) dt (cid:48) (cid:27) . (19)For details of the derivation of Eq. (19) within ACC-TIVE see Appendix A. B. Coulomb wavefunction
As a second simple example and limiting case, we con-sider an unbound electron in the Coulomb field of a pro-ton. In this case the potentials in Eq. (1) are A ( r , t ) = 0 , ϕ ( r , t ) = − k e e r , (20)where e is the elementary charge and k e the electrostaticconstant. Assuming outgoing-wave boundary conditions, −2 0 2−4−2024 z [ Å ] (a) ACCTIVE −2 0 2x [Å] (b) Anal tical 0Real wave function[arb. units](c) ACCTIVEAnal tical
Figure 2. (Color online) Real part of an unbound Coulombwavefunction , subject to the boundary condition given by anoutgoing wave propagating along the z -axis. (a) Numericallycalculated semi-classical 1st order ACCTIVE wavefunction.(b) Analytical Coulomb wavefunction in the y = 0 plane. (c)Real part of the wavefunctions in (a) and (b) along the z -axisfor x = y = 0 . we define the reference wavefunction at a sufficiently largereference time t r as the ‘outgoing’ Coulomb wave Ψ r ( r , t r ) t r →∞ , z → + ∞ −−−−−−−−−−−→ e i (cid:16) kz − (cid:126) k m t r (cid:17) . (21)Here r = ( x, y, z ) and p = (cid:126) k > is the final electronmomentum. In this case the TDSE is solved exactly bythe well-known Coulomb wavefunction Ψ Ck ( r , t ) = e π k Γ(1 − i/k )(2 π ) / F ( i/k, , ikr − ikz ) e i (cid:16) kz − (cid:126) k m t (cid:17) (22)in terms of the confluent hypergeometric function F .Note that for finite distances from the z -axis (i.e., forfinite coordinates x and y ), the asymptotic form of theCoulomb continuum wavefunction for z → + ∞ is just aplane wave (without a logarithmic phase term) [50, 64].Applying ACCTIVE to the outgoing-wave Coulombproblem, t r must be chosen sufficiently long after t , sothat each classical trajectory ˜ r ( t (cid:48) ) propagates far enoughtowards the z → + ∞ asymptotic limit for the referencevelocity to become v r t r →∞ , z → + ∞ −−−−−−−−−−−→ ˆ z p/m, (23)in compliance with Eq. (14b). In this and for the fol-lowing numerical example, we use as reference velocitythe initial trial velocity for points of the spatial numeri-cal grid that are sufficiently far away from the Coulombsingularity at the origin. The correct ‘current’ velocities, v ( r , t ) at the most distant coordinates are subsequentlyused as trial velocities at the nearest neighbor spatialgrid points. This scheme is continued until classical tra-jectories for the entire spatiotemporal numerical grid arecalculated. Further details of the numerical calculationof Coulomb wavefunctions within ACCTIVE are given inAppendix B.Figure 2 shows the very good agreement between thenumerically calculated 1st order ACCTIVE wavefunc-tion (16) and the analytical Coulomb wavefunction (22)for a final electron kinetic energy of p / m = 50 eV. Thecolor/gray scale represents the real part of the wavefunc-tion in the x − z plane. Figures 2(a) and 2(b) show thesame scattering pattern. Good quantitative agreement ofthe 1st order ACCTIVE wavefunction and the analyticalCoulomb wavefunction is demonstrated in Fig. 2(c). C. Coulomb-Volkov wavefunction
A more challenging third example is given by the mo-tion of an electron under the combined influence of apoint charge (proton), located at the coordinate origin,and a spatially homogeneous laser pulse, subject to theboundary condition Eq. (21). In this case, the potentialsin Eq. (1) are (in Coulomb gauge [50]) A ( r , t ) = A ( t ) , ϕ ( r , t ) = − k e e r . (24)Considering a laser pulse of finite duration, t r mustbe chosen such that the laser electric field vanishes at t r . This combination of the two previous examples inSecs. III A and III B constitutes the Coulomb-Volkovproblem, for which merely approximate solutions [65–68],but no analytical wavefunction are known. We assumea laser pulse with 15 eV central photon energy, a cosine-square temporal intensity envelope with a pulse length of0.5 fs full width at half intensity maximum (FWHIM),and × W/cm peak intensity. At time t = 0 , thetemporal pulse profile is centered at z = 0 . We enforcethe