Laser-driven nonlinear cluster dynamics

Laser-driven nonlinear cluster dynamics

Abstract

Laser excitation of nanometer-sized atomic and molecular clusters offers various opportunities to explore and control ultrafast many-particle dynamics. Whereas weak laser fields allow the analysis of photoionization, excited-state relaxation, and structural modifications on these finite quantum systems, large-amplitude collective electron motion and Coulomb explosion can be induced with intense laser pulses. This review provides an overview of key phenomena arising from laser-cluster interactions with focus on nonlinear optical excitations and discusses the underlying processes according to the current understanding. A brief general survey covers basic cluster properties and excitation mechanisms relevant for laser-driven cluster dynamics. Then, after an excursion in theoretical and experimental methods, results for single- and multiphoton excitations are reviewed with emphasis on signatures from time- and angular resolved photoemission. A key issue of this review is the broad spectrum of phenomena arising from clusters exposed to strong fields, where the interaction with the laser pulse creates short-lived and dense nanoplasmas. The implications for technical developments include the controlled generation of ion, electron, and radiation pulses, as will be addressed along with corresponding examples. Finally, future prospects of laser-cluster research as well as experimental and theoretical challenges are discussed.

pacs:
36.40.-c,52.50.Jm
Contents:

I Introduction

Clusters of atoms and molecules frequently appear as a novel state of matter on the one-nanometer scale. For example, different types of bonding or various structural and chemical features can be realized within the same material by just changing the particle size. The opportunity to vary, almost at will, the number of atoms in the clusters thus offers a unique avenue to explore the organization and properties of matter from a fundamental point of view Haberland (1994); Martin (1996); Sugano and Koizumi (1998); Alonso (2006). This also applies to optical phenomena arising from small particles, e.g., due to surface plasmons Kreibig and Vollmer (1995), which fascinated scientists since a long time Rayleigh (1899); Mie (1908). Today’s lasers open an even more exciting perspective of cluster science, i.e., the opportunity to steer and resolve ultrafast dynamics on the nanoscale.

Due to the progress in laser technology Keller (2003); Rullière (2005), well-controlled short and intense laser pulses can be routinely delivered these days. This opens the door to explore light-induced dynamical phenomena far beyond the mere analysis of ground state properties. For example, the real-time analysis of nuclear and even electron motion becomes possible, as in the case of molecules or atoms Zewail (1994); Corkum and Krausz (2007). When applied to clusters, short pulses controlled in amplitude and phase allow one to drive and resolve ion and electron dynamics on their natural time scales and under extreme conditions. For instance, electronic relaxation processes or the time-evolution of collective modes can be studied with laser-excited clusters. As a more violent scenario, strong-field exposure transforms clusters into well-isolated nanometer-sized plasmas, with interesting prospects for pulsed particle, radiation, or even neutron sources. With the advent of VUV free electron lasers Feldhaus et al. (2005) coherent multiphoton inner-shell excitations are accessible with intense femtosecond pulses. Inspired by such opportunities, the subject of laser-cluster interactions has spawned sustained interdisciplinary activities and experienced enormous developments over the last two decades. It definitely holds the promise to deliver unprecedented insights into the nature of light-matter interactions in complex systems and stimulated challenging efforts in experiment and theory.

In this review we focus on the nonlinear response behavior of clusters subject to laser fields, concentrating on the nonrelativistic intensity regime. It is our aim, in close connection between theory and experiment, to discuss signatures and mechanisms for multiphoton as well as for strong-field excitations. Nevertheless, even single-photon absorption can lead to complex dynamics, e.g., due to electron correlations, structural transitions, or competing electronic decay channels. As a result, the response can clearly go beyond a simple and direct mapping of ground state properties. In any case, pronounced nonlinearities emerge when multiphoton absorption is involved. As a typical example within the still photon-dominated regime, above-threshold ionization can be observed with clusters, showing additional finite-size and many-particle effects when compared to atomic systems. At higher intensities in the so-called field-dominated regime, the immediate excitation of several electrons and laser-driven collisions induce avalanche processes of highly nonperturbative nature. As a surprising feature, clusters very efficiently absorb intense laser radiation Ditmire et al. (1997a), with an energy capture per atom much higher than for atoms or bulk material Batani et al. (2001). Moreover, strong-field laser-cluster interactions lead to the emission of fast electrons Springate et al. (2003), multiply charged ions Köller et al. (1999), and high-energy photons McPherson et al. (1994), documenting the excitation of core electrons. When compared to atoms, the appearance intensities for these products are strongly reduced with clusters. The discussion of the underlying dynamics and appropriate theoretical treatments is the central topic of this contribution.

Different aspects of laser-excited clusters have previously been reviewed, such as the electronic structure of simple metal clusters de Heer (1993); Brack (1993); Ekardt (1999), low- and moderate-field dynamics Reinhard and Suraud (2003), ionization mechanisms in strong optical and VUV laser fields Saalmann et al. (2006), and excitations with ultraintense pulses Krainov and Smirnov (2002). The current report aims to deliver a present-day view on cluster dynamics in optical laser fields, with emphasis on the strong-field regime, and incorporates recent findings regarding angular resolved emission, electron acceleration, and processes behind very highly charged ions. Moreover, routes will be reviewed to resolve the cluster response in time by varying the pulse duration or using dual-pulse excitations. Special features of this review are the extensive presentation of experimental and theoretical methods and the attempt of closely combining theory and experiment.

The text is organized in six major parts. Section II offers a quick outlook of the topic and discusses basic physical mechanisms. It thus provides some basic elementary stepping stones on which to build an understanding of the topic. Section III is devoted to a brief survey of available theoretical tools for describing cluster dynamics and tries to show how the various approaches may be linked together in terms of regimes for which they were primarily developed. Section IV focuses on experimental techniques, discussing cluster production and laser sources as a starter. Emphasis is essentially put on the detection of emitted particles. In the ensuing presentation of selected results, Sec. V concentrates on the intermediate intensity domain in which photons still count. In this regime experiments have revealed detailed insight into the quantum nature of clusters and allow one to explore the emergence of nonlinear behaviors. Section VI finally comes to the main topic of the paper and describes highly nonlinear strong-field induced dynamics where quantum effects are partially wiped out. After a brief survey of initial/original results in the field, a detailed analysis of systematic trends and present day more elaborate approaches are presented. This in particular concerns differential cross-sections and time-resolved analyses. Finally, Section VII provides a brief outlook and proposes a few promising future directions of research in the field. We discuss in particular the prospects of laser developments, either in terms of pulse shaping of today’s sources or by considering forthcoming devices like the projected XFEL lasers. We also comment on embedded and deposited clusters, high-energy particle acceleration with clusters, and point out some future challenges for theory.

Figure 1: Five decay channels of laser-excited clusters aside with properties/processes that may be resolved from their analysis (see text). (a) electronic structure of negatively charged gold clusters with 20 atoms (Au) extracted from the photoelectron spectrum, from Li et al. (2003a), with permission from AAAS; (b) optical absorption of Ag and Ag as determined by photofragmentation, adapted from Tiggesbäumker et al. (1993, 1996), with permission from Elsevier; (c) ionization dynamics of Ag in intense laser pulses resolved by measuring the total electron yield as function of pulse width at fixed pulse energy Radcliffe (2004); (d) Coulomb explosion of Pb analyzed by recoil energy spectroscopy of emitted atomic ions, from Teuber et al. (2001), with kind permission of The European Physical Journal (EPJ); (e) inner-shell recombination in strongly excited krypton clusters measured by x-ray spectroscopy, from  Issac et al. (2004), with permission from American Institute of Physics.

Ii General survey of laser-cluster interactions

Laser irradiation of clusters allows the investigation of a broad spectrum of dynamical processes, ranging from single-photon driven ionization to the strong-field induced explosion of a nanometer-scaled plasma. Irrespective of the regime under consideration, the absence of dissipation into substrate material offers a clean analysis of reaction products, i.e., electrons, ions, cluster fragments, as well as photons. Depending on the cluster material and the chosen laser intensity, quite different properties and response mechanisms can be probed, as will be discussed throughout this review. Exemplarily, Fig. 1 illustrates a few response channels and properties that may be analyzed and can be viewed as a rough guideline.

As an example for electron emission in the single-photon regime, Fig. 1a shows an ultraviolet photoelectron spectroscopy (UPS) result on Au obtained with low intensity laser excitation. The photoelectron energy spectrum images the electronic structure, i.e., binding energies and spectral occupation densities of single electron states, and contains comprehensive information on the system. The large band gap in Fig. 1a, for example, reflects the high stability of the tetrahedral Au Li et al. (2003a). Besides structure analysis, photoelectron spectroscopy (PES) is a powerful tool for monitoring excited states and reactions dynamics, see Sec. V.1.

Laser-induced fragmentation may be analyzed, e.g., to determine optical properties. Fig. 1b displays the optical absorption cross-section of size-selected silver clusters measured by photofragmentation Tiggesbäumker et al. (1993, 1996). The spectra exhibit a pronounced resonance, i.e., the Mie surface plasmon, see Secs. II.1 and II.3. Collective excitations, as prime examples for multielectron effects, are not only relevant in the single-photon limit, but are important for the cluster response in the multiphoton and strong-field regime as well, see Secs. V.2.1 and VI.2.

With increasing laser intensity, nonlinear and feedback effects begin to severely influence the cluster response, such as the electron emission. Fig. 1c shows an example for larger silver clusters, where the measured total electron yield, i.e., the average cluster ionization, is plotted as a function of the temporal width of the exciting laser pulse Radcliffe (2004). The strong variation with pulse duration reveals a pronounced ionization dynamics that can be related to the interplay of collective plasma heating and ultrafast relaxation of the ionic structure, see Sec. VI.2.1. In addition, as a result of high charging of cluster constituents, atomic ions are accelerated to high kinetic energies by Coulomb explosion, see Secs. VI.1.2 and VI.1.3. Examples for ion energy spectra from intense laser excitation of lead clusters are displayed in Fig. 1Teuber et al. (2001) and document kinetic energies of up to hundreds of keV as well as a clear cluster size effect in the recoil energy. Within the strong-field induced excitation process a hot and highly ionized nanoplasma is formed. Clear evidence for the presence of energetic electrons is given by the creation of inner-shell atomic vacancies in the cluster constituents, the recombination of which can be monitored by analyzing the extreme ultraviolet (EUV) and x-ray emission, see Secs. VI.1.4 and VI.2.2. The example in Fig. 1e shows energetic -radiation at resulting from irradiation of krypton clusters Issac et al. (2004). A detailed analysis of the EUV and x-ray emission can be used for monitoring ion charge state distributions.

