VOLUME 78, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 17 MARCH 1997

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High Harmonic Generation Spectra of Neutral Helium by the Complex-Scaled (t,t0) Method:Role of Dynamical Electron Correlation

Nimrod Moiseyev* and Frank WeinholdDepartment of Chemistry and Minerva Center of Nonlinear Physics in Complex Systems, Technion-Israel Institute of Tech

Haifa 32000, Israeland Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706

(Received 5 August 1996)

Using highly correlated Hylleraas-type wave functions of1S and 1P symmetry, we show how theharmonic generation spectrum of1 1S He (including short wavelength members previously attributedto He1) arise from a radial-angular “descreening” mechanism. This provides the first indication thatharmonic generation spectrum can arise from strong dynamical electron correlations rather than purelyas a “one-electron phenomenon.” [S0031-9007(97)02676-8]

PACS numbers: 42.65.Ky

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Interest in tunable short-wavelength sources has stimlated numerous experimental and theoretical investigatiof harmonic generation spectra (HGS) of noble gasesintense laser fields [1]. Using an ultra-high-power Klaser (5 eV photons), Sarukuraet al. [2] found the rela-tive intensities of 9th to 23rd-order harmonics of heliufrom a single-shot experiment with peak intensity of2 3

1017 Wycm2. From the dependence of calculated neutatom and ion responses on field intensity [see Fig. (5Ref. [3] ], one can judge that below3.2 3 1015 Wycm2

the field is below saturation intensity, whereas fields eceeding1016 Wycm2 are above it. The3.5 3 1016 Wycm2

can be regarded as the field intensity where the transifrom neutral He to He1 occurs. Although at this intensity100% of helium is ionized after the end of the laser pu(,120 cycles), only 50% of helium atoms are ionized whthe laser pulse reaches 0.905 of its maximal value (a,55 cycles). Indeed, the best fit to the experimental HGwas obtained for intensities near3.5 3 1015 Wycm2.

Previous theoretical fits to the experimental spectrhave involved two distinct models. Fits up to the 13harmonic order were achieved for neutral helium atodescribed by a one-electron “frozen orbital” model Hamtonian or by Hartree-Fock (HF) or density functional metods [3–5]. In such approximations, dynamical electroncorrelations are ignored entirely (frozen orbital, HF)treated in the framework of the homogeneous electron(in which, e.g., the strong “radial” or “angular” correlationof inhomogeneous atomic systems have no direct counpart). The fit to higher harmonics, on the other hand, wachieved with a He1 model [4,5]. These combined fittingprocedures have led to common interpretation of HGSa “one-electron phenomenon” [4,5].

The present work employs highly accurate Hylleraatype wave functions with much greater variational flexibity to correctly describe the dynamical radial and angucorrelation effects in ground and excited resonance stof helium atoms. Many years ago, it was shown by Moseyev and Katriel [6] that coupling of radial and angulcorrelations can actually lead to the electrons being dra

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closer together, rather than pushed apart, toward the lowZend of the helium isoelectronic sequence. We have preously described the critical role of such radial and angulcorrelations in enhancing the lifetime of helium autoionizing states [7]. The simple physical picture underlyinsuch coupling effects can be sketched as follows: In atomthe dominant correlating motions in the1yZ perturbationexpansion are of radial (“in-out”) and angular type. Thformer cause the electrons to separate into inner and ouorbits, while the latter keep the electrons on opposite sidof the nucleus. The effect of the third-order coupling oradial and angular correlations is then to bring the eletrons back closer together, since the outer most electrwill now “see” the other electron tending to liebehindthenucleus, thus “descreening” the nuclear charge. This cofers some He1-like character on the inner part of the wavefunction, permitting the outer electron to move closer to thnucleus and to reduce the radial separation that was estlished in lower order. Although initially drawn for field-free resonances, this picture of dynamical correlations alhas obvious relevance for helium in intense laser fieldOf course, we do not employ perturbative approximations in our actual calculations, but we emphasize ththe variational treatment must have sufficient flexibility tocorrectly incorporate the physical correlation effects thappear in low perturbative order. For this purpose, thHylleraas-type wave functions (with explicit incorporationof interelectron distancer12) are unmatched accuracy andutility.

