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The helium atom and helium-likeions' interaction with XFEL radiationS. Laulan a , H. Bachau a , B. Piraux b , J. Bauer b & G.Lagmago Kamta ca Centre Lasers Intenses et Applications, UMR 5107 duCNRS, Université de Bordeaux I , 351 Cours de la Libération,F-33405, Talence, France E-mail:b Laboratoire de Physique Atomique et Moléculaire,Université Catholique de Louvain , 2 Chemin du Cyclotron,B-1348, Louvain-la-Neuve, Belgiumc Department of Physics and Astronomy , The University ofNebraska , 116, Brace Laboratory, Lincoln, NE68588–0111,USAPublished online: 15 Jan 2009.

To cite this article: S. Laulan , H. Bachau , B. Piraux , J. Bauer & G. Lagmago Kamta (2003)The helium atom and helium-like ions' interaction with XFEL radiation, Journal of ModernOptics, 50:3-4, 353-364, DOI: 10.1080/09500340308233535

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+ Taylor &Francis JOURNAL OF MODERN OPTICS, 2003, VOL. 50, NO. 314, 353-364 0 Taybr hFrancir Gmup

The helium atom and helium-like ions’ interaction with XFEL radiation

S. LAULAN?, H. BACHAU?, B. PIRAUXS, J. BAUERS and G. LAGMAGO KAMTAg t Centre Lasers Intenses et Applications, UMR 5107 d u CNRS, Universite de Bordeaux I, 351 Cours de la Liberation, F-33405 Talence, France; e-mail: bachau6Jcelia.u-bordeaux.fr $ Laboratoire de Physique Atomique et MolCculaire, Universite Catholique de Louvain, 2 Chemin d u Cyclotron, B-1348 Louvain-la- Neuve, Belgium 4 Department of Physics and Astronomy, T h e University of Nebraska, 116, Brace Laboratory, Lincoln, NE68588-0111, USA

(Received 1 March 2002)

Abstract. We study double electron ejection with one- and two-photon absorption in He and He-like ions. We use two different numerical methods aimed at solving the time-dependent Schrodinger equation for two-active electron systems interacting with a strong and short XUV laser pulse. The two approaches, which are relevant to the Configuration Interaction (CI) method, are based on an expansion of the wave function on either B-splines or Sturmian functions for the radial part and bipolar spherical harmonics for the angular part. We treat first double electron ejection with one photon in helium. Then the case of double electron ejection with two photons is studied along the isoelectronic series of He. We calculate the total probabilities for double ionization as well as the energy distribution of the electrons in the double continuum. Emphasis will be put on the role of the correlations along the He isoelectronic series. The pertinence of the distinction between direct and independent-particle (sequential) models for double ionization will be discussed in the context of short pulses.

1. Introduction New X-ray free electron laser (XFEL) facilities open the way to explore light-

matter interaction at high frequency and high intensity, in the short pulse regime. As a result, it will be possible to investigate atomic and molecular processes involving few photon absorption and more than one electron. T h e groups at Bordeaux and Louvain-la-Neuve have developed, in parallel, methods of spectral and configuration interaction type to solve the time-dependent Schrodinger equa- tion (TDSE) for two active atomic systems. Particular attention has been placed on developing approaches which treat properly the electron-electron correla- tions so as to provide an accurate description of many eigenstates simultaneously. We use L2 functions, built as products of either B-splines or Coulomb Sturmians, which permit one to incorporate electron correlations at a high degree of accuracy. In both cases, quantitative information for single and double ionization can

Journal of Modern Opfics ISSN 0950-0340 print/ISSN 1362-3044 online 0 2003 Taylor & Francis Ltd http://www.tandf.co.uk/joumals