The examples highlighted in Fig. 1 illustrate the wide spectrum of phenomena resulting from laser irradiation of clusters. Before analyzing particular response effects in more detail, a few basic facts about ”protagonists” of such processes, i.e., clusters and lasers, will be recalled. In the following we furthermore remind basic mechanisms of energy absorption and ionization relying on both individual atomic and cooperative processes and provide a rough classification of different coupling regimes.

ii.1 Basic cluster properties and timescales

Cluster properties are strongly dependent on the type of their constituents. We consider four typical cluster materials: Na as a simple metal, Ag as a noble metal, C as a covalent material, and Ar as a rare-gas systems. Table 1 recalls a few basic facts of these elements, e.g., the electronic core and valence levels and corresponding energy gaps. Since cluster properties are by nature also size-dependent (number of constituents between a few and several thousand atoms), atomic, dimer, and bulk values are stated, which fixes typical orders of magnitude.

Na Ag C Ar
atom
ionization potential [eV] 5.14 7.58 11.26 15.8
[eV] 26.0 53.9 8.21 -
valence level 3s 5s 2p -
core level 2p 4p 2s 3p
lowest dipole exc. [eV] 2.1 3.66 7.48 11.62
critical laser intensity [] 310 110 610 210
dimer
bond length [Å] 3.08 2.53 1.20 3.83
dissoc. energy [eV] 0.76 1.69 6.3 0.012
bulk
work function [eV] 2.75 4.26 4.8 15.8
cohesive energy [eV] 1.12 2.95 7.8 0.08
Wigner-Seitz radius [Å] 2.10 1.59 1.21 2.21
Table 1: Basic atom, dimer, and bulk properties for four typical cluster materials. Bulk properties for C correspond to graphite which is close to the C cluster and carbon nanotubes. The critical laser intensity is estimated with Eq. (6), see Sec. II.3. The Wigner-Seitz radius characterizes the atomic density. NIST; Weast (1988);Verma et al. (1983);Beutel et al. (1993);Hirschfelder et al. (1954).

For a given element, the atomic ionization potential (IP) and the bulk work function (WF) indicate the electronic stability of a corresponding atomic cluster with respect to optical excitation. Both IP and WF follow a similar trend over the given materials, i.e., increase from Na to Ar. Typically, metal clusters can be ionized or excited much easier, i.e., with lower photon energies or less intense radiation, than covalent or rare gas systems. This trend is also reflected in the first atomic dipole transition (lowest dipole excitation). The IP further indicates the ionization behavior in strong fields as it determines the critical laser intensity required for atomic barrier suppression, see Sec. II.3 for details.

Structural stability is not necessarily linked to that of the electronic system. This becomes evident after comparing dimer dissociation energies or bulk cohesive energies with the IP’s, e.g., for C with Ar. Note that the bulk cohesive energies roughly reflect the binding energy per atom of the cluster, while the atomic Wigner-Seitz radius r may be used to approximate the cluster radius (). The values for the dimer bond length indicate typical interatomic distances.

In the visible and ultraviolet spectral range the optical response is mainly determined by valence electrons. In metal clusters, electron delocalization leads to a strong resonance, the Mie surface-plasmon, as a unique feature of finite objects with sub-wavelength dimension. It corresponds to a collective oscillation of the whole valence electron cloud against the ionic background. When considering schematically a cluster as a metallic drop Mie (1908), the Mie surface plasmon frequency of a neutral system can be estimated as de Heer (1993); Brack (1993)

(1)

with the effective Wigner-Seitz radius of conduction electrons, the elementary charge, the permittivity of vacuum, and the electron mass. For small Na, for example, the plasmon energy is around  Schmidt and Haberland (1999), while Eq. (1) predicts a value of . This indicates that the actual Mie response depends on further details (finite size effects, geometrical structure, excitation, net charge, etc.), but Eq. (1) already provides a reasonable order of magnitude sufficient for many forthcoming discussions.

Figure 2: Typical time scales for the dynamics, taking sodium clusters as a prototype. On the top the ranges associated to fs lasers are depicted . Further, processes related to motion (cycle times) and lifetimes due to relaxation (decay times) are indicated. Approximate expressions for electron-electron collisions () and electron evaporation () are given below.

For considering reaction pathways and energy dissipation it is useful to compare relevant time scales. To that end we consider Na as a typical example for a metal cluster. Fig. 2 provides a schematic overview over times related to laser characteristics, electronic and ionic motion, and lifetimes for relaxation processes. For the moment we ignore the extremely short times associated with core electrons. They certainly play an important role in intense laser fields, but are usually dealt with in terms of simplified rate equations, see, e.g., Sec. III.3. The pulse duration of optical lasers may be varied over a wide range extending from fs to ps or even ns. We focus here on pulse widths of the order a few tens to a few hundred fs.

The shortest time scales in Fig. 2 are related to the electronic motion. The Mie plasmon period as the most basic one is of the order of fs, cf. Eq. (1). In the same range, but with a wider span from sub-fs to several fs, are cycle times for other single-particle excitations and direct electron escape, i.e., single-particle excitation into the continuum. Somewhat slower is the plasmon decay due to Landau fragmentation, in analogy to Landau damping known from plasma physics Lifschitz and Pitajewski (1988). In clusters, Landau fragmentation results from the coupling of plasmons with energetically close single-particle excitations. Viewed in coordinate space, it corresponds to collisions of electrons with the anharmonic potential at the cluster surface. The Landau relaxation time depends on cluster size and has, e.g., for Na, its lowest values for  Babst and Reinhard (1997). For it can be estimated from the time between collisions of an electron with the cluster boundary (“wall friction”) as where is the Fermi velocity Yannouleas et al. (1990). For , however, increases for smaller due to the reduced level density. The relaxation time describes damping due to electron-ion collisions. It is strongly temperature dependent ( fs for Na at 273K) and scales as at low temperature due to electron-phonon scattering Ashcroft and Mermin (1976) and follows in a high-temperature plasma Spitzer (1956).

The most widely varying times are related to the collisional damping from electron-electron collisions and thermal electron evaporation. Both strongly depend on the internal excitation of the cluster, which may be characterized by an electronic temperature . A simple connection between internal excitation energy per electron and temperature can be established by the Fermi gas model. For , can be estimated as , where is the Fermi energy. For the particular case of sodium at bulk density, we have . Electron-electron collisions are the key mechanism for electronic thermalization. The law for the corresponding collision time in Fig. 2 is known from Fermi liquid theory Pines and Nozières (1966); Kadanoff and Baym (1962). For low , collisions are strongly suppressed due to Pauli blocking of energetically available electronic states. At high electron collisions become competitive with Landau damping and sometimes even the dominating damping mechanism. Electron-electron collisions can be described by semiclassical models, see Sec. III.2.3.

An even more dramatic temperature (or excitation energy) dependence appears for the electron evaporation time, whose trend is dominated by the exponential factor , where denotes the value of the ionization potential. The more detailed expression for the evaporation time given in Fig. 2 is based on the Weisskopf formulae Weisskopf (1937) and a cluster size of =100. For this size the crossing point occurs at a temperature of about  eV. This corresponds to a hot (nano-)plasma where finite electron clouds are practically an unstable evaporative ensemble. In general, electron evaporation represents a very (sometimes even the most) efficient cooling mechanism for highly excited clusters.

Ionic motion spans a wide range of long time scales. Vibrations, which may be measured by Raman scattering, see, e.g., Portales et al. (2001), are typically in the meV regime, i.e., have cycle times of 100 fs to 1 ps. In small clusters, ionic vibrations can induce satellites in the optical spectrum Ellert et al. (2002); Fehrer et al. (2006). Strong laser irradiation usually leads to large amplitude ionic motion and cluster explosion due to Coulomb pressure generated by ionization and thermal excitation. Electron-ion coupling due to Coulomb pressure proceeds at the electronic time scale, i.e., within a few fs. The effect on the ions, however, develops at slower scale, typically beyond 100 fs, due to the large ionic mass. The time scale of Coulomb explosion can be estimated by considering sudden ionization of cluster constituents to an average atomic charge state . In this case the cluster expands homogenously and doubles its radius after , where is the ion mass and is the initial atomic Wigner-Seitz radius. For Na this yields . In consequence, strong ionization drives clusters apart quite rapidly, accompanied with strong changes in the optical properties. Corresponding signatures can be analyzed with pump-probe techniques, see Sec. VI.2. For excitations that do not induce explosion, the time scale of electron-ion thermalization reaches up to the ns range Fehrer et al. (2006). Ionic relaxation is even slower, e.g., thermal emission of a monomer can easily last s.

As shown above, cluster dynamics comprises a large span of time scales, making their theoretical description to a great challenge. Ionic motion may require a simulation time up to several ps while electronic times scales down to a small fraction of a fs have to be resolved. Theoretical approaches for a corresponding description are subject of Sec. III. Relaxation processes at the ns scale, however, require more phenomenological approaches.

ii.2 Intense laser fields: key parameters

We proceed with a brief summary of basic facts and key parameters of intense laser fields. In the nonrelativistic regime, laser pulses acting on atoms, molecules, or clusters can usually be described as a homogenous time-dependent electric field of the form

(2)

where denotes linear polarization in z-direction, is the peak field strength, is the normalized temporal field envelope of the pulse, is the photon energy of the carrier, and is an additional temporal phase. Any other polarization (linear or circular) can be described by superposition. The phase can be written as where is the carrier-envelope phase, and denote linear and quadratic chirp, and the last term indicates higher order chirp contributions. Furthermore, the instantaneous frequency reads and the instantaneous pulse intensity is given by , where is the peak intensity and is the vacuum speed of light. Typically, the pulse duration is given as the full width at half maximum (FWHM) of the temporal intensity profile. A common temporal pulse profile is a Gaussian field envelope, which then reads . In absence of chirp, the bandwidth (FWHM) of the corresponding spectral intensity profile is related to the temporal pulse width via the time-bandwidth product . Increasing the pulse duration by dispersive pulse stretching to induces a linear chirp of , where is the stretching factor with respect to the bandwidth-limited pulse. The chirp direction (up or down) depends on the sign of the group velocity dispersion of the optical element. However, it should be noted that the exact forms of and are not always easy to ascertain experimentally. Nonetheless, the pulse duration can nowadays be varied very flexibly over a wide range, e.g., between a few fs up to ns for optical lasers.

In the dipole approximation and using the length gauge, the coupling of the pulse to an electron at position can be described by an external potential

(3)

Therefore the system size has to be well below the wavelength , which is well justified for nm clusters and excitation in the optical domain (). The dipole approximation becomes questionable for UV photons and very large clusters, but will be valid in most cases considered below.

