Extracting continuum information from „t… in time-dependent wave-packet calculations

L. B. Madsen, L. A. A. Nikolopoulos,* T. K. Kjeldsen, and J. FernándezLundbeck Foundation Theoretical Center for Quantum System Research, Department of Physics and Astronomy, University of Aarhus,

8000 Aarhus C, DenmarkReceived 1 October 2007; published 18 December 2007

The theory of measurement projection operators in grid-based time-dependent wave-packet calculationsinvolving electronic continua in atoms and molecules is discussed. A hierarchy of projection operators relevantin their individual restricted configuration spaces is presented. At asymptotically large distances from thescattering or interaction center the projection operators involve plane waves only. To reach this asymptoticregime, however, large propagation times and large boxes may be required. At somewhat smaller distancesfrom the scattering center, the projection operators are expressed in terms of analytical single-center Coulombscattering waves with incoming wave boundary conditions. If propagation of the wave packet to theseasymptotic regimes is impeded, the projection operators involve the exact scattering states which are notreadily available in the wave-packet calculation and hence must be supplied by an additional, typically verydemanding, calculation. The present approach suggests an exact way of analyzing the timely problem of theone-electron continuum in nonperturbative calculations. A key feature is that the propagated wave packetincludes every interaction of the full Hamiltonian. The practicality of the proposed method is illustrated by thenontrivial example of strong-field ionization of the molecular hydrogen ion. Finally, the extension of thepresented ideas to single and double ionization of two-electron systems is discussed.

DOI: 10.1103/PhysRevA.76.063407 PACS numbers: 32.80.Rm, 33.80.Rv, 31.15.p, 02.70.Hm

I. INTRODUCTION

In many ab initio approaches to dynamics in atomic, mo-lecular, and optical physics the time-dependent Schrödingerequation is solved in coordinate space. The result of such anapproach is the space representation of the state of the sys-tem, r t=r , t, where r denotes generally theset of coordinates involved. In practice, such wave packetapproaches have been successfully applied to the strong-fieldionization of atoms and molecules within the single-active-electron approximation see, e.g., the reviews 1–3 and incollisions see, e.g., Ref. 4. In all cases, one of the theo-retical tasks is to relate the theoretical description in terms ofthe wave packet r , t to the experimental observables ofthe system. Restricting for clarity the discussion to the caseof a single active electron, the most specific information instrong-field ionization is the momentum-resolved ionizationprobability dP /dk, i.e., the fully differential ionization prob-ability for a fixed orientation of the nuclei in the molecularcase for ionization into the momentum interval k ;k+dk.Of course it has been known since the early works on scat-tering theory how to obtain this quantity by squaring theprojection of r , t on the exact scattering state k

−r thatasymptotically, at large distances, acquires the momentum k5. There is one major practical problem here: The scatter-ing states are only known exactly analytically for the puresingle-center Coulomb problem and generally it is a demand-ing task to construct these states numerically from first prin-ciples even for the simplest systems such as He, H2

+, and H26,7. It should be noted that quantities of a more integral

nature can be determined without making a projection. Forexample, the total ionization probability, P, can be deter-mined by keeping track of the wave-function norm absorbedby an absorbing boundary. The angular ionization probabil-ity, dP /d, can be obtained by the analysis of the radial fluxin direction k see, e.g., Refs. 8–10 and Sec. VI below.The photoelectron spectrum dP /dE can be obtained by aspectral analysis of the autocorrelation function of r , t11,12, with E=k2 /2 the electron energy. This method ex-tended previous bound state analysis 13,14 to the con-tinuum. Yet an alternative approach for the analysis of thephysics of the final state has been to divide the configurationspace into different regimes and associate the electron den-sity in a particular region with ionization see, e.g., Refs.15,16. As noted, e.g., in Ref. 17, clearly, the fixation ofthese boundaries is associated with a certain degree of arbi-trariness.

It is the purpose of the present work to point out that theavailability of the wave packet r , t in space and as afunction of time in particular at instants of time t= t0+Tlater than the end of the time-dependent interaction, t0 maylead to tremendous simplifications in the analysis of the mea-surable content of the wave packet. All the information aboutthe short-range interaction, i.e., phase shifts, and possiblechannel mixing, e.g., between different angular momentumstates in the case of a nonspherical symmetric molecule, isfully contained in r , t allowing for the analysis of thescattering content by the application of simple asymptoticprojection operators in the long-time limit see Fig. 1. Al-though the time-dependent interaction, inducing the transi-tion into the continuum, can be of general type, we make thediscussion explicit and refer to ionization by an intense laserpulse of short duration in the following.

The paper is organized as follows. In Sec. II, the analysisof the scattering content of t at time t0 just after the end

*Present address: Department of Applied Mathematics and Theo-retical Physics, The Queen’s University of Belfast, Belfast, BT71NN, Northern Ireland, UK.