outgoing-wave boundary condition (21) for an asymp-totic photoelectron kinetic energy of p / m = 50 eV.This energy is reached at a sufficiently large distance ofthe outgoing electron from the proton and long after thepulse has vanished.In Fig. 3 we compare the ACCTIVE-calculatedCoulomb-Volkov wavefunction with Coulomb and Volkovwavefunctions for identical outgoing-wave boundary con-dition and 50 eV asymptotic photoelectron kinetic en-ergy. The Coulomb and Volkov wavefunctions are givenfor a positive elementary charge and the same laser pa-rameters as the Coulomb-Volkov wave, respectively. Thecolor/gray scale represents the real part of the wavefunc-tions. We determined all numerical parameters (numer-ical grid size, spacing and propagation time step) to en-sure convergence of the wavefunctions.Figures 3(a), 3(b), and 3(c), display snapshots at time t = 0 of the Coulomb, ACCTIVE-calculated Coulomb-Volkov, and Volkov wavefunctions, respectively. TheCoulomb-Volkov wavefunction shows a similar (inverse)Coulomb scattering pattern for the incident wave ( z < )as the Coulomb wave. Its outgoing part ( z > ) closelymatches the phase of the Volkov wave. On the otherhand, the time-dependent evolution of the Coulomb-Volkov wavefunction in the y = 0 plane in Fig. 3(e)shows laser-induced wavefront distortions - similar to theVolkov wave in Fig. 3(f). The time evolution of the −2 0 2−4−2024 z [ Å ] (a) Coulomb −2 0 2x [Å](b)
Coulomb-Volkov −2 0 2(c)
Volkov −3 0 3−4−2024 z [ Å ] (d) −3 0 3t [10 −1 fs](e) −3 0 3(f) Figure 3. (Color online) Real parts of (a,d) Coulomb,(b,e) ACCTIVE-calculated Coulomb-Volkov, and (c,f) Volkovwavefunctions in the y = 0 plane. (a-c) Snapshots at time t = 0 , when the laser-pulse center is at z = 0 . (d-f) Timeevolution along the z -axis. ACCTIVE-calculated Coulomb-Volkov wavefunction re-veals the acceleration of the incoming and decelerationof the outgoing wave near the proton at z = 0 of thepure Coulomb wave in Fig. 3(d). An animated version ofthis wavefunction comparison can be found in the Sup-plemental Material [69]. D. Streaked photoemission from hydrogen atoms
As a fourth example, we employ ACCTIVE final-statewavefunctions to calculate IR-streaked XUV photoelec-tron spectra from ground-state hydrogen atoms [6]. Weassume the ionizing XUV and streaking IR pulse as lin-early polarized along the z axis. The relative time delaybetween the centers of the two pulses, τ , is assumed pos-itive in case the IR precedes XUV pulse. The electricfield E X ( t ) of the XUV pulse is characterized by a Gaus-sian temporal profile, 55 eV central photon energy, anda pulse length of 200 as (FWHIM). The IR pulse hasa cosine-squared temporal profile, 720 nm central wave-length, pulse duration of 2 fs FWHIM, and W/cm peak intensity.We model streaked photoemission from the groundstate of hydrogen, | Ψ i (cid:105) , based on the quantum-mechanical transition amplitude [6, 28, 50, 53] T ( k f , τ ) ∼ (cid:90) dt (cid:10) Ψ C − V k f ,τ (cid:12)(cid:12) zE X ( t ) (cid:12)(cid:12) Ψ i (cid:11) , (25)where the IR-pulse-dressed final state of the photoelec-tron, (cid:12)(cid:12) Ψ C − V k f ,τ (cid:11) , is a Coulomb-Volkov wavefunction [55]that we evaluate numerically using the ACCTIVEmethod. In a comparison calculation, we replace theCoulomb-Volkov state by the Volkov state (cid:12)(cid:12) Ψ V k f ,τ (cid:11) andassume otherwise identical physical conditions. As men-tioned in the Introduction, the use of Volkov states [56]in photoionization calculations is referred to as SFA [6]and amounts to neglecting the interaction of the releasedphotoelectron with the residual ion (proton in the presentcase). We scrutinize streaked photoemission spectra ob-tained with ACCTIVE-calculated Coulomb-Volkov finalstates and in SFA against ab initio bench-mark calcula-tions. In these exact numerical calculations we directlysolve the three-dimensional TDSE using the SCID-TDSEtime-propagation code [70].Numerical results are shown in Fig. 4. The streakedphotoemission spectra obtained with ACCTIVE-calculated Coulomb-Volkov final states [Fig. 4(a)],in SFA [Fig. 4(b)], and by direct numerical solutionof the TDSE [Fig. 4(c)] show very similar ‘streakingtraces’, i.e., oscillations of the asymptotic photoelectronenergy with delay τ . For a quantitative comparison, weplot in Fig. 4(d) the centers of energy (CoEs) of thespectra in Figs. 4(a-c). While the three calculationsresult in identical photoemission phase shifts (streakingtime delays) relative to the streaking IR field, withinthe resolution of the graph, the ACCTIVE-calculatedspectra agree with the exact TDSE calculation, whilethe SFA calculation predicts noticeably smaller CoEsdue to the neglect of the Coulomb potential in the finalphotoelectron state [22]. E. Streaked photoemission from metal nanospheres
As a final, fifth, example, we apply the ACCTIVEmethod to model photoelectron states in spatially inho-mogeneous , plasmonically enhanced IR electromagneticfields. For this purpose, we investigate streaked photo-emission [22, 25–27] and the reconstruction of plasmonicnear-fields [28] for gold nanospheres with a radius of R = 50 nm. We represent the electronic structure ofthe nanosphere in terms of eigenstates of a square wellwith a potential depth of V = − . eV and obtain thephotoelectron yield by incoherently adding the transitionamplitudes (25) over all occupied initial conduction-bandstates [6, 53, 57]. For the calculation of the transition am-plitude (25) we closely follow Ref. [26], with the impor-tant difference of employing numerically calculated semi-classical ACCTIVE final photoelectron wavefunctions,while in Ref. [26] the SFA approximation is used, ap-plying heuristically generalized Volkov final states andthus neglecting of the photoelectron interactions with theresidual nanoparticle.For the ACCTIVE calculation we thus solve theTDSE (1) with the potentials A ( r , t ) = (cid:90) ∞ t E tot ( r , t (cid:48) ) dt (cid:48) (26a) ϕ ( r , t ) = (cid:40) V r < R r ≥ R , (26b) P h o t o e l e c t ( o n e n e ( g y [ e V ] (b) SFA35404550 (c) TDSE−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0XUV-IR time delay [fs]39404142 C o E [ e V ] (d) Center of energyACCTIVE SFA TDSE Figure 4. (Color online) IR-streaked XUV photoelectron spec-tra, (a) based on ACCTIVE-calculated Coulomb-Volkov finalstates, (b) in SFA, and (c) obtained by direct numerical so-lution of the TDSE. Red dotted lines in (a-c) indicate therespective centers of energy (CoE). The spectral yields in (a-c) are normalized separately, to their respective maxima. (d)Comparison of the delay-dependent CoE for the spectra in(a-c). and the boundary condition Eq. (21). Here, the asymp-totic wavefunction in Eq. (21) also serves as referencewavefunction for the classical trajectory computation.The net time-dependent inhomogeneous field E tot ( r , t ) isgiven by the superposition of the homogeneous IR fieldof the incident streaking pulse and the inhomogeneousplasmonic field produced by the nanoparticle in responseto the incident IR pulse [27, 28]. For the streaking calcu-lation, we assume an XUV pulse with 30 eV central pho-ton energy and Gaussian temporal profile with a width of200 as (FWHIM). We further suppose a delayed Gaus-sian IR pulse with 720 nm central wavelength, 2.47 fs(FWHIM) pulse length, and × W/cm peak inten-sity.Figure 5 shows simulated streaked photoelectron spec-tra obtained with ACCTIVE-calculated and Volkov fi-nal states for electron emission along the XUV-pulsepolarization direction. In this direction, the effect ofthe induced plasmonic field on the photoelectron isstrongest [28]. The corresponding spectra in Figs. 5(a)and 5(b) show very similar temporal oscillations of thephotoelectron yield and CoE as a function of both,asymptotic photoelectron energy and XUV-IR pulse de-lay τ . As for streaked photoemission from hydrogenatoms discussed in Sec III D above, we find that the SFAshifts the CoE to lower kinetic energies [Fig. 5(b), cf. P h o t o e l e c t r o n e n e r g y [ e V ] (b) SFA XUX-IR time delay [fs]17192123 C o E [ e V ] (c) Center of energy−4 −3 −2 −1 0 1 2 3 4XUV-IR time delay [fs]−2−1012 E l e c t r i c f i e l s [ − V / Å ] (d) Reconstructed electric field strengthACCTIVESFAMie simulated -0.2 0 0.2−2.7−2.3 Figure 5. (Color online) Simulated IR-streaked XUV photo-electron spectra for photoemission along the XUV-pulse po-larization direction (a) using ACCTIVE final-states and (b)in SFA. (c) Corresponding delay-dependent centers of energy.