The fact that only odd-order harmonics were observein the HGS experiments, even when the atoms were eposed to short laser pulses, is an indication that the dnamics of the helium electrons in strong fields is controlleby asingleresonance Floquet state. Indeed, previous nmerical calculations [8] have shown remarkable agreemebetween HGS calculated from direct solutions of the timedependent Schrödinger equation with nonperiodic Hamtonians and the corresponding result from Fourier analysof the time- dependent dipole moment for asingle reso-nance Floquet quasienergy state,provided the duration

© 1997 The American Physical Society

VOLUME 78, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 17 MARCH 1997

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of the laser pulse is larger than 15–50 oscillations ofelectric field (see Fig. 7 in Ref. [8]). We therefore studthe dynamics of helium in an ac field from the Floqustarting point, using the momentum gauge.

As a basis set we used the ground and lowest exc1S and 1P states of field-free helium, the lowest1S and1P autoionizing resonances, and a discrete representaof the continuum which was rotated into the lower halfthe complex energy plane by the complex-coordinatetation technique [9]. The bound states, autoionizing renances (“double excited” states), and rotating continuaall numerical eigenfunctions of the complex-scaled Hamtonian [10]. The advantage of using the complex scing method lies in the dramatic reduction in the numbof states needed to calculate the converged resonancequet state. Upon complex scaling, the autoionization renances (which are present also in the field-free spectrand associated Floquet quasienergy resonances are rsented bysquare-integrablewave functions of pronouncedbound-state character (nodal structure related to quannumber, etc.). One may say that the use of complex sing “collects” information about the decay process froan infinite number of continuum states and “concentratit into a single square-integrable function. The price wpay is that the complex-scaled Hamiltonian is not Hermtian and the associated scalar product should be replaby the more generalc-product [11].

The bound and resonance states are localized in thetential interaction region, whereas the rotating continustates are delocalized functions having nearly zero amtude in this region. Therefore, only the bound (grouand excited), resonance, andlow-lying rotating contin-uum states (i.e., those close to the threshold energwill make large contributions in the basis-set expansionthe resonance Floquet wave function. These basis stwere obtained by diagonalizing a complex-scaled mtrix of higher order for the field-free electrostatic Hamitonian. The Hylleraas-type “primitives” underlying thicalculation are of the forms1 1 P12drl

1rm2 rk

12scosu1dl 3

exps2b1r1 2 b2r2d where P12 permutes electron levelsand the other symbols have their usual meaning [1This basis is in principle complete forany choice of thenonlinear parametersb1, b2 for both S statessl ­ 1d,provided sufficiently high integer powers ofr1, r2, andr12 are included. We included all (138:95 for1S sym-metry states and 43 for the1P ones) possible terms fowhich l 1 m 1 k is less than or equal to 8, and we chob1 ­ b2 ­ 1.0 for Sstates andb1 ­ 0.9, b2 ­ 1.7 for Pstates, sufficient to reproduce the known eigenvalues [of the ground and first 5–10 excited bound states (with zwidth) to at least1025 a.u. accuracy for both symmetrieNote that2-b corresponds approximately to the screeniof one electron by the other, so that theP-type Hylleraasfunctions are chosen withbinner for nearly a bare He1-line charge. The leading autoionizing resonances “s2sd2,”“2s2p,” . . ., were obtained with comparably high accura

