Nonsequential multiple ionization of atoms in intense laser fields is
initiated by a recollision between an electron, freed by tunneling, and its
parent ion. Following recollision, the initial electron shares its energy with
several bound electrons. We use a classical model based on rapid electron
thermalization to interpret recent experiments. For neon, good agreement with
the available data is obtained with an upper bound of 460 attoseconds for the
thermalization time.Comment: 5 pages revtex and 4 figures (eps files

Bauer D

C Figueira de Morisson Faria

Feuerstein B

Forst W

Gradsteyn I S

Hagedorn R

Kulander K C

Landau L D

P B Corkum

Robinson P J

Sacha K

W Becker

X Liu

Zrost K

English

arXiv

Liu, X

de Morisson Faria, CF

Becker, W

Corkum, PB

UCL Discovery

arXiv:physics/0506187v1  [physics.atom-ph]  24 Jun 2005Attosecond electron thermalization by laser-driven electron recollision in atomsX. Liu,1 C. Figueira de Morisson Faria,2 W. Becker,3 and P. B. Corkum41Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA2Centre for Mathematical Science, City University, Northampton Square, London EC1V OHB, UK3Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin, Germany4National Research Council of Canada, 100 Sussex Drive,Ottawa, Ontario, Canada K1AOR6(Dated: February 2, 2008)Nonsequential multiple ionization of atoms in intense laser ﬁelds is initiated by a recollisionbetween an electron, freed by tunneling, and its parent ion. Following recollision, the initial electronshares its energy with several bound electrons. We use a classical model based on rapid electronthermalization to interpret recent experiments. For neon, good agreement with the available datais obtained with an upper bound of 460 attoseconds for the thermalization time.PACS numbers:Atoms exposed to intense laser ﬁelds ionize. The freedelectron and its ionic partner are accelerated by the laserﬁeld away from each other. When the ﬁeld changes sign,it may drive the electron into a recollision with the ion.This simple mechanism, which for high-intensity low-frequency ﬁelds is largely classical, governs many laser-atom processes such as high-order harmonic generation(HHG), high-order above-threshold ionization (HATI),and nonsequential double ionization (NSDI) and explainsthe gross features of the spectra observed [1].The period of the commonly applied titanium-sapphire(Ti:Sa) laser is about 2.7 fs. The recollision physics un-fold on the time scale of a small fraction of the laser pe-riod. Therefore, the analysis of laser-induced recollisionphenomena provides access to the inner-atomic dynamicson the attosecond time scale and, indeed, the focus of re-cent investigations has moved into this temporal domain.For example, it has brought molecular imaging with sub-angstrom spatial and subfemtosecond temporal resolu-tion within reach [2, 3]. The advent of phase-stabilizedinfrared few-cycle pulses and of uv pulses of attosecondduration allows even more control [4], but neither arenecessary for a study of the attosecond dynamics.In this Letter, we analyze recent experiments on nonse-quential multiple ionization (NSMI) of neon [5,6] in whichthe momentum distributions of NeN+ (N = 3,4) weremeasured at two intensities. We use the fact that thetime-dependent laser ﬁeld provides a clock — the ﬁeldaccelerates the ion to a ﬁnal velocity that depends onthe times at which the N electrons ionized. This “streakcamera” [7] therefore measures the range of times of ion-ization. Comparing the measured momentum distribu-tions to those predicted by a classical model, we infer thatthe recolliding electron thermalizes with the N−1 boundelectrons in less than 500 attoseconds. To our knowledge,no other method allows such a low upper bound for ther-malization times within atoms to be measured.Nonsequential double and multiple ionization is de-ﬁned by the fact that it is not sequential, that is, it isnot the product of a sequence of uncorrelated single-ionization events. NSDI and NSMI require electron-electron correlation as a necessary precondition [5, 8].Even for the very simplest such process – NSDI of helium– a