Ultrafast interatomic electronic decay in multiply excited clusters
Alexander I. Kuleff, Kirill Gokhberg, Soeren Kopelke, Lorenz S. Cederbaum
aa r X i v : . [ phy s i c s . op ti c s ] A p r Ultrafast interatomic electronic decay in multiply excited clusters
Alexander I. Kuleff, ∗ Kirill Gokhberg, S¨oren Kopelke, and Lorenz S. Cederbaum
Theoretische Chemie, PCI, Universit¨at HeidelbergIm Neuenheimer Feld 229, 69120 Heidelberg, Germany (Dated: September 28, 2018)An ultrafast mechanism belonging to the family of interatomic Coulombic decay (ICD) phenomenais proposed. When two excited species are present, an ultrafast energy transfer can take placebringing one of them to its ground state and ionizing the other one. It is shown that if largehomoatomic clusters are exposed to an ultrashort and intense laser pulse whose photon energy is inresonance with an excitation transition of the cluster constituents, the large majority of ions will beproduced by this ICD mechanism rather than by two-photon ionization. A related collective-ICDprocess that is operative in heteroatomic systems is also discussed.
PACS numbers: 31.70.Hq, 32.80.Rm, 36.40.-c, 32.80.Wr
The rapid development during the last decades of veryintense light sources with extreme short pulse durationopened a new era in the study of radiation-matter in-teraction. Studying the interaction of intense fields withmatter brought to the discovery of a whole plethora ofnew physical phenomena, like high-harmonic generation,above-threshold ionization, or tunneling ionization, toname only a few. In the same time, the progress in gen-erating extremely short pulses gave the scientific commu-nity a powerful tool to monitor and control the electrondynamics in atomic and molecular systems and to studyprocesses that take place on a time scale in which the elec-tronic motion is still disentangled from the slower nucleardynamics (for recent reviews see, e.g., Refs. [1, 2]). Anumber of free-electron lasers are in operation today pro-viding extremely bright, coherent, and ultrashort pulsesin the VUV regime. Exposed to such highly intensepulses, atomic and molecular systems will absorb a largeamount of photons triggering various dynamical effects.In this letter we will restrict ourselves to situations wherethe single-photon energy in the pulse is not high enoughto directly ionize the system. It is well known that evenin this case the system can be ionized by a multipho-ton ionization mechanism. The multiphoton ionization(MPI) results from the ability of quantum systems toabsorb several and even many photons, whose individ-ual energies are insufficient to ionize the system. Thecombined energy of the absorbed photons, though, suf-fices to eventually eject one or many electrons from thesystem. During the last decade the MPI has been in-tensively studied also in composite systems, like clusters,employing the new powerful laser sources (for a reviewsee, e.g. Ref. [3]). However, little attention was paid toother mechanisms that can lead to a multiple ionizationin an atomic or molecular cluster irradiated by an intenselaser pulse.In this letter we aim at discussing a hitherto unrec-ognized mechanism for producing ionized species in ho-moatomic or homomolecular clusters exposed to an in-tense laser pulse, which in many cases can be by far a)b)
FIG. 1: Schematic representation of the process. a) Two sub-units of the system are excited by absorbing two photons. b)One of the constituents of the system de-excites transferringthe energy to the neighbor which uses it to emit its excitedelectron. the dominating one. For simplicity we will consider anatomic cluster, but we stress that the process is gen-eral and not restricted to atomic systems. Let us takea homoatomic cluster and irradiate it with a short andintense laser pulse with photon energy below the ioniza-tion threshold of the cluster constituents but in resonancewith one of their excited states. A fraction of the clus-ter constituents will be ionized by MPI but, since we areat resonance, the large majority of atoms will be exited.The well known effect of Coulomb blockade will not playa significant role here, since we suppose that the systemis exposed to a very short pulse, i.e. to a broadband ex-citation. Thus, there will be many excited atoms in thecluster whose neighbor is also excited. Having two ex-cited atoms in close proxmity the following interatomicelectronic decay mechanism is conceivable. One of theatoms is de-excited, the energy is transferred to the otherone which uses it to emit its excited electron. Thus, atthe end of the process, one of the atoms is ionized and