Nonsequential double ionization of molecules

Jakub S. Prauzner-Bechcicki,1 Krzysztof Sacha,1 Bruno Eckhardt,2 and Jakub Zakrzewski1

1Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Reymonta 4, PL-30-059 Kraków, Poland2Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Germany

sReceived 25 May 2004; published 15 March 2005d

Double ionization of diatomic molecules by short linearly polarized laser pulses is analyzed. We consider thefinal stage of the ionization process, that is the decay of a highly excited two electron molecule, which isformed after rescattering. The saddles of the effective adiabatic potential energy close to which simultaneousescape of electrons takes place are identified. Numerical simulations of the ionization of molecules show thatthe process can be dominated by either sequential or nonsequential events. In order to increase the ratio ofnonsequential to sequential ionizations very short laser pulses should be applied.

DOI: 10.1103/PhysRevA.71.033407 PACS numberssd: 32.80.Rm, 32.80.Fb, 05.45.Mt

I. INTRODUCTION

High intensity ultrashort-pulse lasers allow experimentalstudies of such effects as a non-sequential double ionization,high order harmonic generation or above threshold ionizationf1,2g. While the single ionization of atoms or molecules, aswell as the high order harmonic generation can be describedwithin a single active electron model, such an approximationin the case of double ionization and laser intensity below thesaturation value gives ionization rates that are much smallerthan experimentally observed, indicating that interactions be-tween electrons are importantf1–4g. In this paper we con-sider the double ionization of molecules within a classicalmodel for electrons in a combined Coulomb and externalfield applying the approach developed inf5–9g for the mul-tiple ionization of atoms.

Multiphoton double ionization of atoms in strong laserfields can be regarded as a three step processf1,2g. In thefirst step one electron tunnels out through the Stark saddleand then is returned back to the nucleusf10,11g. Thereafter,in the second step, a highly excited state of an atom or mol-ecule is formed at the expense of the energy brought back bythe returning electronsup to 3.17Up, whereUp is a pondero-motive energyd. Finally, in the third step, such a highly ex-cited compound state decays in one of several ways, througha single, double or multiple ionization. The starting point inour classical analysis of the double ionization of molecules isthe excited complex, i.e., we assume that we have an initialstate of two highly excited electrons close to the molecularcore. We shall focus on double ionization events that canthen appear as a possible channel of decay.

We would like to stress that sometimes all the doubleionization events that happen after the rescattering processare referred to as being nonsequential. The classical analysis,as evidenced inf5g, clearly suggests to distinguish eventswhere both electrons escape during the same half-cycle ofthe electric field after the formation of the highly excitedstate and those where there is a time delay of one or severalhalf-cycles between escaping electrons. In the following theterm “nonsequential” will be reserved for the simultaneousescape of both electrons during the same half-cycle of theexternal field exclusively.

Experimental investigations of double ionization of di-atomic moleculesf12–15g showed that there are differencesbetween molecular species. For example, the nitrogen mol-ecule, N2, clearly exhibits a “knee structure” in a double-to-single ionization yields ratio, like atoms do, while for theoxygen molecule, O2, such a structure is not visiblef13,14g.Moreover, in the case of N2 it seems that electrons escapewith similar momenta along field polarization axis more of-ten than in the case of O2 f15g. We will here consider thisproblem from the point of view of the classical analysis,which starts after the formation of the highly excited com-pound state, and discuss possible reasons for such differentexperimental observations. In our discourse we will approxi-mate the molecule by two electrons and two charged nuclei,what is standard procedure in a case of highly excited statessinternal structure is neglectedd. On the other hand, the Cou-lomb potentials are taken to describe all particle interactions,not thed potentials as often assumedssee, e.g.f15gd. In aconsequence, what is considered here is a hydrogenlike mol-ecule with different internuclear spacing. Internuclear dis-tance,d=2.067 a.u., is referred to as N2 molecule and inter-nuclear distance,d=2.28 a.u., as O2 molecule, respectively.