fully quantum-mechanical description from ﬁrst prin-ciples, i.e. by solution of the time-dependent Schr¨ odingerequation in six spatial dimensions, has not been accom-plished yet [9], and for the heavier atoms it is clearlyout of the question. This leaves approximate quantum-mechanical approaches, such as density-functional meth-ods [10], S-matrix methods that try to identify the mostrelevant terms of an appropriate perturbative expansion[11, 12], or classical-trajectorymethods [13, 14]. For mul-tiple (triple and higher) ionization, which occurs underthe same conditions as NSDI if the laser intensity is highenough, any description from ﬁrst principles appears tobe utterly out of reach. It seems equally hopeless, forthree electrons and more, to identify the relevant dia-grams in the microscopic S-matrix approach.The model we here propose is in the spirit of the oneof Ref. [12], but essentially classical. We investigate thescenario wherein NSMI is eﬀected by one single recolli-sion. We assume that the pertinent electron tunnels intothe continuum with zero velocity at the ionization time t′according to the time-dependent rate R(t′), for which weadopt the standard quasi-static rate [15]. Thereafter, weturn to an entirely classical description: The laser ﬁeldmay drive the electron back to its parent ion at a latertime t, which is a function t(t′) of the ionization timeand can be easily evaluated. We assume that the en-ergy Eret(t) of the returning electron be completely ther-malized among the ensemble of participating electrons,that is, the returning electron and the N −1 electrons tobe freed. These N electrons then form an excited com-plex with the total energy (with respect to the continuumthreshold) Eret(t) − E(N)0 , where E(N)0 > 0 denotes thetotal ionization potential of the N − 1 (up to the recol-lision time t inactive) electrons. The distribution of en-ergy and momentum over the N electrons is assumed tobe completely statistical and only governed by the avail-able phase space. At the time t + ∆t, the N electrons2become free to move in the laser ﬁeld, which is describedby the vector potential A(t) such that A(t) = 0 outsidethe pulse.The corresponding distribution of the ﬁnal electronmomenta pn (n = 1,...,N) is proportional toF(p1,p2,...,pN) =Zdt′R(t′)δ￿E(N)0 − Eret(t)+12N Xn=1[pn + A(t + ∆t)]2!, (1)where the integral extends over the ionization time t′.The δ function expresses the fact that the total kineticenergy of the N participating electrons at the time t+∆tis ﬁxed by the ﬁrst-ionized electron at its recollision timet. This constitutes the one and only condition on the ﬁnalmomenta pn. The only free parameter of this model isthe time delay ∆t between the recollision time and thetime when the electrons become free. It is the sum of athermalization time ∆tth – the time it takes to establishthe statistical ensemble – and a possible additional “dwelltime”, until the electrons become free. By comparing thepredictions of the model with the data, we will be ableto infer a value of ∆t, which in turn provides an upperlimit for the thermalization time ∆tth.This model is an extension to NSMI of a classicalmodel introduced for NSDI for ∆t = 0 [16, 17]. Suf-ﬁciently high above threshold, it produced momentumdistributions that were virtually indistinguishable fromtheir quantum-mechanical counterparts. Statistical mod-els similar to the one above have been used in many areasof physics. For example, the statistical Rice-Ramsperger-Kassel-Marcus (RRKM) theory [18] describes thermal-ization of molecular vibrational degrees of freedom, andfor high-energy collisions of elementary particles andheavy ions statistical models have been utilized to predictthe momentum spectra of the reaction products [19]. Anexcited complex as the doorway to NSMI was also con-sidered in Ref. [20].A convenient feature of the ansatz (1) is that integra-tion over unobserved momentum components is easilycarried out. To this end, we exponentialize the δ func-tion in Eq. (1) with the help of its Fourier representationδ(x) =Z ∞−∞dλ2πexp(−iλx). (2)Inﬁnite integrations over the momenta pn can then