Metallic clusters in strong femtosecond laser pulses

J. Phys. B: At. Mol. Opt. Phys.30 (1997) 5043–5055. Printed in the UK PII: S0953-4075(97)84778-5

Metallic clusters in strong femtosecond laser pulses

C A Ullrich†, P-G Reinhard‡ and E Suraud§‖† Department of Physics, University of Missouri, Columbia, MO 65211, USA‡ Institut fur Theoretische Physik, Universitat Erlangen, Staudtstrasse 7, D-91058 Erlangen,Germany§ Laboratoire de Physique Quantique, Universite Paul Sabatier, 118 route de Narbonne, F-31062Toulouse Cedex, France

Received 9 June 1997, in final form 4 August 1997

Abstract. We study the electron response of a Na+9 cluster excited by strong femtosecond

laser pulses by means of a time-dependent density-functional method. We investigate the fullelectronic dipolar response and multiphoton ionization processes. A strong correlation betweeninduced electronic dipole oscillations and electron emission is observed, leading to a pronouncedresonant enhancement of ionization when using lasers operating at the frequency of the Mieplasmon. We also examine the probabilities for producing differently charged ionic states of thesystem.

1. Introduction

The experimental and theoretical investigation of the dynamical properties of simple metallicclusters has been an active field of research for several years now [1, 2]. A largenumber of studies have dealt with electronic collective modes in the regime of smalloscillations [3], which are adequately described within linear response theory [4]. Thereis, however, growing interest in the dynamical behaviour of quantum systems under verystrong excitations, such as multiphoton processes in atoms and small molecules irradiatedwith strong laser pulses (peak intensityI is about 1013–1016 W cm−2, which correspondsto electric field amplitudes of orderE0 ∼ 0.5–15 V a−1

0 ) [5–7]. These strong laser pulsestypically have pulse lengths in the range of less than 100 fs up to several picoseconds,with frequenciesω0 extending from the infrared to the ultraviolet. With such powerfulexperimental tools at hand, a wealth of new and sometimes surprising phenomena becomesaccessible, which clearly requires a theoretical description beyond perturbation theory.

Most experiments using laser pulses to investigate simple metal clusters have hithertobeen carried out with relatively moderate intensities. Even if used to prepare clusters in highcharge states (as in the work of Naheret al [8]), the laser intensities are still in a regimewhere the ionization proceeds stepwise via single-photon processes. Since the pulse lengthsused in [8] are of the order of nanoseconds, the ionization processes are accompanied byan additional heating of the clusters, which may lead to the loss of small fragments duringthe charging process. Very recently, however, new experiments with strong femtosecondpulses have been performed which seem to indicate significant differences as comparedwith the behaviour in the long-pulse regime [9, 10]. More experimental activity of that kindmay be expected for the future. It is thus worthwhile to start theoretical investigations of

‖ Membre de l’Institut Universitaire de France.

0953-4075/97/215043+13$19.50c© 1997 IOP Publishing Ltd 5043

5044 C A Ullrich et al

clusters under the influence of strong laser pulses. However, linear response theory ceasesto be applicable to such situations, and one needs to revert to a nonlinear approach suchas that provided within the framework of time-dependent density functional theory [11, 12].Considering experience gathered with laser excitation of atoms [13, 14] as well as nonlineardynamics of the electron cloud of metal clusters [15–19], we present in this paper the firstnumerical investigations of the dynamical behaviour of the valence electron cloud of metalclusters under the influence of strong femtosecond laser pulses. The test case will be theNa+9 cluster, and we consider short laser pulses in an intensity range of 1010–1014 W cm−2.We shall be concerned with two predominant aspects of the electron dynamics: the dipoleresponse of the system and the emission of electrons caused by multiphoton ionizationprocesses.

The paper is organized as follows. In section 2 we present our theoretical and numericalapproach for the treatment of valence electron dynamics and ionic background. Section 3is devoted to a discussion of the dipole response, and in section 4 we consider multiphotonionization, studying the dependence of the total number of emitted electrons on the laserphoton energy and peak intensity. We also evaluate the probabilities of producing differentlycharged ionic states. Finally, in section 5 we give our conclusions.