(d) Comparison of the corresponding reconstructed plasmonicelectric near-fields at the point ( x, y, z ) = (0 , , R ) on thenanoparticle surface with the Mie-theory-calculated electricfield. Fig. 4(d)]. Here, the SFA results in an approximately1.5 eV lower CoE than the ACCTIVE calculation. Thisenergy shift is due to the fact that the SFA, by neglect-ing the potential well of the nanosphere in the final pho-toelectron state, leads to an unphysical enhancement ofthe photoemission cross section at lower photoelectronkinetic energies, thereby increasing the weight of low en-ergy yields in the CoE average [22].Addition comments on the comparison of streaked pho-toelectron spectra within either ACCTIVE or based onVolkov wavefunctions can be found in Appendix C.From streaked photoemission spectra the plasmonicnear-field at the nanoparticle surface can be recon-structed as detailed in Refs. [27, 28]. Figure 5(d) showsthe reconstructed net electric field E tot along the XUV-pulse polarization direction, i.e., at the surface and onthe positive z axis, of the nanosphere. The reconstruc-tion of net plasmonically enhanced near-fields from thesimulated spectra in Figs. 5(a) and 5(b) was performed according to the scheme proposed in Ref. [28]. The ob-tained reconstructed fields are compared in Fig. 5(d) withthe net electric IR near-field obtained within Mie the-ory [71] and used as input in the streaking calculations.As is seen in Fig. 5(d), the ACCTIVE method improvesthe near-field reconstruction in comparison with the SFAcalculation. The least-square deviation between the re-constructed and Mie-theory calculated fields, assembledover the entire IR pulse length, amounts to 1.62% us-ing the ACCTIVE wavefunction and 3.05% using theSFA. A comparative animation of reconstructed and an-alytical electric fields at the surface of Au nanospherescan be found in the Supplemental Material [69]. TheACCTIVE method thus extends the applicability of theplasmonic near-field reconstruction scheme in Ref. [28] tolower XUV photon energies. IV. SUMMARY
In summary, we propose a semi-classical method, AC-CTIVE, to solve the TDSE for one active electron ex-posed to any spatially inhomogeneous time-dependentexternal force field. We validate this method by compar-ing ACCTIVE-calculated electronic wavefunctions withknown Coulomb and Volkov wavefunctions for the elec-tronic dynamics in Coulomb and intense laser fields,respectively, and by scrutinizing ACCTIVE-calculatedCoulomb-Volkov final photoelectron wavefunctions (i)against ab initio numerical solutions of the TDSE and(ii) in streaked photoemission from hydrogen atoms andplasmonic metal nanospheres.For streaked photoemission from hydrogen atoms, wedemonstrate excellent agreement of our ACCTIVE cal-culation with a benchmark ab initio
TDSE calculation,while a comparative calculation using the SFA system-atically deviates from the exact TDSE solution. Forstreaked photoemission from Au nanospheres we findthat ACCTIVE final-state wavefunctions improve the re-construction of plasmonic near-fields over SFA calcula-tions (based on Volkov final states) at comparatively lowphotoelectron energies.
Appendix A: Derivation of Eq. (19)