The paper is structured as follows. In Sec. II we introducea theoretical model on which we base our analysis. SectionIII consists of the identification and description of thesaddles in the potential. Then in Sec. IV we present results ofthe numerical simulations. And finally, in Sec. V we con-clude.

II. MODEL

As we mentioned above, we start from a configurationthat forms as a result of the re-scattering: it has two highlyexcited electrons close to the molecular core in the presenceof linearly polarized laser field. The motion of the molecularcore is frozen due to the fact that for short laser pulses mol-ecules have not enough time to change their orientationf15,16g. Thus the Hamiltonian readssin atomic units, whichare used throughoutd

H =p1

2 + p22

2+ V, s1d

where the potential

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V = V1 + V2 + V12 + VF s2d

consists of the potential energies associated with interactionsof the electrons with the nucleisthe entire structure of themolecular core is approximated by two positively charge nu-cleid,

Vi = −1

ur i − R1u−

1

ur i − R2u, s3d

for si =1,2d, whereRi and r i indicate position of the nucleiand the electrons, respectively, the repulsion between elec-trons,

V12 =1

ur 1 − r 2u, s4d

and the term describing interaction with the fieldspolarizedalong thez axisd,

VF = Fstdsz1 + z2d. s5d

The electric field strengthFstd has an oscillatory componenttimes the envelope from the pulse, namely

Fstd = Ffstdcossvt + fd, s6d

with F, v and f the peak amplitude, frequency, and initialphase of the filed, respectively, and with

fstd = sin2spt/Tdd, s7d

the pulse envelope of duration

Td = n2p

v, s8d

wheren is number of cycles in the pulse.For convenience, we place the origin of our coordinate

system in the center of mass of the nuclei. Then two param-eters appear, namelyd, the distance between nuclei, andu,the angle between the molecular axis and thez axis sthepolarization axisd. Without loss of generality we assume thatthe molecule lies in thex-z plane, so that the potential ener-gies for each electron become

Vi = −1

ÎSxi +d

2sinuD2

+ yi2 + Szi +

d

2cosuD2

−1

ÎSxi −d

2sinuD2

+ yi2 + Szi −

d

2cosuD2

. s9d

Within this model the only difference between different di-atomic species lies in the distance,d, between the nuclei. Asmentioned in Sec. I, what is considered here is a hydrogen-like molecule with different internuclear spacing. Internu-clear distance,d=2.067 a.u., is referred to as N2 moleculeand internuclear distance,d=2.28 a.u., as O2 molecule, re-spectively.

III. LOCAL ANALYSIS

We consider here the evolution of electrons in combinedCoulomb and external fields after a rescattering event oncethe highly excited compound state has formed. For one elec-tron in the Coulomb potential, a superimposed external fieldopens a Stark saddle over which electrons can ionize. In ourcase we have two electrons and if there were no interactionbetween them they could escape simultaneously through thesame saddle on top of each other. But with repulsion betweenelectrons taken into account, the Stark saddle splits into twosaddles that lie on opposite sides of the field polarizationaxis. In the case of atoms the two saddles lie symmetricallywith respect to the polarization axis and the motion of theelectrons can be analyzed in some symmetry subspacef6,7g.For diatomic moleculesssuch as N2 or O2d that does notgenerally occur since they possess their own symmetry axiswhich can be oriented at any angle with respect to the polar-ization axis, thus destroying the global symmetry. Neverthe-less, an external field will introduce saddles over which elec-trons can escape, and the arguments presented inf6–9g againsuggest that electrons that escape simultaneously during onehalf-cycle of the field have to pass close to the interactingelectron saddles.