2. Theoretical framework

2.1. Electron dynamics

Our description of the valence electron cloud is based on time-dependent density functionaltheory (TDDFT) at the simplest level of approximation, the time-dependent local densityapproximation (TDLDA) [11, 12], which has accompanied cluster physics since its earlystages [4] and which has been used successfully beyond the linear regime in several previousapplications [13–19]. In this section we shall only recall the basic formulae necessary forthe forthcoming discussions.

Within TDDFT, the time-dependent density of anN -electron system is obtained, inprinciple exactly, through a set of single-electron wavefunctionsφj (r, t) which satisfy thetime-dependent Kohn–Sham (TDKS) equations

i∂

∂tφj (r, t) =

(−∇

2

2+ v[n](r, t)

)φj (r, t) (1)

(written in atomic units). The TDKS effective potentialv is decomposed into the externalpotentialvext(r, t), a time-dependent Hartree part and a so-called exchange-correlation (xc)potential:

v(r, t) = vext(r, t)+∫

d3r ′n(r′, t)|r − r′| + vxc[n](r, t), (2)

where the electronic density is given byn(r, t) = ∑Nj=1 |φj (r, t)|2. The xc potential

vxc[n](r, t) is a functional of the density and has to be approximated in practice.The simplest choice, which is used in this work, consists in the TDLDA, defined asvTDLDA

xc (r, t) = dehomxc (n)/dn|n=n(r,t), where ehom

xc (n) is the xc energy density of thehomogeneous electron gas. Forehom

xc we use the parametrization of Gunnarsson andLundqvist [20]. By construction, the TDLDA can be expected to be good only if thetime dependence is sufficiently weak. In practice, however, it gives good results even forcases of rather rapid time dependence like the plasmon response in metal clusters [1, 2, 4].

Metallic clusters in strong femtosecond laser pulses 5045

2.2. External field

In our calculations, the external potential which enters the TDKS effective potential (2) ismade up of two parts:

vext(r, t) = vion(r, t)+ E0f (t)z sin(ω0t). (3)

The first part,vion(r, t), accounts for the Coulomb potential caused by the ionic backgroundof the cluster. The excitations of clusters by femtosecond laser pulses are so rapid that, toa very good approximation, ions can be considered as fixed in the course of the excitationprocess and during an early stage of the electronic relaxation (typically up tot . 100 fs). Inaddition, previous investigations of the plasmon response in metal clusters have shown thatdetails of the ionic structure seem to play only a minor role for the electron dynamics andbecome increasingly less important beyond the linear regime [15]. Hence, for our presentexploratory investigations we shall approximatevion by replacing the ionic background ofthe cluster (nuclei plus core electrons) with a static jellium distribution. For our test case,the ‘magic’ Na+9 cluster, we use a spherical jellium density%jel:

%jel(r) = %bulk

1+ exp r−Rrc/√

3

, (4)

where the bulk density is given by%bulk = 3/(4πr3s ), andR is close to the sharp jellium

radius rsN1/3 (R is adjusted so that%jel(r) provides the correct total ionic charge). Thefinite surface widthrc/

√3 is defined via the Ashcroft core radiusrc [21] (rc ∼ 1.73 a0).

This ‘softness’ in the jellium background (4) is equivalent to a folding of the sharp jelliumwith a pseudopotential characterized byrc, and allows us to reproduce the experimentalplasmon resonance energy (2.62 eV for Na+9 ) in an LDA calculation [22].

The second part ofvext(r, t) is the potential of the laser field. The latter is taken to bepolarized along thez-axis and has been written in dipole approximation;E0 is the peak fieldstrength andω0 the frequency of the laser. The envelope of the pulsef (t) has been chosento be Gaussian:f (t) = exp{−(t − t0)2/s2}. In the following we shall consider pulses withFWHM 2

√ln 2s = 25 fs and which peak att0 = 32.2 fs. The pulses will be switched on

at t = 0, and off att = 64.4 fs, and the electrons will be studied up tot = 100 fs.

2.3. Computational aspects

In its initial ground state, the Na+9 cluster contains eight valence electrons in four doublyoccupied Kohn–Sham (KS) orbitals (one s and three p orbitals), thus forming a closed-shellsystem. Due to the linear polarization of the laser field, rotational symmetry of the systemaround thez-axis is preserved for all times. In other words, if working in cylindricalcoordinatesr = (ρ, z, ϕ), the ϕ-dependence of the KS orbitalsφj (r, t) can be treatedanalytically and we only need to perform a discretization of the TDKS equation (2) inthe two spatial dimensions(ρ, z). We work on a finite uniform rectangular grid with amesh spacing of 0.75 a0, representing the kinetic-energy operator in (2) by a three-pointfinite-difference formula in each direction.