We here derive the Volkov wavefunction Eq. (19) usingACCTIVE. Starting from the potentials and initial wave-function in Eq. (18), the velocity field along the classicaltrajectory ˜ r ( t ) is v ( r , t ) = p m + qm (cid:90) tt E ( t (cid:48) ) dt (cid:48) = p − q A ( t ) m . (A1)Therefore, ˜ r ( t ) = r + (cid:90) tt (cid:20) p − q A ( t ) m (cid:21) dt (cid:48) , (A2) ∇ · v ( r , t ) = 0 , (A3)and Eq. (16), applied to the example in Sec. III A, be-comes Ψ( r , t ) = exp (cid:40) i p · r (cid:126) + i (cid:126) (cid:90) tt (cid:20) m (cid:18) p − q A ( t (cid:48) ) m (cid:19) + q (cid:18) p − q A ( t (cid:48) ) m (cid:19) · A ( t (cid:48) ) (cid:21) dt (cid:48) (cid:41) = exp (cid:40) i p (cid:126) · (cid:20) r − (cid:90) tt (cid:18) p − q A ( t ) m (cid:19) dt (cid:48) (cid:21) + i (cid:126) (cid:90) tt (cid:20) m (cid:18) p − q A ( t (cid:48) ) m (cid:19) + q (cid:18) p − q A ( t (cid:48) ) m (cid:19) · A ( t (cid:48) ) (cid:21) dt (cid:48) (cid:41) = exp (cid:40) i p · r (cid:126) + i (cid:126) (cid:90) tt (cid:20) m (cid:18) p − q A ( t (cid:48) ) m (cid:19) − m (cid:18) p − q A ( t (cid:48) ) m (cid:19) (cid:21) dt (cid:48) (cid:41) = exp (cid:26) i p · r (cid:126) − i m (cid:126) (cid:90) tt (cid:2) p − q A ( t (cid:48) ) (cid:3) dt (cid:48) (cid:27) , (A4)which is the Volkov wavefunction Eq. (19). Appendix B: Numerical calculation of Coulombwavefunctions using ACCTIVE
The ACCTIVE method links a quantum-mechanicalproblem of obtaining wavefunctions Ψ( r , t ) to a classicalproblem of determining velocity fields v ( r , t ) . However,in some cases, e.g., for Coulomb wavefunctions, such ve-locity fields are not uniquely defined (Fig. 6). This canresult in interference patterns in the obtained wavefunc-tions, as pointed out by Goldfarb et al. [59].For each event ( r , t ) , two possible classical trajectoriescan be found to satisfy the same boundary condition ofan outgoing plane wave in Eq. (21), as shown in Fig. 6.Goldfarb et al. [59] take this interference into account byapproximating the wavefunction as the superposition ofcontributions from different trajectories, Ψ( r , t ) ≈ (cid:88) l exp (cid:104) i (cid:126) S l (cid:0) ˜ r ( t ) , t (cid:1)(cid:105) , (B1)where each action S l ( r , t ) is associated with a trajectory ˜ r l ( t ) . In this work, we follow a different and simplerapproach.The TDSE is a linear partial differential equation. Itssolution can be expressed as the superposition of a set of −6 −4 −2 0 2 4 6x [a.u.]−505101520 z [ a . u . ] (r,t)+e Traj. 1Traj. 2 Figure 6. (Color online) Two possible classical trajectoriespassing through ( r , t ) satisfying the same outgoing plane waveboundary condition. linearly independent basis functions Ψ l ( r , t ) , Ψ( r , t ) = (cid:88) l C l Ψ l ( r , t ) = (cid:88) l C l exp (cid:104) i (cid:126) S l (cid:0) r , t (cid:1)(cid:105) , (B2)where each S l ( r , t ) is uniquely determined by a velocityfield v l ( r , t ) and the coefficients C l are obtained from theinitial condition, Ψ ( r ) = (cid:88) l C l Ψ l ( r , t ) . (B3)Since two possible trajectories can be obtained for eachgiven event ( r , t ) , we can find two velocity fields, v + ( r , t ) and v − ( r , t ) , which are defined by v + ( r , t ) z → + ∞ , x> −−−−−−−−−→ ˆ z p/m (B4a) v − ( r , t ) z → + ∞ , x< −−−−−−−−−→ ˆ z p/m, (B4b)as illustrated in Fig. 7(a) and 7(b), respectively. Fig-ures 7(c) and 7(d) show the calculated 1st-order ACC-TIVE wavefunctions, Ψ + ( r , t ) and Ψ − ( r , t ) , associatedwith these two velocity fields at t = 0 . Numerical calcu-lation shows that, Ψ + ( r , t ) z → + ∞ −−−−−−→ (cid:40) e ikz x > x < (B5a) Ψ − ( r , t ) z → + ∞ −−−−−−→ (cid:40) e ikz x < x > . (B5b)Therefore, at t , Ψ ( r ) = Ψ( r , can be written as thelinear combination of Ψ + ( r , t ) and Ψ − ( r , t ) and satis-fies the boundary condition (B4), Ψ ( r ) = Ψ + ( r , t ) + Ψ − ( r , t ) . (B6) −5 0 5−505 z [ a . u . ] (a) −5 0 5 (b)−5 0 5−505 z [ a . u . ] (c) −5 0 5 (d)−5 0 5x [a.u.]−505 z [ a . u . ] (e) Figure 7. (Color online) Two possible velocity fields (a) v + ( r , t ) , and (b) v − ( r , t ) . (c) Ψ + ( r , t ) and (d) Ψ − ( r , t ) arethe real parts of the corresponding 1st order ACCTIVE wave-functions at y = 0 plane, respectively, and (d) Ψ( r , t ) is thelinear combination of these two wavefunctions. The wavefunction at any given time t is then obtainedwith the same coefficients, Ψ( r , t ) = Ψ + ( r , t ) + Ψ − ( r , t ) , (B7)as shown in Fig. 7 (e). Appendix C: Comments on streaked photoemissionfrom Au nanospheres