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For the time-dependent problem with external fieldse0yvd sp1 1 p2d sinsvtd, the Hamiltonian becomesHstd ­Ed 2 se0yvdd sinsvtd, whereEd is the diagonalized ma-trix of the time-independent, field-free system when abound, all resonances, and the rotating continua witimaginary parts which are smaller than 0.1 eV were useas a basis set. All together,42 1S and 1P symmetry cor-related eigenstates were kept for the time propagatiod is the matrix representation of the dipole momentumoperator p1 1 p2 (in the same basis of the field-freeHylleraas functions asEd). The corresponding Floquetoperator is represented by the matrixH std ­ 1s2ih≠y≠td 1 Hstd. The st, t0d method [14,15] then enables usto describe the time evolution operator as a single exponential operator,Ust, t0d ­ expf2itH st0dyhg, wherethe physical time evolution operator is a cut in thet, t0

plane,Ust, t0 ­ td. The st, t0d technique can increase thespeed of calculations dramatically, even by several oders of magnitude [16]. By diagonalizing the complexnon-Hermitian matrixUst, t0d, we obtain the quasienergyFloquet statesCQE and the corresponding eigenvaluesL ­ exps2iEQETyhd, whereT ­ 2pyv is the periodic-ity of the field (in our case,hv ­ 5 eV). The free com-plex scaling parameter [9] was optimized to obtain stablHG spectra and not stationary values of the lifetimes of thresonance Floquet states in order to minimize the sizethe basis set. Therefore the ionization rates that we calclated are not precise.

In Fig. 1 we show the squared overlapjk1 1SjQElj2 ofthe longest-lived resonance Floquet state (“QE”) with thcomplex-scaled field-free1 1S ground state as a functionof field intensity I0 (proportional to´0yv2). One cansee that a sharp change in the contribution of the H

FIG. 1. Squared overlapjkQEj1Slj2 of the longest-lived Flo-quet state “QE” with the complex-scaled helium1 1S groundstate as a function of field intensityI0 ­ 335.111´

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VOLUME 78, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 17 MARCH 1997

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ground state to the resonance Floquet occurs nearI0 ­3.5 3 1015 Wycm2. The abrupt change in Floquet resonance composition is due to an avoided crossing phenomnon. Figure 2 shows how the complex eigenvalueL ofthe long-lived “1 1S resonance” (open circles, with width.0.002 a.u.. 0.054 eV) moves clockwise as field inten-sity increases, whereas the “2 1P resonance” (closed cir-cles) which is dominated by the lowest excitedP states,moves counterclockwise. At a very specific field intensity, these two resonance Floquet states move very cloto one another in complex energy space (“crossing” on treal axis). At this point a strongly mixed configurationstate is obtained, having sharply reduced1 1S character.The situation is more complicated than depicted in Fig.since other higher resonance Floquet states (associawith higher excited1P states) also become involved in theoverlapping of multiple avoided crossings in this region.

The association of enhanced high harmonics in thHGS with field-dependent avoided crossings has bepreviously recognized [8]. Moreover, it was shown thaplateau-type HGS is obtained when a nonlinear continously driven system exhibits chaotic classical dynami[17]. Marcus long ago pointed out that overlapping mutiple avoided crossings provide a diagnostic fingerprintclassical chaotic dynamics [18]. Our results in Figs.and 2 therefore suggest that, for a field intensity ne3.5 3 1015 Wycm2, the overlapping of several resonancFloquet states should lead to enhanced high harmonicseven to the appearance of a “plateau”-like feature in higHGS. Indeed, that is what we have found.

Figure 3 depicts the field-dependent changes in the cculated HGS (on a logarithmic intensity plot) near thonset of the plateau-like behavior. When the results apresented on an “absolute” scale (see, e.g., Potvliege aShakeshaft in Ref. [19]), one immediately can see that

FIG. 2. Field-dependent0 trajectories of complex eigenval-ues of the time evolution operator,L ­ exps2iEQE2pyvd,where EEQ are the complex resonance eigenvalues of thcomplex-scaled Floquet operator associated with the1 1Sground state (open circles) and an excited1P state (filled cir-cles), showing the near degeneracy (box) associated with onof plateau-like HGS.