DOI: 10.1080/09500340210152611

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be obtained. Our approaches have been shown to be efficient in the context of ultrashort (fs) pulses where the dynamics have not been studied in detail so far. In particular, we considered the ionization of H- in strong fields [ l , 21 (where the electron correlations play a major role) and the multiphoton excitation of autoionizing states of He with photon energies ranging from 5.4 to 32.6eV [3]. Methods using L2 functions have been shown to be efficient to calculate double photoionization through one-photon absorption (see [4, 51 for recent publications and other references therein). Yet very little quantitative information is known for double photoionization in the multiphoton regime. We note the recent works of Kornberg and Lambropoulos [6] and Parker et al. [7] in the perturbative and the non-perturbative regimes, respectively. Two-photon absorption processes in strong fields resulting in two-electron ionization of He and H- have also been studied by Pindzola and Robicheaux [8]. With the earlier paper of Bachau and Lambropoulos [9 ] , who considered the general features of the He double ioniza- tion photoelectron energy spectrum in the multiphoton regime, these contribu- tions remain the rare relevant studies on the subject. We focus here on processes which involve XUV photons, He atom and He-like ions. At intensities above lo’’ W cmV2 and short pulse duration, we will see below that the interaction of XUV photons with He presents features which deviate from LOPT (lowest order perturbation theory). Therefore, at these intensities, it is necessary to solve the time-dependent Schrodinger equation.

We will present results obtained for one- and two-photon ionization of helium and helium-like ions above the double ionization threshold. We have shown [lo] that our method, already used to study the multiphoton excitation of doubly excited states [3], can be extended to the study of double electron ejection in the multiphoton absorption regime. In contrast with double ionization (or double excitation) through one-photon absorption, two-electron ejection resulting from more than one-photon absorption does not necessarily require electron correla- tions to contribute. In this case, it is therefore justified to perform the calculations by using the perturbation theory in 1/r12. We expect that, for two-electron ejection/excitation through two-photon absorption, the zero order of the perturba- tion theory dominates and it is exact in the limit for heavy ions (in the non- relativistic approximation). The study of the contribution of the different orders along the isoelectronic series of helium should therefore directly provide informa- tion about the role of double continuum correlations in the multiple ionization process. We focus here on calculations where the double continuum is described in the zero order of the perturbation theory.

2. Theoretical approach 2.1. Atomic structure calculations

atom satisfies the equation Let us first briefly describe the method. The electronic structure of the helium

( H - E,)Y, = 0. (1)

H is the non-relativistic Hamiltonian which is given as

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The helium atom and helium-like ions' interaction with XFEL radiation 355

where 2 is the nuclear charge. The labels 1 and 2 refer to electron 1 and 2 respectively. For a given total angular momentum L and projection M, the solution of equation (1) is expanded on a basis of two-electron configurations which are products of one-electron functions as follows

where A is the antisymmetrization operator, li is the angular momentum of electron i and y k ; r ( l , 2) is a bipolar spherical harmonic. The radial functions F, ( r ) refer to either B-spline functions of order k or Coulomb Sturmian functions of the nonlinear parameter K . The B-spline function of order k , denoted by Bb(r), is a piecewise polynomial of degree k - 1 [ l l ] . The Nb B-spline functions are spanned, along the radial axis, in a 'box' defined from r = 0 to R,,,. As usual, we choose the B-spline sequence so that Bt(0) = 0 and B ~ b ( R , , , ) = 0 in order to fulfil the boundary conditions. The Coulomb Sturmian function for a given angular momentum 1 is given by

Stn = Ntnr'+' exp (-tw-)Li3!l ( ~ K Y ) , (4)

where N;,, is a normalization constant and L:3Ll a Laguerre polynomial. K is the so-called nonlinear parameter. In our Sturmian basis, various sets of tc parameters are used simultaneously. This allows one to span efficiently different regions of space and to optimize the construction of the total wavefunction while keeping the basis size within reasonable limits. In addition, by choosing these nonlinear parameters to be complex, it is possible to build in the correct asymptotic behaviour of the total wavefunction [12]. Finally, let us mention that within this basis, it is straightforward to calculate the total wavefunction in momentum space.