bedone by Gaussian quadrature. The remaining integrationover the variable λ is taken care of by the formula [21]Z ∞−∞dλ(iλ + ǫ)ν eipλ =2πΓ(ν)pν−1+ , (3)where xν+ = xνθ(x), with θ(x) the unit step function andǫ → +0.For comparison with the experiments [5, 6], we cal-culate the distribution of the momentum P of the ion.Provided the momentum of the absorbed laser photonscan be neglected, momentum conservation implies P =−PNn=1 pn, so thatFion(P) ≡Z N Yn=1d3pnδ P +N Xn=1pn!F(p1,p2,...,pN)=(2π)3N2 − 32N3/2Γ(32(N − 1))Zdt′R(t′)(∆EN,ion)3N2 − 52+ (4)with ∆EN,ion ≡ Eret(t) − E(N)0 − 12N[P − NA(t + ∆t)]2.In the experiments thus far, the ion momentum trans-verse to the direction of the laser polarization is entirelyor partly integrated over. In the ﬁrst case, the remainingdistribution of the longitudinal ion momentum P  isFion(P ) ≡Zd2P⊥Fion(P)=(2π)3N2 − 12√NΓ((3N − 1)/2)Zdt′R(t′)￿∆Eion ￿ 3N2 − 32+ , (5)where now ∆Eion  ≡ Eret(t) − E(N)0 − 12N[P  − NA(t +∆t)]2. If just one transverse-momentum component(P⊥,2, say) is integrated while the other one (P⊥,1 ≡ P⊥)is observed, the corresponding distribution isFion(P ,P⊥) ≡ZdP⊥,2Fion(P)=(2π)3N2 −1NΓ(3N/2 − 1)Zdt′R(t′)￿∆EN,ion ⊥￿ 3N2 −2+ (6)with ∆EN,ion ⊥ ≡ ∆Eion  − 12NP 2⊥.In Fig. 1 we present calculations of the ion-momentumdistributions for triple (upper panel) and quadruple(lower panel) NSMI of neon according to Eq. (5), forvarious values of ∆t between 0 and 0.2T. The parame-ters are for the experimental data of neon presented inFig. 2 of Ref. [6]. The ﬁgure displays the double-humpstructure of the ion momentum in NSMI of neon. Whenthe time delay ∆t increases from 0, initially, the eﬀect onthe momentum distribution is small. Later, however, thecenter positions of the humps start moving towards zeromomentum and the widths of the humps increase untilthe two humps begin to merge. This behavior can easilybe understood from the recollision kinematics, which areillustrated in Fig. 2. The ﬁnal electrons undergo maxi-mal acceleration if they are released near a zero crossingof the electric ﬁeld, which in the ﬁgure occurs at t = T.For the example of triple ionization of neon, the earliestrecollision with Eret > E(3)0 takes place at t = 0.74T.Already with a delay of ∆t > 0.26T, all electrons will bereleased after the zero crossing, which results in signiﬁ-cantly lower ion momenta.3-20 -10 0 10 20CountsCounts P|| (a.u.):  t = 0:  t = 0.05T:  t = 0.1T:  t = 0.15T:  t = 0.2T   FIG. 1: (Color online) Distribution of the longitudinal ion mo-mentum for triple (upper panel) and quadruple (lower panel)nonsequential ionization of neon at 2 PWcm−2 calculatedfrom Eq. (5) for various delays ∆t as indicated in the lowerpanel. Note that in the upper panel the curves for ∆t = 0and ∆t = 0.1T almost completely overlap.0.3 0.4 0.5051015E0(4) = 7.41 a.u.  E0(3) = 3.84 a.u.Eret (a.u.)t’ / T0.60.91.2 0.30 T   t / T 0.41 TFIG. 2: The kinetic energy Eret(t) (lower panel) and recol-lision time t (upper panel) of electrons liberated at the tun-neling time t′. The classically allowed intervals ∆trec of therecollision time, for which the corresponding electron returnenergy Eret(t) is greater than E(3)0 (and E(4)0 ), are shown inthe upper panel. The calculation is for neon at 2.0PWcm−2.In Fig. 3 we compare the results of the statisticalmodel with the data of Ref. [6]. The intensities are thosegiven for the experiment. We display momentum dis-tributions calculated from Eq. (5) for zero delay and for∆t = 0.17T. The latter value was chosen to yield optimalagreement for the entire set of data. We notice, in partic-ular, that with this nonzero delay the model reproducesthe maxima of the experimental ion-momentum distribu--20 -10 0 10 20  CountsP|| (a.u.)Counts   Counts  FIG. 3: (Color online) Distribution of the longitudinal mo-mentum of nonsequential triple ionization of neon at 1.5PWcm−2 (upper panel), 2.0 PWcm−2 (middle panel), andof nonsequential quadruple ionization of Ne at 2.0 PWcm−2(lower panel). The rugged (black) curves represent the data ofFig. 2 of Ref. [6]. The outermost smooth (red) curve and theinnermost smooth (green) curve are calculated from Eq. (5)for ∆t = 0 and ∆t = 0.17T.