First of all, thestatic KS equations for Na+9 (in the local density approximation (LDA))are solved on the grid using a Lanczos method. The propagation in time of the KS orbitals isthen carried out with the Crank–Nicholson algorithm [7, 23]. To account for the nonlinearityof the TDKS equations, each time step involves a predictor/corrector scheme [24]. Thisimplicit procedure ensures unitarity and second-order accuracy in the time step. In ourcalculations we use a constant time step of 0.2 au (1 au corresponds to 0.024 fs), andwe have implemented absorbing boundary conditions [25]. The Hartree potential entering


equation (2) is obtained by solving the Poisson equation on the grid by means of a conjugategradient method.

2.4. Observables

Let us now define the relevant observables for the phenomena we are interested in, namelyelectron emission and dipole response. The evaluation of electron escape within TDDFTrelies on the basic relation

N(t) =∫V

d3r n(r, t), (5)

which associates the number of electrons remaining in a bound state,N(t), with theelectronic density within a finite volumeV centred around the ionic background. FromN(t)we can then calculate the total number of escaped electrons asNesc(t) = N(t = 0)−N(t).It turns out thatN(t) is not very sensitive to the actual choice ofV, in particular in thelimit of large t (for more details see [19, 25]). In practice, we add a stripe of 2rs to thecluster radius in the radial as well as inz direction of our cylindrical coordinate system,thus definingV as a cylinder of length 32a0 and radius 16a0. The dipole momentd(t)with respect to thez-axis is evaluated inside the same volumeV [17]:

d(t) =∫V

d3rz n(r, t). (6)

Fourier transformation of the dipole signald(t) then yields the power spectrum|d(ω)|2.An important link with experiment may be established by calculating probabilities of

finding the clusters at a timet in one of the possible charge statesk to which they canionize. Explicit expressions for theP k(t) can be obtained [14] in terms of bound-stateoccupation probabilitiesNj(t) associated with the single-particle KS densitiesnj ,

Nj(t) =∫V

d3r |φj (r, t)|2 =∫V

d3r nj (r, t). (7)

Since the single-particle KS orbitals have no rigorous physical meaning, one must considertheP k[{nj }](t) obtained here only as a reasonable approximation to the exact probabilities.To derive the expressions for theP ks, we start with the simple example of a system which att = 0 has only one doubly occupied orbital, such as the helium atom [13, 26] or a Na2 clusterin a spherical jellium model. In this simple case, theP k(t) are, in fact, explicit functionalsof the total density. If the bound-state occupation probability (7) for these systems isgiven byN1s(t), then the probabilities for the possible charge states areP 0(t) = N1s(t)

2,P+1(t) = 2N1s(t)(1 − N1s(t)), P+2(t) = (1 − N1s(t))

2. These expressions have beenconstructed to fulfil the requirement that the probabilities must sum up to unity. The squarein P 0 andP+2 and the factor of 2 inP+1 account for the degeneracy, as we work with aspin-unpolarized system.

In the case of Na+9 we start with the following relation:

1=+9∑k=+1

P k(t) =occ∏j

[Nj(t)+ (1−Nj(t))]2 =occ∏j

[Nj + Nj ]2. (8)

We then work out the right-hand side of equation (8) and rearrange the resulting terms,collecting terms containing(k − 1) factorsNj = (1− Nj) and (9− k) factorsNj . Theseare then identified with the ion probabilitiesP k(t) [14]. To give an example, the first two


–4

–2

0

2

4

d(t)

0

0.005

0.01

0.015

0.02

0.025

0 10 20 30 40 50 60 70 80 90 100

Nes

c(t)

Figure 1. Upper part: full time-dependent dipole response of Na+9 (full curve) and envelope

of the response of a classical oscillator (broken curve) to a 25 fs Gaussian laser pulse (peak att = 32.2 fs, peak intensity: 1011 W cm−2, photon energy: 2.95 eV). Lower part: number ofescaping electrons for the same laser parameters.

ionization probabilities become

P+1(t) = N21sN

21p0N4

1p1(9)

P+2(t) = 4N21sN

21p0N3

1p1N1p1 + 2N2

1sN1p0N1p0N41p1+ 2N1sN1sN

21p0N4

1p1. (10)

A more detailed derivation will be published elsewhere [27]. It is interesting to note thatthis strategy to derive ionization probabilities from a combinatorial identification has beenused before in a somewhat different context, namely within stationary scattering theory [28].