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of the pulse is discussed in the case of a one-electron wavepacket Fig. 1a. In Sec. III, we present some formal andmathematical details which prove firmly the validity of thepresent approach based on asymptotic projection operators.In Sec. IV, we discuss how to perform the analysis in theasymptotic free-electron zone at times t0+T, where T is cho-sen such that all momentum components of interest havereached the asymptotic regime Fig. 1c. In Sec. V, wediscuss how to do the analysis in the somewhat closer regimewhere the Coulombic monopole dominates: The Coulombzone Fig. 1b. In Sec. VI we compare numerical results onstrong-field ionization of H2

+ obtained by projection on exactscattering states and with projection in the Coulomb zone attimes t= t0+T and illustrate the accuracy and practicality ofthe method. In Sec. VII, we discuss an extension of thepresent ideas to the two-electron system: Single-ionizationand excitation, as well as double ionization. In Sec. VIII, themain conclusions are presented.

Atomic units me==e=1 are used throughout unlessstated otherwise.

II. SCATTERING ANALYSIS JUST AFTER THE END OFTHE PULSE

When the short intense laser pulse has ended at time t0,the time-dependent wave packet r , t given in terms ofcomplex amplitudes in points in space is available in a the-oretical grid-based calculation. Though not readily at hand inthis case, the wave packet may formally be thought of asbeing represented by an expansion in a basis of energyeigenstates r , t=ncnnrexp−iEnt+dkckk

−rexp−ik2 /2t, with Hnr=Ennr, and where the ex-pansion includes a sum over bound and an integral over con-tinuum scattering states of the field-free Hamiltonian, H. It isclear that the probability of being in a particular eigenstatedoes not change in time once the driving pulse is over. Theanalysis of the scattering content of r , t may therefore beperformed immediately after the end of the pulse with theprojection operator

Pk = k−k

− , 1

where k−r is the momentum normalized, k

− k− =k

−k, scattering state that conforms with incoming-waveboundary condition and develops asymptotic momentum k.Accordingly, the probability for ionization with asymptoticmomentum k is TrPk, = t0t0, i.e.,

dP

dk= k

−t02. 2

In scattering theory, k−r is called the out or outgoing state.

The advantage of this approach where the projection is per-formed with the exact scattering state is that no propagationof the wave packet is required after the end of the pulse, andcompared with the methods discussed below it may hencepose less demands to the box size and avoid associated nu-merical problems in propagating the wave packet. An addi-tional advantage is that the scattering states and the boundenergy eigenstates making up the formal expansion of the

r0

r0

R

a) Reaction zone

t0

t0+T

t0+T

b) Coulomb zone

c) Free zone

r0

r0

R

r0

r0

R

k

k

k

FIG. 1. Separation of configuration space into spatial zones witheach of their characteristic Hamiltonian. The features moving awayfrom the scattering center going from a to c illustrate a compo-nent of the wave packet moving with central momentum k. In a,the wave-packet component moving away is shown just after theend of the pulse at time t0. The spatial region corresponding to thissituation we refer to as the reaction zone and discuss it in Sec. II. Inb, an additional time T has passed and the packet is at interme-diate distances r0rr0, where the Coulombic monopole from thescattering center dominates. This situation we refer to as the Cou-lomb zone and discuss it in Secs. V–VII. In c, the time T haspassed by since the end of the pulse, and the packet is in the freezone at asymptotic large distances, rr0. In neutral atomic andmolecular systems the situation in c is, in a strict theoretical sense,not encountered since the Coulomb tail of the potential gives alogarithmic phase distortion of the plane-wave asymptotic state atall distances. We include c, since it is of relevance in negative ionsand other branches of physics. We discuss the projections corre-sponding to c in Sec. IV. The radius R denotes the boundary of thesphere defining the part of space that is accounted for in a numericalcalculation. In practical calculations, to prevent reflection from theboundary, an absorbing boundary may be used.

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wave packet are solutions to the same field-free Hamiltonian.This means that eigenstates are orthogonal and the projectionintegral in Eq. 2 is readily performed over all space: Noprior projecting out of any bound state part is necessary.Additionally, in Eq. 2 there is in principle nothing that fa-vors particular k values, i.e., application of Eq. 2 is equallyvalid and “easy” for low, intermediate, and high k values. Inpractice, of course, the “easiness” will typically depend onthe dimensions of the grid. The disadvantage in the resolu-tion in Eq. 2 is that the analysis requires per default theexact scattering states of the Hamiltonian H, the constructionof which in general is an elaborate affair. The hydrogen atomand hydrogenlike ions are the only cases where the scatteringstates are known analytically in terms of Coulomb wavefunctions. For atoms described in the single-active-electronapproximation the scattering states can be found relativelysimply numerically by exploiting the spherical symmetry ofthe problem. In molecules the multicenter character of theremaining molecular ion means that the angular momentumof the ejected electron is no longer a constant of motion andthe construction of the scattering states becomes a multi-channel problem in terms of angular momenta 7,18,19.Hence, in general the scattering states have to be provided byan additional structure calculation typically performed in aneigenstate basis. This computational overhead is a clear dis-advantage. In Sec. VI we find the scattering states for H2

+ ina B-splines basis-state approach and compare the results withthe asymptotic analysis in the Coulomb zone.