-12 -9 -6 -3 0 3 6 9 12-4-2024    P (a.u.)P|| (a.u.)024.0048.0072.0080.00-4-2024    P (a.u.)030.0060.0090.00100.0FIG. 4: Ion-momentum distribution of nonsequential tripleionization of neon at 1.0 PWcm−2 for ∆t = 0 (upper panel)and ∆t = 0.17T (lower panel), calculated from Eq. (6), to becompared with the data of Ref. [5].4tion. This removes a longstanding discrepancy betweenmodels of the type discussed in Refs. [12, 16, 17] and thedata. For triple ionization, the calculated momentumdistributions are wider than those of the data, in partic-ular for the higher intensity, even though the data mayinclude a contribution of the partly sequential channelNe → Ne+ → Ne3+, or the contribution of a recollision-excitation channel, both of which are, of course, not partof the model.Figure 4 exhibits the results of our thermalizationmodel for N = 3 for the conditions of Ref. [5]. The dis-tribution of two components of the ion momentum is pre-sented, the component parallel to the laser ﬁeld and onetransverse component, which therefore provides a morestringent test of the model. The third component is in-tegrated over in the data, which corresponds to Eq. (6).The intensity given in the experiment is 1.5PWcm−2;we obtain good agreement with the data for the lowerintensity of 1.0PWcm−2 and ∆t = 0.17T as before [22].The nonzero delay ∆t has little eﬀect on the transversewidth of the distribution, but it causes an elongation inthe longitudinal direction and moves the centers to lowermomenta, markedly improving the agreement with thedata.Encouraged by the good agreement between the modeland the data, we interpret the delay for which we ob-served optimal agreement as an upper bound of thethermalization time ∆tth, as argued above. This yields∆tth < ∆topt = 0.17T ≃ 460 as. A lower limit of thethermalization time should be given by the inverse ofthe plasma frequency for an electron density ρe = N inatomic units. This produces ∆tth <pπ/N, which is ofthe order of the atomic unit of time.All of the above discussion has been for neon. NSDI ofargon appears to be governed by a recollision-excitationscenario [23]. We note in passing that for (quadruple)NSMI of argon we get good agreement of our statisticalthermalization model with the data [6] for the experi-mental intensity and ∆t = 0.35T, twice as long as forneon. Details will be given elsewhere.To test the model further, and possibly to set a tighterupper limit on the thermalization time, it is necessary torestrict the time range ∆tCM of recollision (Fig. 2). Ifthermalization is more rapid than 460 as, then the widthof the ion-momentum distribution will decrease as we re-strict ∆trec. There are a number of ways to minimize∆trec. Within limits, all that is needed is to lower thelight intensity or to increase the laser frequency. How-ever, the best way to minimize ∆trec and to control thetime of recollision is to use a second harmonic ﬁeld, po-larized perpendicular to the fundamental. The combinedrequirement that the electron and ion recollide in bothdirections allows the time of recollision to be preciselydetermined and controlled via the relative phase of thetwo beams.The limit on the thermalization time that we deter-mine (as well as the much tighter bounds that seem fea-sible in future experiments) should apply to stationaryelectron-atom scattering in general. From a collision-physics perspective, as a result of streaking, the time-dependent laser ﬁeld reveals information that is hard toobtain by other means. The streaking principle shouldalso be applicable for studying nuclear dynamics. As inNSMI, nuclear processes also can be initiated by laser-controlled re-collision [24] (of course, at much higher in-tensity). Any nuclear decay process that results in a massor charge change of the fragments will be streaked by thelaser ﬁeld just as electrons and ions are streaked in ourcase.To summarize, by comparison of experimental datafor triple and quadruple nonsequential ionization of neonwith a simple statistical recollision model where the re-turning electron thermalizes with a subset of the boundelectrons, we have been able to conclude that (i) forneon such a model appears to contain the most relevantphysics, and (ii) the time for this thermalization to occuris extremely fast, well below one femtosecond.We gratefully acknowledge stimulating discussionswith G.G. Paulus, H. Rottke, and W. Sandner.