To classify coupling regimes it is useful to consider a freely oscillating electron (pure quiver motion, no drift velocity) in the laser field. The cycle averaged kinetic energy defines the ponderomotive potential, which reads

(4)

at the pulse peak. It can be expressed more conveniently by .

Figure 3: Intensity-frequency regimes attainable with different high intensity laser systems (shaded blocks). Corresponding wavelengths and electric field strengths are displayed on the additional scales. Lines indicate regions of constant ponderomotive potential . The transition from photon- to field-dominated coupling is roughly given by , as schematically depicted for an IP of a few eV. VUV-FEL: vacuum ultraviolet free electron laser; X-FEL: x-ray free electron laser.

Fig. 3 displays the dependence of in the frequency-intensity plane aside with the characteristic parameter regions which can be realized with high intensity laser sources. As a rule of thumb, regimes of photon- and field-dominated coupling are separated by a that equals the typical electron binding energy in the considered system, as schematically displayed in Fig. 3. This condition is related to the Keldysh parameter, as discussed in more detail in Sec. II.3. Figure 3 further illustrates the enormous flexibility of optical lasers to produce high intensities up to the relativistic limit where becomes nonnegligible compared to the electron rest energy. In this review, however, we focus on intensities for which relativistic effects and the magnetic field of the pulses may be neglected. Compared to optical lasers, vacuum ultraviolet and x-ray free electron lasers (VUV-FEL/X-FEL) cover a fundamentally different regime, i.e., photon-driven dynamics at high intensities due to the low ponderomotive potential, see Saalmann et al. (2006) and Sec. VII.2.

ii.3 Ionization and heating mechanisms in clusters

Several basic ionization and energy absorption mechanisms are of relevance for describing laser irradiated particles and will be briefly introduced below. Departing from concepts for atomic and molecular systems we move on to cooperative and collective effects which stem from the many-particle nature of clusters.

On the atomic level, two fundamentally different photoionization processes may be considered. The first is vertical excitation of a bound electron by single- or multiphoton absorption in a rapidly oscillating laser field, see multiphoton ionization (MPI) in Fig. 4a.

Figure 4: Schematic view of ionization mechanisms in atoms molecules and clusters. Panels (a) and (b) display potentials of the unperturbed ions , the laser , and their effective sum. In panel (a) the pathways for MPI and OFI of a bound electron are indicated, while panel (b) depicts the charge-resonance-enhanced ionization (CREI). The vertical arrows in (b) indicate the Stark shift. Panel (c) illustrates inner and outer ionization of a cluster based on an effective potential.

This mechanism proceeds over many laser cycles and prevails for weak and moderate fields in the so-called perturbative domain. A MPI process of order is characterized by the reaction rate , where is the corresponding cross-section. MPI, which may be enhanced when intermediate resonant states are available, can promote electrons far beyond the continuum threshold, leading to characteristic peaks separated by units of the photon energy in the electron energy spectrum. This effect, termed as above-threshold ionization (ATI), is well-known from atoms and also appears in clusters, see Sec. V.2.2. The second mechanism is optical field ionization (OFI). Here the laser acts as a quasistationary electric field. For sufficiently strong fields, bound electrons tunnel through the barrier emerging from the combined potential of the residual -charged ion and the laser field, i.e., , with . This is schematically depicted in Fig. 4a (dashed curve). The probability for atomic tunneling ionization can be described by the well-known ADK rates found by Ammosov and Delone and Krainov Ammosov et al. (1986).

A useful measure for the significance of MPI over OFI is the Keldysh adiabaticity parameter Keldysh (1965)

(5)

which compares the IP with the peak kinetic energy of a freely quivering electron (2). Single- or multiphoton ionization dominates for , where the quiver energy is small compared to the IP. For , the binding energy can be overcome within a single laser cycle and OFI is promoted. An equivalent expression for the Keldysh parameter is , which gives a ratio of the tunneling time and the optical period. Optical field ionization dominates if the tunneling time is comparable to or smaller than the optical period; MPI is the leading process otherwise.

Within the tunneling regime (), the ionization probability in one optical cycle approaches unity if the potential barrier can be fully suppressed. For an atomic system, this so-called barrier suppression ionization (BSI) roughly sets in at the threshold intensity

(6)

which reasonably predicts ion appearance intensities in atomic gases Augst et al. (1989). Note that Eq. (6) was used to determine the critical intensities in Tab. 1.

The above considerations apply to isolated atoms where the laser parameters govern the dynamics. For extended systems, i.e., from the molecular level on, structural details become increasingly important. Ionization barriers are influenced by the fields from neighboring ions, which, for example, gives rise to charge-resonance-enhanced ionization (CREI) well-known from strong-field ionization of diatomic molecules Seideman et al. (1995); Zuo and Bandrauk (1995). Within this process, an appropriate internuclear separation results in a simultaneous lowering or suppression of inner and outer potential barriers with respect to the Stark-shifted electronic states (see Fig. 4b), giving rise to an enhanced ionization rate. For larger or smaller separations either the inner or the outer barriers increase and the ionization probability is reduced. As a truly cooperative effect, CREI has been considered also for very small clusters Véniard et al. (2001); Siedschlag and Rost (2002), cf. Sec. VI.2.1.

Very convenient for describing charging dynamics in larger systems is the concept of inner and outer ionization Last and Jortner (1999). As indicated in Fig. 4c, electrons in the cluster may be classified into tightly bound, quasifree, and continuum electrons. Within this picture, inner ionization describes the excitation of tightly bound electrons to the conduction band, i.e., electrons are removed from their host ion but reside within the cluster. Correspondingly, the final excitation into the continuum is termed outer ionization, which contributes to the net ionization of the system. At moderate laser intensities, systems with initially delocalized electrons, like metallic particles, may undergo outer ionization only. In any case, however, the energy span between the thresholds for inner and outer ionization grows with cluster charge, cf. Fig. 4c, underlining the growing importance of quasifree electrons for the interaction dynamics. Besides purely laser-induced MPI and OFI, ionization can be driven by cluster polarization (field amplification) or cluster space-charge fields, e.g., subsequent to strong ionization. In addition, quasifree electrons can drive electron impact ionization (EII), as may be described by semiempirical cross-sections Lotz (1967). The onset and self-amplification of such additional processes is frequently termed ionization ignition Rose-Petruck et al. (1997).

The presence of a nanoplasma, i.e., of quasifree electrons and (multi-)charged atomic ions in the cluster, has substantial impact on the energy capture from a laser pulse. If collective effects are negligible, electrons can acquire energy from the laser field via Inverse Bremsstrahlung (IBS), i.e., by absorbing radiation energy during scattering in the Coulomb field of the ions. IBS relies on the conversion of laser-driven electron motion into thermal energy because of directional momentum redistribution within elastic collisions and is a basic volume-heating effect in underdense plasmas Krainov (2000). Considering a fixed collisional dephasing time (inverse collision frequency), the IBS heating rate per electron in terms of the ponderomotive potential reads

(7)

Whereas the heating rate becomes independent of in the low-frequency case (dc-limit), a -dependence is found for . It should be noted that the collisional relaxation time, which is a function of electron temperature (cf. Sec. II.1) and becomes frequency dependent () for short-wavelength laser excitation, is in general difficult to obtain. For laser-irradiated clusters, pure IBS heating dominates the energy capture of quasifree electrons only at laser frequency far above the Mie plasmon frequency. If the laser frequency becomes comparable to or smaller than , the collective response of quasifree electrons in the cluster has to be taken into account. Surface charges from the laser-driven collective electron displacement induce polarization fields, that strongly modify the effective field in the cluster in amplitude and phase. For a spherical plasma and sufficiently small displacements the corresponding restoring force is linear, i.e., the absorption rate per electron for collective IBS heating is described by a Lorentz profile

(8)

This expression is equivalent to the heating rate assumed in Ditmire’s nanoplasma model, cf. Sec. III.3. Whereas the absorption rates in Eqs. (7) and (8) meet in the high-frequency limit, IBS heating is strongly suppressed for due to efficient screening of the external field by the collective electron displacement. Most importantly, excitation with leads to plasmon-enhanced energy absorption in Eq. (8), cf. the cross-sections in Fig. 1b. Resonant collective driving of cluster electrons can produce strong field amplification that supports cluster ionization and direct acceleration of electron Reinhard and Suraud (1998); Fennel et al. (2007a).

In the above discussion the absorption rates have been assumed to scale linearly with intensity (), cf. Eqs. (7) and (8). This requires that the dephasing time and the plasmon frequency are constants. In strong fields, however, the large quiver amplitudes actively modify the nanoplasma properties. Hence, both the dephasing time and the plasmon frequency become functions of intensity which introduces additional nonlinear terms.

Another very important aspect for the cluster response to strong optical laser fields is the time dependence of the plasmon energy. It scales as , where is the ion-background charge density. In early stages of the interaction is usually too high for being in resonance with the driving IR-field, i.e., the system is overcritical. This is the case in metal- and, already after moderate inner ionization, in rare-gas clusters and leads to strongly suppressed IBS heating as explained above. Less efficient surface heating effects like vacuum heating or Brunel-heating Brunel (1987); Taguchi et al. (2004) remain active in this overcritical state. Therefore electrons that are pulled away from the surface by the laser field are accelerated outside and contribute their acquired energy upon recollision with the cluster. In any case, as a result of moderate charging and heating, Coulomb forces and thermal electron pressure eventually induce an expansion of the cluster. Corresponding time scales are typically between a few tens of fs to some ps, see Sec. II.1 for an estimate of the radius doubling time for pure Coulomb explosion. With cluster expansion the frequency of the collective mode decreases and transiently matches the laser frequency at a certain time, producing a short-lived but strong absorption enhancement, cf. Eq. (8). This idea is a central element of the hydrodynamic approach from Ditmire, see Sec. III.3, however, characterizing the resonance condition in terms of a critical electron density. The latter is justified only for nearly charge neutral systems, such as very large clusters. Since, according to the harmonic potential theorem Dobson (1994), the ionic background creates the restoring force for quasifree electrons, the background charge density is the more general parameter applicable also to charged systems. Nonetheless, for sufficiently long pulses the transient resonance induces efficient heating of quasifree electrons and, as a consequence, strongly supports outer ionization and cluster Coulomb explosion. At high laser intensity, this delayed resonant coupling is important irrespective of the cluster material and leaves clear signatures in the absorption as well as in emission spectra, see Sec. VI.2.1.

ii.4 Classification of coupling regimes

While the relative importance of the above mechanisms depends on the specific scenario, regimes can be identified where particular processes prevail. However, such classification cannot be achieved based on a single parameter like laser intensity. While very low intensities lead to linear and very high ones to nonlinear behavior, other laser characteristics or cluster properties determine the nature of the response for intermediate cases. We briefly discuss a rough sorting of regimes used throughout this article.