We note in passing that Eq. 2 as it stands is the naturalstarting point for analysis in the perturbative regime. Forexample, for perturbative single-photon ionization, the time-dependence drops out and essentially in Eq. 2 t0→ ·p 0, where is the polarization vector, p the momen-tum operator, and 0 the field-free initial bound state. Inthis case the perturbation introduces a transition between ex-act energy eigenstates of H and no wave packet is formed.Accordingly there is no possibility to propagate toasymptotic regions and perform the scattering analysis there.For femto- and attosecond pulses, however, the creation of awave packet makes the approach of Secs. IV and V natural.

III. PROPERTIES OF THE OUT STATE ANDIMPLICATIONS FOR THE SCATTERING ANALYSIS

A fundamental property of the out state k−r is that it

asymptotically acquires the momentum k. Although this iswell-known, we dwell a little on the way this is proved to beable to understand and describe clearly the implications inthe present context.

We write the total field-free time-independent Hamil-tonian H=H0+V, where H0 is assumed to be simple suchthat all its eigenstates n and eigenvalues En are known,i.e., H0 n=En n. Here the index n discriminates betweendifferent, possibly degenerate, channels. Following Ref. 20,we call the system with energies En and states n the ref-erence system. The term V in H is the interaction that causesperturbations of the unperturbed reference system. We mayfind the stationary continuum states that solve the true prob-lem including V, H−Ek k

−=0, at the particular energy Ek

by the usual integral equation technique leading to the fol-lowing Lippmann-Schwinger equation for k

−:

k− = k + lim

→0+

1

Ek − H0 − iVk

− , 3

where the subscript k specifies the channel, e.g., energy Ekand asymptotic momentum k.

To obtain an understanding of the physical meaning ofk

−, we form a wave packet centered around this stationarystate 20

k−t =

k

cke−iEktk

− , 4

where the index k specifies energy and asymptotic momen-tum, and where each k

− is a solution of Eq. 3. The coef-ficients ck only attain nonvanishing values for subscripts kclose to k. Likewise, we construct a wave packet from thesolutions of the reference system centered also at k and withexpansion coefficients as in Eq. 4,

kt = k

cke−iEktk , 5

with k eigenstates of the reference system, H0 k=Ek k. Now, following the procedure in Ref. 20, weinsert the complete set of reference states in Eq. 3 betweenthe inverse operator and V and we take the limiting proce-dure t→ using the relation limt→ lim→0+ eit / + i=0 to obtain the late time behavior of the out wave packet

limt→

k−t = kt . 6

Equation 6 shows explicitly that at large times the wavepacket centered around the scattering state k

− turns into apacket made up of eigenstates of the reference HamiltonianH0. In particular the distribution over amplitudes remainsunaffected: The coefficients ck are equal in Eqs. 4 and 5.This is a result which has important consequences for thepresent discussion. Going back to Eq. 2, we see why. Theindex k specifies energy and asymptotic momentum, and wehave shown that the distribution over this index is the samein the wave packet constructed from a superposition of exactscattering states and in the wave packet formed by the un-perturbed reference states. Hence, we are naturally led to theassumption that by letting the exact wave packet solution ofthe problem in the right-hand side of Eq. 2 evolve to largetimes we may perform the projection on reference states inorder to obtain the energy and angle resolved spectrum, i.e.,one might suspect that dP /dk= k

− t02

=limt→ k t0+ t2. This relation, however, is incorrectbecause it does not account correctly for the nonorthogonal-ity of the bound-state spectrum of H and the continuumstates of H0. What is true, however, is that the exact scatter-ing state is obtained from the reference state by applicationof the Møller or wave operator20–24

k− = −k , 7

where −=limt→ eiHte−iH0t projects the states in the con-tinuum of the reference Hamiltonian H0 onto the correspond-

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ing elements in the continuum of H. To stress this fact, weinclude explicitly the projection on the continuum scatteringstates of H, PH

cont=dk k−k

− in Eq. 7, i.e.,

k− = PH

cont−k . 8

Inserting this expression and the definition of the wave op-erator into Eq. 2, we obtain

dP

dk= lim

t→keiH0te−iHtPH

contt02. 9

Finally, we use the completeness relation 1=PHcont+, with

=n nBn

B the projection on the bound state part of thespectrum of H, to obtain

dP

dk= lim

t→keiH0te−iHt1 − t02. 10

The goal we are pursuing is to move the time evolutionoperator to the right and evolve the exact wave-packet for-wards in time from t0. Clearly exp−iHt ,PH

cont= exp−iHt ,1−=0. Accordingly, we propagate the fullwave packet from time t0 to t and project out the bound statepart of t0+ t to obtain the continuum part t0+ t= 1− t0+ t of the wave packet, and the differentialionization probability reads

dP

dk= lim

t→kt0 + t2, 11

which is the formally exact relation justifying the approachproposed in the present paper. Basically, Eq. 11 shows thatif the continuum part of the exact wave-packet solution maybe propagated long enough after the end of the short laserpulse, then there is no need to construct the exact outgoingscattering states. The angle and energy resolved spectrummay be obtained by projection on the much simpler referencestates k.