Before the double ionization escape both electrons passclose to the nuclei where they interact strongly with eachother and with the nuclei. For that reason we may assumethat all memory of the previous motion is lost. Then it ispossible to assume that the compound state which decays toa doubly charged molecule can be modeled classically by astatistical distribution for two electrons close to the nuclei.Furthermore, the classical motion of the electrons is fastcompared to the field oscillations and an adiabatic approxi-mation, keeping the field fixed, becomes useful in the analy-sis of the ionization channels. Using this adiabatic assump-tion we will identify and describe saddle points near whichthe electrons can escape nonsequentially.

For a molecule oriented along the field axis the problempossesses axial symmetry. Then switching to the cylindricalcoordinatessri, wi, zi for ith electrond one can easily define asymmetry subspace of electron motion. Restricting the elec-trons to a planesi.e., w1−w2=pd their coordinates in theC2vsymmetry subspace arer1=r2=R, z1=z2=Z and the potentialenergy reduces to

V = −2

ÎR2 + SZ −d

2D2

−2

ÎR2 + SZ +d

2D2

+1

2uRu+ 2FstdZ.

s10d

The saddle configuration in this plane then corresponds to aring of saddles around the field axis in the full configurationspace.

With a molecule oriented perpendicular to the field thereare twoC2v symmetry subspaces. One subspace is defined inthexzplane, the other in theyzplane. In the former case, the

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electron coordinates in the subspace arex1=X, y1=0, z1=Zandx2=−X, y2=0, z2=Z and the potential energy reads

V = −2

ÎSX −d

2D2

+ Z2

−2

ÎSX +d

2D2

+ Z2

+1

2uXu+ 2FstdZ.

s11d

For saddles which are in theyz plane coordinates arex1=0,y1=Y, z1=Z and x2=0, y2=−Y, z2=Z and the potential en-ergy is

V = −4

Îd2

4+ Y2 + Z2

+1

2uYu+ 2FstdZ. s12d

The potential energies Eqs.s10d–s12d and are shown in Fig. 1for a set of parameters corresponding to the nitrogen mol-ecule with an internuclear distance ofd=2.067 a.u. and forthe field Fstd=0.07 a.u., corresponding to an intensity of1.731014 W/cm2. The saddles are clearly visible.

The case of a general orientation of the molecule is simi-lar to the one with a perpendicular orientation: there are ingeneral two saddles for nonsequential ionization, one in theplane defined by the molecular axis and the field, and a pairof saddles outside this plane. These two channels becomeequivalent when the molecule is parallel to the field axis andthe axial symmetry is restored.

For a fixed external field the local stability analysis of thesaddles reveals a few stable and several unstable directions.One unstable direction corresponds to the “reaction coordi-nate” for double ionization. The other unstable directionsreflect interactions that will push electrons away from thedouble ionization path; if successful, they will push one elec-tron back to the molecular core. If only one electron escapesthe remaining one will typically be in a highly excited stateand will have a chance to singly ionize during another fieldcycle. But following the distinction introduced earlier, wewould not call such an event nonsequential.

The cross section behavior for nonsequential double ion-ization events can be obtained from the Lyapunov exponentsthat characterize the different directions. As in the case ofdouble ionization without a field, analyzed many years agoby Wannier f17–20g, the competition between the variousunstable directions gives rise to an algebraic variation of thecross section with energy close to the threshold, namely,

ssEd ~ sE − VSda, s13d

whereVS is the saddle energy and the exponent contains theLyapunov exponents,

a =oi

li

lr; s14d

lr is the Lyapunov exponent of the unstable direction corre-sponding to the nonsequential double ionization path, andliare the Lyapunov exponents of all other unstable directionsof the saddlef8,21g.

The properties of the saddles depend on the internucleardistance,d, as shown in Fig. 2. Starting withd=0 sin whichcase the orientation angleu is meaningless since a moleculereduces to an atomd and increasingd the energyVS of thesaddle corresponding to the parallel orientationsu=0d is al-ways the lowest and it decreases. For the other extremalorientation, i.e.,u=p /2, the energies of the saddlessthereare two saddles because of the axial symmetry breakingdincrease and their values are the highest. Analyzing the de-pendence ofa on the internuclear distance we see that in-creasingd the cross section exponent of one of the saddlesscorresponding tou=p /2d goes up while the other goesdown. The exponent foru=0 increases only slightly.