theother one has returned to its ground state. The processis pictorially represented in Fig. 1 and can be written inshort as: A ∗ · · · A ∗ → A · · · A + + e − . This process bears similarities with the interatomic (in-termolecular) Coulombic decay (ICD) predicted theoret-ically more than ten years ago [4] and since then stud-ied very intensively both theoretically and experimen-tally (see, e.g. Refs. [5–10]). The ICD is a very efficientelectronic decay mode of inner-valence ionized atoms ormolecules embedded in an environment. Inner-valenceionized states usually have energies below the double ion-ization threshold and, thus, cannot autoionize. However,here the environment plays a critical role. When theinitially ionized atom or molecule has neighbors, like ina cluster, an electron from a higher level may fill thevacancy and the released energy can be transferred toa neighbor form which a secondary electron is emitted.Thus, the creation of a single hole in one of the subunitsin the system leads to the formation of two positivelycharged subunits that repel each other typically leadingto a Coulomb explosion that disintegrates the system.The process is ultrafast with typical lifetimes of few tofew tens of femtoseconds, quenching all other energeti-cally allowed relaxation modes of the system. The dis-covery of the ICD revealed a whole zoo of related phe-nomena, involving both energy and electron transfer andinitiated by single or multiple ionization, as well as byinner- or outer valence excitation (for recent review, seeRef. [6]). Although these processes have different namesand acronyms, we will refer here to all these phenomenaas ICD in order to make the text more transparent. Theonly, but important difference of the process proposedhere and the ICD phenomena studied until now is thatthe ICD assumes an excited system interacting with anon-excited environment, while in the process sketchedin Fig. 1 the distinction between system and environmentis not possible. On the contrary, both constituents areequally suitable to undergo an electronic decay. However,we will refrain from giving a new name to the process dis-cussed in this letter and will refer to it as ICD.The important question is, of course, whether this ICDprocess is efficient enough and can compete with theother possible de-excitation modes (e.g. photon emis-sion) in the dimer or, even more interesting, in a largecluster. To estimate that we have to calculate the rateof the process, or the decay width Γ. The easiest wayto estimate the decay width is to consider the processwithin the simplified but insightful picture of interactionbetween two dipoles via a virtual-photon exchange. Avirtual photon is emitted as a result of the de-excitationof one of the excited atoms and then absorbed by theother excited atom causing its ionization. The virtual photon exchange picture, which is correct at large in-teratomic distances, enables the derivation of analyticalformulae for the decay width [11]. Such formulae exhibit1 /R dependence ( R being the interatomic separation)with a prefactor specific for the emitting and absorbingconstituents and accounting for the dipole selection rulesof the involved transitions. The derivation of such an ex-pression is straightforward using the procedure explainedin detail in Ref. [12].In order to illustrate the efficiency of the ICD mecha-nism proposed here we consider a concrete example. Letus take a neon dimer in which both of the neon atomsare in their first excited state Ne ∗ (2 p − s )-Ne ∗ (2 p − s ).The energy of Ne ∗ (2 p − s ) is about 16.7 eV above theground state while the ionization potential (IP) of theneon atom is about 21.6 eV. Thus, the 3 s → p tran-sition in one of the neons will release enough energy toionize the other one, emitting an electron with kinetic en-ergy of about 11.8 eV. Averaging over the multiplicities ofthe initial states and summing over the final states we ob-tain for the total decay width of the system Ne ∗ (2 p − s )-Ne ∗ (2 p − s ) as a function of the internuclear distance R the following expression (in atomic units)Γ( R ) = 3 c f σπω R , (1)where f is the oscillator strength of the 3 s → p transi-tion, σ is the ionization cross section of Ne ∗ (2 p − s ), c is the speed of light, and ω is the virtual-photon energy.The values of the quantities entering Eq. (1) are knownfrom the literature – the oscillator strength for the 3 s → p transition in neon is 0.16 [13] and the photoionizationcross section of Ne ∗ (2 p − s ) is about 0.18 Mb [14]. Atthe Ne equilibrium distance of 3.1 ˚A the decay widthfor the process is 0.24 meV, which implies