Figure 5 in the main text shows the comparison of sim-ulated streaked photoelectron spectra using either ACC-TIVE wavefunctions as final states or Volkov wavefunc-tion in SFA. ACCTIVE wavefunctions are more accurateat low photoelectron energy, but entail higher CoEs thanVolkov wavefunctions [Fig. 5(c)]. In comparison withFig. 4(d), this might appear as counter-intuitive. An ex-planation is given below.Figure 8(a) shows the real part of the 1st-order AC-CTIVE wavefunction near the Au nanosphere surface, R e a l w a v e f un c t i o n s [ a r b . un i t s ] (a) 1st order ACCTIVE state(b) SFA final state(c) Initial state (at Fermi level)5 10 15 20 25 30 35Photoelectron energy [eV]10 σ ( f ) [ a r b . un i t s ] (d) ACCTIVESFA Figure 8. (Color online) Real parts of photoelectron final-state wavefunctions near the surface of Au nanospheres alongthe XUV polarization direction: (a) 1st order ACCTIVEwavefunction and (b) SFA modeled wavefunction in Ref. [28],for the electron detection along the XUV polarization direc-tion and asymptotic photoelectron energy E f = 5 eV. (c) Ini-tial state wavefunction, modeled as bound state in a sphericalsquare well potential, at the Fermi level. The vertical dashedline indicates the nanosphere surface. (d) Simulated XUVphotoemission cross sections. and Fig. 8(b) the corresponding Volkov wavefunction inSFA [28]. Both are calculated for photoelectron detec-tion along the XUV polarization direction and outgoingphotoelectron energy E f = 5 eV. Inside the nanosphere,the Volkov final-state wavefunction neglects the sphericalwell potential. It therefore has a longer wavelength thanthe ACCTIVE wavefunction and more strongly overlapswith the initial-state wavefunction shown in Fig. 8(c).Thus, the cross section, calculated following Ref. [50], islarger in SFA than if based on ACCTIVE final states.This effect becomes less significant a larger photoelec-tron kinetic energies, where both, ACCTIVE and SFAwavefunctions have shorter wavelengths and overlap lesswith initial-state wavefunction. Figure 8(d) shows thatthe energy-dependent photoemission cross sections calcu-lated with ACCTIVE and Volkov final states converge atlarge photoelectron energies, while at small energies theSFA leads to larger cross sections. The net effect of thiscross-section difference is to put more weight on photo-electron yields at lower energy and thus to shift streakingtraces and CoEs in SFA photoemission spectra to lowerenergies as compared to ACCTIVE-calculated spectra.0 ACKNOWLEDGEMENT
This work was supported in part by the Chemical Sci-ences, Geosciences, and Biosciences Division, Office ofBasic Energy Sciences, Office of Science, US Department of Energy under Award DEFG02-86ER13491 (attosecondinterferometry), NSF grant no. PHY 1802085 (theory ofphotoemission from surfaces), and the Air Force Office ofScientific Research award no. FA9550-17-1-0369 (recolli-sion physics at the nanoscale). [1] S. Hüfner,
Photoelectron Spectroscopy. Principles andApplications (Springer, Berlin, 2003).[2] M. Hentschel, R. Kienberger, C. Spielmann, G. A. Rei-der, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann,M. Drescher, and F. Krausz, Nature , 509 (2001).[3] Z. Chang, Phys. Rev. A , 043802 (2004).[4] G. Sansone, E. Benedetti, F. Calegari, C. Vozzi,L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Al-tucci, R. Velotta, S. Stagira, S. De Silvestri, andM. Nisoli, Science , 443 (2006).[5] F. Krausz and M. Ivanov, Rev. Mod. Phys. , 163(2009).[6] U. Thumm, Q. Liao, E. M. Bothschafter, F. Süßmann,M. F. Kling, and R. Kienberger, in The Oxford Handbookof Innovation , edited by D. Andrew (Wiley, New York,2015) Chap. 13.[7] P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé,P. Balcou, H. G. Muller, and P. Agostini, Science ,1689 (2001).[8] K. Klünder, J. M. Dahlström, M. Gisselbrecht,T. Fordell, M. Swoboda, D. Guénot, P. Johnsson,J. Caillat, J. Mauritsson, A. Maquet, R. Taïeb, andA. L’Huillier, Phys. Rev. Lett. , 143002 (2011).[9] R. Locher, L. Castiglioni, M. Lucchini, M. Greif, L. Gall-mann, J. Osterwalder, M. Hengsberger, and U. Keller,Optica , 405 (2015).