A direct diagonalization of equation (1) gives the eigenenergies and the expansion coefficients cf;: of the bound and continuum states. By using complex nonlinear parameters for the Coulomb Sturmian functions, we also have access to the energy and width of the autoionizing states. Since the basis has a finite dimension, we obtain a discretized representation of the whole atomic spectrum. B-spline expansions have been shown to be extremely efficient in many situations in atomic and molecular physics: see [13] for a recent review. Coulomb Sturmian functions prove themselves very useful in reproducing accurately the energy and width of high-lying Rydberg bound and doubly excited states [14].

2 .2 . Time-dependent calculations Let us now consider the interaction between the atom and the laser field.

Within the dipole approximation, the time-dependent Schrodinger equation to be solved is given in the velocity gauge by

The vector potential, polarized along the z axis, is defined as

A(t) = A o ( ~ o s ~ t ) ~ s i n ( w t ) e , , T

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where T is the total pulse duration and w is the photon energy. Similarly, in the length gauge, the time-dependent Schrodinger equation reads as

with the electric field defined as

(8) d at

The time-dependent total wavefunction is expanded on the basis of the field-

E(t) = --A(t).

free atomic eigenstates (see equations (l), (2) and (3)), normalized to unity

n m , L

Equations (5) and/or (7) are integrated over the total pulse duration T. The initial conditions are given by

Yl(rl,r2,t = -T/2) = Yu(rl,rZ,t = -T/2) = YkM=’(rl,r2), (10) where YiL,’‘=o(rl, r2) is the initial state of the field-free system, i.e. the ground state of He or He-like ion in the present case. In the following we will only mention M in the angular parts; it is assumed that M = 0. It is also useful to define, at t = T / 2 , the wavefunction Y;(rl, r2, t = T/2) which is the part of the total wavefunction Ya(rl, r2, t = T/2) located above the double ionization threshold (we subtract all negative energy components in Y”(r1, r2, t = T/2) [lo])

In order to obtain quantitative information about the probability for double ionization, we have used two different approaches. In the first one, we project the total wavefunction (or, alternatively, the function Y;(rl, r2, t = T/2)) onto a product of antisymmetrized hydrogenic orbitals @(r) = &(r) Y;t(B, 4 ) at the end of the pulse:

where i = 1,2 and Ho(i) is defined in equation (2). As in the two-electron case, the above equation is solved by expanding the function 4E(ri) either on B-splines or Coulomb Sturmian functions. For the sake of coherency, we use a B-spline/ Coulomb Sturmian basis set similar to the one used to solve equation (3). To calculate the electron energy distributions in the double continuum, we need to evaluate the density of probability dPL(kZk’Z’)/dkdk’; it is given by

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The helium atom and helium-like ions’ interaction with XFEL radiation 357

= P(klk’l’)p( kl)p(k’l‘), dPL(klk’l‘)

dk dk’ where p(k1) and p(k’1’) are the density of states in the continua k1 and k’l’, respectively. As usual, we calculate directly these densities from the eigenenergies Ek, [13]. T o calculate the total probability of double ionization, we simply sum the probabilities PL(klk’l’) for all possible pairs (kl, k’l’) with Ek,Ekt > 0 and all L values. Note that it is easy to calculate the contributions of the different angular pairs (I, 1’) and/or of total angular momentum L values. In the second approach, we consider the total wavefunction in momentum space and subtract all negative energy components. The corresponding ionizing wavepacket is then written as follows:

yI(PI,P,) = ~&rn(Pl)@nlrn(P2) -k ~An/rn(P2)@nlm(Pl) -k ydc(Pl,P,), (17) n , b nJ,m

where Qnlrn(p) is a bound state of the He’ ion and the third term Ydc(p1,p2) is an antisymmetrized wavefunction describing the double continuum states of helium. The first two terms on the right-hand side of the above equation are associated with the single continua of helium and describe the single ionization with or without excitation of the residual Hef . Since the ionizing wavepacket Yy~(p,, p2) is known at the end of the pulse, we obtain the double continuum wavefunction by subtracting from the total ionizing wavepacket, the single continuum wave- function.