The linear regime is the domain of weak laser fields associated with single-photon processes and large values of (cf. Eq. 5). Mechanism are sensitive predominantly to the laser frequency. The prevailing examples are optical response spectra. As this is a key tool, there is a huge body of reviews and books see, e.g., Kreibig and Vollmer (1995); Brack (1993); de Heer (1993); Haberland (1994). Early cluster experiments often used ns pulses for studies on structure or low-energy dynamics Haberland (1994); Näher et al. (1997). Another typical process is single-photon ionization which can be analyzed by photoelectron spectroscopy, see Fig. 1a and Sec. V.1.

The multiphoton regime is associated with moderate laser intensities where multiphoton processes begin to show up ( W/cm depending on material and frequency). Each laser parameter, i.e., frequency, field strength, and pulse profile, becomes equally important. Typical examples are second harmonic generation Götz et al. (1995); Klein-Wiele et al. (1999) and multiphoton ionization. Of particular interest are cases where a multiple of the photon energy can excite an intermediate state of the system. Then, besides direct MPI, a sequential ionization from the (long-living) intermediate state becomes possible Pohl et al. (2001). Another example is above-threshold ionization. Processes emerging in the multiphoton regime are subject of Sec. V.2.

At sufficiently high intensity the laser irradiation produces large ionization and strong heating ( W/cm). The excitation of many electrons and strong feedback effects on the response indicate the so-called strong-field domain where the dynamics cannot be treated perturbatively. Typically, the excitation leads to cluster Coulomb explosion, accompanied by emission of energetic particles, i.e., electrons and ions, as well as photons. The emitted ions usually carry higher charges than in the case of irradiation of single atoms which underlines the impact of cooperative processes. Moreover, the reactions proceed somehow similar for very different cluster materials (from metals to rare gases) since electrons from atomic shells are activated and the transient nanoplasma determines the dynamics. Such highly nonlinear processes are in the focus of Sec. VI.

Figure 5: Ionization of Na as function of laser intensity for excitation by 70 fs -shaped laser pulses for two frequencies (as indicated). The ionization potential is eV. Three photons of eV are required to lift an electron into the continuum (multiphoton ionization) while one photon suffices for eV (linear behavior). At high intensity both cases become nonperturbative, indicating strong-field conditions. Note, that  eV is close to the Mie plasmon of Na, which leads to the early onset of the strong-field response in this case. Calculations are done in TDLDA.

A possible marker for the actual regime is the total ionization yield as function of laser intensity. Lowest order perturbation theory predicts that the yield scales with , where is the number of photons required to overcome the ionization potential. Figure 5 gives an example for Na excited with 70 fs laser pulses and shows the intensity-dependent electron yield for two different laser frequencies. The slope at low intensities agrees nicely with the law, yielding (multiphoton) for the lower- and (single-photon) for the higher frequency. However, the curves turn over at higher intensities where sorting in orders of photon becomes obsolete (breakdown of perturbation theory). One approaches the ”strong-field domain”. Note, that the two laser frequencies perform in a very different way. With  eV excitation, the yield follows the linear behavior and becomes nonperturbative at rather large intensities. With the lower frequency the ionization is a three-photon process and the transition to the nonlinear regime evolves at a much lower intensity. Two effects contribute in the latter case: the near-resonance excitation of the Mie plasmon Reinhard and Suraud (1998) and the stronger impact of optical field effects at lower Keldysh parameters.

Iii Theoretical tools for cluster dynamics

iii.1 Approaches in general

Approximations for the electron system
approach scheme system [eV] regime examples
ab initio full TDSE He 2 S D Parker et al. (2003)
QMC C 0 S (D) Ceperley and Alder (1980)
pure e Needs et al. (2002); Parker et al. (1996)
CI any S E Krause et al. (2005); Schlegel et al. (2007)
MC-TDHF any S E Nest et al. (2005); Caillat et al. (2005)
quantum basis expansion, any 0 S E Guan et al. (1995); Matveev et al. (1999)
DFT all electrons
basis expansion, any S E D Saalmann and Schmidt (1996); Matveev et al. (1999)
pseudopotentials
coord. space grid, any S E D Calvayrac et al. (2000); Yabana and Bertsch (1996)
pseudopotentials
semiclassical Vlasov clusters S D Feret et al. (1996); Fennel et al. (2004)
DFT VUU S D Domps et al. (1998a); Köhn et al. (2008)
Thomas-Fermi any S D Blaise et al. (1997); Domps et al. (1998b)
classical MD any D Haberland et al. (1993); Rose-Petruck et al. (1997)
rate equations any D Ditmire et al. (1996); Milchberg et al. (2001)
Approximations for the ionic system
quantum full TDSE 1+2 any D Saugout et al. (2007)
nonadiabatic MD any any S E D Calvayrac et al. (2000)
BO MD, SE any D Bréchignac et al. (1994)
Table 2: Hierarchy of approaches for the description of electrons and ions in a cluster. Acronyms are defined in the text. The range of applications is listed in the column regime where structure is abbreviated as S, excitation spectra (optical response) as E, and dynamics as D. The label D indicates the capability to describe electron emission and stands for excitation energy.

Clusters are complex systems and their theoretical description requires approximations to the full quantum-mechanical many-body problem - the more so for truly dynamical situations. As approximations are always a compromise between feasibility and demands, there exists a rich spectrum of methods. Table 2 tries to provide a brief overview of commonly applied methods - in the upper part for electrons and in the lower part for the ions. Keywords, numbers, and citations are guidelines and by no means exhaustive. They should be understood as examples and estimates of orders of magnitude. For ab-initio methods some entries for typical sizes and excitation energies are left open as they have, in principle, a huge range of validity, but are, in practice, very limited by quickly growing numerical expense. We add a few remarks while going through the table.

The class of ab initio theories covers a huge range of treatments depending on the size of the underlying basis space, in particular for the configuration interaction (CI) and the multiconfiguration time-dependent Hartree-Fock (MC-TDHF) approach. The most general methods, i.e., exact time-dependent Schrödinger equation (TDSE) and Quantum Monte-Carlo (QMC), are still restricted to very few electrons and presently not applicable to clusters. The vast majority of theoretical investigations of cluster dynamics with quantum aspects relies on density-functional theory (DFT) based methods, with quantum mechanical (QM) or semiclassical propagation, where the latter means Vlasov- or Vlasov-Uehling-Uhlenbeck (VUU) schemes. These will be reviewed in Secs. III.2.1, III.2.2 and III.2.3. Very violent processes exceed the capability of DFT methods and are treated in a purely classical manner, either with molecular dynamics (MD) or, more simple, with rate equations. We will briefly sketch both methods in Secs. III.2.4 and III.3.

The large ionic mass usually permits their classical propagation by MD. This may be performed simultaneously with the (nonadiabatic) electron cloud or in Born-Oppenheimer (BO) approximation, if the electrons follow adiabatically the ion field. Light elements (particularly H and He) often call for a quantum mechanical treatment also for the ions. A full quantum treatment for both, all electrons and ions, is extremely demanding and has not yet been applied to clusters. However, a QM treatment of He atoms has been widely used for He clusters Serra et al. (1991); Weisgerber and Reinhard (1992) and for He material in contact with metal clusters Ancilotto and Togio (1995); Nakatsukasa et al. (2002).

Figure 6 complements Tab. 2 in sketching the regimes of applicability of theoretical models in the plane of excitation and particle number. As the decision for a method depends on several other aspects (e.g., demand on precision, material, time span of simulation), the boundaries of the regimes are to be understood as very soft with large zones of overlap between the models. Note also the two intensity scales on top in Fig. 6, which indicate that limitations are also sensitive to the nature of the system response, i.e., resonant or nonresonant. The distinction has to be kept in mind when discussing specific systems.

Figure 6: Schematic view of applicability regimes for different approaches in a landscape of system size vs. excitation energy. The excitation energy can be loosely related to typical laser intensities in the optical range, as indicated by the intensity scales on top for resonant or nonresonant conditions.

The limitations for CI (and other ab-initio methods) are purely a matter of practicability. Time-dependent local density approximation (TDLDA) is limited in system size for practical reasons and in excitation energy for physical ones, because of the missing dynamical correlations from electron-electron collisions. The upper limits of VUU are also of purely practical nature while the lower limits are principle ones, e.g., the negligence of shell effects, tunneling, and interference. The same holds for MD and rate equations. The upper limits in energy and/or laser intensity are given by the onset of the relativistic regime, where retardation effects within the coupling begin to severely influence the dynamics. For the particle size, a general upper limit results from the application of the dipole approximation, which typically breaks down beyond some ten thousand atoms. In larger systems the field propagation effects (attenuation, diffraction, reflection) need to be taken into account.

iii.2 Effective microscopic theories

Since a fully ab initio treatment of cluster dynamics is hardly feasible, simplifications are necessary by eliminating details of many-body correlations. This naturally leads to a description in terms of single-particle states which is well manageable and still maintains crucial quantum features. The eliminated degrees of freedom are moved to an effective interaction to be used in the reduced description. This leads into the realm of DFT Dreizler and Gross (1990). TDDFT, i.e., its dynamical extension Runge and Gross (1984); Gross et al. (1996), is widely employed in cluster dynamics Reinhard and Suraud (2003) and still under development, see, e.g., Marques et al. (2006). This section provides a brief overview over the typical approaches used for cluster dynamics these days. We begin with the discussion of the energy functionals, proceed with quantum- and semiclassical DFT methods, and end up with the most simplified treatment, i.e., molecular dynamics.

The energy functional

Table 3: Composition of the basic energy-density functional for electrons, ions and their coupling . The ions are described as classical particles with coordinates and momenta , . They correspond to the nuclear centers and the deeper lying, inert core electrons. The coupling to the electrons is mediated by pseudopotentials which are designed to incorporate also the impact of the core electrons on the active electrons. The counterweight the Coulomb singularity of point charges (see Coulomb coupling term) and install effectively a soft inner charge distribution for the ion. We show here for simplicity a local pseudopotential which applies throughout all approaches. Nonlocal versions are often used in connection with QM electron wavefunctions. The electrons can be treated at various levels of approximation. The QM stage employs single-electron wavefunctions where . The semiclassical Vlasov description replaces an orbital based treatment by a phase-space function . In both cases, the Coulomb exchange term and correlations are approximated by effective functionals, usually in local density approximation (LDA) and optionally augmented by a self-interaction correction (SIC). The fully classical level treats electrons as point particles with specifically tuned effective interaction potentials, e.g., by assuming a charge distribution having a finite width. The total electronic density is computed differently when going from the QM over Vlasov to the MD approaches. Note that the current is defined analogously to the density.