FIG. 1. Section through equipotential surfaces of the adiabaticpotential,V=V1+V2+V12+VF, for fixed timet and for two symmet-ric orientation of the molecule, namely parallelfu=0, top panelcorresponding to Eq.s10dg and perpendicular to the field polariza-tion axis, Z fu=p /2, middle and bottom panels corresponding toEq. s11d and Eq. s12d, respectivelyg; the molecular parameterd=2.067 a.u. and the external fieldFstd=−0.07 a.u.

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Variations of the parameters of the saddles with the ori-entationu are shown in Fig. 3 for the internuclear distancescorresponding to nitrogen, N2, with d=2.067 a.u., and oxy-gen, O2, with d=2.28 a.u. The energyVS of all saddles in-creases and the exponenta of one member of the saddlepairs increases while the other decreases. Figure 3 shows thatthe energies and exponents for the N2 and O2 molecules arequite similar—the largest differences between the species areof the order of few percent. Taking into account that theexperimental resultsf15,16g are for a statistical mixtures ofdifferent molecular orientations, we may conclude that fromthe point of view of our local analysis of the nonsequentialdecay channels one should not expect differences betweenN2 and O2.

IV. NUMERICAL SIMULATIONS

So far we have discussed the local analysis of the poten-tial within the adiabatic assumption. Now we go one stepfurther and perform numerical simulations of the dynamicsof the ionization process.

Basic model of the rescattering processf10,11g assumesthat the electron, after tunneling, interacts only with the ex-ternal field and is returned, by this field, back to the nucleus.From that model follows that the electron returns to thenucleus most probably when the field is zero. On the otherhand, our local analysis implements that in order to havedouble nonsequential ionization event the saddle point in po-tential, through which electrons may escape simultaneously,is needed. From that fact it is evident that events when elec-

tron is returned back to the nucleus while the field is nonzeroseem to dominate nonsequential ionization process. Similarconclusion is obtained from the analysis of experimental datafor atoms; see Ref.f22g. In the following we perform nu-merical simulations starting with random initial phases of thefield.

To avoid numerical divergences in the integration of theequations of motion, we introduce a smoothing factore inthe Coulomb potentials between the electrons and nuclei, Eq.s9d f23,24g. Then the potential terms read

Vi = −1

ÎSxi +d

2sinuD2

+ yi2 + Szi +

d

2cosuD2

+ e

−1

ÎSxi −d

2sinuD2

+ yi2 + Szi −

d

2cosuD2

+ e

.

s15d

We choosee=0.01; it introduces negligible changes of thesaddle energiesVS and exponentsa compared to the pureCoulomb values shown in Figs. 2 and 3.

During the final stages of a rescattering event both elec-trons are highly excited and they pass close to the nucleiwhere they interact strongly with them and with each other.Therefore, it is reasonable to assume, as mentioned in Sec.III, that all the memory of the earlier motion is lost. Then theinitial state of the final stage for double ionization is a sta-tistical distribution of two electrons close to the nuclei.

FIG. 2. Energy of the saddlestop paneld and cross section ex-ponentsbottom paneld as a function of internuclear distance,d, fororientations of the molecule parallel and perpendicular to the fieldaxis,u=0 andu=p /2, respectively. The dashed line corresponds tothe saddle in thexz plane and the dash-dotted line to the saddle inthe yz plane. The external field isFstd=−0.07 a.u.