3. Dipole response

We first consider the excitation of our Na+9 cluster with a 25 fs Gaussian pulse of peakintensity I = 1011 W cm−2 and with two different photon energies ¯hω0. The spectrumof Na+9 is dominated by the Mie plasmon peak at ¯hωM = 2.6 eV [17]. The excitationfrequencies are 2.95 eV (off resonance) and 2.725 eV (closer to resonance). Results aredisplayed in figures 1 and 2.

The full curve in the upper panel of figure 1 shows the dipole responsed(t) calculatedfrom equation (6) for a photon energy of ¯hω0 = 2.95 eV. The broken curves indicatethe envelope of the corresponding dipole response of a classical oscillator with the springconstant tuned to the plasmon frequency (2.6 eV). The electronic dipole moment almost


-15

-10

-5

0

5

10

15

d(t)

0

0.5

1

1.5

2

0 10 20 30 40 50 60 70 80 90 100

Nes

c(t)

Figure 2. The same as figure 1, but for a laser photon energy of 2.725 eV, i.e. only 0.125 eVhigher than the Mie plasmon for Na+9 .

perfectly follows the classical model over the whole time interval considered. After theend of the laser pulse, the oscillations both in the electronic as well as in the classicaldipole moment have died off, as expected in a situation off resonance. Furthermore, in thisoff-resonant case the electron emission is negligible (Nesc' 0.02 at t = 100 fs, lower partof figure 1). Note also that emission takes place predominantly in a time slot of about 20 fsaround the peak of the pulse.

This nice quasi-classical behaviour of the electron cloud off resonance experiencesa radical change as soon as the laser is tuned sufficiently close to the Mie resonancehωM = 2.6 eV, as can be seen in figure 2, whered(t) and Nesc(t) are plotted for aphoton energy of ¯hω0 = 2.725 eV. For the first 25 fs, the electronic dipole momentd(t)

again perfectly follows the classical model. Then, however,d(t) strongly falls back belowthe classical envelope even before the pulse reaches its peak. This reflects the fact that,at this instant, the system starts to ionize rapidly, as can be seen from the lower part offigure 2. Due to this quite abrupt loss of electrons (Nesc= 1.8 at the end of the pulse), thedipole oscillations are strongly damped. Aroundt = 40 fs, however, the rate of electronemission becomes weaker and we observe a new phenomenon: as the charge state of thecluster increases, the collective oscillation of the residual electron cloud is swept towardshigher frequencies. As a consequence, after having gone through a minimum att = 40 fs,d(t) now increasesagain and reaches a local maximum att = 50 fs. This corresponds tothe instant when the collective oscillation of the residual electron cloud is just in resonancewith the laser (2.725 eV). As the collective oscillation frequency continues to increase, the


amplitude ofd(t) decreases again. After the pulse is switched off at 64.4 fs and electronemission has come to an end, we find that the remaining electrons continue to performcollective oscillations (in contrast to the classical model) at the new plasmon frequency of2.9 eV.

This example clearly shows how the spectral features of the electron cloud aredynamically shifted by strong excitations combined with electron loss. Similar, thoughless drastic, effects have been observed in earlier studies [17] when increasing excitationenergies were found to induce a systematic broadening of the resonance.

4. Multiphoton ionization

In this section we investigate how the ionization behaviour of Na+9 depends on the

wavelength and peak intensity of our 25 fs Gaussian pulses. Using a laser operating inthe visible or IR, it takes several photons to ionize Na+

9 (see table 1). In the following weshall restrict ourselves to such cases.