Since DFT relies on a variational formulation, it aims at well-controlled approximations. The starting point is an expression for the total energy of electrons and ions from which all static and dynamic equations can be derived. Approximations are made only at one place, namely within this energy functional, and everything else follows consistently. Typical energy functionals used in cluster physics (and many other fields) are summarized in Tab. 3. We comment additional aspects briefly.

Key to success (or failure) is the choice of a reliable functional for exchange and correlations. There are several well-tested functionals within local density approximation (LDA) around, see, e.g., Perdew and Wang (1992). These are the workhorses in cluster dynamics. Higher demands, e.g., in describing molecular bonding of covalent materials require more elaborate functionals including gradients of the density, as in the generalized gradient approximation (GGA) Perdew et al. (1996). And even these turn out to be insufficient in some dynamical situations. The spurious self-interaction spoils ionization potentials and related observables. This can be cured to some extent by a self-interaction correction (SIC) or an appropriate approximation to it (for a discussion in the cluster context see Legrand et al. (2002)). Recent developments in TDDFT employ the full exchange term and try to simplify that by optimized effective (local) potentials (OEP) Della-Sala and Görling (2003); Kümmel and Kronik (2008). This is still in an exploratory stage and schemes applicable in large-scale dynamical calculations have yet to be developed.

Another source of effectiveness are pseudopotentials for ions containing inert core electrons Szasz (1985) – a well-settled topic for static problems. Dynamical applications require to consider the polarizability of core electrons, e.g., in noble metals Serra and Rubio (1997). This can be done by augmenting the pseudopotentials with polarization potentials as done in mixed quantum mechanical molecular dynamics approaches Gresh et al. (1999), for a cluster example see Fehrer et al. (2005).

Table 3 finally includes the step down to a fully classical treatment (MD for electrons). This level develops its effective interactions on an independent route, i.e., by explicit adjustment of the effective interactions to basic molecular and/or bulk properties, see Sec. III.2.4.

Time-dependent density-functional theory

The time-dependent Kohn-Sham (KS) equations coupled with ionic MD are derived by variation of the given energy (see Tab. 3) with respect to the single-electron wavefunctions and to the ionic variables, for details see, e.g., Reinhard and Suraud (2003). They read

(9a)
(9b)

Since by far most applications employ the local density approximation (LDA), the electronic part is coined time-dependent LDA (TDLDA). It is coupled to MD for the ions, yielding together TDLDA-MD. This treatment where electronic and ionic dynamics is propagated simultaneously is compulsory for strong electronic excitations.

There are many situations where rather slow ionic motion dominates and the electron cloud acquires only very little excitation energy. For then, one can switch to the adiabatic Born-Oppenheimer (BO) picture:

(10a)
(10b)

It is assumed that the electronic wavefunctions are always relaxed into the (electronic) ground state for the given ionic configuration and its energy expectation value produces a Born-Oppenheimer energy which depends effectively only on ionic variables, see Eq. (10a). That ionic energy is then used in a standard ionic MD, see Eq. (10b). The method allows one to use larger time steps because only the slow ionic motion is to be propagated. On the other hand, full electronic relaxation takes many static steps. It depends very much on the particular application whether BO-MD is advantageous or not.

The stationary limit of TDLDA (electronic part) is obvious - it is given by Eq. (10a). The situation is more involved at the side of the ions. A stationary point is defined by and may be reached by simply following the steepest gradient of the potential field. However, the ionic energy landscape is swamped by competing local minima. A straightforward gradient path will end up in some minimum, but not easily in the lowest one, i.e., the ground state. One needs to employ stochastic methods, such as simulated annealing and Monte-Carlo sampling, to explore the high-dimensional landscape of the ionic energy surface, for details see Press et al. (1992).

The most time consuming part in TDLDA-MD, i.e., Eqs. (9), is the electron propagation. There are basically two different approaches: Basis expansion or coordinate-space grid representation, see Tab. 2 and references therein. Basis expansions are more efficient in handling different length scales, as typical for covalent systems. Coordinate-space grids, on the other hand, are more adapted for the treatment of highly excited systems where electron emission plays a crucial role. In the latter, absorbing boundary conditions can easily be implemented to avoid unphysical backscattering for the analysis of photoelectron spectra and angular distributions, see, e.g., Calvayrac et al. (2000); Pohl et al. (2004b). A very efficient means to find the electronic ground state is the accelerated gradient iteration Blum et al. (1992). Time stepping is usually based on a Taylor expansion of the time evolution operator. An efficient alternative is the time-splitting method which proceeds by interlaced kinetic and potential evolution Feit et al. (1982); Calvayrac et al. (2000). The ionic MD usually employs the Velocity-Verlet-algorithm, see, e.g., Press et al. (1992). Ground state configurations are best searched for by stochastic methods as mentioned above.

Semiclassical approaches

As particle number and excitation energy grow, an orbital-based treatment of the electronic degrees of freedom becomes practically unfeasible and further approximations have to be made. Less demanding are semiclassical time-dependent density-functional methods, which describe the evolution of the one-body electron phase-space distribution or the electron density and average local currents. The price for such simplification is the loss of the quantized electronic level structure, interference effects, and single electron-hole excitations. However, as these contributions become less important for larger systems with sufficiently narrow energy levels and high excitations, semiclassical methods provide a powerful tool to explore strongly nonlinear laser-cluster dynamics.

A semiclassical equation of motion for the one-particle electron phase-space density as an approximation to quantal mean-field dynamics can be found from the well-known expansion, see, e.g., Bertsch and Das Gupta (1988); Domps et al. (1997); Plagne et al. (2000); Fennel et al. (2004); Fennel and Köhn (2008). This, to lowest order, yields the Vlasov equation

(11)

which is widely used in plasma physics. The effective electron mean-field interaction potential in Eq. (11) follows from the variation of the potential energy , cf. Tab. 3, with respect to the local electron density , i.e., by . Ionic motion may be described in the same way as for TDLDA-MD, see Eqs. (9). Quantum effects, such as exchange and correlation in LDA, are now solely contained in the effective potential and the initial conditions for the distribution function. The latter can be determined from the self-consistent Thomas-Fermi ground state Thomas (1927); Fermi (1928) according to where the Heaviside function, is the local Fermi momentum, and the chemical potential. The Thomas-Fermi-Vlasov dynamics resulting from the propagation of the initial distribution according to Eq. (11) constitutes the semiclassical counterpart of TDLDA.

A generic limitation of mean-field approaches, such as TDLDA and Vlasov, is the negligence of electron-electron collisions. This deficiency may become significant for strong departure from the ground state because of considerably weakened Pauli-blocking. In the semiclassical formulation, binary collisions can be incorporated by a Markovian collision integral of the Uehling-Uhlenbeck (UU) type Uehling and Uhlenbeck (1933), see Bertsch and Das Gupta (1988); Calvayrac et al. (2000); Köhn et al. (2008). This results in the Vlasov-Uehling-Uhlenbeck (VUU) equation

(12)

The collision term embodies a local gain-loss balance for elastic electron-electron scattering determined by the differential cross-section , the local phase-space densities , and the Pauli blocking factors in parenthesis as functions of the relative phase-space occupation for paired spins . The velocity-dependent scattering cross-section can be calculated for a screened electron-electron potential using standard quantum scattering theory Domps et al. (2000); Köhn et al. (2008). Since the collision term in the VUU description vanishes in the ground state because of the blocking factors, the Vlasov dynamics is recovered asymptotically in the limit of weak perturbation. Commonly, the Vlasov as well as the VUU equation are solved by the test particle method only for valence electrons, while core electrons are described by ion pseudopotentials, see, e.g., Giglio et al. (2002); Fennel et al. (2004); Köhn et al. (2008).

Further simplifications can be deduced from hydrodynamic considerations Bloch (1933); Ball et al. (1973), i.e., by assuming local equilibrium and a slowly varying irrotational velocity field. In this case, the electronic dynamics can be solely described by the time-dependent electron density and a velocity field . The corresponding equations of motion follow from a variational principle Domps et al. (1998b), leading to a standard hydrodynamic problem for an inviscid fluid

(13a)
(13b)

where and are the potentials of the internal kinetic energy characterizing the local equilibrium and the interaction energy. The continuity equation Eq. (13a) and the Euler equation Eq.(13b) describe the conservation of mass and momentum explicitly, while the equation of state is implicit in the self-consistent potentials. Analogous to , results from variation of the now density-dependent internal kinetic energy. Within the time-dependent Thomas-Fermi (TDFT) approach, the internal kinetic energy is described in Thomas-Fermi approximation by . TDTF represents the most simple semiclassical time-dependent density-functional approach. The reduction to the propagation of four scalar fields tremendously simplifies the numerical treatment, which is particulary appealing for the study of large systems. For an application to metal clusters see, e.g., Domps et al. (1998b). However, as deformations of the local Fermi sphere are neglected (local equilibrium), TDTF is not capable to describe thermal excitations or highly nonlinear dynamics.

Classical molecular dynamics

A basic limitation of DFT treatments, quantum or semiclassical, lies in the fact that they are of mean-field nature and thus neglect the effect of fluctuations, even if thermalization due to electron-electron collisions can be accounted for approximately in the semiclassical case. While mean-field treatments provide a fully acceptable approach for moderately perturbed systems, they cannot account for the large microfield fluctuations arising from strong-field laser excitation. Exploring these fluctuations on a microscopic basis requires the construction of a statistical ensemble of possible trajectories, which exceeds standard mean-field capabilities. However, even if the approximate description of strong-field induced cluster dynamics with the instantaneous ensemble average provided by mean-field DFT methods may be sufficient, technical difficulties hamper their application to realistic systems in this case. The problem arises if energetic quasifree electrons and strongly bound electrons become involved at the same time, which is the typical situation in cluster ionization dynamics in strong fields where highly charged ions are produced. Hence, very different sets of scale in terms of distances and energies need to be resolved numerically, which quickly becomes prohibitive.

Presently, the single practical solution to microscopically resolve ionization dynamics leading to high atomic charge states are classical MD techniques. Numerous groups have developed corresponding methods over the years where quasifree electrons and ions are described purely classically way Rose-Petruck et al. (1997); Ditmire et al. (1998); Last and Jortner (1999, 2000); Ishikawa and Blenski (2000); Toma and Muller (2002); Siedschlag and Rost (2002); Saalmann and Rost (2003); Siedschlag and Rost (2004); Jurek et al. (2004); Bauer (2004a); Jungreuthmayer et al. (2005); Belkacem et al. (2006b, a); Fennel et al. (2007b).