FIG. 3. Energy of the saddlestop paneld and cross section ex-ponentsbottom paneld as a function of orientation angle,u, for N2

sd=2.067 a.u.d and O2 sd=2.28 a.u.d molecules. For both moleculesthe top line corresponds to the saddle in theyz plane, whereas thebottom line corresponds to the saddle in thexz plane. The externalfield is Fstd=−0.07 a.u.

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Hence we choose initial values of positions and momentawith respect to the microcanonical distribution for a giveninitial energyE. The energy should be in the range between−I and −I +3.17Up, where I is the ionization energy. Thepositions are chosen microcanonically, butsfor conveniencedwith the additional constraints thatzi =0 and Îxi

2+yi2

,85 a.u.ssee discussion inf6gd and we start all simulationsat the peak of the laser pulse with the phase of the fieldchosen randomly. In all simulations an ensemble of 105 tra-jectories is used.

To simulate a real experiment one should consider a dis-tribution of different initial energy of the complex and alsodifferent moments in a pulse when the complex is created.Within our model a fully quantitative description of the ion-ization process is not possible. We restrict ourselves to analy-sis how final distributions of electron momenta change witha change of the initial energy and pulse duration. We start allsimulation at the peak of the pulse—starting at different mo-ments changes the range but not the structure of the distri-butionsf6,7g.

Distributions of the final electron momenta parallel to thefield axis for different initial energyE and for different pulsedurations are presented in Fig. 4. All data in the figure cor-respond to the N2 molecule oriented along the field axis. Forvery short laser pulsessn=2 cyclesd signatures of simulta-neous electron escape are clearly visible—the distributionsare localized along the diagonals indicating that the electronsescape predominately by passing close to the saddles ana-lyzed in Sec. III. For longer pulses the distributions changetheir character. The first and fourth quadrants of the panels

become strongly populated implying that a number of se-quential decays significantly increases. The reason for that isquite obvious. After rescattering when a highly excited two

FIG. 5. Final distribution of ion momenta parallel to the fieldaxis corresponding to Fig. 4.F=0.07 a.u., v=0.057 a.u., d=2.067 a.u. and u=0. sad E=−0.3 a.u., n=2—solid line;n=26—broken line; sbd E=−0.6 a.u., n=2—solid line;n=26—broken line.

FIG. 4. Final distribution ofthe parallel momenta for differentinitial energy, E, and differentpulse length si.e., number ofcycles in the pulse,nd in thedouble ionization of the nitrogenmolecule parallel to the field axis.The field strengths isF=0.07 a.u.,the frequencyv=0.057 a.u., theinternuclear distance d=2.067 a.u., and the orientationparallel to the field,u=0. Theplots differ in initial energy Eand the number of field cyclesn:sad E=−0.3 a.u., n=2; sbd E=−0.3 a.u., n=26; scd E=−0.6a.u., n=2, and sddE=−0.6 a.u.,n=26.

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electron complex is created, both a nonsequential double es-cape or a single ionization are possible. During the nextcycles, a singly ionized molecule can emit their second elec-tron, thus also contributing to double ionization. As the pulsebecomes longer, the number of such sequential ionizationevents increases, even to the point of overwhelming thenumber of direct, nonsequential double ionization events.Thus for sufficiently long pulses, even though the rescatter-ing scenario is involved in the double ionization process, themomentum distributions will also show signatures of the se-quential electron escape. If the initial energyE is muchhigher than the minimal energy of a saddlesVS<−1.2 a.u.dthen the probability of nonsequential ionization is bigger andeven if the pulse duration is quite longfe.g.,n=26 in panelsbd of Fig. 4g the signatures of nonsequential process remainnoticeablefcontrary to panelsdd where forE=−0.6 a.u. suchsignatures are not visibleg.

Figure 5 shows the distributions of ion momenta parallelto the field that correspond to the data presented in Fig. 4. Asexpected, the distributions are much narrower in the casewhen sequential ionization dominates than in the case whennonsequential processes dominate.