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is far from being a plateau. (After the submission of thispaper Prestonet al. [20] published new experimental datawith HG spectra up to the 35th harmonic, which enable onto see the deviation from a plateau-like behavior, even oa logarithmic scale.) There is, however, a clear enhancment of the high harmonics in the harmonic generation, aexpected. Whereas weaker fields give unstable HGS wiexponentially diminishing intensities for higher harmon-ics, one sees that near3.2 3 1015 Wycm2 (closed circles)a highlystableplateau-like behavior (i.e., enhancement othe probability to observe high harmonics) begins to sein. Closer study shows that the HGS is most stable witrespect to small variations in the maximum field amplitudfor 3.46 3 1015 Wycm2, in very satisfying agreement withfield intensities inferred from previous “best” theoreticafits to experimental data [3]. However, whereas previouneutral helium treatments [3–5] (based on frozen orbitaHF, or density functional approximations) could fit experi-mental data only up ton ­ 13, our results exhibit excellentagreement with experiment for the full range of observeharmonics up ton ­ 23, as shown in Fig. 4.

All HGS were obtained by repropagating the “1 1S”resonance Floquet state [i.e., the eigenfunctionCQEstd ­Ust, t0dCQEst ­ 0djt0­t] of the time evolution operatorUst, t0d having at least 50% contribution from the field-free He1 1S ground state. Thenth order membersnmndof the HGS is obtained from thenth Fourier componentsmnd of the time-dependent dipole expectation valuekCQEstd j p1 1 p2jCQEstdlc, where k. . .lc denotes thec-product [11]. Note that there is a simple relationbetween the time derivative of the dipole moment in thlength gauge,Ùd, and the time-dependent dipole moment inthe momentum gauge,Ùd ­ kCjp1 1 p2 2 s´0yvd sin3

vtjClc, so that the dipole acceleration is given by≠y≠tkCjp1 1 p2jClc 2 ´0 cosvt ­

Pn ivnmn exp3

sivntd 2 ´0y2seivt 2 e2ivtd.

FIG. 3. Field-dependent HGS for thesingle longest-lived(“1 1S-like”) resonance Floquet state, showing the onset otransition to stable plateau-like HGS near3.2 3 1015 Wycm2.sI0 ­ 3.2 3 1015siy4d2; i ­ 1, . . . , 8d. The HG were scaled tomake theV ­ 9v of the 4th spectra to coalesce with theexperimental results.

VOLUME 78, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 17 MARCH 1997

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FIG. 4. Experimental HGS (filled circles, Ref. [2]) compared with present theoretical work for field intensity3.46 31015 Wycm2 (open circles).

We can obtain further insight into the role of dynamcal electron correlations in generating high HGS by rducing the flexibility of the Hylleraas-type basis (e.gomitting powers ofr12) to “control” the degree of cor-relation allowed into the description. If we first sel ­ k ­ 0 in all Hylleraas terms, forcing one electron tbe “1s-like” in every term, the resonance Floquet sta(neglecting spin dependence) is necessarily of the foCQEstd ­ 1ss$r1dfs$r2, td 1 1ss$r2, tdfs$r1, td, where f isa linear combination of termsrm exps2btd, m ­ 0 5.Such a function is of unrestricted Hartree-Fock (UHFform, simulating the much lower correlation treatmeof previous models [3–5]. This UHF-like model reproduces the experimental harmonics up ton ­ 15, but failsthereafter. As more terms restored to the full Hylleraexpansion (allowing more correlation flexibility) successive harmonics “pop out” to the experimentally observeplateau. Thus, we find that excellent agreement wthe entire experimental high HGS up ton ­ 23 can beobtained if electron correlation effects are treated wsufficient accuracy. (Note that our results, like thoseprevious theoretical studies, are for anisolated atomex-posed to an intense laser field, and therefore cannotcorporate collective effects that may be present in texperimental spectra.)