When discussing ionization of atoms and molecules in strong fields [5–7], one usuallydistinguishes two ionization mechanisms [29–32]: the first one is conventional multiphotonionization, whereas the second mechanism is tunnelling through the Coulomb barrier whenthe potential is suppressed by the laser field. Which one of the two mechanisms prevailscan be seen from the Keldysh adiabaticity parameterγ [33], defined (in atomic units) as

γ =√

2W0ω20/I . Here,W0 is the (field-free) ionization potential of the system under study,

ω0 is the frequency andI the intensity of the laser. For values ofγ much greater thanunity, multiphoton ionization is the dominating mechanism. Forγ � 1, on the other hand,the ionization proceeds via tunnelling. For the ionization of Na+

9 discussed in detail inthe following, we haveW0 = 5.52 eV (the binding energy of the highest occupied orbitalobtained within LDA), and we work with laser parameters such thatγ is always significantlygreater than one. We are thus safely in a regime where the multiphoton picture of ionizationapplies.

4.1. Variation of the photon energy

In the upper part of figure 3 we plot the number of emitted electrons of Na+9 at t = 100 fs,

as a function of photon energy and at three different peak intensities,I = 1011, 1012 and1013 W cm−2. In all three cases we find thatNesc (t = 100 fs) has a maximum for photonenergies around ¯hωM . At I = 1011 W cm−2, the peak value ofNesc = 1.8 is found at2.725 eV (see also figure 1). ForI = 1012 W cm−2, the peak is located at 2.825 eV witha value ofNesc= 5.8. For the highest intensity considered,I = 1013 W cm−2, we find avery broad peak with numerous substructures around 2 and 3 eV. The width of the peaksincreases dramatically with increasing intensity from about 0.3 eV at 1011 W cm−2, 0.6 eVat 1012 W cm−2 to almost 2 eV atI = 1013 W cm−2. Furthermore, the overall numberof emitted electrons rises drastically with intensity. It is to be noted, however, that whenNesc becomes extremely large (close to eight, the total number of valence electrons of Na+

9 ),we are presumably somewhat beyond the safe grounds of our approach since for very highintensities the inner electrons could be ionized as well.

Additional interesting features appear when comparing the curves forNesc with thedipole power spectrum in the linear regime [17] as is shown in the lower part of figure 3.There is an obvious correlation between the spectral features of the cluster and electronemission. The maximum of ionization is situated at the position of the Mie plasmon,


0

1

2

3

4

5

6

7

Nes

c(t=

100

fs)

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7

Pow

er

Figure 3. Upper part: number of escaped electrons att = 100 fs for Na+9 irradiated with a25 fs Gaussian laser pulse (peak att = 32.2 fs) versus photon energy. The peak intensities are1011 W cm−2 (triangles), 1012 W cm−2 (squares) and 1013 W cm−2 (diamonds). Lower part:dipole power spectrum of Na+9 in the linear regime.

i.e. there is a strong resonant enhancement of ionization due to excitation of collectiveoscillations by the laser pulse. Note that the peak of emission is slightly blue-shifted withrespect to the plasmon at 2.6 eV. This shift is due to the fact that the remaining electronsare more strongly bound when the charge of the cluster increases, as discussed in section 3.For I = 1011 W cm−2 (Nmax

esc = 1.8) the blue-shift is 0.125 eV, and forI = 1012 W cm−2

(Nmaxesc = 5.8) it is 0.225 eV.

The Mie plasmon, however, is not the only spectral feature that is reflected in electronemission. For the cases ofI = 1012 andI = 1013 W cm−2, we find clear signals at 1.8 eVand around 4 and 5 eV (corresponding signals forI = 1011 W cm−2 are not resolved onthe scale of this plot). The former can be explained by resonance with a one-particle–one-hole excitation at that energy, whereas the latter signals may be attributed to a resonantenhancement of ionization with the volume plasmon. A further interesting feature in theupper part of figure 3 is the asymmetry of the peaks.Nescrises steadily from low frequenciesto the resonance peaks, but then drops rather abruptly at the right flank of the peaks. Thisleads us to expect that ionization proceeds differently for laser frequencies above or belowthe plasmon. When we study ionization from a different point of view in the followingsubsection, we will find evidence supporting this conjecture.


4.2. Variation of the peak intensity

We now keep the laser wavelength fixed and vary the peak intensity of the laser pulses. Ithas been shown theoretically and experimentally in atomic physics that ionization via long(picosecond) laser pulses of moderate intensities proceeds at a rate varying asσνI

ν , whereν is the minimum number of photons to ionize the system andσν is a generalizedν-photoncross section [7, 29–32]. It is to be noted, however, that the derivation of this trend requiresthat no resonances with excited states of the system come across. In our case it is moredifficult to predict the ionization behaviour of Na+9 since (i) we use extremely short 25 fspulses, (ii) the laser-induced electron dynamics of metal clusters is dominated by resonanceswith the collective Mie-plasmon. It is, nonetheless, interesting to compare the results withthe trendI ν as a guideline.