When we change the orientation of the molecule the dis-tributions do not change significantly as one can see by com-paring Fig. 6 and Fig. 4. For the field amplitude used in thesimulationsswhich corresponds also to the experimentf15gd

the saddles for nonsequential process are far away from thenuclei and the positions and other parameters of the saddleschange only slightly with the angle of orientationu—compare Fig. 3.

Finally, in Fig. 7 we show data for the O2 molecule. Asexpected, from the local analysis presented in Sec. III for thesame initial energiesE and pulse durations as in the case ofthe N2 molecule, the observed momentum distributions arevery similar, Fig. 4. In the experiment of Ereminaet al. f15g,the results for O2 differ considerably from those for N2. Thelatter shows signatures of simultaneous electron escape whilefor O2 it seems that sequential processes dominate. Ouranalysis indicates that there is practically no difference be-tween N2 and O2 if similar initial conditions for the highlyexcited complex are used. It thus strongly suggests that thedifferences between the observations in for the two mol-ecules are due to differences in the early stages of the exci-tation process and the nature of the compound state beforethe final decay towards multiple ionization.

This conclusion agrees well with results off25,26g. Thelatter indicate that the symmetry of the valence orbitalsN2hassg, bonding symmetry and O2 haspg, antibonding sym-metryd has an influence on the above-threshold ionizationspectra. Thus it may affect also the excitation process and, inparticular, the initial conditions for decay of highly exitedcomplex for different molecular species. That, in turn, may

FIG. 6. Final distribution of the parallel momenta for the nitro-gen molecule at nonzero angle to the field axis.F=0.07 a.u.,v=0.057 a.u.,d=2.067 a.u.,E=−0.3 a.u., andn=2. The angles aresad u=p /4 andsbd u=p /2.

FIG. 7. Final distribution of the parallel momenta for the oxygenmolecule parallel to the field axis for different pulse length.F=0.07 a.u.,v=0.057 a.u.,d=2.28 a.u.,E=−0.3 a.u., andu=0. sadn=2 andsbd n=26.

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modify the final parallel momenta distributions.Our analysis also indicates that very short laser pulses

should increase the ratio of non-sequential to sequential ion-izations. It opens a possibility for observation of clear signa-tures of simultaneous electron escape both for N2 and O2molecules, but in particular for oxygen, where the sequentialevents seem to dominate the nonsequential one.

V. SUMMARY

To summarize, we have performed a purely classicalanalysis of the final stage of the nonsequential double ion-ization of molecules in the strong laser field. It is based onthe fact that all trajectories leading to the nonsequential es-cape of electrons have to pass close to the saddles in thepotential that is formed when the laser pulse arrivesf6–9g.

We have started with the local analysis of the potentialwithin the adiabatic approximation, and we have identifiedand described the saddles. Thereafter, we have shown resultsof numerical simulations. The later allow us to draw twoconclusions.sid From the point of view of classical analysis,within the considered model, there is no difference betweennitrogen and oxygen molecules in the sense that both of them

can show signatures of simultaneous double escape.sii d Ori-entation of the molecule with respect to the field axis doesnot influence significantly the final momentum distributionfor the initial energy range considered in the model. Theseconclusions are based on classical analysis of decay ofhighly excited two electron complexsthat is formed in arescattering eventd where a molecular core is approximatedby two positively charged nuclei.

The numerical results and their interpretation suggestsstrongly that shorter laser pulses should lead to an increase ofthe ratio of nonsequential to sequential ionizations. That sug-gestion could be tested experimentally. Higher relative effi-ciency of nonsequential process would be visible in the mo-mentum distributions as a more pronounced symmetricalescape of the electrons.

ACKNOWLEDGMENTS

We are grateful to E. Eremina for making the experimen-tal results accessible before publication. This work waspartly supported by the Polish Ministry of Scientific Re-search Grant PBZ-MIN-008/P03/2003 and by the DeutscheForschungsgemeinschaft.

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