a life time ofabout 2.8 ps. This is 3 orders of magnitude faster thanphoton emission, which is known to be about 2 ns [15],and thus ICD is by far the dominant relaxation pathwayin the dimer. The results obtaind by the virtual-photonmodel are correct at large interatomic distances R . Theyare very promissing, in particular, since it is known fromprevious studies [11] that such kind of asymptotic for-mulae underestimate the decay rates around equilibriumdistances due to neglecting the orbital overlap.In order to have more reliable values for the decay rate,we used the L ab initio method, known as Fano-Stieltjesapproach [16]. In this method the boundlike and thecontinuumlike components of the wave function of thedecaying state are constructed using the Green’s func-tion formalism, and the problem of the normalization ofthe continuum wave function is addressed by using theStieltjes imaging technique (see Ref. [16] for details).The ab initio results are shown in Fig. 2 together withthe predictions of the virtual-photon model, Eq. (1). Fora reference, the atomic fluorescence decay width is alsoshown in the figure. We see that up to about 9 ˚A of in- −4 −3 −2 −1
10 8 6 4 2 G [ m e V ] R [Å]
FIG. 2: Total ICD width Γ for the system Ne ∗ (2 p − s )-Ne ∗ (2 p − s ) compared to the prediction of virtual photonmodel, Eq. (1). The atomic fluorescence decay width is indi-cated by a horisontal line, while the equilibrium interatomicseparation by a vertical one. Note the double logarithmicscale used. ternuclear separation, i.e. about 3 times the equilibriumdistance, the asymptotic formula largely underestimatesthe ICD decay width. When the width becomes verysmall (i.e., at large internuclear separation) the ab initio method suffers fron numerical instabilities and cannot besafely employed. It is also around 9 ˚A distance where theradiative decay becomes competitive. At the equilibriumdistance of the neon dimer the ab initio computation pre-dicts a total decay width of 5.4 meV which is more than20 times larger than the virtual photon result. This decaywidth corresponds to a life time as short as 122 fs, whichmeans that the ICD sets in before the nuclear dynamicsplay a role.Let us now comment on larger clusters. Most impor-tantly, since the total decay width is a sum of the partialwidths of all possible decay channels, it is clear that ifwe have more than two interacting excited atoms theICD process will become dramatically faster [5, 17]. In(Ne ∗ ) , for example, there are 12 open channels, whichsuggests that the ICD life time in this cluster will be 6times shorter than that for (Ne ∗ ) . Thus, in big clus-ters, where a resonant intense laser pulse will produce alarge number of excited atoms, the ICD mechanism willbe extremely efficient.Once we have seen that the ICD process is ultrafastand can be expected to outperform other possible waysof relaxation, let us return to the question of the compe-tition between the ICD and the MPI in the productionof positive ions in a cluster irradiated by a laser pulsewith high density of photons. For that purpose, it is il-luminating to consider again a concrete example. Let aNe cluster be exposed to a short and intense laserpulse with photon energy of 16.7 eV, i.e. resonant to the 2 p → s excitation of the neon atom. We can estimatethe number of excited atoms and the number of thoseionized by two-photon ionization in the cluster after thepulse by solving the following system of rate equations dN ( t ) dt = − σ Φ( t ) N ( t ) − σ Φ ( t ) N ( t ) ,dN ( ∗ ) ( t ) dt = σ Φ( t ) N ( t ) − σ Φ( t ) N ( ∗ ) ( t ) ,dN (+) ( t ) dt = σ Φ( t ) N ( ∗ ) ( t ) + σ Φ ( t ) N ( t ) . (2)In Eqs. (2) N ( t ), N ( ∗ ) ( t ), and N (+) ( t ) are the numberof neutral, excited, and ionized by two-photon ionizationatoms as a function of time, respectively, while σ is theabsorption cross section, σ is the photoionization crosssection of Ne ∗ (2 p − s ), σ is the two-photon ionizationcross section, and Φ( t ) denotes the photon flux whichcontains the information on the temporal profile of thepulse. In order to obtain quantitative results, one has toconsider also the spatial profile of the pulse and the ge-ometry of the irradiated cluster. However, we aim hereat making only an estimate of the ratio between N ( ∗ ) and N (+) after the pulse and that is why we will usethe rather simplified picture of a rectangular pulse withintensity 10 W/cm and duration 50 fs, ignoring thedependence of the laser-cluster interaction on the spa-tial profile of the pulse and the geometry of the cluster.In this case, Eqs. (2) can be easily solved and using theatomic data, σ ≈