The difficulty of disentangling single and double ionization contributions from the final wavefunction remains a very challenging problem. In addition to the fact that the two-electron single and double continuum states may be degenerate in energy, the numerically built positive energy states contain necessarily both single and double continuum components because our finite basis does not allow one to describe either the single or the double continuum wavefunction in the asymptotic region. Let us stress that the two approaches we developed to extract the double ionization probability are in principle equivalent, but this is only true if our two bases are complete.

2.3. Scaling of the time-dependent Schrodinger equation for He-like ions In the case of hydrogenic ions it is easy to show that, provided that the laser

parameters are changed according to table 1, it is possible to scale the LOPT equations or TDSE with 2 [15, 161. This also applies to He isoelectronic series if (i) we follow the 2-scaling law shown in table 1 and (ii) we neglect the correlation term l/r12, i.e. in zero-order perturbation theory (PT) with 1/Z as a parameter (zero-order PT). If the two latter conditions are fulfilled, the time-dependent coefficients in equation (9) and the associated probabilities (excitation, ionization, etc.) do not depend on 2. In contrast with one-photon absorption, double

Table 1 . Z-scaling law for laser and B-spline basis set parameters.

Parameters Nuclear charge: Z = 1 Nuclear charge: Z

Photon frequency Pulse duration Intensity

W

T I

w x z2 z x z6 T x Z-2

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ionization probability through two-photon absorption is different from zero in the zero-order PT. Note that, strictly speaking, equation (1 2) is exact in the zero-order P T since the double continuum is described as a product of hydrogenic functions. Therefore the approach described above to calculate the double ionization prob- ability should give the dominant contribution for low 2 and it is exact in the limit of high 2 (in the non-relativistic approximation). It is easy to show [9] that double photon absorption from ls2 in zero-order PT leads to the production of electrons with I = 1’ = 1 at the end of the pulse. The presence of other angular channels is due to electron correlations. At this point it must be stressed that correlations are included during the numerical solution of the TDSE (as well as in the description of the initial state). Therefore, the accuracy of our results is mainly limited by the description of the final double continuum state. In other words, the comparison of our results with ‘exact’ calculations (or experimental results) should provide a direct evaluation of the importance of the correlation effects in the final double continuum state.

3. Results and discussion 3.1. Double ionization of helium by single photon absorption

In figure 1, we give the probability of total ionization, single ionization with and without excitation of the residual ion and double ionization of He as a function of the peak intensity of the laser pulse. We consider here a very short (sine square shape) laser pulse whose full width at half maximum is 5 optical cycles. The frequency w = 3.27 au (89 eV). These results have been obtained by using a basis of Coulomb Sturmian functions. About 4000 atomic states are included per total angular momentum L which varies from 0 to 4. With this basis, we obtain a ground

: T.I.. RC cqwkmr A : 7.1. : l l B E

- : D.I.:nCcquMkru - a , D.I.:TDSE

I 1 l 1 I l 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I 1 1 1 1

0.01 0.1 1 10

Peak Intensity (a.u.)

Figure 1. Probability of total (TI), single (SI) and double ionization (DI) as a function of the peak laser intensity in au for helium exposed to a sine square pulse whose full width at half maximum is 5 optical cycles and the frequency 3.27 au (89 eV). TDSE results are compared to those obtained from the rate equations.