[10] M. Drescher, M. Hentschel, R. Kienberger, M. Uib-eracker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh,U. Kleineberg, U. Heinzmann, and F. Krausz, Nature , 803 (2002).[11] R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Bal-tuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher,and F. Krausz, Nature , 817 (2004).[12] P. Johnsson, J. Mauritsson, T. Remetter, A. L’Huillier,and K. J. Schafer, Phys. Rev. Lett. , 233001 (2007).[13] H. Wang, M. Chini, S. Chen, C.-H. Zhang, F. He,Y. Cheng, Y. Wu, U. Thumm, and Z. Chang, Phys.Rev. Lett. , 143002 (2010).[14] M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon,M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri,Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pa-zourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer,R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Gouliel-makis, F. Krausz, and V. S. Yakovlev, Science , 1658(2010).[15] C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Ev-ers, C. H. Keitel, C. H. Greene, and T. Pfeifer, Science , 716 (2013).[16] B. Bernhardt, A. R. Beck, X. Li, E. R. Warrick, M. J.Bell, D. J. Haxton, C. W. McCurdy, D. M. Neumark,and S. R. Leone, Phys. Rev. A , 023408 (2014).[17] H. Niikura, F. Légaré, R. Hasbani, M. Y. Ivanov, D. M.Villeneuve, and P. B. Corkum, Nature , 826 (2003). [18] M. F. Kling, C. Siedschlag, A. J. Verhoef, J. I.Khan, M. Schultze, T. Uphues, Y. Ni, M. Uiberacker,M. Drescher, F. Krausz, and M. J. J. Vrakking, Science , 246 (2006).[19] A. Staudte, D. Pavičić, S. Chelkowski, D. Zeidler,M. Meckel, H. Niikura, M. Schöffler, S. Schössler, B. Ul-rich, P. P. Rajeev, T. Weber, T. Jahnke, D. M. Vil-leneuve, A. D. Bandrauk, C. L. Cocke, P. B. Corkum,and R. Dörner, Phys. Rev. Lett. , 073003 (2007).[20] S. R. Leone, C. W. McCurdy, J. Burgdörfer, L. S. Ceder-baum, Z. Chang, N. Dudovich, J. Feist, C. H. Greene,M. Ivanov, R. Kienberger, U. Keller, M. F. Kling, Z.-H. Loh, T. Pfeifer, A. N. Pfeiffer, R. Santra, K. Schafer,A. Stolow, U. Thumm, and M. J. J. Vrakking, Nat. Pho-ton. , 162 (2014).[21] B. Förg, J. Schötz, F. Süßmann, M. Förster, M. Krüger,B. Ahn, W. A. Okell, K. Wintersperger, S. Zherebtsov,A. Guggenmos, V. Pervak, A. Kessel, S. A. Trushin,A. M. Azzeer, M. I. Stockman, D. Kim, F. Krausz,P. Hommelhoff, and M. F. Kling, Nat. Commun. , 11717(2016).[22] J. Li, E. Saydanzad, and U. Thumm, Phys. Rev. A ,051401 (2016).[23] L. Seiffert, Q. Liu, S. Zherebtsov, A. Trabattoni, P. Rupp,M. C. Castrovilli, M. Galli, F. Süßmann, K. Winter-sperger, J. Stierle, G. Sansone, L. Poletto, F. Frassetto,I. Halfpap, V. Mondes, C. Graf, E. Rühl, F. Krausz,M. Nisoli, T. Fennel, F. Calegari, and M. Kling, Nat.Phys. , 766 (2017).[24] J. Schötz, B. Förg, M. Förster, W. A. Okell, M. I. Stock-man, F. Krausz, P. Hommelhoff, and M. F. Kling, IEEEJournal of Selected Topics in Quantum Electronics ,77 (2017).[25] E. Saydanzad, J. Li, and U. Thumm, Phys. Rev. A ,053406 (2017).[26] J. Li, E. Saydanzad, and U. Thumm, Phys. Rev. A ,043423 (2017).[27] E. Saydanzad, J. Li, and U. Thumm, Phys. Rev. A ,063422 (2018).[28] J. Li, E. Saydanzad, and U. Thumm, Phys. Rev. Lett. , 223903 (2018).[29] M. Lucchini, A. Ludwig, L. Kasmi, L. Gallmann, andU. Keller, Opt. Express , 8867 (2015).[30] C. Chen, Z. Tao, A. Carr, P. Matyba, T. Szilvási,S. Emmerich, M. Piecuch, M. Keller, D. Zusin,S. Eich, M. Rollinger, W. You, S. Mathias, U. Thumm,M. Mavrikakis, M. Aeschlimann, P. M. Oppeneer,H. Kapteyn, and M. Murnane, Proc. Natl. Acad. Sci.USA , E5300 (2017).[31] S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell,G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst,M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel,J. Burgdörfer, P. Feulner, F. Krausz, and R. Kienberger,Nature , 342 (2015). [32] Z. Tao, C. Chen, T. Szilvási, M. Keller, M. Mavrikakis,H. Kapteyn, and M. Murnane, Science , 62 (2016).[33] F. Siek, S. Neb, P. Bartz, M. Hensen, C. Strüber,S. Fiechter, M. Torrent-Sucarrat, V. M. Silkin, E. E.Krasovskii, N. M. Kabachnik, S. Fritzsche, R. D. Muiño,P. M. Echenique, A. K. Kazansky, N. Müller, W. Pfeiffer,and U. Heinzmann, Science , 1274 (2017).