In the next sections, we shall discuss from a more practi-cal point of view how the above ideas can be implemented ina numerical calculation. For example, the bound state part isexcluded by reference to spatial separation. As we shall seethe reference projection states of H0 can be chosen as ana-lytically known functions. All the phase shifts and informa-tion about the short-range behavior of the interaction poten-tial V is in the wave packet t0+ t.

IV. SCATTERING ANALYSIS IN THE ASYMPTOTICREGION: FREE ZONE

In this section we write the total field-free Hamiltonian asH=H0+V, where H0= p2 /2 is the kinetic energy part andwhere V denotes the interaction potential see Fig. 1c. It isassumed that the wave packet is propagated for a time T afterthe pulse large enough to ensure that even the slowest con-tinuum components of interest have reached asymptotic dis-tances where the potential V can be neglected. In a laboratoryexperiment long-range Coulomb forces are shielded by theenvironment at sufficiently large distances so the assumptionof a finite range is automatically fulfilled. In a theoretical

grid calculation, there is an asymptotic regime where theCoulomb field is small compared with the kinetic energy ofthe outgoing electron. Slightly more quantitatively, one mayconsider the drop-off of the Coulomb-induced logarithmicphase distortion of the phase of the outgoing wave, ln2krcompared with the free kr behavior, with =Z /kand Z the nuclear charge. If this ratio = ln2kr /kr issmall, the interaction V may be neglected compared with H0.For example, =0.13 for k=0.1 a.u., Z=1, and r=3200a0.This means that large boxes and long propagation times areneeded to reach the asymptotic free zone. In gauging thepracticality one should of course also consider the complica-tions in determining the exact scattering states entering Eq.2. Note also that recently box radii of 1000a0 were con-sidered for strong-field ionization of atoms described in thesingle-active electron model 25. Hence large boxes are of-ten used in practical calculations and therefore theasymptotic regimes may be available on the numerical grid.

A momentum component k of the wave packet r , twill reach the asymptotic regime, say beyond r0, where theHamiltonian is described by H=H0 at a time r0 /k. In thisregion rr0, the eigenstates corresponding to the Hamil-tonian H0 are simply momentum eigenstates r k= 2−3/2 expik ·r and the associated projection operator isPk= kk. This means that in this free zone, at a time Tr0 /k sufficiently large that the momentum component ofinterest has arrived, the differential ionization probability isfound by projection on plane wave states

dP

dk= kt0 + T2. 12

The prime on the time-dependent wave packet indicates thatthe part of the wave packet not in the asymptotic regime withwell-defined plane wave projection operators has been pro-jected away. This is necessary since the plane wave and theeigenstates formally forming the wave packet belong to thespectra of two different Hamiltonians, H0 and H, respec-tively, and hence the eigenfunctions are not orthogonal, i.e.,k n is generally different from zero. The formal justifi-cation for this projection method was given in Sec. III interms of the wave or Møller operator, which acts only on thecontinuous spectrum. In Eq. 12 it is the superposition ofeigenstates in the wave packet r , t0+T that ensures thespatial localization of the continuum components atasymptotic distances, rr0 at large times, and hence theunderlying physical reason why the proposed scatteringanalysis is well justified and exact. Without details and for-mal justification the approach discussed here was used re-cently in the high-frequency regime to described ionizationof H2

+ 26.The partial wave expansion of the momentum normalized

plane wave kr describing the free final state reads 27

kr = =0

m=−

2

ijkrYm

kYmr , 13

where jkr is a spherical Bessel function and Ym arespherical harmonics. If as in, e.g., our recent approaches

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9,11,12, the wave packet is expanded in partial waves,(r , t= t0+T)=,mfmr , t /rYmr, the insertion of Eq.13 into Eq. 12 leads to

dP

dk=

m

2

i−Ymk

r0

jkrfmr,t0 + Trdr2

.

14

Equation 14 is easily evaluated numerically since thefmr , t are available on the grid and only a one-dimensionalintegral is involved. As seen from Eq. 14 the projection outof bound states is implemented by integrating radially onlyfrom r0 and outwards.

V. SCATTERING ANALYSIS IN THE MONOPOLEREGIME: COULOMB ZONE

In this section we discuss the most practical way of per-forming scattering analysis in grid-based wave-packet calcu-lations. For single ionization of atoms, it is clear that theregion where the full Hamiltonian is well described by H0= p2 /2−Z /r is reached when the continuum electron is out-side the electronic cloud associated with the other boundstate electrons. In many cases this already happens at 10a0.For molecules, the situation is slightly more complicated dueto the multicenter character of the problem. The generic fea-tures of the problem introduced by the additional nuclei areillustrated already for the molecular potential correspondingto the H2