Figure 4 showsNesc (t = 100 fs) versus intensity in a double-logarithmic plot for fourselected photon energies: ¯hω0 = 0.87, 1.17, 2.33 and 3.10 eV. In all four cases, the curvesrise almost linearly at the beginning and level off towards the saturation valueNesc= 8.The behaviour in between differs from case to case. The slope of the curves at 0.87 and1.17 eV is essentially constant at the beginning and begins to decrease monotonically ataroundNesc = 0.5. The two other curves experience anincreaseof the slope forNesc

between about 0.1 and 1 before levelling off. This is probably due to some resonanceeffect. In the following, we shall mainly consider thelinear part of the curves at lowintensities where theNesc are still small. Note that in this caseNesc is identical with theprobability to produce Na2+9 [27]. Higher ionized states will be discussed in section 4.3.

The slopes characterizing the linear trend of the curves in figure 4 for small valuesof Nesc are summarized in table 1. We find them to be very close to integer numbers.Non-resonantlowest-order perturbation theory, as mentioned above, predicts a trend∝ I ν ,whereν is the smallest integer bigger thanW0/hω0 (usingW0 = 5.52 eV for Na+9 ). Theν corresponding to the laser frequencies chosen in our simulations are listed in the second

0.0001

0.001

0.01

0.1

1

10

1e+10 1e+11 1e+12 1e+13 1e+14

Nes

c(t

=10

0 fs

)

Intensity (W/cm )

0.87 eV1.17 eV2.33 eV3.10 eV

2

Figure 4. Number of escaped electrons att = 100 fs for Na+9 irradiated with a 25 fs Gaussianlaser pulse (peak att = 32.2 fs) versus peak laser intensity. The photon energies are 0.87 eV(diamonds), 1.17 eV (plus signs), 2.33 eV (squares) and 3.10 eV (crosses).


Table 1. Comparison of the minimum number of photonsν to overcome the ionization thresholdof Na+9 (5.52 eV in LDA) with the slopes obtained from the numerical results shown in figure 4for the four frequencies considered.

Photon energy (eV) ν Numerical slope

0.87 7 4.01.17 5 3.02.33 3 1.83.10 2 2.0

column of table 1.As mentioned at the end of the previous subsection, we see here clear indications that

the ionization mechanism is different depending on whether the photon energy isaboveorbelow the plasmon energy ¯hωM = 2.6 eV. Forhω0 = 3.10 eV we see that the numericalslope coincides withν, which indicates a straightforward two-photon ionization process.Below the plasmon frequency, on the other hand, the numerical slopes aresmaller than thecorresponding values forν. In fact, we find that the numerical slopes for these cases mayagain be explained with a simple∝ Iµ law, whereµ is now the smallest integer bigger thanW 0/hω0, with W 0 given byW0− hωM = 2.92 eV. For the photon energies 0.87, 1.17 and2.33 eV we obtainµ = 4, 3 and 2, respectively, which exactly reproduces the numericalslopes given in table 1. In other words, ionization seems to proceed via resonance with theMie plasmon, which effectively reduces the emission threshold by ¯hωM . This simple pictureis corroborated by earlier observations showing the plasmon to be extremely harmonic androbust, hence little coupled to any other mode [17]. For the photon energy 2.33 eV and toa lesser extent also for 1.17 eV, this mechanism is additionally favoured by the fact that weuse extremely short pulses which consequently have a non-negligible spectral width. Thetail of the frequency distribution of these pulses then extends into the region of resonancewith the plasmon. We furthermore notice that ¯hω0 = 0.87 eV is just one third of theplasmon energy. In this case, the Mie resonance may be reached directly via a three-photonprocess.

4.3. Ion yields

It was explained in section 2.4 how to deduce ion probabilities from the informationcontained in a TDLDA calculation. In figure 5, we plotP+1 throughP+9 at t = 100 fsversus peak intensity for a laser photon energy of 1.17 eV. The corresponding total numbersof emitted electrons can be found in figure 4. Qualitatively similar curves for the ionprobabilities are obtained for the other photon energies considered.