273 Mb [13], σ ≈ .
18 Mb [14], and σ ≈ × − cm s [18], one obtains that in the Ne cluster after the pulse 991 atoms will be excited and only3 will be ionized by a two-photon ionization. Since, as wesaw, the ICD process is very efficient, one would expectthat every pair of Ne ∗ (2 p − s ) will undergo ICD produc-ing about 495 neon ions. Thus, the ratio of the neon ionsproduced by ICD and those produced by a two-photonionization is about 166:1. It is clear that by increasingthe laser intensity one will produce more ions by two-photon ionization, while decreasing it will favor the ionproduction via ICD mechanism. For example, with alaser intensity of 10 W/cm this ratio is 13.7:1. Wesee that even at these relatively high intensities, the ICDmechanism is still by far the dominant source of ionizedspecies in the cluster. It is clear that even at higher inten-sities, the production of ions via ICD has to be taken intoaccount when interpreting experimental results. Indeed,the peak intensity is achieved only in the focal point ofthe laser which usually is much smaller than the interac-tion region. A large fraction of the clusters, thereby, willbe exposed to a less intense field where the ICD is thedominant ion-production source.At the end we would like to comment briefly on an-other possibility to create ionized species in multiply ex-cited clusters which will be operative in the case of het-eroatomic systems. In the case when the de-excitation A AB
A*...A* → A... B + ...B A... + e- FIG. 3: Schematic representation of the collective process dis-cussed in the text for heteroatomic clusters. Two constituentsof the system de-excite simultaneously transferring the energyto a third species and ionizing it. energy of an excited atom is insufficient to ionize an-other atom, a process related to the recently discussedcollective-ICD [19] can take place. In the collective-ICDprocess two inner-valence ionized species de-excite simul-taneously transferring their “collective” energy to a thirdneighbor and ionizing it. In analogy, one can think abouta collective-ICD where two excited atoms or moleculesde-excite simultaneously and the released energy is usedby a third atom or molecule to eject one of its electrons,see Fig. 3. An important point to note is that, in contrastto the former case, in the case of collective-ICD from ex-cited species the process will not have to compete withthe Coulomb explosion dynamics of the two neighbor-ing ions. It is clear that the collective-ICD from excitedspecies will be energetically open when 2 E ( A ∗ ) > IP ( B ).This implies that A should be different from B since wesupposed that the ICD process of Fig. 1 is energeticallyclosed. Of course, if the ICD channel is open, the collec-tive decay can also take place, but since it involves threeelectrons, its importance compared to the two-electronICD process will be low.Let us conclude. In this letter we proposed a hithertounrecognized mechanism for producing ionized speciesin multiply excited atomic or molecular clusters. Themechanism belongs to the family of interatomic (inter-molecular) Coulombic decay phenomena and consists ofan ultrafast energy transfer between two excited species,bringing one of them to its ground state and ionizingthe other. We showed that the process is ultrafast (inthe femtosecond time regime) and as such is extremelyefficient compared to other possible relaxation modes. Moreover, we showed that if large clusters are exposedto an ultrashort and intensive laser pulse (10 − W/cm in the present example) which is in resonancewith an excitation transition of the cluster constituents,the large majority of ions will be produced by this ICDmechanism rather than by two-photon ionization. In ad-dition, we proposed a collective ICD process that cantake place in heteroatomic or heteromolecular systemsalso yielding ionized species. We hope that our work willtrigger more theoretical and experimental investigationsof these ICD effects in systems exposed to ultrafast laserpulses with high density of photons.The authors thank K. Ueda for stimulating discus-sions and for sharing with us his experimental dataprior to publication which triggered the present work.The research leading to these results has received fund-ing from the European Research Council under the Eu-ropean Community’s Seventh Framework Programme(FP7/2007-2013) / ERC Advanced Investigator Grantn ◦ ∗ E-mail: alexander.kuleff@pci.uni-heidelberg.de[1] F. Krausz and M. Ivanov, Rev. Mod. Phys. , 163(2009).[2] M. Nisoli and G. Sansone, Prog. Quantum Electron. ,17 (2009).[3] U. Saalmann, Ch. Siedschlag, and J. M. Rost, J. Phys.B , R39 (2006).[4] L. S. Cederbaum, J. Zobeley, and F. Tarantelli, Phys.Rev. Lett. , 4778 (1997).[5] R. Santra and L. S. Cederbaum, Phys. Rep. , 1(2002).[6] V. Averbukh et al. , J. Electr. Spectr. Relat. Phen. (2010),doi:10.1016/j.elspec.2010.03.003.[7] S. Marburger et al. , Phys. Rev. Lett. , 203401 (2003).[8] T. Jahnke et al. , Phys. Rev. Lett. , 163401 (2004).[9] T. Jahnke et al. , Nature Phys. , 139 (2010).[10] M. Mucke et al. , Nature Phys. , 143 (2010).[11] V. Averbukh, I. B. M¨uller, and L. S. Cederbaum, Phys.Rev. Lett. , 263002 (2004).[12] K. Gokhberg et al. , Phys. Rev. A , 013417 (2010).[13] W. F. Chan, G. Cooper, X. Guo, and C. E. Brion, Phys.Rev. A , 1420 (1992).[14] R. Kau, I. D. Petrov, V. L. Sukhorukov, and H. Hotop,J. Phys. B , 5673 (1996).[15] D. A. Verner, E. M. Verner, and G. J. Ferland, At. DataNucl. Data Tables , 1 (1996).[16] V. Averbukh and L. S. Cederbaum, J. Chem. Phys. ,204107 (2005).[17] G. ¨Ohrwall et al. , Phys. Rev. Lett. , 173401 (2004).[18] C. McKenna and H. W. van der Hart, J. Phys. B , 457(2004).[19] V. Averbukh and P. Kolorenˇc, Phys. Rev. Lett.103