[1] P.B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).[2] H. Niikura, F. Legare, R. Hasbani, A.D. Bandrauk,M.Yu. Ivanov, D.M. Villeneuve, and P.B. Corkum, Na-ture (London) 417, 917 (2002).[3] S.N. Yurchenko, S. Patchkovskii, I.V. Litvinyuk, P.B.Corkum, and G.L. Yudin, Phys. Rev. Lett. 93, 223003(2004).[4] A. Baltuˇ ska, Th. Udem, M. Uiberacker, M. Hentschel, E.Goulielmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev,A. Scrinzi, T.W. H¨ ansch, and F. Krausz, Nature (Lon-don) 421, 611 (2003).[5] R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn,C.D. Schr¨ oter, J. Ullrich, H. Rottke, C. Trump, M.Wittmann, G. Korn, K. Hoﬀmann, and W. Sandner,Phys. Rev. Lett. 84, 447 (2000).[6] A. Rudenko, K. Zrost, B. Feuerstein, V.L.B. de Jesus,C.D. Schr¨ oter, R.Moshammer, and J. Ullrich, Phys. Rev.Lett. 93, 253001 (2004).[7] E. Constant, V.D. Taranukhin, A. Stolow, and P.B.Corkum, Phys. Rev. A 56, 3870 (1997).[8] Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger,O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M.Vollmer, H. Giessen, and R. D¨ orner, Phys. Rev. Lett.84, 444 (2000).[9] K.T. Taylor, J.S. Parker, K.J. Meharg, and D. Dundas,Eur. Phys. J. D 26, 67 (2003).[10] D. Bauer and F. Ceccherini, Opt. Express 8, 377 (2001).[11] A. Becker and F.H.M. Faisal, Phys. Rev. Lett. 84, 3546(2000); ibid. 89, 193003 (2002).[12] R. Kopold, W. Becker, H. Rottke, and W. Sandner, Phys.Rev. Lett. 85, 3781 (2000).[13] L.B. Fu, J. Liu, S.G. Chen, Phys. Rev. A 65,021406(R)(2002); J. Chen and C.H. Nam, Phys. Rev.5A 66, 053415 (2002).[14] P.J. Ho, R. Panﬁli, S.L. Haan, and J.H. Eberly, Phys.Rev. Lett. 94, 093002 (2005).[15] L.D. Landau and E.M. Lifshitz, Quantum Mechan-ics (Nonrelativistic Theory) (Pergamon Press, Oxford,1977).[16] C. Figueira de Morisson Faria, X. Liu, W. Becker, andH. Schomerus, Phys. Rev. A 69, 012402(R)(2004).[17] C. Figueira de Morisson Faria, H. Schomerus, X. Liu, andW. Becker, Phys. Rev. A 69, 043405 (2004).[18] See, e.g., W. Forst, Theory of Unimolecular Reactions(Academic, New York, 1973); P.J. Robinson and K.A.Holbrook, Unimolecular Reactions (Wiley-Interscience,New York, 1972).[19] R. Hagedorn, Nuovo Cimento Suppl. 3, 147 (1965).[20] K. Sacha and B. Eckhardt, Phys. Rev. A 64, 053401(2001); J. Phys. B 36, 3923 (2003).[21] I.S. Gradsteyn and I.M. Ryzhik, Table of Integrals, Se-ries, and Products (Academic Press, New York, 1980).[22] A delay of the same order has also been observed inclassical-trajectory simulations of NSDI of helium [14].[23] V.L.B. de Jesus, B. Feuerstein, K. Zrost, D. Fischer, A.Rudenko, F. Afaneh, C.D. Schr¨ oter, R. Moshammer, andJ. Ullrich, J. Phys. B 37, L161 (2004).[24] N. Milosevic, P.B. Corkum, and Th. Brabec, Phys. Rev.Lett. 92, 013002 (2004); S. Chelkowski, A.D. Bandrauk,and P.B. Corkum, Phys. Rev. Lett. 93, 083602 (2004).

Attosecond electron thermalization by laser-driven electron recollision in atoms

Nonsequential multiple ionization of atoms in intense laser fields is initiated by a recollision between an electron, freed by tunnelling, and its parent ion. Following recollision, the initial electron shares its energy with several bound electrons. We use a classical model based on rapid electron thermalization to interpret recent experiments. For neon, good agreement with the available data is obtained with an upper bound of 460 as for the thermalization time. This may be the shortest thermalization time measured to date.Peer reviewed: YesNRC publication: Ye

Liu, X.

Faria, C. Figueira de Morisson

Becker, Wilhelm

Corkum, Paul

NRC Publications Archive

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Attosecond electron thermalization by laser-driven electron recollision
  in atoms

Attosecond electron thermalization by laser-driven electron recollision in atoms

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