+, VMr ;R=−1 / r−R /2 −1 / r+R /2. A multipoleexpansion of VM at distances r larger than the internuclear

separation R gives VMr ;R=−2L even

R/2L

rL+1 PLcos , so theleading terms are VMr ,R−(2 /r1− R / 2r)2P2cos ;where PLcos are the Legendre polynomials and the polaraxis is parallel to the internuclear axis. For more generalmolecules, also odd multipoles are allowed and therefore theleading correction to the monopole in the Coulomb zone isthe dipole1 /r2. The parameter =1 /r is hence a measureof the strength of the dipole compared with the monopole. Ata radius of r0=100a0 it appears to be a quite accurate ap-proximation to ignore higher-order corrections; for H2

+, r0=100a0 makes the approximation excellent. It is hence clearthat there exists an intermediate Coulomb zone in space rr0, with r0r0 where the monopole from the atomic ormolecular system dominates see Fig. 1b. In this zone theprojection operators corresponding to the asymptotic mo-mentum k are known analytically in terms of Coulombwaves k

−,C, Pk= k−,Ck

−,C, and for rr0, the ionizationprobability may be calculated as

dP

dk= k

−,Ct0 + T2. 15

Here, as in Eq. 12, the prime on the wave packet indicatesthat only the part of the wave packet localized at distancesrr0 with HH0= p2 /2−Z /r is involved. The prime on theadditional propagation time T indicates that now, for thesame problem, TT where T corresponds to the propaga-tion time needed to reach the asymptotic free zone. The for-mula in Eq. 15 is very general and highly applicable since

in practice quite small r0 values 20–50 a0 are large enough,which is a fact that limits the demands on computationaltime and box size. This is a huge advantage compared withthe scattering analysis discussed in Sec. II where new scat-tering wave functions have to be constructed for each systemat hand—the Coulomb waves are analytical and universal.Explicitly, in an expansion in partial waves, the outgoing partof the momentum normalized Coulomb wave functionk

−,Cr, normalized to the k scale, reads 27

k−,Cr =

m

2

ie−i

Fkr,kr

Ym kYmr , 16

with F the regular Coulomb function, and k=arg+1+ i the Coulomb phase shift expressed interms of the complex Gamma function . With the con-tinuum wave packet r , t expanded in partial waves as inSec. IV, the momentum resolved probability is

dP

dk=

m

1

k 2

i−eiYmk

r0

Fkr,fmr,t0 + Tdr2

. 17

We illustrate the usefulness of Eq. 17 in Sec. VI, where weconsider the case of strong-field ionization of H2

+. In Eq. 17the radial integral is from r0 and effectuates the projectionout of the bound states. Note that in the limit →0: Fkr→krjkr, l→0, and Eq. 17 turns into Eq. 14 corre-sponding to the high-energy limit and/or large r or negativeions.

VI. RESULTS AND DISCUSSION

To illustrate the method, we consider strong-field ioniza-tion of H2

+. The field is linearly polarized along the internu-clear axis with peak intensity 21013 W /cm2, photon en-ergy 0.8 a.u., and a duration of 10 cycles. The envelope ofthe field is taken to be a sine square and the carrier-envelopephase difference CEPD is zero. For the ten-cycle pulse con-sidered here the results are insensitive to the value of theCEPD. Note that the ionization potential is 1.1 a.u. corre-sponding to the two-photon ionization regime. The frequencyof the source can, e.g., be obtained from a free-electron lasersource. We expand the time-dependent wave function inspherical harmonics and solve the time-dependentSchrödinger equation by the split-step method discussed indetail elsewhere 9–11. The maximum angular momentumis max=11, the radius of the sphere in which the calculationis performed is rmax=300 a.u., the number of grid points inthe radial coordinate is 2048, and the time increment is t=0.005 a.u.

To construct the scattering states of H2+ at equilibrium

distance R=2, we expand the continuum states as

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k−r =

=0

m=−

ie−i,m

k,− rr

Ymk , 18

with defined after Eq. 16. The function ,mk,− r is ex-

panded in spherical harmonics and the radial part in turn isexpanded in a B-splines basis set 28,29. The scatteringstates are calculated applying the inverse iteration methoddescribed in Refs. 7,30. In this way we first obtain wavefunctions ,m

k r with the K-matrix asymptotic boundarycondition

,mk r

m

Fkr,, + K,m EGkr,Y,mr ,

19

with F and G the regular and irregular Coulomb wavefunctions. With the sign convention followed here, theasymptotic forms of the Coulomb waves are Fsin andGcos , =kr− ln 2kr− /2+. To obtain the cor-rect out state with incoming spherical wave boundary condi-tion, we change to the S-matrix representation

,mk,− r =

m

I + iKmE,−1 ,m

k r . 20

In terms of the B-splines basis, the result in Eq. 20 is ex-pressed as

,mk,− r =

m

i

cm−,i EBirY,mr , 21

where Bi denotes the ith B-spline of order 10 defined in a boxof size of 300 a.u. with a linear breakpoint sequence of 2048equidistant points in which the first B-spline is deleted fromthe basis set and the last one is retained i.e., N=2055 Bsplines per angular momentum. The spatial grids coincide inthe wave packet and in the B-spline calculation. The partialwave expansion in the B-spline calculation also includes an-gular momenta up to max=11. In order to perform the pro-jection Eq. 2 over the continuum states of the H2