We see that for small intensities (less than about 1012 W cm−2), P+1 stays close to 1and the otherP ks are very small. This means that there is a very high probability that ourNa+9 cluster does not ionize at all. We find that in this regimeNesc= P+2, as mentionedbefore.

As the intensity grows, higher charge states come into play. It is interesting to note thatthey show up successively, with previous charge states getting decreasing weight when anew, more favoured state reaches its peak. Note that it is guaranteed by construction thatthe probabilities sum up to unity. Thus we can determine the dominating charge states foreach intensity. Finally, at very high intensities (around 2× 1014 W cm−2), P+9 approaches1, i.e. the valence shell of the system becomes completely ionized. In this case, we have


0

0.2

0.4

0.6

0.8

1

1e+11 1e+12 1e+13Intensity (W/cm2)

1e+14 1e+15

Prob

abili

ty1

23 4 5 6 7

8

9

Figure 5. Probabilities for the ionic states+1 through+9 of Na9 at t = 100 fs versus peakintensity (25 fs laser pulses with peak att = 32.2 fs). Photon energy: 1.17 eV.

Nesc = 8P+9. One has to keep in mind, however, that the production of such highlycharged states probably goes beyond the safe grounds of the presently used TDLDA. Finalstate correlations may significantly come into play and the inner core electrons may nolonger be kept frozen at these high laser intentities. Such effects can affect the quantitativepredictions. But the general pattern, as e.g. this typical sequence of interlaced productionmaxima, is most probably correctly reproduced by our present simulations.

5. Conclusion

In this work we have studied various nonlinear effects occurring when a Na+9 cluster is

irradiated with a strong femtosecond laser pulse. The cluster was described within thespherical jellium model and a time-dependent density-functional approach for the valenceelectrons. As observables we have considered the dipole moment of the valence electroncloud with its associated spectral characteristics, the number of emitted electrons, and thedetailed ionization probabilities for the production of discretely charged ionic states of thecluster. These probabilities may be compared with experiment if the spatial pulse profile ofthe laser is appropriately accounted for. The laser pulse was shaped with a Gaussian profileof width 25 fs.

We first considered the dipolar response of Na+9 at two different frequencies on and

off resonance with the Mie plasmon. It was shown that the electron cloud off resonancebehaves as a classical oscillator, whereas resonance with the plasmon leads to a pronouncedinternal excitation with subsequent emission of electrons which, in turn, induces a blue-shiftof the collective mode. Electron emission is accompanied by a strong damping of the dipoleoscillations.

In a second step, we have studied the mechanisms involved in multiphoton ionizationof the cluster. Varying the photon frequency for fixed laser intensities revealed a strongcorrelation between the spectral distribution of the dipole oscillations and the total numberof emitted electrons. Multiphoton ionization is strongly enhanced if the laser is in resonance


with the collective mode, so that an almost complete stripping of the valence shell can beachieved with intensities of the order of 1013 W cm−2. As a complementary point of view,we have looked at electron emission as a function of laser intensity at fixed photon frequency.The emerging curves look similar to the well known experimental findings in atomic physics:the number of emitted electrons first rises linearly in a doubly logarithmic plot and beginsto saturate when the number of emitted electrons exceeds one. If the frequency of thelaser is less than the plasmon frequency, the slope of the initial linear growth turns out tobe smaller than expected from a simple nonresonant perturbative estimate of multiphotonionization. As a reason for this we propose that resonant excitation of the Mie plasmonleads to a reduced effective emission threshold for these cases. This again points out theall dominating role of the dipole plasmon in strongly out-of-equilibrium electron dynamicsin clusters.

Finally, we have evaluated the detailed probabilities for producing the different chargestates to which the Na+9 cluster can ionize. We found that 25 fs laser pulses are capableof generating highly charged states, up to Na+9

9 , for moderately high peak intensities. Theprobability for a fixed charge state drawn versus laser intensity has one clear cut maximum,and the peak positions for the various charge states are interlaced such that they follow inincreasing order and one charge state clearly dominates all others near its maximum. Thesetypical patterns should allow us to produce fairly well defined charge states by tuning thelaser intensity appropriately.

Acknowledgments

The authors thank the French–German exchange program PROCOPE 95073 and InstitutUniversitaire de France for financial support during the realization of this work. CAUthanks CERFACS for financial support and the use of its computational facilities.

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