[34] L. Kasmi, M. Lucchini, L. Castiglioni, P. Kliuiev, J. Os-terwalder, M. Hengsberger, L. Gallmann, P. Krüger, andU. Keller, Optica , 1492 (2017).[35] M. J. Ambrosio and U. Thumm, Phys. Rev. A , 043431(2018).[36] M. J. Ambrosio and U. Thumm, Phys. Rev. A ,043412 (2019).[37] M. Ossiander, J. Riemensberger, S. Neppl, M. Mitter-mair, M. Schäffer, A. Duensing, M. S. Wagner, R. Heider,M. Wurzer, M. Gerl, M. Schnitzenbaumer, J. V. Barth,F. Libisch, C. Lemell, J. Burgdörfer, P. Feulner, andR. Kienberger, Nature , 374 (2018).[38] C.-H. Zhang and U. Thumm, Phys. Rev. A , 063403(2011).[39] S. H. Chew, F. Süßmann, C. Späth, A. Wirth, J. Schmidt,S. Zherebtsov, A. Guggenmos, A. Oelsner, N. Weber,J. Kapaldo, A. Gliserin, M. I. Stockman, M. F. Kling,and U. Kleineberg, Appl. Phys. Lett. , 051904 (2012).[40] M. Lupetti, J. Hengster, T. Uphues, and A. Scrinzi,Phys. Rev. Lett. , 113903 (2014).[41] C. Lemke, C. Schneider, T. Leißner, D. Bayer, J. W.Radke, A. Fischer, P. Melchior, A. B. Evlyukhin, B. N.Chichkov, C. Reinhardt, M. Bauer, and M. Aeschlimann,Nano Letters , 1053 (2013).[42] Q. Liao and U. Thumm, Phys. Rev. A , 031401 (2015).[43] A. E. Schlather, A. Manjavacas, A. Lauchner, V. S.Marangoni, C. J. DeSantis, P. Nordlander, and N. J.Halas, J. Phys. Chem. Lett. , 2060 (2017).[44] M. T. Sheldon, J. van de Groep, A. M. Brown, A. Pol-man, and H. A. Atwater, Science , 828 (2014).[45] E. Le Ru and P. Etchegoin, Principles of Surface-Enhanced Raman Spectroscopy: And Related PlasmonicEffects (Elsevier, Oxford, 2008).[46] A. V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren,G. A. Wurtz, R. Atkinson, R. Pollard, V. A. Podolskiy,and A. V. Zayats, Nat. Mater. , 867 (2009).[47] C. Ayala-Orozco, C. Urban, M. W. Knight, A. S. Ur-ban, O. Neumann, S. W. Bishnoi, S. Mukherjee, A. M.Goodman, H. Charron, T. Mitchell, M. Shea, R. Roy,S. Nanda, R. Schiff, N. J. Halas, and A. Joshi, ACSNano , 6372 (2014). [48] F. Krausz and M. I. Stockman, Nat. Photon. , 205(2014).[49] ‘Free-Electron Lasers’. A collection of recent articles onFEL generation and characterization and their applica-tion in fundamental studies of light-matter interaction.Nature Photonics Collection (January 23, 2019).[50] E. Merzbacher, Quantum Mechanics , 3rd ed. (Wiley,1998) pp. 115, 315ff, 491, 496.[51] B. Obreshkov and U. Thumm, Phys. Rev. A , 012901(2006).[52] Q. Liao and U. Thumm, Phys. Rev. Lett. , 023602(2014).[53] Q. Liao and U. Thumm, Phys. Rev. A , 033849 (2014).[54] S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell,G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst,M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel,J. Burgdörfer, P. Feulner, F. Krausz, and R. Kienberger,Nature , 342 (2015).[55] C.-H. Zhang and U. Thumm, Phys. Rev. A , 043405(2010).[56] D. M. Wolkow, Zeitschrift für Physik , 250 (1935).[57] C.-H. Zhang and U. Thumm, Phys. Rev. Lett. ,123601 (2009).[58] M. Boiron and M. Lombardi, J. Chem. Phys. , 3431(1998).[59] Y. Goldfarb, J. Schiff, and D. J. Tannor, J. Chem. Phys. , 164114 (2008).[60] H. Goldstein, C. Poole, and J. Safko, Classical mechan-ics , 3rd ed. (Addison-wesley, 2001) pp. 22–23,342,433.[61] C. G. Broyden, Mathematics of Computation , 577(1965).[62] K. E. Atkinson, An introduction to numerical analysis (John Wiley & Sons, 2008).[63] G. K. Batchelor,
An introduction to fluid dynamics (Cambridge University Press, 1967) p. 74.[64] L. D. Landau and E. M. Lifshitz,
Quantum mechanics:non-relativistic theory , 3rd ed. (Elsevier, 1977) p. 570.[65] L. Rosenberg and F. Zhou, Phys. Rev. A , 2146 (1993).[66] H. R. Reiss and V. P. Krainov, Phys. Rev. A , R910(1994).[67] P. A. Macri, J. E. Miraglia, and M. S. Gravielle, J. Opt.Soc. Am. B , 1801 (2003).[68] J. Dubois, S. A. Berman, C. Chandre, and T. Uzer,Phys. Rev. A , 053405 (2019).[69] See Supplemental Material at [url] for animations ofthe (a) wavefunction comparison and (b) Mie-theory-calculated and reconstructed electric near-field distribu-tions for Au nanospheres.[70] S. Patchkovskii and H. Muller, Comp. Phys. Commun. , 153 (2016).[71] G. Mie, Ann. Phys. (Berlin, Ger.)330