+ mol-ecule we rewrite the time-dependent radial wave functionsfmr ; t0 just after the end of the pulse at t0 in terms of thesame B-splines basis set, i.e,

fmr;t0 = i

N

bmi t0Bir , 22

where bmi t0 are complex amplitudes since fmr ; t0 is com-

plex. The evaluation of the scalar products is performed inthe B-spline basis and involves matrix-vector operations. Thefully differential ionization probability obtained by projec-tion on numerical scattering states of H2

+ is given by

dP

dkt0 =

m

i−eiI m− E;t0Y,mk2

, 23

where

I m− E;t0 =

max

i

N

j

N

cm−,i EOi,jbm

j t0 24

is the continuum projection matrix element associated withpartial wave normalized to the S-matrix, consistent withthe time-dependent representation of the photoionizationprocess 18,31,32, Oi,j =0

rmaxBirBjrdr is the overlap ma-trix, cm

−,i E are the coefficients that define the continuumpartial wave Eq. 21, bm

j t0 the coefficients enteringEq. 22, and m designates the value of the projection of theangular momentum over the molecular axis. We note that theresults are insensitive to the time t0 since the field-freeHamiltonian does not induce coupling between the field-freeeigenstates. We keep the argument t0 in the differential prob-ability as a parameter stressing that the projection on eigen-states is performed immediately after the end of the pulse attime t0. Also note that the spatial integrals are over the entireregion r 0,rmax.

In Fig. 2 we show the photoelectron angular distribution,i.e., the quantity dP /d=dP /dkk2dk, for two-photon ion-ization of H2

+ by the field described previously. For thepresent orientation of the molecular axis with respect to thefield there is rotational symmetry around the molecular axisand the angular distribution is the same for all values ofazimuthal angle. In the figure we have chosen this angle tobe 0. The full curve shows the angular distribution obtainedby i projection on the exact H2

+ scattering states Eqs. 2

0 30 60 90 120 150 180θ (degrees)

0.0

2.0×10-7

4.0×10-7

6.0×10-7

8.0×10-7

1.0×10-6

1.2×10-6

dP/d

Ω| ϕ=

0(P

roba

bilit

y/Sr

)

T = 0.01T = 5.90T = 11.8T = 17.6T = 20.6T = 26.4H

2

+

Flux

FIG. 2. Color online Photoelectron angular distribution forstrong-field two-photon ionization of H2

+ at the equilibrium distanceby a linearly polarized laser field with polarization along the direc-tion of the internuclear axis. The azimuthal angle is zero see text.The field has peak intensity 21013 W /cm2, photon energy0.8 a.u., and a duration of 10 cycles. The envelope of the field istaken to be a sine square. The full curve shows the angular distri-bution obtained by flux analysis Eq. 25 and by projection on theexact H2

+ scattering states Eqs. 2 and 23. The other curvesshow snapshots of the projection on single-center Coulomb wavesZ=2 Eq. 17 at different instants of time T after the end of thepulse. The inner boundary of the Coulomb zone is r0=40 a.u. Onthe scale of the figure there is no difference between the resultsobtained by the single-center Coulomb projector applied at T=17.6,20.6,26.4 a.u., by the flux analysis and by the H2

+ projector.

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and 23 and ii flux analysis Eq. 25 12. The latter caseii is analyzed with reference only to the grid. We simplyevaluate the probability current at the distance rflux

=200 a.u. in the direction k, and integrate over time up totime T=500 a.u. after the end of the pulse when all radialoutgoing flux has passed the point of observation see, e.g.,Ref. 12. We obtain the angular distribution from the for-mula

dP

d,T =

t0

t0+T

dtrflux2 jrrflux,,t , 25

with jrr , t=Imr , tr ·r , t. On the scale of Fig. 2we see no difference between the results obtained by fluxanalysis and the scattering-state-projection method confirm-ing that both methods are accurate. The other curves in Fig.2 show snapshots of the projection on single-center Coulombwaves with Z=2 Eq. 17 at different instants of time Tafter the end of the pulse as detailed in the inset. In thiscalculation we followed the procedure discussed in Secs. IIIand V and projected out the bound state part of the wavepacket. The inner boundary of the Coulomb zone is r0=40 a.u., i.e., the radial integrals are performed from 40 to300 a.u. Eq. 17. We see, in accordance with the discus-sion in the preceding sections, that the results based on theanalysis in the Coulomb zone converges to the exact result:On the scale of the figure there is no difference between theresults obtained by the single-center Coulomb projector ap-plied at T=17.6,20.6,26.4 a.u. and the flux and theH2

+-projector results. Stability of the quantity of interest withrespect to time is hence a sign of convergence in the appli-cation of the asymptotic projector method. Compared withthe flux analysis where a propagation time of T=500 a.u.was needed much less propagation time is required with theprojection method as seen from the insert in Fig. 2—theresult is fully converged at T=17.6 a.u.