Our results may seem to contradict the theoretical resupresented in Refs. [4,5], which found it necessary to eploy excited He or He1 models to reproduce higher HGSpeaks. However, as pointed out above, the actual colated behavior of the inner electron is indeed rather He1-like, and the previous treatments of electron correlatiwere simply too crude to exhibit this aspect of the neutatom directly. Moreover, it is clear that neutral He atommust be abundantly present under the experimental cditions, since at3.5 3 1015 Wycm2 the density of atomsis estimated [21] to equal that of ions (10–20 torr, coresponding to about1017 1018 atomsycm3). Our resultstherefore support the possibility that theentireHGS spec-

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In conclusion, we find that (a) HGS in He can btreated as asingle Floquet mode phenomenon, (b) aexplicit connection can be drawn between stable Hto small variations in the field intensity and overlappinavoided crossings of resonance quasienergy states, mated by a critical field intensity of,3.5 3 1015 Wycm2,and (c) strong radial-angular coupling gives rise“correlation-induced descreening” in neutral helium thaccounts for the increased He1-like character leading tothe shortest-wavelength HGS emissions.

*Electronic address: nimrod@chem.technion.ac.il[1] A. L’Huillier, K. J. Schafer, and K. C. Kulander, Phys

Rev. Lett.66, 2200 (1991); A. L’Huillier and P. Balcou,Phys. Rev. Lett.70, 774 (1993) and references therein.

[2] N. Sarukura, K. Hata, T. Adachi, and R. Nodomi, PhyRev. A 43, 1669 (1991).

[3] A. Sanpera, P. Jonsson, J. B. Watson, and K. BurnPhys. Rev. A51, 3148 (1995).

[4] Xu Huale, X. Tang, and P. Lambropoulos, Phys. Rev.46, R2225 (1992).

[5] C. A. Ullrich, S. Erhard, and E. K. U. Gross, inSuperIntensity Laser Atomic Physics IV,edited by H. G. Muller(Kluwer, Boston, 1996).

[6] N. Moiseyev and J. Katriel, Chem. Phys.10, 67 (1975).[7] N. Moiseyev and F. Weinhold, Phys. Rev. A20, 27

(1979).[8] N. Ben-Tal, N. Moiseyev, R. Kosloff, and C. Cerjan

J. Phys. B26, 1446 (1993).[9] W. P. Reinhardt, Annu. Rev. Phys. Chem.33, 223 (1982);

N. Moiseyev, Isr. J. Chem.31, 311 (1991).[10] N. Moiseyev, P. R. Certain, and F. Weinhold, Mol. Phy

36, 1613 (1978).[11] N. Moiseyev, in Letropet Symposium View on

Generalized Inner Product, edited by E. Brän-das and N. Elander, Lecture Notes in Physics Vol. 3(Springer-Verlag, Berlin, 1989), p. 549.

[12] E. A. Hylleraas, Adv. Quantum Chem.1, 1 (1964).[13] C. L. Pekeris, Phys. Rev.126, 143 (1968).[14] U. Peskin and N. Moiseyev, J. Chem. Phys.99, 4590

(1993).[15] N. Moiseyev, Comments At. Mol. Phys.31, 88 (1995).[16] U. Peskin, R. Kosloff, and N. Moiseyev, J. Phys. Che

100, 8849 (1994).[17] V. Averbukh and N. Moiseyev, Phys. Rev. A51, 3911

(1995).[18] R. A. Marcus, inQuantum Chaos,edited by G. Cassat

and J. Ford (Academic, New York, 1984).[19] P. M. Potvliege and R. Shakeshaft, Phys. Rev. A40, 3061

(1989).[20] S. G. Preston, A. Sanpera, M. Zept, W. J. Blyth, C.

Smith, J. S. Wark, M. H. Key, K. Burnett, and M. NakaPhys. Rev. A53, R31 (1996).

[21] A. L’Huillier (private communication).

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