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The helium atom and helium-like ions’ interaction with XFEL radiation 359

state energy equal to -2.9035 au which compares very well with the accurate result of Pekeris [17] (-2.903 724au). Various sets of (real) nonlinear parameters have been used to ensure a sufficient density of continuum states for a broad range of energies above the double ionization threshold. The time-propagation has been carried out by means of a 5th order embedded Runge-Kutta formula [18]. We compare our results with those obtained by solving the rate equations and then averaging the corresponding probabilities on the time profile of the pulse. We used the accurate one-photon direct double ionization rates and total ionization rates of Pont and Shakeshaft [19]. In order to include sequential double ionization, we also calculated the rate of ionization of He+ initially in the Is, 2s or 2p state. Our TDSE results and those obtained from the rate equations for total and single ionization agree perfectly. In the case of double ionization, and on the basis of the rate equation results (full line), we can define three regimes. In the first one below a laser peak intensity of 0.1 au, the double ionization is dominated by the direct process. Note that the slope of the curve is about 1 as expected since perturbation theory is assumed to be valid. Between 0.1 and 1 au for the peak laser intensity, the slope of the curve increases by a factor of 2: this means that we reach the regime where sequential double ionization begins to be dominant. Beyond 1 au, the curve saturates quickly as soon as the single ionization probability starts to decrease significantly. Our TDSE results agree with the rate equation results only when the sequential double ionization dominates. In the low peak laser intensity regime, there are significant discrepancies. We used here the second approach (projection of the total wavefunction onto the bound states of Hef) to extract the double ionization probability. But at low peak laser intensity, the direct double ionization probability is almost two orders of magnitude smaller than both the total and the single ionization probabilities, which almost coincide. This means that the total and the single ionization probability have to be calculated to a very high accuracy, which is not possible because of the numerics, and also because the way in which the single continuum component of the total wavefunction is extracted is not exact. Our two-in principle equivalent-approaches to extract the double ionization probability are based on the assumption that the final state the total wavefunction is projected on is uncorrelated. To what extent the correlation in the final state plays a significant role remains an open question. The analysis which follows sheds some light on this problem.

The results presented in the following tables and figures have been obtained by using a basis of B-splines. We use a basis set of 50 B-splines B ) ( r ) of order k = 7. They are placed on a linear knot sequence with R,,, = 50.0au. In expansion (9) we include the total angular momentum L = 0, 1 , 2 and 3. We show in table 2 the pairs (11,12) of angular momenta considered for each total angular momentum and the total number of combination terms in equation (3).

The eigenvectors and eigenenergies are calculated through a standard diag- onalization method. The fundamental state is found at -2.902au, a value which compares very well with the result of Pekeris [17] (-2.903 724au). We choose a photon energy of 3.6 au and a full pulse duration of 55.85 au (32 optical cycles). The maximum intensity is 3.51 x 1014 W cmP2. We assume a cosine squared pulse envelope, see equation (6), and a linear polarization of the field. The TDSE equation is solved in the length gauge and within the interaction picture by means of an explicit Runge-Kutta method of order 4. We include in equation (9) the 1500 lowest eigenstates for each total angular momentum L, therefore there is a total

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Table 2. ( 1 , , 1 2 ) pairs of angular momenta and total number of combination terms in equation (3) for each total angular momentum L.

Number of combination Angular momentum L ( I I , 1 2 ) angular pairs terms

L = O L = l L = 2 L = 3

number of 6000 eigenstates in the expansion. We have verified that, for each momentum L, the upper energy is well above the double ionization threshold. We have also verified that the density of continuum states in the ionization region is large enough to ensure that the wavepacket does not reach the limit of the box at the end of the pulse (see Bachau and Hasbani [lo], chapter 3). According to the previous discussion, at an intensity of 3.51 x 1014 W ~ m - ~ , the double electron ejection process should be close to the perturbative regime. Therefore, it is meaningful to compare our results with the theoretical double ionization cross- section calculated by Proulx and Shakeshaft [20] and Pont and Shakeshaft [19]. There are also recent experimental results where the distribution of energy between the electrons has been measured [21]. We found a double ionization cross-section of 7.4kb, a value that compares well with the result of Pont and Shakeshaft [19] which is close to 9.0 kb. We have plotted in figure 2 the density of probability for double ionization as a function of the electron energies El and E2. The wavefunctions being antisymmetrized, the electron energy distribution spec- trum is symmetric around the diagonal El = E2. We see that the distribution is almost flat, in agreement with the theoretical [20] and experimental [21] results. The calculation of the probabilities have been performed by using, in equation (12), Yo(rl,r2,t = T/2) or !PA(rl ,rZ, t = T/2) (see equations (9) and ( l l ) ) , with similar results.

3.2. Double ionization of helium by two-photon absorption We have performed calculations for different systems (He, B3+ and Ne8+) and

different laser parameters. Two-photon absorption populates the L = 0 and L = 2 symmetries. For each case, we have calculated the total double ionization prob- ability and the contributions of the L = 0 and L = 2 channels. The results are reported in table 3.

Figure 2. Electron energy distribution, see equation (16), for w = 3.6au, 2 = 2, Z = 3.51 x 1014 W cm-* and a pulse duration of 32 optical cycles.

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The helium atom and helium-like ions’ interaction with XFEL radiation 361

Table 3. Double ionization (DI) probability for different systems. Z is the nuclear charge of the atom (He) or hydrogenic ion. In all cases the pulse has a duration of 16 cycles. We report the total DI and the contributions of channels L = 0 and L = 2.

w ( a 4 z (4 Intensity (W ~ r n - ~ ) DI probability (Total, L = 0, L = 2)

1.9 2.0 3.5 1 (1 4) 2.0 2.0 3 . 5 1 (1 4)

12.5 5.0 8.57( 1 6) 50.0 10.0 5.48(18)

0.21(-3), 0.62(-4), 0.15(-3) 0.26(-3), 0.79(-4), 0.18(-3) 0.36(-3), O.ll(-3), 0.25(-3) 0.39(-3), 0.13(-3), 0.26(-3)

Figure 3. Electron energy distribution in channel L = 2, see equation (16), for w = 1.9 au, 2 = 2, Z = 3.51 x 1014 W cm-2 and a pulse duration of 16 optical cycles.

We consider first the case of He with a photon energy w = 1.9au. The laser parameters are given in table 3. We show in figure 3 the probability density for the electron energy distribution in the double continuum of symmetry L = 2, which dominates. Note that two-electron ejection is not permitted in the sequential model since the ionization of He+ is not possible through one-photon absorption. (Below we also use the term ‘independent-particle’ instead of ‘sequential’ which may be misleading [9].) Nevertheless, with the frequency bandwidth AW being of the order of 0.24au, the one-photon ionization of He+ is not strictly forbidden. The energy distribution of the electrons shows that one of the electrons is likely to be emitted with maximum energy while the other electron has an energy close to zero. The angular channel I = 1’ = 1 (see equation (13)), denoted (l , l) , dominates in the final state. In between the peaks the probability decreases, with a dip at

We study now the cases of He with w = 2.0au, B3+ and Ne*+ with the laser parameters scaled according to table 2. We see in table 3 that the channel L = 2 dominates in all cases. The distributions for He and w = 2.0au is more peaked at the edges of the spectrum than for w = 1.9 au (see figures 3 and 4). The difference comes from the fact that, with w = 2.0au, we have reached the limit where one- photon ionization of He+ is energetically possible and one electron has an energy close to zero while the other electron is left in the continuum with maximum energy. This is in agreement with the results of our earlier work [9] where we predicted that two peaks, located at the positions El and E2, should dominate the electron energy spectrum on top of a continuous contribution coming from the correlation effects. These two peaks are related to two-photon double ionization in the independent-particle model (i.e. in the zero-order PT). The separation energy IEl - E2I is given by the electron correlation energy in the initial state. In the limit of a fully independent-particle energy in the initial state, we would have only one peak. In the electron energy distribution spectrum, this situation corresponds to

El = E2.

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Figure 4. Electron energy distribution, in channel L = 2, for w = 2.0au, Z = 2, I = 3.51 x 1014 W cm-' and a pulse duration of 16 optical cycles.