We note that the angular distribution and the full momen-tum distribution are much more delicate tests of the accuracyof the theory than, e.g., the photoelectron energy spectrum.The reason is that the former quantities depend on the rela-tive phases between the different partial waves making upthe continuum wave functions while the latter is insensitivethereof.

We now turn to the differential momentum distribution.This quantity cannot be obtained by the flux analysis usedabove nor through the autocorrelation function 11. A pro-jection method seems to be the only practical method. Figure3 shows the differential momentum distribution dP /dk forstrong-field two-photon ionization of H2

+ by the same pulseused to obtain the results shown in Fig. 2. The horizontalaxis shows the component k parallel with the linear polar-ization. The vertical axis shows a component k perpendicu-lar to the polarization axis. The momentum distribution isrotationally invariant with respect to the k=0 axis. Figure3f shows the momentum distribution obtained by projec-tion on the H2

+ scattering states Eqs. 2 and 23. The otherdistributions show snapshots of the projection on single-center Coulomb waves Z=2 at different instants of time Tafter the end of the pulse Eq. 17. We see that the projec-

tion on the single-center Coulomb wave functions in theCoulomb zone converges to the correct numerical result ofFig. 3f for increasing T, and hence the figure provides an-other example for the practicality and accuracy of the presentmethod.

VII. THE TWO-ELECTRON PROBLEM

In this section we propose the extension to two-electronsystems. An accurate description of two-electron systemsand in particular the helium atom under time-dependentstrong fields has been a quest for theory in the last 10–15years 33.

A. Single ionization and excitation

The channel corresponding to single ionization and exci-tation of He is readily resolved by the projection operatorPk1

Pn2where Pk1

= k1 k1

is the projection operator ofthe appropriate reference system see Sec. III that “mea-sures” one of the electrons in the asymptotic momentum k1.The operator Pn2

= n2 n2

projects on the nth eigenstateof the He+ subsystem. The advantage here is that both Pk1

FIG. 3. Color online Differential momentum distributiondP /dk for two-photon ionization of H2

+ by the linearly polarizedlaser field used in Fig. 2 with polarization along the direction of theinternuclear axis. The horizontal axis shows the component k par-allel with the linear polarization. The vertical axis shows a compo-nent k perpendicular to the polarization axis. The distribution isinvariant with respect to rotations around the k=0 axis. The fieldis as in Fig. 2. Panel f shows the momentum distribution obtainedby projection on the H2

+ scattering states Eqs. 2 and 23. Theother distributions show snapshots of the projection on single-centerCoulomb waves Z=2 Eq. 17 at different instants of time T afterthe end of the pulse a T=0.01 a.u., b T=5.9 a.u., c T=11.8 a.u., d T=20.6 a.u., and e T=26.4 a.u.. The inner bound-ary of the Coulomb zone is r0=40 a.u.

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and Pn2are known analytically. So, the proposed formula

giving the photoelectron angular distribution with the ion leftin the analytical n2

bound state of He+, and working in thespatial regime where the continuum electron moves in the−1 /r Coulomb potential from the screened nuclei, reads

dP

dk1n2 = k1

C,− n2t0 + T2, 26

where k1

C,− is a Coulomb wave function with asymptoticmomentum k1. In Eq. 26, only the bra or the ket needs to beproperly antisymmetrized since the antisymmetrization op-erator A is idempotent A2=A.

B. Double ionization

Recently, works have appeared that focus on the problemof two-photon double ionization for a recent paper thatgives a detailed survey of the literature see Ref. 17. Mul-tiphoton double ionization is a computationally very hardproblem: It involves six spatial dimensions, and infinitelymany coupled channels. Also the extraction of scattering in-formation is very complicated. One approach 15,16 is toperform the analysis of the numerical data based on where inspace the electron density is localized. An alternative is togenerate the multichannel double-continuum wave function19. Another method was developed in connection with thetime-dependent close-coupling method 34 where the final-state wave function after the pulse was projected on a prod-uct of uncorrelated Coulomb wave functions. With thismethod fully differential cross sections can be obtained. Veryrecently, an alternative ab initio approach relying on the con-struction of the correlated multichannel scattering wavefunction by means of the so-called J matrix was developed

17. In that approach the probability for double ionizationwas obtained by subtracting from the total wave function thebound-state and single-continuum parts, and no angular dif-ferential information was obtained. Hence, it is fair to saythat the extraction of scattering information in the doublecontinuum reached by multiphoton absorption is a very chal-lenging problem.

In this section, it is discussed how a straightforward gen-eralization of the ideas of the preceding sections may help inthe scattering analysis. An advantage of the proposed methodis that only controllable approximations in the measurementprojection-operator step are introduced. If propagation tolarge distances of the wave packet is possible, the presentedformula is exact and the final state for the analysis of thetwo-electron continuum is analytical. Hence, also in thiscomplicated case the grid-based wave-packet calculationneed not be supplemented in any way.