Once innerionized, electrons are explicitly followed according to classical equations of motion under the influence of the laser field and their mutual Coulomb interaction. A striking advantage of the classical treatment is the account of the classical microfield and many-particle correlations. Nevertheless, there are some difficulties to be circumvented. First, the Coulomb interaction has to be regularized in order to restore the stability of the classical Coulomb system and to avoid classical electron-ion recombination below the atomic energy levels. This is usually done by smoothing the Coulomb interaction, e.g., by inserting a cutoff Ditmire et al. (1998) or by attributing an effective width to the particle Fennel et al. (2007b); Belkacem et al. (2006b). The second problem concerns the computational costs. Standard MD simulations scale with the square of the particles number due to the direct treatment of the two-body interactions. For clusters beyond a few thousands of atoms this may easily become prohibitive and more elaborate algorithms such as hierarchical tree codes or electrostatic particle-in-cell (PIC) methods can be used Pfalzner and Gibbon (1996); Barnes and Hut (1986). Such methods indeed allow the treatment of large clusters on sufficiently long times Saalmann and Rost (2003); Jungreuthmayer et al. (2005); Saalmann and Rost (2005); Saalmann et al. (2006); Krishnamurthy et al. (2006); Petrov and Davis (2008); Kundu and Bauer (2006). Another option for describing large clusters, even at very high laser intensity including relativistic effects, are electromagnetic PIC codes, see, e.g., Jungreuthmayer et al. (2004); Fukuda et al. (2006).

Inner ionization can be treated in various nonexplicit ways. Since deeply bound electrons are associated to large energies and short time scales (typically in the attosecond domain), they are not propagated explicitly in most cases. An exception can be found in Belkacem et al. (2006b, a). In general, however, statistical approaches relying on probabilistic estimates of inner ionization are used. Common strategies for describing atomic field ionization are the consideration of barrier-suppression ionization or the application of tunnel ionization rates, see Sec. II.2. Collisional ionization may be modeled with the semiempirical Lotz cross sections Lotz (1967). However, this implies that ionization rates, which may be altered by many-particle effects in the systems, become a crucial ingredient of the dynamics.

iii.3 Rate equations and the nanoplasma model

The last step in the hierarchy of approaches from the most microscopic to the most macroscopic ones are the rate equations models, which describe the system in terms of a limited set of averaged global variables. Their time evolution is obtained from a few equations accounting for the major couplings, i.e., the interactions with the laser field and the internal electronic and ionic processes. Such description is based on a continuum picture and thus requires the clusters to be sufficiently large.

The original formulation of a corresponding model for strong-field cluster dynamics was done by Ditmire et al. (1996) and is known as the nanoplasma model. This name reflects the assumption that rapid inner ionization of clusters exposed to intense laser fields creates a strongly charged but quasi-homogeneous plasma. The typical cluster size domain for which such picture applies is the nanometer range, whence the denomination. The assumption of a homogeneous plasma requires clusters of sizes larger than the Debye length of the system. Typical density cm and temperature  keV lead to .

The basic dynamical degrees of freedom in the nanoplasma model are: the number of ions in charge state , the number of “free” (innerionized) electrons, the internal energy of the electron cloud, and the radius of the cluster. The global character of these variables implies that ions, electrons, and energy are distributed homogeneously in a sphere of radius . The evolution of ion numbers follows the rate equation

(14)

where is the ionization rate for ions in charge state accounting for tunneling and impact ionization. While tunnel ionization dominates early stages of the evolution, collisional ionization takes the lead at later times. The electron number evolves according to

(15)

where is the total net charge of the cluster whose change is determined by the integrated net flow through the cluster surface. The evolution of the cluster radius is determined by the total pressure

(16)

which is composed of Coulomb pressure due to net charge and thermal pressure of the hot electron gas (treated as an ideal gas of temperature and internal energy ). Here, and denote the number density and the mass of the ions.

The internal energy of the electron cloud follows

(17a)
(17b)

due to absorption of electromagnetic energy (), to cooling through global expansion ( term), to ionization processes ( term), and to energy loss by electron flow through the cluster surface (). Here are ionization potentials of ions with charge state . The cycle-averaged heating rate involves the volume and the internal electric field amplitude in a dielectric sphere , where is the vacuum laser field envelope. The dielectric constant is usually taken from the Drude model with the plasma (or volume plasmon) frequency and the collision frequency for electron-ion scattering. With these assumptions, the cycle-averaged heating rate is equivalent to the expression given in Eq. (8) in Sec. II.3 and exhibits a resonance when the electronic density fulfills , where is termed critical density. This condition reflects the Mie plasmon resonance of a neutral spherical particle Kreibig and Vollmer (1995), see the discussion in Secs. II.1 and II.3.

The Eqs. (14) through (17b) constitute the dynamics of the nanoplasma model. In spite of its simplicity, the model contains the basic competing processes in the dynamics of the irradiated cluster in a nanoplasma state. It is technically simple, but requires several empirical ingredients, such as, e.g., the various ionization rates. It also involves strong simplifications such as a thermal electron distribution, an intensity independent heating rate and very crude treatment of space-charge effects and electron emission. Nevertheless, it was applied to many experimental results with some successes and its original formulation was extended in several respects.

The original model may be questioned at various places, e.g., regarding to the assumption of homogeneous distributions of all species in the cluster. This constraint was relaxed in Milchberg et al. (2001) by considering a radius-dependent distribution. Further, the damping effect of the cluster surface is neglected. It can be introduced by using a modified collision frequency , which then contains electron-ion collisions through () and an additional term for surface-induced Landau damping ( - average electron velocity). The surface contribution has been shown to play an important role for energy absorption Megi et al. (2003). More recently, detailed cross-sections were computed to include high-order ionization transitions involving intermediate excited states for describing the x-ray emission from Ar clusters Micheau et al. (2007). Another important contribution is the lowering of ionization thresholds in the cluster due to plasma screening effects Gets and Krainov (2006), which was shown to significantly alter the ion charge distribution as well as the heating dynamics Hilse et al. (2009).

One should further remind that the nanoplasma model, as a statistical continuum picture, may only describe the gross features of the interaction of intense lasers with clusters. In particular, it cannot access experimental results beyond average values. The model may thus fail in describing the profiles or far tails of, e.g., ion charge state or energy distributions. More detailed insight can, for example, be gained from MD simulations. Nonetheless, even in its crudest version the nanoplasma model may serve as an acceptable starting point to get first insights into the time evolution of charging or the explosion dynamics for large (nanometer) clusters.

Iv Experimental methods

With the modern molecular beam machines, the variety of radiation sources from the infrared to the x-ray regime, and the multiply parallel detection and data processing possibilities, challenging and highly sophisticated experiments on clusters can be performed. It is possible to prepare targets with narrow size distribution or even completely size-selected, partially at low or ultralow temperature. Vast literature exists on cluster production, e.g., Echt and Recknagel (1991); Haberland (1994); Milani and Ianotta (1999); Pauly (2000); Whaley and Miller (2001). Optical single or many-electron excitation, in some cases also being followed by a probing ultrashort light pulse, has led to far-reaching insight into fundamental processes of the light-matter interaction in clusters. In this chapter, rather than covering the vast multitude of experimental methods, we review selected current techniques used for probing dynamics on free clusters.

iv.1 Generation of cluster beams

Rare-gas or molecular clusters are produced from an adiabatic expansion through a continuously working or pulsed nozzle with nozzle diameters ranging from a few to m, usually restricted by the pumping speed of the apparatus. Mixed clusters are generated by a co-expansion of a gas mixture or by using a pick-up technique with a cross-jet. The cluster size may be varied by changing the nozzle temperature or the stagnation pressure. Typically, the width (FWHM) of the size distribution roughly equates the average number of atoms per cluster. According to semiempirical scaling laws Hagena (1974, 1987, 1981) derived from general considerations about condensation kinetics, scales with the ”condensation parameter”

(18)

where is the stagnation pressure, is the nozzle temperature, is the effective nozzle diameter. The gas constants (in units of ) can be calculated from the molar enthalpy at zero temperature and the density of the solid according to Hagena (1987), ranging from for Ne, over for Ar and for Kr, to for Xe. Equation (18) holds for monoatomic gases; otherwise the exponents of and are different. For conical nozzles, has to be replaced by an equivalent diameter that depends on the half opening cone angle. The scaling laws developed for rare gases have been modified afterwards for metal vapors.

For experiments at ultra-low temperatures, helium droplet pick-up sources prove to be very versatile Bartelt et al. (1996); Goyal et al. (1992); Tiggesbäumker and Stienkemeier (2007). A sketch of a typical setup is shown in Fig. 7. He droplets are produced by the supersonic expansion of precooled helium gas with a stagnation pressure of 20 bar through a 5 m diameter nozzle. By choosing the temperature at the orifice (9-16 K), the log-normal droplet size distributions can be adjusted in the range of = 10-10 atoms. After passing differential pumping stages the beam enters the pickup chamber containing a gas target or a heated oven, where atoms are collected and aggregate to clusters inside the He droplets. With this setup it is possible to record clusters with up to 150 silver atoms Radcliffe et al. (2004) or 2500 magnesium atoms Diederich et al. (2005), respectively. Downstream another differential pumping stage, laser light or an electron beam ionizes the doped droplets. The benefits of pick-up sources rely on the feasibility to embed clusters into a well-controlled environment. In the case of He, the embedding medium is superfluid, weakly interacting, and ultracold with a temperature of about 0.4 K Hartmann et al. (1995), being an ideal nanomatrix for spectroscopic studies Whaley and Miller (2001). Similarly, droplets or particles of other elements might serve as pick-up medium, e.g., Ar, Kr, or Xe. Subsequent atom agglomeration also can lead to the formation of electronically excited species Ievlev et al. (2000). While in particular in the case of helium the nanomatrix is mostly transparent under low laser intensity conditions, it may become an active part in the interaction process under strong laser fields that substantially alters the clusters dynamics. Subsequent to plasma formation in the embedded cluster, the nanodroplet may be ionized as well, giving rise to a core-shell type nanoplasma.