Figure 5 . Electron energy distribution, in channel L = 2, for w = 12.5 au, Z = 5, I = 8.57 x 10l6 Wcm-* and a pulse duration of 16 optical cycles.

two symmetric peaks located at the positions ( E l , E2), where El + E2 is the energy available above the double ionization threshold and [El - E21 is the correlation energy in the initial state. This is illustrated in figure 4 where the peaks are well separated from the diagonal El = E2. Their width, given by Aw (e 0.25au), is much smaller than the correlations in l s2 (1.1 au in the helium case). We compare now the energy distributions for He (figure 4) and B3+ (figure 5); we note that the peaks broaden in figure 5. The frequency bandwidth is now about 1.5au while the correlation energy in the ground state is about 3.0au. As a consequence the distinction between the direct and independent-particle models loses its perti- nence. We also clearly note the dip at El = E2 in figure 5 . These effects are even more pronounced in the case of Ne8+ (not shown here) where the bandwidth is of the order of the correlation energy in the l s2 state (Aw M 6.0 au). In fact, we have observed that as the pulse duration shortens, the peaks broaden and get closer to the diagonal El = E2. We examine now the double ionization probabilities for the cases Z = 2 with w = 2.0au, 2 = 5 and 2 = 10. In agreement with section 2.3, we note in table 3 that the probabilities for double ionization tend to a constant value when Z increases. We have also found that, in agreement with zero-order PT, the angular momentum pair (1,l) dominates in the double continuum. On the basis of angular algebra, it is easy to show that the pair (1,l) gives a relative contribution of 1/2 for channels L = 0 and L = 2. Table 3 shows that this ratio is reached as 2 increases. As expected, for 2 = 5 and 2 = 10, the contribution of the angular pairs ( O , O ) , (2,2), (3,3) (0,2) and (1,3) in the double continuum is negligible.

4. Conclusion We have presented two numerical methods to treat the problem of double

ionization in the context of strong XUV laser and short pulses. Electron correla- tions are taken into account in the initial state and they are also present in the numerical solution of the TDSE. The double ionization probability is extracted from the final wavefunction either by projecting this function onto a product of

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uncorrelated hydrogenic continuum states or indirectly after projecting this function on the eigenstates of Het. In the case of one-photon direct double ionization, the comparison between our TDSE results and those obtained with the rate equations indicates clearly that correlation in the final state plays a significant role. The rate equation results allowed us to distinguish between three laser intensity regimes for the double ionization probability; a low intensity regime where the direct process is dominant, an intermediate regime where the sequential process begins to take over and a high intensity regime, beyond 1 au, where sequential double ionization becomes more important than single ionization. Two- electron ejection of helium with two photons (with w M 2.0 au) exhibits an electron energy spectrum with two symmetric peaks. This is in contrast with double ejection with one photon that exhibits a spectrum which is more or less flat. Double electron ejection with two-photon absorption proceeds through uncorre- lated intermediate states to an independent-particle final state. In the long pulse regime, the peaks are located at the positions (El, E2), where El + E2 is the energy available above the double ionization threshold and IEl - E21 is the correlation energy in the initial state. As the nuclear charge increases, the probability for double ejection follows the scaling law given by the zero-order perturbation theory in 1 /Z. We observe a broadening of the electron energy distributions and the peaks shift toward the diagonal El = E2. Therefore, as the frequency bandwidth becomes of the order of the electron correlations in the initial state, the notions of direct or independent-particle models for double ionization lose their perti- nence. This opens the way to explore the multiple ionization dynamics in multi- electron systems (atoms, metals, etc.) during ultrashort laser-atom interactions.

Calculations of the final double continuum state, including corrections beyond the zero-order PT, are underway.

Acknowledgments JB who is on leave of absence from the Department of Nuclear Physics and

Radiological Safety of the University of Lodz, Poland, thanks the ‘Fonds National de la Recherche Scientifique de la Communautk Francaise de Belgique’ for his post-doctoral position at the University of Louvain, Belgium.

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