In the case of double ionization, a fully differential mea-surement determines the two momenta k1, k2 of the ejectedelectrons. Now, as discussed already in connection with theeffective two-body Coulomb problem in Sec. V, it is wellknown that an asymptotic plane wave is inconsistent with thesolution to the Schrödinger equation pertaining to the Cou-lomb problem. The Coulomb tail of the potential generates alogarithmic phase that distorts the plane wave so that theasymptotic momentum k is only reached slowly as r in-creases. Generalizing to the case of the three-body Coulombproblem with a He++ nucleus and two continuum electrons,we expect such logarithmic phase distortions in the relativemotion of each pair. Indeed it has been known for a longtime see, e.g., 35 and references therein that the exactform of the wave function describing two electrons in thecontinuum of an infinitely heavy He++ nucleus when all in-terparticle distances are large is given by

−k1,r1;k2,r2 =1

23eik1·r1+k2·r2e−i1 lnk1r1+k1·r1e−i2 lnk2r2+k2·r2e−i12 lnk12r12+k12·r12, 27

where j =−Z /kj j=1,2, Z=2, 12= /k12, k12=k1−k2,r12=r1−r2. Equation 27 shows that the asymptotic formconsists of two phase-distorted plane waves for the two out-going electrons and an additional phase last factor stem-ming from the electron-electron interaction. Clearly this lastphase factor is just as important as the other two, and inparticular it has to be included in order to conform with thecorrect boundary conditions. This means that projectionsbased on a product of two uncorrelated Coulomb wave func-tions 34 should only be applied with care see also thediscussion in 17. As in Eq. 26, the projection state needsnot be symmetrized.

We note in passing that in the 90’s much work was in-vested into the construction of an analytical three-body Cou-lomb wave function that satisfied the correct boundary con-dition of Eq. 27 36. The interest was, e.g., in single-photon double ionization in the perturbative regime. The

analysis was made in time-independent theory and, accord-ingly, the wave functions that entered in the construction ofthe appropriate matrix elements had to be known and definedin all of space: There was no possibility to propagate a wavepacket to large interparticle distances and to make the scat-tering analysis in the asymptotic part of space with the exactstate in Eq. 27. In this work, however, this possibility formsthe grounds of the proposed method, and therefore we sug-gest that the analysis of the double continuum may be stud-ied by evaluation of the formula

dP

dk1dk2= k1,k2

− t0 + T2, 28

with r1 ,r2 k1,k2

− given in Eq. 27 and with the continuumpart of the wave packet t evaluated at time t0+Twhere all the particles are far away from each other r1 ,r2

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and r12 large. The asymptotic form of Eq. 27 is straight-forwardly generalized to the N-particle problem 37 and ageneralization to the N-particle breakup of the present ap-proach is therefore formally possible.

The exclusion of the two-electron bound states implied bythe prime on the wave packet in Eq. 28 ensures that noartificial result from the single-ionization channel appears inthe double-ionization signal. This is easily proved by notingthat for bound state components of the singly ionized speciesof a spatial range similar to the range of the two-electronbound states, the bound state wave function in the single-ionization channel is zero in the integration volume underconcern in Eq. 28 and thus implies an overall vanishingoverlap between the double ionization projector and thesingle-ionized channels.

A nice feature of the present approach is that the wavepacket includes every interaction of the full Hamiltonian. Itis at the level of the measurement projection operator that theapproximations are introduced and in a controlled way: Thelonger propagation time, the more exact result. When themomentum components of interest are in the asymptotic re-gime then Eq. 27 is valid, and the angle and energy re-solved ionization probability of Eq. 28 remains constant intime; a prerequisite for a converged calculation.

VIII. SUMMARY AND CONCLUSION

In conclusion, the theory of measurement projection op-erators in time-dependent grid-based wave-packet calcula-tions involving continua was discussed. The mathematicalfoundation of the approach relies on the properties of the

wave operators that act on continuum states. A hierarchy ofprojection operators relevant in their individual restrictedconfiguration spaces was presented. At large distances fromthe scattering or interaction center the projection operatorsare simply given in terms of momentum eigenstates. If thewave packet is only propagated to intermediate distanceswhere the Coulombic monopole is significant, the projectionoperators are generated from Coulomb waves with incomingscattering wave boundary conditions. If propagation of thewave packet to these asymptotic regimes is impeded, theprojection operators involve the exact scattering states whichare not readily available in the wave-packet calculation andhence must be supplied by an additional calculation. Thepresent work demonstrates that the Coulomb and plane wavestates forming the projection operators are conveniently ex-pressed in a spherical coordinate partial wave expansion.However, since the Coulomb and plane wave states areknown analytically, it is possible to use the present method inany coordinate grid representation. Therefore, in generaltime-dependent grid calculations, the present method is po-tentially very practical since a whole range of processes maybe analyzed directly by making the appropriate projectionsover restricted parts of a numerical grid. There is no need forfinding the scattering states for each individual subsystemsunder study. The accuracy of the method was illustrated byconsidering strong-field ionization of H2

+ where the exactscattering states were found by a numerical ab initio method.

ACKNOWLEDGMENT

This work was supported by the Danish Research AgencyGrant No. 2117-05-0081.

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