Figure 7: Schematics of a He droplet pick-up cluster beam machine. Atoms from the vapor in the pick-up cell can be loaded into the droplets at 0.4 K. After Diederich et al. (2005)

In these days pure metal clusters are mainly produced with laser vaporization or plasma-based methods. In both cases the material is vaporized, being partially ionized, and then undergoes cooling and expansion in a rare gas. This can be pulsed, allowing for a hard expansion of the seeded clusters into vacuum, or continuously streaming at lower pressure. In a laser vaporization cluster source a rotating target rod or plate of the desired material is mounted close to a piezo or magnetically driven pulsed gas valve. Usually He pulses with an admixture of Ne or Ar at backing pressures of 2-20 bar serve as seeding gas. Intense ns laser light pulses with about 50 to 100 mJ/pulse erode target material by producing a plasma plume, which is flushed by the seeding gas through an about 1 mm diameter channel and a nozzle into high vacuum. The close contact with the cold gas leads to supersaturation and efficient aggregation already in the source channel. The nozzle - often elongated by an extender - can be cone-shaped or merely be a cylinder. In some cases an additional small mixing chamber between source body and extender might increase the intensity within a desired mass range. Depending on material and operation conditions, different types of nozzles are in use, partially with very long extenders of 10 cm or more. There is no optimal photon energy, but the intensity must be sufficient to induce vaporization or create a plasma. However, frequency doubled Nd:YAG laser are often used, as its green color facilitates the beam adjustment. With laser vaporization sources practically all solid materials can be vaporized. As a significant fraction () of the clusters is charged, no additional ionization is necessary for studies on mass selected species.

Several types of plasma-based sources are commonly used, the most prominent being the magnetron sputtering cluster source, going back to developments in the group of Haberland Haberland et al. (1992). The basic erosion process is high pressure (1 mbar) magnetron sputtering. This versatile tool operates with a few cm in diameter plane solid target mounted close to an axial permanent magnet, see Fig. 8. In the presence of the seeding gas, a high voltage between a ring-shaped electrode and the target initiates and drives a discharge, efficiently eroding the material and producing a circular well after several hours of operation. The mainly charged vapor is cooled by the seeding gas and transported through a nozzle. Conducting materials can be sputtered by this source, whereas ferromagnets may cause difficulties.

Figure 8: Plasma plume of an uncovered Haberland-type magnetron sputtering cluster source during operation. The ion and electron motion is guided by permanent magnets behind the target. Right: picture of a used silver target.

In contrast to the magnetron sputtering source which operates with a high voltage discharge, arc cluster ion sources make use of high current arcs. Such are known as vacuum arcs, self-stabilizing at about 40 Volts and 40 Amps. The discharge can be sustained in vacuum once a spark has initially brought some metal into the vapor phase. It is important that the discharge is carried by the metal vapor rather than by the seeding gas. In order to accomplish this, the temporal development of the high voltage-driven sparc needs special care. Once the metallic component in the source rules the conductivity, the discharge voltage switches to a low level so that the seeding gas will not directly be ionized. Two variants of the arc sources are in use, pulsed ones and continuously working ones. The concept of the pulsed arc cluster ion source PACIS Siekmann et al. (1991); Cha et al. (1992) is very similar to the laser vaporization cluster source, only that the laser is replaced by a pulsed high-current arc between two electrode rods at about 1 mm separation. An offspring of the PACIS uses one rotating electrode, then being called ”Pulsed Microplasma Source” Barborini et al. (1999). When operated continuously we obtain the Arc Cluster Ion Source ACIS Methling et al. (2001); Kleibert et al. (2007). Here the target is a water-cooled hollow cathode, a water-cooled counter electrode serves as anode. Magnet coils around the hollow cathode help to control the arc. Again, the plasma is flushed by an inert seeding gas into vacuum, producing a cluster beam with a high amount of charged species (about 80%, depending on the material). The beams from the ACIS can be focussed by aerodynamical lens systems. These are sets of orifices and/or confining tubes connected to the nozzle. By choosing appropriate dimensions the on-axis intensities increase, which goes along with a narrowing of the particle size distribution Passig et al. (2006). This type of source can generate large metal particles from 2 to 15 nm in diameter, an interesting size range for future studies of the intense laser-cluster interactions.

All cluster sources described above are housed inside well-pumped vacuum chambers in order to reduce the gas load at the point of investigation. Ideally, only the central filament of the jet passes a narrow skimmer and enters as collimated cluster beam the photoexcitation chamber. Further differential pumping can lead to sufficiently low pressure for the spectroscopy on isolated species. However, many strong-field experiments do not make use of single cluster excitation. In particular for rare gas clusters, the laser is often focussed onto the beam in the high pressure zone close to the nozzle. In such cases many interacting clusters are simultaneously excited, thus the observed signal might originate from a dense cluster ensemble rather than from isolated systems.

iv.2 Sources for intense radiation

Within the last 20 years ultrashort-pulse lasers have undergone dramatic improvements with respect to pulse width, power, and repetition rate. This was first enabled by the technique of colliding pulse modelocking (CPM) within a ring dye laser Fork et al. (1981) and later by the invention of the chirped pulse amplification (CPA) scheme by Maine et al. Maine et al. (1988). Nowadays, the broadband fluorescent (690-1050 nm) laser crystal Ti:sapphire operating at a central wavelength of 800 nm is the working horse in delivering ultrashort and intense optical radiation. Laser pulse durations as short as some femtoseconds Brabec and Krausz (2000); Keller (2003) or attoseconds Corkum and Krausz (2007) as well as pulse powers in the Petawatt regime Ledingham et al. (2003) are available. To avoid damage of the optical components, the pulses from a modelocked femtosecond laser oscillator is first stretched to some ps before amplification and then re-compressed in the final step Maine et al. (1988). For energy enhancement regenerative amplifiers or bow-tie shaped multi-pass configurations are typically used. Stretching as well as compression of the pulse is achieved by introducing diffractive elements, e.g., reflection gratings Strickland and Mourou (1985) in the optical path. High energy pulses in other wavelength regions can be realized, e.g., by amplification of the third harmonic in a KrF amplifier operating at 248 nm Bouma et al. (1993). Due to the limited bandwidth of the transition the pulse duration in this type of laser is limited to some hundreds of femtoseconds. With high harmonics (HH) generated by focussing intense pulses into atomic gases the short wavelength regime becomes accessible opening up the route towards attosecond pulses Papadogiannis et al. (1999). Pulse intensities as high as  W/ cm have been reported for the 27th harmonic Nabekawa et al. (2005). Only recently the vacuum ultraviolet free electron laser (VUV-FEL) FLASH at DESY has been setup, currently delivering pulses with wavelengths down to 6.5 nm at peak energies up to 100 Ayvazyan et al. (2006).

In the optical domain single-shot autocorrelators or more sophisticated setups Trebino (2002) are applied for pulse characterization. In many experiments only the pulse width is varied by detuning the compressor length. This introduces a linear chirp (Sec. II.2) and allows continuous variation of the pulse duration between sub-100 fs to many ps. To generate dual-pulses with variable optical delay (pump-probe) the initial pulse may be split into two replica, e.g., by a Mach-Zehnder setup. Moreover, liquid crystal spatial light modulators, acousto optical modulators, and deformable mirrors allow one to modify the pulse structure at will Weiner (2000). Besides amplitude and phase, also the polarization can be altered, e.g., to drive reactions selectively into a desired channel in coherent control experiments Brumer and Shapiro (1995); Tannor et al. (1986). This scheme connected to a feedback algorithm Judson and Rabitz (1992) is capable of optimizing the laser-matter coupling, see e.g. Assion et al. (1998) and Sec. VII.1.

For pulse focussing, lenses or parabolic mirrors can be used. The latter avoids pulse modification due to the propagation through optical elements, i.e., pulse broadening, self-focussing, or phase modulation. The waist radius of a Gaussian beam at the focus is , where the -number relates the size of the unfocussed beam diameter to the focal length of the lens by , and is the wavelength. Typical spot sizes are a few tens of m. For a qualitative description of nonlinear laser-matter interactions the intensity profile in the focal region has to be taken into account. For a given peak intensity , the intensity profile is given by Milonni and Eberly (1988)

(19)

where and are the axial and transverse distances to the focus and specifies the Rayleigh length, where the beam radius has increased to . The focal intensity profile leads to volumetric weighting, which has been used to determine intensity thresholds in the strong field ionization of atoms Hansch et al. (1996); Goodworth et al. (2005); Bryan et al. (2006) and molecules Benis et al. (2004). Applied to clusters, this intensity-selective scanning method has revealed a dramatic lowering of the threshold intensities for producing highly charged ions when compared to atoms Döppner et al. (2007b); Döppner et al. (2009).

iv.3 Particle detection techniques

Optical excitation of clusters can lead to extensive fragmentation. Usually fragment mass spectra are analyzed in terms of stabilities, similar to nuclear fission processes Schmidt et al. (1992). In strong fields, however, dedicated techniques are needed to resolve the emission spectra of ions and electrons in detail.

Determination of charge state distributions

The most straightforward method to determine charge state distributions of clusters and their fragments is ion mass spectrometry. Irrespective of the particular method, the mass separation will always be connected to the charge-to-mass ratio. In particular time-of-flight (TOF) methods with accelerating electrical fields are widely used for analyzing charged products after photoionization. Fig. 9 shows an example of highly charged atomic ions emerging from silver clusters embedded in He droplets after irradiation with intense fs laser light.

Figure 9: Charge state spectrum from a time-of-flight analysis of Ag in He droplets with  = 40, exposed to 400 fs laser pulses at and 800 nm. The resulting Ag signals from the Coulomb explosion are highlighted. Ions with up to are been detected. The occurrence of He stems from charge transfer with the Ag ions at the chosen laser intensity. From Döppner et al. (2005).

The TOF spectrum exhibits contributions of He and Ag clusters with high masses (not shown here). At short flight times a situation appears like in Fig. 9. Whereas the background peaks are signatures of the He droplet fragments, the highlighted series can uniquely be assigned to atomic ions in high charge states from the Coulomb explosion of Ag. As a matter of fact, the Ag ions carry high recoil energies due to the violent expansion. Therefore TOF methods that use an acceleration of the ionic ensemble by electric fields in the few kV range loose part of their resolution and transmission. Consequently, the TOF spectra only prove the occurrence of the ions but do usually not image the real charge state distribution.

Acquisition of ion recoil energy spectra

A simple and versatile tool to investigate ion recoil energies is the acceleration-free TOF spectroscopy. Two preconditions have to be met in order to allow a unique interpretation of the results: First, there has to be a defined source point for the ion emission. Second, the nature (mass) of the ions must be known, which often is a point difficult to achieve. However, the excitation of single-element clusters with sufficiently strong laser fields leads to complete fragmentation into atomic ions with known mass. In this case, the kinetic energy is determined by TOF measurements through a field free drift tube of about 0.5 m, without initial electric field. For reducing noise caused by secondary electrons and, moreover, to restrict the ion detection to the Rayleigh region of the laser focus, an adjustable narrow slit confines the ion trajectories. Resulting TOF spectra can then directly be converted into kinetic energy spectra, see, e.g., Fig. 1d.

Figure 10: Sketch of the Thomson analyzer. Ions enter a region of parallel electric and magnetic fields trough a tiny hole. The resulting deflection gives characteristic parabolas from which the charge state selective recoil energy can be deduced. A multi channel plate detector with an imaging system serves to record the data. From Döppner et al. (2003), with kind permission of The European Physical Journal (EPJ).

The field-free ion TOF yields recoil energies irrespective of the ion charge states. For a detailed analysis it is necessary to resolve charge state dependent recoil energies. To this end two methods have successfully been applied, both of which simultaneously measure the ion charge state and energy. The first one uses magnetic deflection time-of-flight (MD-TOF) mass spectrometry Lezius et al. (1998). This technique bases on TOF measurements at different positions behind a magnetic field. With the MD-TOF, highly energetic (up to 1 MeV), multiply charged ions could be recorded.

The second method is of static nature and bases on a principle first applied by Thomson Thomson (1907). Fig. 10 sketches the Thomson analyzer for the simultaneous measurement of energy and charge of ions expelled from an exploding cluster. It consists of parallel electric and magnetic fields, followed by a field free drift zone in connection with a position-sensitive detector. The experimentally obtained raw data reflect momentum and energy per charge and have to be transformed to energy vs. charge spectra. For Ag the charge state resolved ion energy distribution is rather narrow and the maximum energy grows almost linearly with ionization stage Döppner et al. (2003).

Energy and angular resolved electron detection

The experimental challenge in photoelectron spectroscopy results from the notoriously low densities in mass-selected charged cluster beams. To cope with this, time-of-flight electron spectroscopy has been developed with a magnetic field gradient. When the clusters are ionized at a certain spot within an electron magnetic bottle spectrometer the complete photoelectron spectrum can be recorded by time-of-flight measurements with up to 100% detection efficiency Kruit and Read (1983); Taylor et al. (1992); Ganteför et al. (1988); Arnold et al. (1991). Whereas this method turned out to be extremely fruitful to reveal the electronic level structure of many mass selected cluster anions, the magnetic fields involved hamper the retrieval of satisfying angular information. In the case of a neutral cluster beam, the target density can be sufficiently high in order to get a spectrum even without the magnetic field. Electron emission and drift occur within a field-free tube, equipped with a time-resolving detector. By rotating the polarization direction of the laser, angular resolved photoelectron spectra are obtained. An increasing length of the drift tube increases the energy resolution on the expense of signal intensity. Acceptable results can be achieved with magnetically well-shielded tube of about 0.5 m length.

In contrast to the electron TOF method, where kinetic energy release information is contained in the electron drift times, imaging techniques extract energy and angular distributions from spatially resolving detection. The striking advantage of this method is that the full emission characteristics can be reconstructed from the 2-D image by means of an Abel inversion. The energy resolution is limited by the quality of the 2-D detector Heck and Chandler (1995). An improvement of the 2-D imaging technique has been obtained by introducing a lens optics which maps all particles with the same initial velocity vector onto the same point on the detector Eppink and Parker (1997). So far, this technique has mainly been used to record low-energy electron spectra. With modified electrode configuration energetic electrons from clusters driven to Coulomb explosion are accessible as well Skruszewicz et al. (2009).

V Single- and multiphoton processes in clusters

The previous sections have provided basic tools for the description and analysis of laser-induced cluster dynamics. In the following presentation of specific examples we begin with single-photon processes in Sec. V.1 and move on to multiphoton effects in Sec. V.2. In both cases clear signatures of the photon energy persist. Single-photon excitations are typically investigated by photoelectron spectroscopy (PES), which is usually interpreted as a static image of the density of states and so indirectly of the underlying geometry. When carried out with angular resolution, PES reveals structural details of the electronic orbitals being excited. However, even single-photon photoemission goes beyond a mapping of system properties in a static and direct way, as it reflects a dynamical process. Pump-probe studies, as a time-resolved version of PES, give access to ultrafast structural dynamics and energy redistribution pathways. Additional reaction channels emerge with the absorption of multiple photons, as will be the subject of Sec. V.2. Besides above-threshold ionization, as a prime example for multiphoton signatures, thermalization and its effect on electron spectra will be discussed. Another issue are plasmons, which often govern the response of metal clusters and become broadened by nonlinear contributions at higher intensity. However, they remain a dominant doorway process up to the strong-field domain, which is subject of Sec. VI.

v.1 Single-photon electron emission

Probing the density of states

For studying single-electron excitations by photoemission it is often useful to assume, motivated by Koopmann’s theorem Weissbluth (1978), that the essential structures of the electron and ionic systems do not change significantly upon electron emission. The photoelectron energy spectrum thus basically images the density of states (DOS). Based on this assumption, PES has become a powerful tool to explore the electronic structure of mass-selected clusters. Figure 11 displays an example from Na. The measured data (black curve) exhibits pronounced peaks at binding energies between 1.8 and 3.5 eV. Such electronic fingerprints reveal details of the quantum confinement and change dramatically with cluster size or structure. With DFT calculations it has become possible to obtain theoretical DOS for comparison with experimental PES spectra. Fig. 11 displays an attempt to identify the cluster ground state geometry out of theoretically suggested candidates by matching the DOS.

Figure 11: PES spectrum of Na: experimental result (black curve) from nanosecond laser excitation with  eV at  100 K and theoretical DOS calculated by DFT using different ground state structures (as shown). From the matching of the spectra the left structure is favored while the right ones show less agreement. After Kostko et al. (2007)

A vast amount of photoelectron spectra on different systems has been accumulated since first successful experiments Leopold et al. (1987); Ho et al. (1990); Pettiette et al. (1988); Cheshnovsky et al. (1990); McHugh et al. (1989); Ganteför et al. (1988, 1996). During the course of time, developments in cluster production and electron detection have made it possible to cover large size ranges at high energy resolution. For instance, in Wrigge et al. (2002) PES spectra of Na for N=31-500 show peaks that can be assigned to the electronic shell structure. For small systems a higher level of theoretical understanding can be obtained from ab-initio quantum chemical methods Bonačić-Koutecký et al. (1991).

To date, most PES studies rely on low-energy photon excitations, i.e., valence-band PES. Inner shell photoionization, i.e., core-level PES, has been demonstrated as well Wertheim (1989); Siekmann et al. (1993); Eberhardt et al. (1990). These studies, however, dealt with deposited clusters excited with high photon energy lamps or synchrotron radiation. Common results are shifts of core levels with cluster size. Due to the surface contact a thorough understanding remains difficult since core-hole screening, chemical shifts, electronic relaxation or charge transfer dynamics contribute to the spectra.

With third-generation synchrotron sources, experiments on free neutral (not mass-selected) clusters became possible. One issue of such studies is the absorption site as a probe of the local environment von Pietrowski et al. (2006); Hatsui et al. (2005). In rare gas clusters the measured line profiles Tchaplyguine et al. (2004) show well-separated features that can be attributed to the ionization of surface and volume atoms, respectively Amar et al. (2005); Bergersen et al. (2006). Such analyses can also provide an indirect size measurement, as has recently been shown for neutral nanometer clusters of various metals, i.e., Na Peredkov et al. (2007a), Pb Peredkov et al. (2007b), Cu, and Ag Tchaplyguine et al. (2007).

Latest progress in core-level PES has been achieved at the free electron laser FLASH which delivers intense pulses with up to 200 eV photon energy. The energy range and high brilliance open new possibilities to interrogate both the complete valence regions as well as shallow core levels of numerous systems. For example, PES on free mass separated Pb revealed a pronounced -dependent shift of the 5 core level Senz et al. (2009) which is in accordance with the metallic droplet picture for large . However, strong deviations starting below indicate a transition from metallic to nonmetallic bonding due to less efficient core-hole screening.

A solid theoretical understanding of the photoionization process requires the complete toolbox of computational many-particle physics. One example where DFT calculations for Na are compared to experimental PES was shown above in Fig 11. In the same spirit, Si for have been investigated theoretically in Guliamov et al. (2005) and compared to data from Hoffmann et al. (2001). In both cases, not all peaks could be fully reproduced by theory, especially for deeply bound electronic states. Nevertheless, from comparison of the calculated DOS with the experiment the ground state geometry can be identified and discriminated against competing isomers in many cases. Remaining discrepancies reflect that static DFT calculations based on the Kohn-Sham eigenvalues are insufficient to fully describe the photoemission. It is well-known that the interpretation of eigenvalues as single-particle energies requires attention Kümmel and Kronik (2008); Mundt et al. (2006). This concerns the meaning of single-particle eigenvalues itself as well as dynamical aspects, as Koopmann’s theorem does not hold in a strict way. In other words, photoemission is a highly correlated process. The photoelectron interacts with the residual system during its removal and may substantially modify the level structure. The effect becomes important with low energy electrons and dramatic in the zero electron kinetic energy measurements.

Figure 12: Comparison of measured PES spectra for Na (lower panel) Moseler et al. (2003) and two different theoretical predictions. The upper panel shows the single-electron levels from a (static) Kohn-Sham calculation applying ADSIC. The middle panel presents the theoretical result deduced from the excitation spectrum of neutral Na, the final product after photoemission. The excitations were computed with TDLDA Mundt and Kümmel (2007).

The question whether PES reflects parent or daughter cluster DOS or a dynamical mixture of both has been tackled in the case of sodium cluster anions, see Fig. 12. The comparison between the experimental spectrum Moseler et al. (2003) and the Kohn-Sham eigenvalues of the (parent) cluster anion calculated with average-density self-interaction correction (ADSIC) is clearly not satisfying Legrand et al. (2002). A way to circumvent the use of the Kohn-Sham eigenenergies is to perform a time-dependent DFT calculation of the response to a small pertubation. In Mundt and Kümmel (2007), the energies of excited states of the neutralized daughter cluster are extracted from the time evolution of the dipole and quadrupole moments and are related to the photoelectron kinetic energies by energy conservation (middle panel). While some discrepancies still remain there is a clear improvement over mere static considerations which points out the key role of final state interactions.

Angular distributions

Besides pure energy spectra, which reflect the electronic level structure, photoemission may also reveal details of the involved orbitals and thermalization phenomena. To this end the emission has to be analyzed with angular resolution, a subject that still is in its early stage. The directionality of the photoelectron angular distribution (PAD) can be quantified by a Legendre expansion: