RESEARCH ARTICLE Electron-electron correlation in

Journal of Modern Optics Vol. 00, No. 00, 00 Month 200x, 1–65

RESEARCH ARTICLE Electron-electron correlation in strong laser fields C. Figueira de Morisson Faria Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom X. Liu State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China (Received 00 Month 200x; final version received 00 Month 200x) In the context of strong field-matter interaction, an increasing number of phenomena has been found to be not understood within the single-active-electron approximation. For such phenomena, electron-electron correlation plays an important role in the underlying dynamics. In this review, we will provide a broad overview of electron-electron correlation and examine two distinct cases of its manifestation in strong field physics. In the first example, we examine nonsequential double and multiple ionization of atoms, discussing the experimental manifestation of the electron correlation and the theoretical models that have been developed to describe the effect. The second case examines the interaction of larger systems with intense laser field, for which multielectron effects have to be invoked for an accurate description of the dynamics.

1.

General overview

Until the late 1990s, the key features of most phenomena occurring in the context of the interaction of matter with intense laser fields, of the order of 1013 W/cm2 or higher, could be described by considering a single active electron, which, after being released by tunneling or multiphoton ionization, is accelerated by the field and, subsequently, either rescatters or recombines with its parent ion. Elastic rescattering leads to high-energy photoelectrons, or explicitly, high-order above threshold ionization (HATI). Recombination causes the release of the kinetic energy acquired by the electron from the field in form of high-frequency, XUV radiation whose frequency is a multiple of that of the driving field, i.e., high-order harmonic generation (HHG). In fact, early computations of such phenomena in multielectron atoms using, for instance, time-dependent density functional theory (TDDFT), which accounts for collective effects arising from electronic correlation, have shown at most light discrepancies from the single-active electron models. For instance, in (1), TDDFT studies of high-order harmonic generation in neon propagating all valence electrons revealed only quantitative differences from frozen-core computations, apart from isolated resonances in this latter case (see also (2), in which both multiphoton ionization and HHG have been studied). Furthermore, in (3), an increase in the high-order harmonics were reported, as compared with the single-active electron approximation. All these studies, however, did not alter more fundamental features such as the existence of a plateau and a cutoff in the HHG spectra. There exist, however, phenomena for which electron-electron correlation has turned out to be extremely important, and even essential for the understanding of the underlying dynamics. Concrete examples are laser-induced nonsequential double and multiple ionization and the interaction of complex systems with intense laser fields. The former constitute the traditional realm in which strong-field electron-electron dynamics has been observed. The latter are becoming increasingly important due to the ultrafast imaging of matter, which has become an important ISSN: 0950-0340 print/ISSN 1362-3044 online c 200x Taylor & Francis ⃝ DOI: 10.1080/09500340xxxxxxxxxxxx http://www.informaworld.com

Electron-electron correlation in strong laser fields

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aim in strong-field physics.

1.1.

Nonsequential double and multiple ionization

Electron-electron correlation, has been observed since those early days of strong-field laser physics, in the context of laser-induced double and multiple ionization. The first evidence of electron-electron correlation in intense field - matter interaction came as early as in 1983 (4). The study of multiphoton ionization of xenon atoms suggested that multiply charged ions may result from a nonsequential physical mechanism. For this nonsequential ionization, both electrons leave the atom in a coherent process. Already in those early years, the underlying mechanisms behind double ionization in strong laser fields have raised a great deal of debate. In particular the existence of a direct process, in which both electrons are released simultaneously, and of a stepwise, cascade-like process, in which the electrons are released one after the other, have been extensively investigated for alkali-earth atoms (5–7). It has been found that, while in the optical and XUV frequency regime the cascade-like process was dominant, for the infra-red regime the collective process prevailed. In this latter regime, several collective resonances with Stark shifted bound states have also been identified (8). One decade later, it was observed that the double ionization yield of helium, as a function of the driving-field intensity, deviated in orders of magnitude from predictions of sequential models for a broad intensity range (9, 10). These models are based on the underlying assumption that the electrons are ripped off one by one without interacting with each other. This deviation has become known as “the knee” in the nonsequential double ionization yield as a function of the intensity. Apart from the simplest case involving two active electrons in an atomic system, a prominent nonsequential channel was also observed for double ionization of small molecules (11, 12) and furthermore, evidence has been accumulated that nonsequential triple, quadruple and multiple ionization may occur in a certain range of laser intensities (13–16). Even more striking evidence has been found with the advent of the cold-target recoil ion momentum spectroscopy (COLTRIMS) technique (17), which allowed one to measure electron momentum distributions as functions of the electron momenta parallel to the laser-field polarization (18, 19). Such distributions, especially for neon, were peaked at nonvanishing parallel momenta. This indicated that both electrons left preferentially at the crossing of the external laser field, instead of at its maximum. The latter would happen for a sequential process, in which each electron would reach the continuum by tunneling ionization at peak field. In this case, the electrons would leave most probably with vanishing momenta. These features were explained by a three-step physical mechanism (20) in which the first electron is released by tunneling ionization. Subsequently, this electron propagates in the continuum, being accelerated by the field. Finally, it is driven back by the field towards its parent ion, with which it collides. In this collision, the first electron transmits part of its kinetic energy to the core, releasing a second electron. Hence, laser-induced nonsequential double ionization (NSDI) can be understood as the result of laser-induced inelastic rescattering. For nonsequential multiple ionization, a similar mechanism has been suggested, with the difference that there is a time delay, of the order of hundreds of attoseconds, between the time the first electron hits the core and the release of the other active electrons. This time delay has been related to a thermalization mechanism in which this kinetic energy is redistributed (21). The three-step mechanism, which has been extensively shown to play a major role in the interaction of matter with laser fields in the long wavelength (e.g., near-infrared wavelengths) and high-intensity (e.g., I > 1014 W/cm2 ) limit, however, may not apply in other intensity and frequency regimes. For example, in nonsequential ionization processes with very low intensity, near-infrared laser pulses, the kinetic energy acquired by the first electron is not enough to kick off the second electron and the tunneling picture may not be valid (22–25). Another example can be found in nonsequential processes occurring in the vacuum ultraviolet (VUV) wavelength regime. Recent advances in free-electron lasers (FEL) delivering VUV photons at unprecedented

C. Figueira de Morisson Faria and X. Liu

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Figure 1. (Color online) The timeline of relevant experimental progress in the study on electron-electron correlation in intense laser field. The pictures in the figure have been taken from Refs. (10), (123), (19) and (126).

intensities, open the door for unveiling the electron-electron correlation in the few-photon regime. Several experiments performed on noble gas atoms have demonstrated that, similar to the case with intense near-infrared laser pulses, a nonsequential ionization mechanism dominates in the interaction of laser with atoms (26, 27). Due to the very high frequency of the FEL photons involved in the process, however, the above mentioned laser-induced inelastic rescattering scenario is expected to break down. In the two situations discussed above, the specific structure of the targets may become very important. In particular the resonance of the multiple photons with the intermediate excited states may be responsible for the enhanced multiple ionization yields. The target is also expected to be important for nonsequential double and multiple ionization of molecules, even in the tunneling regime. The electron momentum distributions, for instance, may contain interference patterns which can be traced back to electron rescattering at spatially separated centers, as it is the case with other strong-field phenomena such as HHG or ATI. In fact, experimental evidence that the peaks and the shapes of the NSDI electron momentum distributions change according to the molecular orbital symmetry (28) and orientation (29) have been reported. This fact has been confirmed by several theoretical studies, which suggest that two center interference patterns may be present (30). Apart from that, for these systems, recollision-excitation with subsequent tunnel ionization (RESI) is expected to occur (29). The electron momentum distributions determined by this latter mechanism also strongly depend on the state in which the second electron was initially bound, and on the state from which it subsequently tunnels (31). This mechanism is also important for the so-called below-threshold intensities, for which the maximal energy of the returning electron, upon return, is not sufficient to make the second electron overcome the binding energy of the singly ionized target. In this case, RESI is expected to be the dominant channel. The potential of NSDI for ultrafast imaging goes hand in hand with the fact that laser-induced rescattering and recombination phenomena are powerful tools for resolving, or even controlling dynamic processes in matter at ultrashort timescales, namely hundreds of attoseconds. This is related to the emerging field of attosecond science, for which this control is very important. Figure 1 shows a brief timeline of important milestones for nonsequential double and multiple ionization in intense laser fields.

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1.2.


Complex systems in strong fields

Another manifestation of electron-electron correlation in strong laser fields is the influence of collective processes on high-order harmonic generation or above-threshold ionization. These effects have become particularly relevant in the past few years, due to the fact that more complex systems, such as molecules, clusters or even solids have been used as high-order harmonic or photoelectron sources. Worldwide, complex targets in intense fields are attracting a great deal of attention due to the fact that strong-field phenomena are, in principle, powerful tools for resolving or even controlling matter in the subangstrom and subfemtosecond domain. This is a consequence of the fact that the laser-induced rescattering or recombination processes behind such phenomena take place within hundreds of attoseconds. This is roughly the time it takes for an electron to travel through atomic distances. Hence, strong-field phenomena, in principle, allow one to resolve and even steer electronic motion in real time. A very important application of this fact is the attosecond imaging of matter. Indeed, in recent years, strong-field phenomena have been employed as tools in the reconstruction of molecular orbitals (32), coherent control of molecular wavepackets (33), or the ultrafast probing of molecular vibration (34). In this context, one must take into account structural and collective effects. Firstly, when an electron interacts with an extended system such as a molecule in order to generate harmonics or release photoelectrons, it may rescatter or recombine with more than one ion. This will lead to interference maxima and minima in the spectra which are due to the high-order harmonic or photoelectron emission at spatially separated sites. Furthermore, the bound-state wavefunctions in molecules will also exhibit a particular spatial structure, which will be embedded in the high-order harmonic or photoelectron spectra. Concrete examples are the suppression of tunnel ionization due to the nodal planes in π orbitals (35, 36), or the s-p mixing in the 3σg orbital (36–38). These static, structural features may be observed in a single-active electron framework. However, they may also be present in a two-electron framework, as patterns in the NSDI electron momentum distributions (30). It is not clear, however, how a complex system such as a molecule or a solid would respond to an intense field within a fraction of a cycle, i.e., hundreds of attoseconds. In fact, these systems possess many electrons, which may interact with each other, with the field and with the ions in a non-trivial fashion. Hence, collective effects and the structure of the target may play a very important role. For instance, multielectron effects may influence the high-order harmonic polarization (39), there may be excitations with electron-hole creation, even for relatively small molecules (35), or Mie resonances for clusters (40). These effects may considerably alter highorder harmonic or photoelectron spectra, and thus be an obstacle to the identification of the structural minima. If, on the one hand, these collective effects may pose an obstacle to ultrafast imaging, they may bring many important questions, and unveil completely new physics. A particular challenge in this context is that the energy redistribution in the target after recollision may take place within a very short time interval. In this context, it is not at all clear whether the usual collective effects in complex systems will have enough time to build up. One also expects that the structure of the target will be increasingly important. One should note that the two manifestations of electron-electron interaction discussed above are not mutually exclusive. In principle, there may also be nonsequential double and multiple ionization of complex targets, for which the structure of the core and collective effects involving other active electrons may be important. This is very likely to be the case for multiple ionization, in which there is a time delay between the recollision of the first electron and the ionization of the remaining electrons. Furthermore, when an atomic or molecular system interacts with a free-electron laser, one can no longer assume that there is only a single active electron, as the driving-field photons have enough energy to rip off electrons from the inner shells. Hence, a multielectron problem must be considered. In this work, we intend to address both of them. They will be the two main sections of this work. We will commence by discussing nonsequential double and multiple ionization (Sec. 2 ), starting from the general features encountered and the physical mechanisms behind them (Secs. 2.1), the models employed to deal with such features


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(Sec. 2.2) and the key experiments (Sec. 2.3). Subsequently in this section, we will provide a brief overview of our own work on nonsequential double and multiple ionization, focusing on the method employed (Sec. 2.4) and on the results obtained by us employing such methods (Sec. 2.5 ). In Sec. 3 we will discuss complex systems, including molecules and clusters, and put emphasis on the multielectron response and collective effects in the interaction of complex systems with intense laser fields. This will be a much shorter Section, whose main objective is to give a general idea of the main developments currently taking place in the field. The review will end in Sec. 4 with a summary and a brief outlook.

2. 2.1.

Laser-induced nonsequential double and multiple ionization Main features and physical mechanisms

There are several rescattering mechanisms by which the second electron can be dislodged, depending on the species and on the driving-field intensity. For instance, in the first COLTRIMS experiments (18,√19) electron-impact ionization is responsible for the double-peak structure near p1∥ = p2∥ = ±2 Up , where pn∥ , n = 1, 2 denote the electron momentum components parallel to the laser-field polarization and Up the ponderomotive energy. Physically, this means that the first electron provides the second electron with enough energy to be able to overcome the second ionization potential of the species in question and reach the continuum. Both electrons then leave simultaneously. For the typical intensities employed in these seminal experiments, it was found that electronimpact ionization is dominant for neon. This mechanism is the simplest possible rescattering scenario, and has been extensively investigated since the early 2000s. It allows, in principle, √ electron momenta extending far beyond 2 Up . There has been, however, some controversy of whether this specific momentum is the maximum or the most probable momentum the electrons may have. An argument for the former statement was based on the assumption that an electron leaves exactly at a field maximum and returns at a field crossing and that the remaining energy of the returning electron is zero after recollision. In√this case, the maximal momentum the electrons acquire from the laser field is limited to 2 Up . This argument is, however, only a rough estimate, and exhibits a series of flaws. In fact, if the electron left exactly at a field maximum, it would return (if at all) at the other maximum of the field with vanishing momentum. An electron returning near a crossing √ would leave slightly after the maximum and the overall momentum transfer would not be 2 Up (see, for instance, the discussions in (41)). A classical cutoff has been derived and it has been shown that the electron momentum from electron√ impact ionization may extend far beyond 2 Up , depending on the driving-field intensity (42). Hence, in principle, the electron-momentum distributions may extend beyond that. This has been observed first in (43) and also confirmed by several theoretical approaches, as diverse as the time-dependent Schrödinger equation (44), classical-trajectory methods (45, 46) and the strong-field approximation (47). For other species, such as argon or helium, other mechanisms such as recollision-excitation with subsequent ionization (RESI) also play an important role (48, 49). In this case, the first electron, upon return, promotes the second electron to an excited bound state, from which it subsequently reaches the continuum. This would imply that there is a time delay between the recollision of the first electron and the ionization of the second electron. These are the most important recollision mechanisms for laser-induced NSDI in the tunneling regime1 . In Fig. 2, we provide a schematic representation of both mechanisms, as functions of the ponderomotive energy. For higher energies, the system is in the above-barrier regime and other ionization mechanisms take √ this regime, the Keldysh parameter γ = Ip /(2Up ), where Ip and Up give the ionization potential and the ponderomotive energy, is smaller than unit. Physically, this means that a quasi-static picture can be adopted for the external field and one can describe the process of ionization as tunneling through an effective potential barrier.

1 In

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place. The figure should be understood as a guideline, as the prevalence of either electron-impact ionization or RESI is dependent on the target employed. Electron-impact ionization has been extensively studied in the past decade, while the RESI mechanism is considerably less understood. This may be attributed to two main reasons: (1) Electron-impact ionization is sufficient to explain the most striking features in the electron momentum distributions observed experimentally. These features are the peaks at nonvanishing momenta revealed by the COLTRIMS measurements, and the recently reported V-shaped structure (50, 51), which extends to very high momenta and is the signature of a long-range electron-electron interaction (44, 46, 52–55). In contrast, RESI has always been related to much lower momenta, around p1∥ = p2∥ = 0, and, hence, less spectacular features (48, 49, 56). (2) From a more theoretical perspective, electron-impact ionization is much simpler to model. This is specially true for semi-analytical approaches such as the strong-field approximation. This is a consequence of the fact that electron-impact ionization in this framework leads to a simpler Feynman diagram, while, for RESI, the excitation and the time delay must be incorporated. This is particularly difficult for such models. Apart from that, it is an established fact that electron-impact ionization has a classical counterpart. This allows one to define a classical limit for the SFA-based approaches (see Sec. 2.4 for details). It is not clear, however, whether RESI can be treated entirely classically, whether it is intrinsically quantum mechanical, or whether it can be modeled quasi-classically by incorporating quantum mechanical ingredients in classical models. Furthermore, there may be more than one single physical mechanism behind RESI, depending on the target and on the driving-field parameters. We will discuss classical models in more detail in Sec. 2.2.2. One could then ask the question whether RESI is worth investigating if it leads to less spectacular features than electron-impact ionization and it is far more difficult to model. In the last few years, however, RESI has become increasingly important. This is due to the utilization of more complex targets, such as molecules, and/or below-threshold intensities in NSDI experiments. This coincided with the shift of focus in strong-field laser physics towards attoscience and ultrafast imaging. For below-threshold intensities, the maximal kinetic energy upon return acquired by the first electron is no longer sufficient to release the second electron in the continuum by electron-impact ionization. This implies that, instead, the first electron excites the second, which, subsequently, reaches the continuum. For NSDI of molecules, footprints of the RESI mechanism have been identified recently. Indeed, in (29) it has been reported that this mechanism is prominent for molecules aligned perpendicular to the laser-field polarization. One should note, however, that, apart from early results in which RESI has been computed using the strong-field approximation (56), most existing results are the outcome of computations, either classical (54, 57–60) or quantum mechanical (61–63), for which the different rescattering mechanisms cannot be easily disentangled. In fact, in many of the above-cited studies, RESI is obscured by electron-impact ionization, or multiple collisions. In this context, it is worth mentioning that time-delayed double ionization pathways for NSDI are currently generating a great deal of debate. Several open questions are: • Is there a single, or several physical mechanisms behind RESI?. In the outcome of several classical or semiclassical computations, a time delay between the rescattering of the first electron and the ionization of the second electron has been identified. It is not clear, however, whether this delay and the resulting electron-momentum distributions may be attributed to a single, or to several physical mechanisms. In fact, the interplay between the Coulomb potential, which temporarily traps the second electron, and the external laser field, which causes ionization, is a rather complex problem (64–66). It has been suggested that the release of the second electron in RESI may be attributed to multiple field-assisted recollisions with the trapping potential (22, 67), which would correspond to electron-impact ionization


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with a time delay, or, for lower intensities, to tunneling ionization (68). Another possible physical mechanism is over-the-barrier ionization (57). Apart from that, the rescattering time of the first electron and the ionization time of the second electron are not entirely agreed upon. In fact, it could well be that the picture of an electron returning at a crossing and leaving at a field maximum, put across by several groups, does not account for the whole dynamics of the laser field and the binding potential. For instance, recent investigations suggest another time-delayed pathway in which the first electron returns close to a field maximum and leaves close to a field minimum. This further RESI mechanism has been identified for intensities in which the second electron leaves by over-the-barrier ionization (57), and also for lower intensities (69), and leads to anti-correlated electron momentum distributions. Finally, it has even been shown that excitation is not necessarily a pre-requisite for obtaining electron momentum distributions localized in the second and fourth quadrant of the parallel momentum plane (70). • Is RESI classical or quantum mechanical? On the one hand, one could argue that, if the second electron is promoted to an excited bound state, and leaves by tunneling ionization, RESI is quantum mechanical as tunneling has no classical counterpart. In fact, several classical models, in order to incorporate the tunneling of the second electron, employed several quantum-mechanical ingredients in order to match the electron motion in different regions, corresponding to bound or continuum dynamics (68, 71). A counter-argument is that the electron is excited to a quasi-continuum of states, for which classical models are expected to be valid. This argument is applicable insomuch as one is close to the ionization threshold. With decreasing laser-field intensity, however, it is expected to become less reliable. Apart from that, the second electron could leave through other ionization mechanisms, such as over-the-barrier or multiphoton ionization. In this case, there would be no classically forbidden region for the second electron to overcome in order to reach the continuum. Some authors even adopt the rather radical viewpoint that NSDI is a completely classical phenomenon, and treat even the ionization of the first electron without recurring to tunneling (67). • What momentum regions does RESI populate? It is commonly argued that, for RESI, the first electron leaves near a crossing of the driving field, whilst the second electron is freed near a field maximum. Due to this time delay, both electrons were expected to leave with opposite momenta (22, 60), and thus populate the second and fourth quadrant of the plane spanned by the electron-momentum components parallel to the laser-field polarization. Several investigations, however, suggest that this may be an oversimplified picture, based on electron-impact ionization with a time delay, for which such momentum regions should be populated. These constraints would be in agreement with simplified thermalization models, for which multiple collisions have been taken into account statistically for multiple ionization (21). On the other hand, if the second electron is released by tunneling ionization, all four quadrants of the parallel momentum plane may be populated by RESI. These constraints have been recently derived in (31, 72), and are to some extent in agreement with early estimates in (48). RESI electron momentum distributions occupying the four quadrants has also been observed for Argon in (71) and for H2 in (57) (see also the so-called “RESIa mechanism” in (69)). If, however, other mechanisms such as multiple collisions with the core, or over-the-barrier ionization are dominant, the momentum regions populated by RESI will change. In this context, it is worth mentioning that, even though now there is an overall consensus that rescattering is the mechanism behind NSDI and NSMI in the tunneling regime, until the early 2000s there was considerable debate about what this mechanism should be. For instance, it was suggested that nonsequential double ionization occurred due to a “shake-off” process, in which, due to the very fast rearrangement of the core after the first electron has been ripped off, the second electron was promoted to an excited state, from which it ionized (9). Furthermore, it has also been suggested that NSDI owed its existence to a collective tunneling process, in which

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Figure 2. (Color online) Schematic representation of the dominant physical mechanisms behind laser-induced double ionization, for increasing ponderomotive energy. We consider driving fields of low frequencies and high intensities, so that the tunneling and over-the-barrier physical pictures are valid.

both electrons reached the continuum simultaneously by tunneling ionization (73). These physical mechanisms, however, could not explain two important features observed experimentally: the decrease of the NSDI yield with increasing ellipticity of the driving field (74), and the peaks at nonvanishing momenta in the electron momentum distributions (18, 19). In (74), it was shown that the ellipticity dependence of NSDI was strikingly similar to that obtained for high-harmonic generation. For both phenomena, there was a decrease in the yield as the ellipticity of the driving field increased, which could be explained using simple classical rescattering models: if an electron, upon return, misses its parent ion, the high-harmonic or NSDI signal is expected to decrease1 . The agreement with the experimental yield also improves if the focusing of the electron trajectories towards the core due to the presence of the Coulomb potential is accounted for. This effect, known as ”Coulomb focusing”, has been first studied in detail in (75) and led to an increase of over one order of magnitude in the calculated NSDI yields. Further studies in which Coulomb focusing have been accounted for, and turned out to be important, employ semi-empirical ionization and excitation cross sections for NSDI, an ensemble of classical trajectories and semisudden preturbation theory (76). The peaks at non-vanishing momenta reported in (18, 19) were even more evidence that rescattering is the physical mechanism behind NSDI. Indeed, if the electrons were created by a shake-off or collective tunneling mechanisms, they would reach the continuum at times for which the electric field is near its peak. This would imply electron momentum distributions peaked at vanishing momenta. Rescattering, in contrast, means that at least one electron is released close to a field crossing. Hence, at least one of them will have non-vanishing momentum. Specifically, for electron-impact ionization, we expect both electrons to be released close to a field crossing. √ Hence, the most probable momenta will be near ±2 Up . For RESI, on the other hand, we expect that one electron will leave at a field crossing and the other at a field maximum. Hence, the final 1 Recently,

however, it has been shown that there may be a recollision of the first electron with its parent ion, and hence NSDI, for highly elliptically (or even circularly) polarized fields. This recollision has been explained in (66, 122)using classical models.


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√ momentum of the first electron will be near ±2 Up and that of the second electron near zero. This issues will be discussed in more detail in Sec. 2.4. Although, for relatively high laser intensities, electron impact ionization plus RESI can provide an intuitive picture to account for the NSDI process, it remains a puzzle to understand the physical mechanism of NSDI for lower laser intensities, at which the maximal return energy of the recolliding electron is too small to excite the remaining bound electron and even the RESI pathway is forbidden. For example, very recent experiments on NSDI of argon at such low intensity regime have revealed a strongly correlated back-to-back emission of the electrons along the polarization direction (22), which is in striking contrast to the previous observations. The authors pointed to the possible role of multiple recollisions in production of this back-to-back emission. However, the underlying mechanism remains a puzzle. At such a low intensity, one enters the multiphoton regime and the tunneling picture may not be valid. In this case, one may have to resort to an intrinsically quantum mechanical picture. Recent studies on NSDI of xenon have addressed this issue. It was found that, in the recorded electron kinetic energy spectra correlated with the doubly charged ion, sharp resonant structures were present (23). This may point to the fact that a transition in the dynamics of strong-field double ionization, i.e., from the tunneling to multiphoton regime, occurred. Further studies (24, 25) have revealed a wavelength dependence of the ratio between doubly charged ion and singly charged ion and a physical mechanism related to resonant enhancement was proposed to explain the experimental observation. As the other extreme with respect to multiphoton ionization in the low-frequency limit, multiple ionization upon absorption of a single photon with very high frequency, was considered to be well understood within the validity of dipole approximation (77). As a bridge between the single- and multiphoton regime, few-photon multiple ionization has attracted much attention in recent years, for targets subject to free-electron lasers. In this shorter XUV wavelength regime, however, the semiclassical rescattering scenario, which is the basis for the understanding of multiphoton nonsequential ionization in the long wavelength regime, breaks down. A full quantum mechanical description, as in the single-photon limit, may be considered. In combination with the sophisticated COLTRIMS technique, recent studies (26, 27) performed on noble gas atoms have revealed that, similar to its multiphoton counterpart, a nonsequential ionization channel is present in the two-photon double ionization process. However, in contrast to a double peak structure at nonzero momentum in the multiphoton regime, the recoil ion momentum distribution from two-photon double ionization exhibited a single peak centered at zero. This feature suggested that both electrons are preferentially emitted back-to-back into opposite hemispheres with similar energies. The calculation based on the perturbative theory with dipole approximation reproduced well the experimental observation and revealed the important role of electron-electron interaction in the final continuum state in the two-photon double ionization process.

2.2.

Existing theoretical approaches

The theoretical modeling of the interaction of a multielectron system with a strong laser field is a highly non-trivial matter, especially as far as nonsequential double and multiple ionization is concerned. In fact, even if one considers the simplest physical system with two active electrons, i.e., helium, the evolution of such a system in a strong, time-dependent field has no analytical solution. If this problem is tackled numerically, it will require the solution of the time-dependent Schrödinger equation in six spatial dimensions. This has only been achieved very recently, and has turned out to be very demanding (44). For other species, such as argon and neon, the picture is even more complex as, even if only two electrons are released in the continuum, there exist other electrons which also interact with the laser field and with each other, and, in principle, may also contribute to the dynamics of the system. In the past two decades, since the first experimental results on nonsequential double ionization

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Figure 3. Outcome of the first ab-initio computations reproducing the knee in laser-induced nonsequential double ionization. The upper panels correspond to Figs. 1 and 3 of Ref. (85), while the the lower panels reproduce Fig. 1 of Ref. (86). In the upper panels, the triangles and squares correspond to the He single and double ionization yields, respectively, and the circles give the single ionization yield for He+ . In the lower panels, the single and double ionization signals are given by squares and circles, respectively, and the dashed lines give the single-ionization signals from He+ .

were available, a multitude of theoretical approaches have been developed in order to tackle this challenge. These approaches are as diverse as the fully numerical solution of the timedependent Schrödinger equation, semi-analytical quantum mechanical models in which laserinduced rescattering processes are studied within the strong-field approximation, classical models in which the behavior of a quantum-mechanical two electron wavefunction has been mimicked by ensemble computations, or other approaches, such as the time-dependent density functional theory. Below, we will provide a brief overview of these approaches, and place particular emphasis on their strengths, their weaknesses, and on the results obtained with them. 2.2.1.

Ab initio computations

In order to study the evolution of a two-electron system in a strong laser field, one may solve the time-dependent Schrödinger equation (TDSE) numerically. This approach has the main advantage of being fully ab-initio. This implies that there are no physical approximations. For that reason, it constitutes a good benchmark, against which other models can be tested and improved. It also exhibits the best agreement to date with the experimental findings. One pays, however, a high price for this accuracy. The first drawback of the TSDE is that the full dynamics of the interaction of a two-electron system with an intense field is so complex that the physical mechanisms behind the outcome of such computations are very hard to extract, or disentangle. In fact, in many situations the features obtained from TDSE computations are interpreted employing ad hoc arguments. Apart from that, this approach is computationally very demanding. Especially for a realistic, three-dimensional model, the task of solving the timedependent Schrödinger equation numerically has only been accomplished very recently. This was the result of a huge effort extending over one and a half decade (78). The first efforts to solve the time-dependent Schrödinger equation for a two-electron system started in the early 1990s, for reduced-dimensionality models and soft-core potentials (79, 80). These investigations addressed probability densities as functions of the electron coordinates, the


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temporal evolution of the double ionization yields, or photoelectron spectra. Other examples include comparisons with other methods (81), autoionization (82) or stabilization (83). Extensions of such methods for models with more than one spatial dimension have been reported in (78), and, shortly afterwards, in (84). It is worth mentioning that all these early computations considered high-frequency driving fields. This is related to the fact that the excursion amplitudes of the active electrons in the continuum are smaller in this case and convergency is achieved more easily. In fact, it was only in the late 1990s that one-dimensional ab initio models for the tunneling regime became computationally feasible (85, 86). Instead of integrating the two-electron probability density over all space, in both computations this integration has been restricted to an adequate choice of the coordinate range (see also (81) for similar coordinate choices in the high-frequency regime). These early computations focused on the total NSDI yield, as a function of the driving-field intensity, and reproduced the experimentally observed “knee”. In Fig. 3, we are showing the outcome of these pioneering computations, obtained by Lappas and van Leeuwen (85) (upper panels), and by Liu, Eberly and co-workers (86) (lower panels). Both research groups observed deviations of more than one order of magnitude between the double ionization signal of He and the single-ionization signals of He+ . This deviation occurs for intensities up to 1015 W/cm2 , when both signals start to coincide.

Figure 4. The two-electron momentum distributions for double ionization shown in Fig. 1 of Ref. (87), for a one-dimensional model of Helium subjected to an intense field of frequency ω = 0.057 a.u.. The driving-field intensities have been taken as: (a) 1 × 1014 W/cm2 , (b) 3 × 1014 W/cm2 , (c) 6.6 × 1014 W/cm2 , (d) 1 × 1015 W/cm2 , (e)1.3 × 1015 W/cm2 , and (f) 2 × 1015 W/cm2 .

Differential electron-momentum distributions employing one-dimensional models were obtained slightly later, by Lein and collaborators (87). Their investigations shed additional light on the mechanisms behind NSDI, and provided additional support for the early findings in (74), and also showed that rescattering was responsible for this phenomenon in the tunneling regime. Fig. 4 shows the electron-momentum distributions obtained in (87), for increasing intensities


12

of the external laser field. In the figure, one may identify a V-shaped structure in the first and third quadrants of the p1 p2 plane, peaked at nonvanishing momenta. This structure is particularly prominent for a driving-field intensity of 6.6 × 1014 W/cm2 (panel (c)) and becomes less pronounced with increasing driving-field intensity. Finally, for 2 × 1015 W/cm2 (panel (f)), this finger-like structure is absent and the electrons leave most probably with vanishing momenta. This structure can be traced back to the physical mechanism of electron-impact ionization, and is due to the fact that, by leaving simultaneously, both electrons repel each other, and electronelectron repulsion favors unequal momenta. This has been demonstrated in subsequent years, in semi-analytical computations (47, 88). Furthermore, unequal momenta have also been associated with the long-range Coulomb interaction by which the second electron is dislodged (47, 53, 88). For many years, however, it was believed that, in (87) the reduced dimensionality in the model overestimated the influence of the Coulomb tail. This was due to the fact that the existing experimental data suggested that the electrons left most probably with equal momenta, centered √ at ±2 Up . Recently, however, the V-shaped structure has been identified experimentally (50, 51). Other mechanisms which may be identified in the figure are RESI, in panels (a) and (b), and over-the-barrier ionization in the panels (d) and (e). The former mechanism populates all quadrants of the momentum plane at much lower momenta (31), and the latter the second and the fourth quadrants (57). Only in recent years have they been understood in more detail and properly identified. Finally, towards the mid and late 2000s, the time-dependent Schrödinger equation has been solved in a more realistic, three-dimensional framework for a two-electron atom. To our knowledge, a full ab-initio computation has only been achieved at the Queen’s University Belfast. The outcome of such computations is depicted in Fig. 5. In this figure, a finger-like structure is clearly observed in the electron-momentum distributions, in excellent agreement with existing experiments (50, 51). They also find a cutoff of the order of 5Up for the electron momentum distributions. This is in agreement with the predictions of semi-analytical models such as the strong-field approximation, and disagrees with simple estimates which suggest that the cutoff √ should be around 2 Up 1 . Apart from that, rings for constant electron kinetic energy are also observed. These rings have been also reported in similar TDSE computations for helium (89) and molecular hydrogen (61) which went beyond one-dimensional models, but considered a restricted motion for the system’s center of mass. 2.2.2.

Classical models

In many situations, one may mimic the evolution of the quantum-mechanical wavepacket using a classical ensemble model. One assumes that each electron in this ensemble is released in the continuum following a specific probability distribution and that it follows classical equations of motion which includes the residual binding potentials, the external laser field and electronelectron correlation. In general, two different approaches have been employed to construct the electron ensemble: (1) Classical approaches including quantum mechanical ingredients. Most NSDI classicaltrajectory computations in the literature employ essentially classical methods, but, still, borrow some ingredients from quantum mechanical models. An example are, for instance, the set of initial conditions for the first electron. One may assume that the first electron is tunnel ionized through the barrier formed by the Coulombic and external laser electric field and its initial conditions are determined by its quantum mechanical wavefunction at the tunneling time. The initial conditions for the the second, i.e., the bound electron are subsequently determined by assuming that the electron is in the ground state of the √ constraints determined by SFA-based models suggest that 2 Up should correspond to the most probable, but not the maximum electron momentum in NSDI electron-momentum distributions. In fact, depending on the intensity of the driving field, this momentum can go beyond 4Up . This is in agreement with this recent outcome of TSDE computations. More detailed discussions on this constraint will be provided in Sec. 2.4. 1 Kinematic


13

Figure 5. Electron-momentum distributions as functions of the electron momentum components parallel to the laser-field polarization computed with the fully numerical solution of the time-dependent Schr¨ odinger equation (Fig. 4 in (44)). The driving-field intensity and frequency were taken as I = 1.0PW/cm2 and ω = 0.057 a.u., respectively. The vertical line constrains the kinetic energy of electron 1 to 1.9Up and the white circular arc indicates when the total kinetic energy equals 5.3Up .

singly charged ion and its initial phase space is modeled by a microcanonical distribution. This procedure is essentially in the spirit to Corkum’s seminal rescattering model (see, e.g., (46, 52, 57, 75, 90)). Further quantum-mechanical ingredients incorporated in quasiclassical models are, for instance, ionization and excitation cross sections, which have been incorporated considering the external field in the theory of semisudden perturbation and approximate quantum mechanical time evolution operators with the passive electron (76). Apart from that, more recently, WKB-type expressions have been employed to account for the sub-barrier motion of the second electron (68). This was done in order to account for tunnel ionization of the second electron in RESI. (2) Purely classical approaches. Another approach, adopted mainly by Eberly and co-workers, sets up the initial conditions of position and velocity for both electrons from microcanonical ensembles (59, 60, 67, 91) and hence discards all quantum-mechanical ingredients. Therefore, it treats NSDI as a completely classical phenomenon. Compared to numerical solution of the time-dependent Schrödinger equation, classicaltrajectory methods are much easier to implement and have the great advantage of including all the residual binding potential, the electron-electron correlation and the external laser field. Within the past two decades, classical computations have proven to be very powerful and reproduced important features in NSDI. Concrete examples are the “knee” in the total double ionization yield (67, 92), the peaks at nonvanishing momenta for the ion and electron momentum distributions (67, 90) and, more recently, the finger-like structure observed in the experiments and in TDSE computations (46, 52). It is however not clear to which extent the full NSDI dynamics can be reproduced by classical models. In fact, they suffer from four major drawbacks, compared to ab initio computations: (1) Classical models do not take into account quantum interference effects. (2) They do not account for the Heisenberg uncertainty relation and the Pauli exclusion principle. (3) The spreading of the electronic wavepacket when it propagates in the continuum is underestimated. (4) Processes involving atomic bound states, such as excitations, can only be incorporated to a certain extent.

14


Drawback (1) is not really a problem as, in most NSDI computations, it is of interest to integrate over several degrees of freedom, such as the momentum components perpendicular to the laser-field polarization, in order to compare the theoretical results with existing experiments. In this case, such effects get washed out. This could, however, be problematic if one is willing to obtain fully differential distributions, for which quantum interference effects are expected to be important. Drawback (2) means that, in quantum mechanical computations, the Coulomb singularity may be smoothed out by the fact that the electronic wave packet has a finite width. Classically, however, the electron is a point charge and this singularity is ever-present. Apart from that, classical models do not account for the spreading of the electronic wave packet in the continuum, which, clearly, is embedded in its quantum mechanical counterpart. Thus, they overestimate the collision of the first electron with its parent ion. In the early days of classical simulations for NSDI this problem has been overcome by considering effective forces in order to account for the uncertainty relation and the Pauli exclusion principle (93, 94). Subsequently, the same initial conditions, such as wave packet spreading and tunnel ionization probabilities found in the quantum mechanical computation have been incorporated in its classical counterpart (90). Drawback (3) means that, while an ensemble of classical trajectories can mimic the spreading of a quantum mechanical wavepacket to some extent, there is still an extra time factor, inversely proportional to (t − t′ )3/2 in the quantum mechanical transition amplitudes, where t and t′ refer to the rescattering and ionization time, respectively. This factor is also present in saddle-point computations for each trajectory and is related to the spreading of the wave, and can be included by hand in classical-orbit approaches. This has been done recently in (95, 96) for NSDI with elliptically polarized fields. Drawback (4) is a direct consequence of the fact that classical models are expected to work as long as the rescattering process to be studied has a classical counterpart. This is the case if the driving-field intensity is far above the threshold, for which electron-impact ionization is dominant. As the intensity decreases, classically, the returning electron may not have enough energy to make the second electron overcome the atomic binding forces. This is the so-called “below-threshold regime”, for which the quantum-classical correspondence for NSDI becomes less reliable. In this case, excitation and thus the structure of the target in question, is expected to play an important role. In this context, it is worth mentioning that it is a very difficult task to consider excitation within a classical framework. On the one hand, it has been shown by Sacha and Eckhard (64) that the combined effect of the external laser field and the atomic potential forms a saddle, which, in principle, may trap the active electrons and even allow a time delay between the ionization of the first and second electron. This saddle is also important when analyzing NSDI in elliptically polarized fields, as recently shown by Uzer and co-workers (66). The recollision dynamics has also been investigated rather extensively in (97) by using simplified Hamiltonians in different dynamical regions. There is still, however, a great deal of controversy with regard to delayed pathways, such as excitation followed by tunneling, multiphoton or over-the-barrier ionization (see discussion in Sec. 2.1). Excitation can be modeled classically as soon as the excited bound state can be approximated by a quasi-continuum. This is, however, not always the case, especially as the driving-field intensity decreases. Furthermore, tunneling ionization is an intrinsically quantum mechanical mechanism, and this would pose a problem for classical models when describing the RESI pathway. Apart from that, most classical models consider electron-impact ionization with a time delay between the rescattering of the first electron and the release of the second electron (29, 58–60). This time delay is caused by multiple collisions between the second bound electron and the core. The main difference between this process and plain electron-impact ionization is that the second electron cannot escape through a single collision and leave at the same time as the first electron, but remains trapped and gains the necessary kinetic energy in order to escape by recolliding several times. The kinematic constraints delimited by this process will populate mainly the sec-


15

ond and fourth quadrant of the plane spanned by the electron momentum components parallel to the laser-field polarization. These are rather different constraints from those delimited by a model in which the explicit quantum mechanical Feynman diagram for excitation with subsequent tunneling ionization is considered. This latter process is expected to populate the four quadrants of this plane. It could be, however, that tunneling is not a very important ionization mechanism for the second electron in many parameter ranges of interest. For instance, several ionization mechanisms have been recently associated with time-delayed NSDI pathways (see, e.g., (21, 57, 69, 70, 98, 99) and discussion in Sec. 2.1). Recently, excitation with subsequent tunneling has also been incorporated in by splitting the problem in distinct spatial regions, corresponding to the bound and continuum dynamics. The dynamics in each region was considered classically, and they were connected employing WKBtype formulae (68). Whilst this model reproduced experimental results for NSDI below the threshold even quantitatively (71), it was necessary to incorporate several quantum ingredients in this model. Apart from that, for Argon electron-momentum distributions extending over the four quadrants of the momentum plane have been found. This issue will be discussed in more detail in Sec. 2.4. These drawbacks appear in addition to the fact that, in classical models, several rescattering mechanisms are present, and are therefore difficult to disentangle. Progress in this direction has been made recently by restricting the ionization and rescattering times in a classical-ensemble computation (57). We will now provide a brief review of the main achievements obtained using classical ensemble computations in the modeling of NSDI. Classical ensemble computations started to be employed in the context of NSDI relatively early, and they led to quite important breakthroughs. As for the ab-initio approaches, classical models started by trying to compute the NSDI yield as functions of the driving-field intensity, for relatively high frequencies (81, 93, 94). These investigations found deviations from the predictions of sequential models which resembled the “knee” observed experimentally. They also found, by a direct inspection of the electron orbits, that there was a time lag between the rescattering of the first electron and the tunneling of the second. A further breakthrough was made by Chen and collaborators (90) when this model was applied to the tunneling regime. In contrast to the above-mentioned classical methods, the first electron was assumed to be tunnel ionized through the field-lowered Coulomb potential in their treatment, in the spirit to the seminal rescattering model. The tunneled electron momentum distribution was chosen such as to mimic the electron wavepacket spreading. While for the second bound electron, the initial conditions are, as usual, determined by a microcanonical distribution. With this model, a double-hump structure peaked at nonvanishing ion momenta parallel to the laserfield polarization was obtained. Apart from that, for the ion momentum perpendicular to the field polarization axis, a distribution peaked at vanishing momentum was found. Both features are in agreement with existing experiments. Furthermore, the authors were able to trace the doublepeak structure to electron trajectories rescattering inelastically with its parent ion. This was additional evidence that the rescattering mechanism was responsible for NSDI in the tunneling regime. The results obtained in this paper are displayed in Fig. 6. The same group have also been able to obtain the knee for the NSDI yield as a function of intensity (92) and relate this feature to the inelastic rescattering of the first electron with its parent ion. This shed additional light on the mechanism behind it, which was still an object of debate at the time. In the meantime, NSDI has been studied within a completely classical framework. Snapshots of the distributions of electron trajectories which participate in NSDI have been taken in (91). These snapshots showed that basically two types of collisions exist: the speed-up collisions, in which the electrons, upon return, are accelerated by the field, and the slow-down collisions, for which the electrons return to the core against the field. Furthermore, the pronounced doublehump structure observed experimentally could be ascribed to two groups of classical electron trajectories (67): one group gives a distribution of ions peaked at zero momentum, and the other

16


(c)

(d)

Figure 6. Figs. 1 and 2 from (90). Panels (a) and (c) give the experimental results for the ion momentum components of He2+ parallel and perpendicular to the laser-field polarization, while panels (b) and (d) give their theoretical counterparts obtained with the classical-trajectory model. The dotted lines give the calculated momentum and the solid lines the smoothed results. The laser-field intensity and frequency were I = 6.6 × 1014 W/cm2 .

gives the distributions peaked at nonzero momenta. In recent years, the classical method has also been used to compute electron momentum distributions in a three-dimensional framework. Such computations √ revealed a V-shaped structure for the electron-momentum distributions extending beyond 2 Up , in agreement with existing experiments and ab-initio computations. These results are shown in the left panel of Fig. 7. In (46), it has been shown, by a careful choice of the type of interaction by which the electron was dislodged and of the ionic potentials, that the V-shaped structure is due to the interplay between the long-range type electron-electron interaction and the hard-core nature of the ionic potential. In fact, these V-shaped features are not present if a short-range, Yukawa interaction is taken for the former, or a soft-core potential is taken for the latter. In the right panel of Fig. 7, we are showing the outcome of another classical trajectory-based computation (52), for which the V-shaped structure was also obtained. Recently, classical models have also been applied to molecules, in order to investigate how the electron-momentum distributions behave with the alignment angle between the molecule and the field (58), and also to nonsequential triple and quadruple ionization in our recent work (21, 98, 99). More details will be provided in Sec. 2.4. 2.2.3.

The strong-field approximation

Another widely used approach to model laser-induced nonsequential double ionization is the strong-field approximation (SFA). This approach consists in neglecting the atomic binding potential when the active electrons are in the continuum and the influence of laser field when the electrons are bound. A field-dressed plane wave can be computed exactly, both for a one- and two-electron system. Such waves are known as Volkov waves. Hence, this approach allows one to deal with NSDI analytically to a very large extent. This has several advantages. First, the SFA is relatively easy to implement, as far as the numerical effort is concerned. Second, it allows a high degree of


17

Figure 7. Left and right panels, respectively: Fig. 1(a) in (46) and (52). Both figures exhibit the V-Shaped structure in the electron-momentum distributions obtained employing classical methods. In the left panel, the driving-field intensity is I = 4.5 × 1014 W/cm2 and its frequency is ω = 0.057 a.u., while in the right panel I = 3 × 1014 W/cm2 and ω = 0.055 a.u.

transparency. This is due to the fact that the physical mechanism involved, for instance, electron impact ionization or RESI, is defined and singled out very clearly from the start when the model is constructed. In contrast, the outcome of ab initio methods such as those mentioned in the previous sections resemble numerical experiments, in which several physical mechanisms are present and must be disentangled. This is not always an easy task and may lead to ambiguities and ad hoc interpretations. Finally, the SFA allows a direct connection with the classical trajectories of an electron in an external laser field and, yet, it retains quantum-interference effects. This is in particular true if saddle-point methods are employed. For all the above-stated advantages, however, one pays a price. The fact that several approximations are made in order to render the NDSI problem analytically tractable means that important physics may be lost. For instance, the fact that bound states are distorted by the field, or the influence of the residual ionic potentials may bring important physics, which is being neglected by the SFA. Apart from that, the SFA is not gauge independent. This means that computations in different gauges lead to different answers, which must be tested by comparing with other methods. All in all, the SFA constitutes a very powerful tool for gaining physical insight and for a rigorous analytical modeling of nonsequential double ionization, or strong-field phenomena in general, in intense laser fields. Below we will provide a brief historical overview of this method and how it has been applied to NSDI. Electron-electron correlation, in the context of a semi-analytic, S-matrix framework, was first investigated by Becker and Faisal in (100). Therein, the authors have shown that, for strong-field nonsequential double ionization, electron-electron correlation in the final two-electron state will favor unequal final electron momenta. These conclusions have been reached by investigating the emission angle of one of the electrons, and assuming that the kinetic energy is equally shared between the two ejected electrons. Apart from that, they have also investigated high-frequency, weak-field double ionization. Efforts to reproduce the knee in the NSDI yield, and also to identify the most relevant Feynman diagram in this context, have been made subsequently by the same group (101). In this early work, it was found that the intermediate state of the two-electron system could be approximated by a product state of a one-electron Volkov state for the first electron, and a field-free bound state for the second electron. Subsequently, the electrons interact with each other and emerge in a final two-electron continuum state. This Feynman diagram was able to reproduce the knee observed in experiments using an approximate expression for the NSDI transition amplitude (102). At this early stage, however, no connection has been made between this diagram and the classical rescattering model advocated in (20). Instead, an energy sharing mechanism was mentioned.

18


Figure 8. Fig. 1 from Ref. (56) for distributions of the total electron momentum P in NSDI of neon in a field of intensity I = 8 × 1014 W/cm2 and ω = 0.057 a.u.. Panel (a) displays the electron momentum distribution as a function of the momentum components P∥ and P⊥ parallel and perpendicular to the laser-field polarization. The horizontal line at P⊥ = 0 and the four vertical lines at various values of P∥ give the positions of cuts along which the distribution is plotted in the remaining panels. Each horizontal arrow in Panel (b) indicates the corresponding transverse cut in (c).

This connection has been established a few years later by Kopold and co-workers (56). Therein, ion-momentum distributions have also been computed, and it has been shown that the quantummechanical counterpart of an inelastic rescattering mechanism, in which an electron, initially bound, tunnel ionizes, propagates in the field and, upon return, releases a second electron by electron-impact ionization, leads to distributions peaked at nonvanishing momenta. In this context, it is worth noticing that the Feynman diagram used in (56) to describe electron-impact is closely related to that in (101). This is in agreement with the findings of TDSE computations and classical methods, as previously discussed. These distributions are shown in Fig. 8. The NSDI double-peak structure has also been reproduced in (103) and (104), in agreement with the above-stated results. Shortly afterwards, differential electron-momentum distributions have been computed employing the SFA in a realistic, three-dimensional situation (105, 106). This could be performed with relatively little numerical effort, and allowed one to investigate the influence of different types of electron-electron interaction on the distributions. The distributions obtained using a contact-type interaction placed at the position of the ion, as functions of the momentum √ components pn∥ (n = 1, 2) parallel to the laser-field polarization, are peaked at p1∥ = p2∥ = ±2 Up . In contrast, if a long-range, Coulomb type interaction is taken, the electron momentum distributions are peaked at unequal momenta. On may even identify a “butterfly-shaped” structure as a footprint of the long-range, Coulomb-type interaction, in agreement with the previously discussed theoretical approaches, experimental results and more recent S-matrix computation in which the Coulomb potential has been incorporated perturbatively (53). In those early days, however, the experimental data were much less precise and it was believed that the electrons left more probably with equal parallel momenta. Since then, several aspects of the electron-impact ionization mechanism has been studied. In particular it has been shown that several physical features may influence the shapes and the center of the electron-momentum distributions, for both a contact-type and a Coulomb-type interaction. Concrete examples are the influence of the final-state electron-electron repulsion (47, 88), the initial bound states of the first and second electron (55), the carrier-envelope phase of a few-cycle driving pulse (107, 108) or the two-center interference in diatomic molecules (30). Furthermore, the SFA model supports a classical limit, which, if the driving-field intensity is high


19

enough, exhibits a high degree of agreement with its quantum-mechanical counterpart (47, 88). A great extent of this work has been developed by us, and will be discussed in detail in Secs. 2.4 and 2.5. Other important studies address other rescattering mechanisms, such as RESI, and incorporate the Coulomb potentials in the continuum for NSDI (53). In particular, as far as RESI is concerned, this pathway has been considered in the early paper (56). Therein, it was shown that this physical process yields contributions in the low-momentum regions, which can be prominent for species such as argon. This mechanism is considerably less understood, especially in the framework of the strong-field approximation. In fact, apart from these early studies, only very recently has a detailed analysis of RESI been performed by us in this context (31, 72). This issue will be addressed in Secs. 2.4 and 2.5. 2.2.4.

Other approaches

Especially in the early days of NSDI, a multitude of approaches have been employed in order to model this phenomenon. Two such approaches are the time-dependent density functional theory (TDDFT) and the time-dependent Hartree-Fock method. They are closely related as, in both cases, the many-particle interacting Hamiltonian can be reduced to that of non-interacting particles subjected to an effective potential. The main difference is that the Hartree-Fock method just accounts for electron exchange, while in TDDFT exchange and correlation are included. This is a key advantage if compared to purely ab initio methods such as the TDSE. This means that the physical observables are then described as functions of the time-dependent density. Since the density is a three-dimensional quantity, the computational effort does not increase as critically with the number of particles as for the TSDE. Apart from that, these methods are gauge-invariant and include the residual ion in the electron propagation in the continuum. These are advantages over the strong-field approximation. Time-dependent Hartree-Fock methods have been applied to NSDI in the late 1990s for few-photon ionization (109, 110) or NSDI in both tunneling and multiphoton regimes (85). Unfortunately, it completely failed to reproduce the knee in the NSDI yield. As pointed out in (85), this is possibly related to this approach, at least in its standard form, not being appropriate for describing correlated electronic wave functions. Recently, progress has been made in the context of ionization and excitation in extended systems, such as molecules, performing multiconfiguration time dependent Hartree-Fock computations (111, 112). The first attempts to employ TDDFT to model NSDI as a function of the driving-field intensity, however, have shown that the treatment of electron-electron correlation poses a great challenge. In fact, such attempts, in the late 1990s, completely failed to reproduce the knee structure in the NSDI yield as a function of the laser intensity (81). This was attributed to the fact that the electrons behaved as free and almost independent electrons. Similar results were also obtained in (85), and shortly thereafter in (113). All these early publications stressed the importance of finding more accurate density functionals for the exchange and correlation potentials. In fact, in (113) a wide variety of functionals has been tested, and none of them could account for the essential aspects of electron correlation in NSDI. In fact, the approximation of considering an effective potential works far better for a system with many electrons, such as cluster or solid, than for a two-electron system. Only in the mid 2000s have adequate functionals been found by Lein and Kummel (114) and Wilken and Bauer (115). In the former paper, the authors constructed the exchange-correlation potential employing direct data from a reduced-dimensionality TDSE computation, whereas the latter focused on the functional choices for the observables. Both publication, however, have shown that, in order to obtain the knee, a stepwise behavior must be incorporated in the density of the two-electron system. An even more difficult task is to reproduce the double-hump structure in the ion-momentum distribution. This has been achieved by the latter group in (116). Therein, a very important issue turned out to be the functionals for computing the electron-momentum distributions. If these functionals are uncorrelated, the distributions exhibit a single peak at vanishing ion momentum, i.e., as in a sequential process. A double peak structure has only been obtained for correlated functions. The theory, however, still requires a great degree of


20

106

101 100

Ion signal (arb.units)

104

10-1 102 10-2 100

10-3 10-4

10-2

10-5 10-4 10-6 10-6

10-7 1014

1015 Intensity (W / cm2)

1016

1013

1014 1015 Intensity (W / cm2)

1016

Figure 9. Measured ionization yields as a function of laser intensity. Left panel: Double ionization of helium at 780 nm (Fig. 1 in (10)). Right panel: Multiple ionization of xenon at 800 nm (Fig. 3 in (13)).

development until a realistic, three-dimensional model is obtained. Another method employed to describe double ionization in intense laser fields is the R-matrix theory. Roughly speaking, this method consists in dividing the problem in two spatial regions, close and far from the core. In the outer region, i.e., far from the core, the indistinguishability of the ionized electron is neglected. This means that the exchange interaction between the outer electron and those left close to the core is neglected, and simplifies the problem considerably. In contrast, if the first electron enters the inner region, indistinguishability cannot be neglected and the fully correlated problem must be treated. Both regions are connected by the R-matrix. Furthermore, if one is dealing with long enough laser fields, its periodicity may be employed to transform the time-dependent problem into a time-independent one. This is the essence of the R-matrix Floquet approach. In the context of nonsequential double ionization, this approach has been employed in the high-frequency regime, in order to investigate one- (117) and two-photon double ionization (118) of helium. With the advent of free-electron lasers, photons of frequencies as high as 45 eV became available, and this problem became particularly interesting from the experimental viewpoint. Finally, another approach which has recently been employed to compute NSDI yields is the coupled-coherent-state method. It consists in expanding the time-dependent Schrödinger equation on a coherent-state basis, which reduces the full time dependent problem to solving coupled differential equations. The solutions to these equations can be related to the orbits of a classical electron in the continuum (119). 2.3.

2.3.1.

Key experiments

Total ionization yields

The first experiment that suggested the important role of electron-electron correlation in intense laser fields was the measurement of the double ionization yield of xenon as a function of


21

Figure 10. (Color online) Electron kinetic energy distribution from single (solid line) and double ionization (circles) of helium at 780 nm and two laser intensities of (a) 8 × 1014 W/cm2 and (b) 4 × 1013 W/cm2 (Fig. 3 in (123)).

the driving-field intensity (4). It was found that, in a certain range of laser intensities, doubly charged xenon ions are produced in a direct (or nonsequential) process. Further experiments on double ionization of helium and multiple ionization of xenon showed a much higher ionization yield than the prediction of the single-active-electron (SAE) model by many orders of magnitude (9, 10, 13). A so-called “knee” structure was characterized in the double/multiple ionization yields. Characteristic experimental data is shown in Fig. 9 in plots of the ionization yields as a function of the laser peak intensity. In order to investigate the physical mechanism behind this striking ”knee” structure, the ellipticity dependence of ion yield has been measured by Dietrich et al. (74), soon after the extensive ion yield measurements with linearly polarized light. It was found that the doubly charged ion yield decreases dramatically with the laser ellipticity, exhibiting very similar features as those of high harmonic generation. This measurement suggested a common underlying mechanism behind these two closely related processes, i.e., high harmonic generation and nonsequential double ionization, and for the first time, provided experimental support for electron rescattering as the physical mechanism of nonsequential double ionization. While most experiments performed on noble gas atoms exhibited the common feature that the NSDI probability decreases rapidly with the increase of laser field ellipticity, recent measurements on more complex systems, such as diatomic molecule of NO (120) and alkali-earth atom of Mg (121), found that the ”knee” structure, the signature of nonsequential process, persists even under circularly polarized light. This behavior is seemly in striking contrast to the intuitive rescattering picture and has raised a great deal of controversy until very recently. Interestingly, it has been demonstrated that this seemly abnormal behavior in ionization yield measurements may also be understood as the consequence of electron rescattering in the context of classical model (66, 122).

22


Figure 11. (Color online) Recoil ion momentum distribution along laser polarization direction for double ionization of helium (left panel, Fig. 3 in (18)) and neon (right panel, Fig. 3 in (19)).

2.3.2.

Electron kinetic energy distribution

To gain more insight into the mechanism behind the nonsequential ionization process, elaborate differential measurements have become necessary. In recent years, electron kinetic energy spectra correlated with differently charged states of ion have been measured for noble gas atoms, including helium (123), argon (23, 124) and xenon (23, 125). The laser intensities were all chosen in the regime such that the “knee” structure appeared. A common feature from those measurements is that the electron spectra correlated with doubly charged ion extend to much higher energies than those from single ionization. Typical spectra of helium is shown in Fig. 10. The observed hotter electron production from NSDI was found to be in good agreement with the prediction from the rescattering model and gave a hint that the rescattering mechanism was responsible for the NSDI process. 2.3.3.

Recoil ion momentum distribution

The advent of COLTRIMS (COLd Target Recoil Ion Momentum spectrometer) technique has to a large extent facilitated the exploration of the physical mechanism behind nonsequential ionization. The first measurements of recoil ion momentum distributions have been performed for double ionization of helium (18) and neon (19). A common feature in these data is a distinct double hump structure in the ion momentum distribution along the laser polarization direction and a single peak structure perpendicular to the laser polarization. The double peak structure in the parallel ion momentum distribution gave a strong support to the rescattering mechanism which suggests that the first tunneled electron returns to its parent ion and both electrons are emitted into the continuum at the crossing of the external laser field. Both electron acquire the most energy from the laser field in this process. These measurements thus rule out the shakeoff and the collective tunneling mechanisms, since for both one expects a single peak structure centered at the origin.


23

Figure 12. (Color online) Recoil ion momentum distribution along laser polarization direction for double ionization of helium, neon and argon at various laser intensities (Fig. 2 in (49)).

On the other hand, a closer comparison of the ion momentum distributions for different noble gas atoms suggested that the distribution is target specific. It has been found that the double hump structure along the laser polarization direction is pronounced for neon, while much less apparent or even absent for the other two species, i.e., helium and argon. In Fig. 12 experimental data for the ion momentum distribution along the field direction for doubly charged helium, neon and argon, measured by Jesus et al (49), are shown. This different feature for various noble gas atoms may be understood by considering two different mechanisms in the context of the rescattering scenario, i.e., electron impact ionization and RESI. For the latter mechanism, the doubly charged ions are produced with small momenta. Depending on the relative strength of contribution from RESI and electron impact ionization mechanisms, different recoil ion momentum distributions could be present for different noble gas atoms (49). Momentum distributions of multiple charged ions have been measured by Rudenko et. al. (16) for neon and argon up to quadruply charged ions and are shown in Fig. 13. Note that the change of the double hump structure to a single peak structure for N e+ 2 with the increase of laser intensity indicates a switch of the ionization mechanism from a nonsequential to a sequential process. Similar to the doubly charged ion, the differences between the momentum distributions of the multiply charged ion of neon and argon are prominent: neon exhibits a clear double hump structure with almost no events created at zero momentum, while, for argon, there is only a shallow minimum at zero momentum (for Ar3+ ) or no double hump structure (for Ar4+ ) at all, indicating that, similar to the case of double ionization, the direct electron impact ionization can not be the dominant mechanism for multiple ionization of argon. 2.3.4.

Correlated electron momentum distributions

More complete insight into the NSDI mechanism can be obtained by analyzing the momentum correlation between the two emitted electrons. The most straightforward way to obtain the momenta of two electrons would be to measure both electrons in coincidence. However, due to

24


Figure 13. (Color online) Recoil ion momentum distribution along laser polarization direction for double, triple and quadruple ionization of neon (left panel, Fig. 2 in (16)) and argon (right panel, Fig. 3 in (16)) at various laser intensities.

the limitation of detection efficiency and the dead time of the electron detector, most experiments so far were performed by measuring only one electron in coincidence with the doubly charged ion. In the limit of negligibly small momentum transfer by the absorbed photons, the momenta of the second electron can be deduced using the momentum conservation. The first experimental data on electron-electron momentum correlation, as shown in the left panel of Fig. 14, has been obtained by Weber et al (126). The data recorded at lower laser intensity of 3.6 × 1014 W/cm2 , at which nonsequential ionization dominates, shows that both electrons are emitted preferentially into the same hemisphere, i.e., first or third quadrants. The preferential emission of both electrons with the identical momentum gave a strong support to the rescattering mechanism for NSDI. For the higher laser intensity (15 × 1013 W/cm2 ), at which double ionization proceeds sequentially, the electron correlation becomes lost. In the right panel of Fig. 14, we show the electron momentum correlation distributions from NSDI of helium recently measured by two groups (50, 51). An eye-catching feature in these two high resolution measurements is a V-shaped or finger-like pattern. This feature may be understood as a consequence of the final state electron-electron Coulomb repulsion and typical electron impact ionization kinematics. A common feature of the correlated electron momentum distribution obtained so far is the dominant population in the first and third quadrants, each centered at non-vanishing momentum position. This feature has been well understood as a consequence of the rescattering mechanism (127). However, a very recent experiments on NSDI of argon, performed at very low laser intensities such that the maximal return energy of 3.17Up was below the field-modified second ionization potential, has revealed a strongly correlated back-to-back emission of the electrons along the polarization direction (22). Possibly, this intensity is even too low for RESI to occur. The authors ruled out Coulomb repulsion effects and point to the possible significance of multiple recollisions. The mechanism responsible for this back-to-back emission is not yet entirely clear and calls for more theoretical efforts.


25

Figure 14. (Color online) Correlated electron momentum distribution along laser polarization direction from NSDI of argon at laser intensities of 3.8 × 1014 W/cm2 (upper left panel) and 15 × 1014 W/cm2 (lower left panel) (Fig. 2 in (126)), and of helium at laser intensities of 15 × 1014 W/cm2 (upper right panel, Fig. 2 in (51)) and 4.5 × 1014 W/cm2 (lower right panel, Fig. 1 in (50)).

2.4.

The S-matrix approach and its classical limit

In this section we will describe the approach employed in our previous publications (30, 31, 47, 55, 72, 88, 107, 108), in order to investigate laser-induced NSDI. We based our approach on the strong-field approximation and used saddle-point methods. This approach has the advantage of providing a clear physical picture of the physical processes involved, and, due to the fact that it is analytical to a great extent, it is computationally less demanding than ab-initio methods. In particular, we consider NSDI in the tunneling regime and assume that the second electron, upon recollision, is either immediately released by electron-impact ionization, or promoted to an excited state, from which it subsequently tunnels. We approximate the continuum by Volkov states and neglect the presence of the laser field in the electronic bound states. The Feynman diagrams utilized by us are provided in Fig. 16. In all examples discussed here, we are neglecting the interaction of the electrons with the residual ion. We are also employing atomic units throughout. 2.4.1.

Electron-impact ionization

The transition amplitude corresponding to electron impact ionization, in the strong-field approximation, reads M

EI

′

(p1 , p2, t, t ) =

∫

∫ t ⟨ ⟩ (V ) (0) dt dt′ ψp(V1 ,p) 2 (t) V12 U1 (t, t′ )V U2 (t, t′ ) ψg (t′ ) ,

∞

−∞

(1)

−∞

where U1 (t, t′ ) and V denote the Volkov time-evolution operator and the atomic potential (0) acting on the first electron, U2 (t, t′ ) gives the field-free time evolution operator for the second (V )

26


9h1013 W / cm2

7h1013 W / cm2

4h1013 W / cm2

Figure 15. (Color online) Correlated electron momentum distribution along laser polarization direction from NSDI of argon at different laser intensities of 9 × 1013 W/cm2 (a), 7 × 1013 W/cm2 (b) and 4 × 1013 W/cm2 (c) (Fig. 2 in (22)) .

Figure 16. (Color online) Feynman diagram of the main rescattering processes behind nonsequential double ionization in the tunneling regime. Part (a) describes recollision-excitation with subsequent tunnel ionization, in which an electron, (1) initially in a bound state |ψg >, is released by tunneling ionization into the Volkov state at a time t′′ , returns at a time t′ and excites a second electron from |ψg >, to the state |ψe >, from which it subsequently tunnels at a later time t, reaching a Volkov state, while part (b) depicts the electron-impact ionization process in which the first electron is also bound (1) at |ψg >, is released by tunneling ionization into the Volkov state at a time t′′ , returns at t and gives enough energy to the second electron to overcome the second ionization potential. The electron -electron interaction is indicated in the figure by V12 , the initial bound states by the dark blue lines, the excited bound state of the second electron by the thick black line and the Volkov states by the double red lines. (2)

(2)


27

electron, and V12 the electron-electron interaction. The initial state of the two-electron system ⟩ is (1) the two-electron ground state |ψg (t′ )⟩, which is approximated by the product-state ψg (t′ ) ⊗ ⟩ ⟩ ⟩ (2) ′ (n) ′ (n) ′ , n = 1, 2, where Egn gives the ψg (t ) of the one-electron states ψg (t ) = exp[iEgn t ] ϕg ⟩ (V ) first or second ionization potential. The final state ψp1 ,p2 (t) is either taken to be a correlated two-electron Volkov state, or the product state of one-electron Volkov states, with asymptotic momenta p1 , p2 . In the former case, final-state electron-electron repulsion is considered. Considering the explicit expressions for the Volkov time-evolution operator, Eq. (1) may be written as M (EI) (p1 , p2, t, t′ ) =

∫

∫ t ∫ (EI) ′ dt dt′ d3 kVpn k Vk0 eiS ({pn },k,t,t ) ,

∞

−∞

(2)

−∞

with the action S

(EI)

′

({pn }, k, t, t ) = −

2 ∫ ∑ n=1 t

∫ −

t

t′

∞

[pn + A(τ )]2 dτ 2

[k + A(τ )]2 dτ − Eg2 t − Eg1 t′ 2

(3)

and the prefactors ⟨ ⟩ ˜ ′ ) V ϕ(1) Vk0 = k(t 0

(4)

⟩ ˜ Vpn k = ⟨˜ p1 (t) , p ˜ 2 (t)| V12 k(t), ϕ(2) , g

(5)

and

where A(τ ) is the vector potential. Eq. (2) describes the physical process in which an electron, ⟩ (1) initially in a bound state ϕg , is released by tunneling ionization at a time t′ into a Volkov ⟩ ˜ state k(t) . Subsequently, this electron propagates in the continuum from t′ to a later time t with intermediate momentum k. ⟩At this time, it is driven back by the field and frees a second (2) electron, which is bound at ϕg , through the interaction V12 . Finally, both electrons are in Volkov states. The final electron momenta are described by pn (n = 1, 2). The form factors (4) and (5) contain all the information about the binding potential, and the interaction by which the second electron is dislodged, respectively. In the length gauge, p ˜ n (τ ) = pn + A(τ ) and ′ ˜ ˜ ) = k. This is k(τ ) = k + A(τ ), with τ = t, t , while in the velocity gauge p ˜ n (τ ) = pn and k(τ due to the fact that the gauge transformation to the length to the velocity gauge, when applied to a plane wave |p⟩, causes the shift p → p − A(t). This shift cancels out with those present in the Volkov states. This implies that such prefactors will be different in each gauge. This will lead to different two-center interference conditions for molecules, for instance (30). Unless otherwise stated, we will work within the length gauge. The can be seen even more explicitly if the transition amplitude (2) is solved by saddle-point methods. In this case, one looks for values of k, t and t′ for which the action (3) is stationary, i.e., for which ∂t S (EI) ({pn }, k, t, t′ ) = ∂t′ S (EI) ({pn }, k, t, t′ ) = 0 and ∂k S (EI) ({pn }, k, t, t′ ) = 0. The conditions upon t′ , k and t lead to the saddle point equations [ ]2 k + A(t′ ) = −2Eg1 ,

(6)


28

k=−

1 t − t′

∫

t

A(τ )dτ

(7)

t′

and 2 ∑ [ ]2 [pn + A(t)]2 = k + A(t′ ) − 2Eg2 ,

(8)

n=1

respectively. Physically, Eq. (6) can be interpreted as the energy conservation of the first electron at the time t′ , when it reaches the continuum by tunneling ionization. This equation has no real solution. This is expected, as the tunneling process has no classical counterpart. This leads to a nonvanishing imaginary part for t′ , which, loosely speaking, is related to the width of the barrier through which the electron must tunnel. Eq. (7) provides a constraint upon the intermediate momentum of the electron, and guarantees that it returns to the site of its release. This equation can be associated with the classical equations of motion of an electron in the presence of the driving field only, if the electron is assumed to leave from and return to the origin. Finally, the saddle-point equation (8) gives the conservation of energy at the instant of rescattering: The first electron returns with a kinetic energy Eret = [k + A(t′ )]2 /2, which is transferred to the second electron. Both electrons are released simultaneously and reach the detector with momenta pn , n = 1, 2. The kinetic energy Eret may or may not be enough to make the second electron overcome the second ionization potential Eg2 . In case Eret > Eg2 , electron-impact ionization is classically allowed. If, on the other hand, Eret < Eg2 , this process has no classical counterpart, and the corresponding transition probability will be exponentially decaying. Eq. (8), if written in terms of the electron-momentum components parallel and perpendicular to the laser-field polarization, reads 2 ∑ [ ]2 [pn|| + A(t)]2 + p2n⊥ = k + A(t′ ) − 2Eg2 .

(9)

n=1

This is the equation of a six-dimensional hypersphere in momentum space centered at pn|| = −A(t), pn⊥ = 0, n = 1, 2. Since A(t) varies, the classically allowed region will be defined by all the possible hyperspheres defined by Eq. (9). A rough estimate can be made if one considers that the electron returns at a field crossing. This means that the center is at approximately √ pn|| = ±2 Up . One should note that, for electron-impact ionization, this is not the maximal parallel momentum expected in the distributions, but the most probable momentum value. The radius of the hypersphere is defined by the right-hand side of Eq. (9), and, for high drivingfield intensities, may extend far beyond that, namely up to almost 5Up . This has been observed in SFA computations, classical models (46) and also the full solution of the time-dependent Schrödinger equation (44). If, on the other hand, the driving-field intensity decreases, the radius of the hypersphere shrinks until it collapses. In this latter case, the maximal kinetic energy of the first electron upon return is of the order of the second ionization potential. This specific intensity is known as “threshold intensity”. Even more information may be obtained from Eq. (9), if one considers fixed transverse momentum. In this case, one may define circles in the parallel-momentum plane and the kinematic constraints are defined by the equation of a circle, namely 2 ∑ [ ]2 ˜g2 , [pn|| + A(t)]2 = k + A(t′ ) − 2E n=1

(10)


p2||/[Up]1/2

29

-4

-2

2

4

p1||/[Up]1/2

Figure 17. (Color online) Classically allowed region for electron-impact ionization, as a function of the electron momentum components pn∥ (n = 1, 2) parallel to the laser-field polarization. The different concentric circles correspond to different fixed √ transverse momenta pn⊥ (n = 1, 2). In order to simplify the picture, we assumed A(t) = ±2 Up and a monochromatic driving field.

˜g2 = Eg2 + where E

2 ∑

p2n⊥ /2 gives an effective ionization potential. This means that the regions n=1 √ in the parallel-momentum plane are more localized around pn|| = ±2 Up for non-vanishing electron momentum. It is as if the second electron had to overcome a higher second ionization potential. From the above-stated equation, it is also clear that electron-impact ionization populates the first and third quadrants of the parallel momentum plane. If A(t) = ±A(t ± T )/2), ( where T = 2π/ω is a cycle of the laser field, the distributions are symmetric upon p1|| , p2|| → ( ) −p1|| , −p2|| . This is the case, for instance, for monochromatic fields. If, on the other hand, this property does not hold, an asymmetry between the first and third quadrant is expected. This asymmetry occurs for few-cycle pulses, and has been verified both theoretically (107, 108) and experimentally (128). In Fig. 17, we present a schematic representation of the classically allowed region according to the previously discussed constraints. This rough estimate agrees with the solutions encountered for the start time t′ and the return time t, obtained solving the saddle-point equations. This has been studied in detail in Ref. (41) and is displayed in Fig. 18 for equal parallel momenta for both particles, i.e., p1∥ = p2∥ = p∥ , for the shortest pair of orbits. These orbits will provide the dominant contributions to the NSDI distributions, as they exhibit the least wave-packet spreading. These solutions are complex. Their real parts are associated to the classical start and return times of both electrons, and the imaginary part is related to a process being classically allowed or forbidden. If the physical process in question has a classical counterpart, the imaginary part of the saddle-point variable is nearly vanishing. This is the case for Im[t], but not for Im[t′ ], which is a consequence of the fact that tunneling has no classical counterpart. If the imaginary parts of the variables exhibit a local minimum, this means that the probability of a process occurring is maximal. An inspection of the real parts of the start and return times, displayed in Panels (a) and (b), shows that the electron may return after or before √ the field crossing. These two times are furthest apart if the electron momentum is around 2 Up and almost coalesce at two specific momenta. These momenta give the minimal and the maximal classically allowed momenta. In fact, an inspection of Im[ωt] shows that this quantity is nearly vanishing in this region (see panel (d) in the figure). Furthermore, as expected from Eq. (10), this region decreases for increasing

30


Figure 18. Upper and middle panels of Fig. 1 in (41). Panels (a) and (b) give the real parts of the return and start times, respectively, obtained by solving the saddle-point equations (6)-(8), while panels (c) and (d) depict the corresponding imaginary parts. In the figure, we considered p1∥ = p2∥ = p∥ . The numbers in the curves give the transverse momenta √ (p1⊥ , p2⊥ ) in units of Up .

transverse momenta until it collapses. This latter case corresponds to the situation for which the returning electron can no longer √ enable the second electron to overcome the effective ionization ˜ potential Eg2 . Interestingly, 2 Up is not the maximal momentum, but the most probable one. This can also be seen in the imaginary parts of the return times, even if there is no classical region. Indeed, |Im[t]| exhibits minima around this momentum (see Figs. 18 (c) and (d)). All this implies that, in the high parallel momentum regions, instead of returning at a time such that its kinetic energy is maximal, the first electron returns at a time which leads to the highest possible drift momentum. For a detailed discussion see our previous publication (41). Finally, the rescattering model discussed in this section supports a classical limit, which occurs if the first ionization potential Eg1 vanishes. The electron momentum distributions, in this limit, are given by the expression, up to a constant factor (EI) Fcl (p1 , p2 )

∫ =

′

′

(

dt R(t )|Vpn k | δ 2

) 1 2 2 (p + p2⊥ ) − ∆E , 2 1⊥

(11)

{ } where R(t′ ) ∼ |E(t′ )|−1 exp −2 (2Eg1 )3/2 / [3|E(t′ )|] is the quasi-static tunneling rate (129), E(t′ ) = −dA(t′ )/dt′ is the electric field at the instant of tunnel ionization, and ∆E ≡ Eret (t) − Eg2 − 21 [p1|| + A(t)]2 − [p2|| + A(t)]2 . In this model, we assume that the first electron appears in the continuum with zero velocity and according to the quasi-static rate R(t′ ). The kinetic energy Eret (t) of the first electron at the instant of recollision is computed employ-


31

ing the classical equations of motion for the first electron in the presence of the field only. In practice, one mimics the quantum-mechanical electron momentum distribution considering a classical ensemble computation, in which each electron is released with vanishing drift velocity. Some of these electrons return and release a second ensemble, whose momenta satisfy the condition determined by the argument of the δ function in Eq. (11). This condition represents the energy conservation upon electron-impact ionization, i.e., the saddle-point equation (8) with real t, t′ and k. One should note that, outside the classically allowed region for electron-impact ionization, this argument is non-vanishing and the yield is zero. In contrast, the quantum-mechanical transition amplitude is exponentially decaying outside this region. This leads to small differences at the boundary of the classically allowed region. Other features which are not present in this model are quantum-interference effects and the spreading of the electronic wave packet from the tunnel ionization time t′ to the rescattering time t. The former will be washed out if one integrates over the transverse momenta, and the latter may be incorporated by including an additional prefactor. We have observed, however, that, if the intensity is high enough, the agreement between the classical and quantum mechanical models is very good, leading to almost identical distributions (47, 88, 108). Only near the boundary of the classically allowed region have small discrepancies been observed. For details on the electron-impact ionization transition amplitude and its classical limit, see (47). 2.4.2.

Recollision-excitation with subsequent tunneling ionization

We will know write down the SFA transition amplitude describing the recollision-excitation with subsequent tunneling ionization (RESI) mechanism. This transition amplitude reads ∫ M (RESI) =

∫

∞

t

dt

dt′′

−∞

−∞

∫

t′′

dt′

(12)

−∞

˜ (t, t′′ )V12 U (t′′ , t′ )V |ψg(1) (t′ ), ψg(2) (t′ ) >, < p1 (t), p2 (t)|Vion U ˜ (t′′ , t′ ) denote the time evolution operator of the two-electron system, where U (t′′ , t′ ) and U (1) ′ (2) ′ |ψg (t ), ψg (t ) > is the two-electron initial state, and |p1 (t), p2 (t)⟩ the final two-electron continuum state. The interactions V, V12 , and Vion correspond to the atomic binding potential, the electron-electron interaction and the binding potential of the singly ionized core, respectively. When defining Eq. (12), we imposed that V and Vion are connecting different subspaces, namely the bound states and the continuum. This is necessary due to the lack of orthogonality of the Volkov solution (for details see (72)). When defining these operators, we assume that the system is initially in a product state of one-electron ground states, i..e., (1) (2) (1) (2) (n) (2) |ψg (t′ ), ψg (t′ ) >= |ψg (t′ ) > ⊗|ψg (t′ ) >, with |ψg (t′ ) >= exp[iEng t′ ]|φg >. The timeevolution operator of the system from the tunneling time t′ of the first electron to the recollision (1) (2) (1) time t′′ was approximated by U (t′′ , t′ ) = UV (t′′ , t′ ) ⊗ Ug (t′′ , t′ ), where UV is the Gordon(2) Volkov time-evolution operator for the first electron and Ug is the field-free time evolution operator for the second electron in the ground state. Subsequently to the recollision, the time ˜ (t, t′′ ) = U (1) (t, t′′ ) ⊗ Ue(2) (t, t′′ ), where U (1) evolution operator of the system was taken to be U V V (2) is the Gordon-Volkov time-evolution operator for the first electron and Ue is the field-free time evolution operator for the second electron in the excited state of the singly ionized ion. Using the explicit expressions for the Gordon-Volkov time evolution operator we find ∫ M (RESI) =

∫

∞

t

dt −∞

−∞

dt′′

∫

t′′

−∞

dt′

∫

d3 kVp2 e Vp1 e,kg Vkg exp[iS (RESI) ({pn }, k, t, t′ , t′′ )]

(13)


32

with the action S

(RESI)

′

∫

′′

({pn }, k, t, t , t ) = −

∞

t

∫ −

[p2 + A(τ )]2 dτ − 2

′′

t

dτ t′

∫

∞

dτ t′′

[p1 + A(τ )]2 2

[k + A(τ )]2 + E2e (t − t′′ ) 2

+E2g t′′ + E1g t′ .

(14)

Thereby, A(τ ) is the vector potential, the energy E1g denotes the first ionization potential, E2g the ground-state energy of the singly ionized atom and E2e the energy of the state to which the second electron is excited. The intermediate momentum of the first electron is given by k and the final momenta of both electrons by pn (n = 1, 2). Eq. (13) describes a physical process in which the first electron leaves the atom at a time t′ , propagates in the continuum with momentum k from t′ to t′′ , and upon return, gives part of the kinetic energy to the core so that a second electron is promoted from a state with energy E2g to an excited state with energy E2e . This electron then reaches the detector with momentum p1 . At a subsequent time t, the second electron tunnels from the excited state, reaching the detector with momentum p2 . All the dependence on the binding potential will be embedded in the prefactors ⟨ ˜ ′′ ) V Vkg = k(t ∫ ×

⟩ (1) = ψ g

1 (2π)3/2

˜ ′′ ) · r1 ]ψg(1) (r1 ), d3 r1 V (r1 ) exp[−ik(t

(15)

and Vp2 e

⟩ = ⟨˜ p2 (t)| Vion ψe(2) = ∫ ×

1 (2π)3/2

d3 r2 Vion (r2 ) exp[−i˜ p2 (t) · r2 ]ψg(2) (r2 ),

(16)

and the dependence on the electron-electron interaction V12 will be contained in the prefactor ⟨ ( ) ⟩ ˜ ′ Vp1 e,kg = p ˜ 1 t′ , ψe(2) V12 k(t ), ψg(2) = ∫ ∫ ×

1 (2π)3

d3 r2 d3 r1 exp[−i(p1 − k) · r1 ]

×V12 (r1, r2 )[ψe(2) (r2 )]∗ ψg(2) (r2 ).

(17)

Thereby, V and Vion correspond to the atomic binding potential of the system as seen by the first and second electron, respectively. In particular Eqs. (16) and (17) will determine the shape of the electron-momentum distributions, and will be very much influenced by the initial and final bound states occupied by the second electron. If the electron interaction is only dependent on the difference between both electron coordinates, i.e., if V12 (r1, r2 ) = V12 (r1 − r2 ), one may rewrite Eq. (17) as Vp1 e,kg =

V12 (p1 − k) (2π)3/2 ∫ × d3 r2 e−i(p1 −k)·r2 [ψe(2) (r2 )]∗ ψg(2) (r2 ),

(18)


33

with 1 V12 (p1 − k) = (2π)3/2

∫ d3 r exp[−i(p1 − k) · r]V12 (r)

(19)

and r = r1 − r2 . Clearly, Vkg and Vp2 e are gauge dependent. In the length gauge p ˜ n (τ ) = ˜ ) = k + A(τ )(τ = t′ , t′′ ), while in the velocity gauge p pn + A(τ ) and k(τ ˜ n (τ ) = pn and ˜ ) = k. This is a consequence of the momentum shifts caused by the gauge transformation k(τ from the length to the velocity gauge. These shifts cancel out in Eq. (18), so that Vp1 e,kg remains the same in the length and velocity gauges. In practice, however, since the second electron will leave close to the maximum of the field, A(t) ∼ 0 and the prefactors (16) are almost identical in the velocity and length gauges (see (31) for details). Similarly to what happens for electron-impact ionization, Eq. (13) may be solved by saddlepoint methods, by finding the values for t′ , t′′ , t and k so that ∂t′ S (RESI) ({pn }, k, t, t′ , t′′ ) = ∂t′′ S (RESI) ({pn }, k, t, t′ , t′′ ) = ∂t S (RESI) ({pn }, k, t, t′ , t′′ ) = 0 and ∂k S (RESI) ({pn }, k, t, t′ , t′′ ) = 0. This leads to the saddle-point equations [ ]2 k + A(t′ ) = −2E1g ,

1 k = − ′′ t − t′

∫

(20)

t′′

dτ A(τ )

(21)

t′

[ ]2 [p1 + A(t′′ )]2 = k + A(t′′ ) − 2(E2g − E2e ).

(22)

[p2 + A(t)]2 = −2E2e .

(23)

and

The first saddle-point equation is identical to that found for the action S (EI) ({pn }, k, t, t′ ), i.e., Eq. (6). It expresses the fact that, at the instant t′ , the first electron tunnels and reaches the continuum. Identically to its counterpart in Sec. 2.4.1, Eq. (21) imposes a constraint upon the intermediate electron momentum, so that the first electron can return to the site of its release. The main difference lies in the remaining saddle-point equations. Eq. (22) gives the conservation of energy upon the inelastic rescattering of the first electron at a time t′′ . Thereby, this electron gives part of its kinetic energy Eret (t′′ ) = [k + A(t′′ )]2 /2 to the core and excites the second electron from a state with energy E2g to a state with energy E2e . The first electron leaves immediately and reaches the detector with momentum p1 . Finally, Eq. (23) describes the tunneling ionization of the second electron from an excited state of energy E2e at a subsequent time t. The second electron then reaches the detector with momentum p2 . One should note that there is now a time delay between the rescattering of the first electron and the ionization of the second electron. The saddle-point equations (22) and (23) allow one to delimit constraints for the final momenta of the first and the second electron, and thus obtain useful information about the momentumspace regions populated by the RESI mechanism. Eq. (23), for instance, is formally identical to the saddle-point equation describing the low-energy electrons in above-threshold ionization (ATI). These electrons are known as the “direct electrons”, as, after being tunnel ionized, they reach the detector without rescattering. It is known that the cutoff energy for direct ATI is 2Up .


34

This implies that −2

√

√ Up ≤ p2 ≤ 2 Up ,

(24)

( )1/2 where p2 = p22∥ + p22⊥ . Furthermore, since the second electron is expected to leave most likely at field peak, this implies that the distributions will have a peak for vanishing momentum. If, to first approximation, we neglect the momentum components perpendicular to the laserfield polarization, the momentum of the second electron, in the plane p1∥ p2∥ spanned by the electron momentum components parallel to the laser-field polarization, is expected to be centered √ around vanishing momentum p2∥ and be limited by the bounds p2∥ = ±2 Up . The latter are the maximal momenta the second electron can have. One should note that there is no classical counterpart for the tunneling process described by Eq. (23). This means that nonvanishing transverse momenta would effectively widen the potential barrier through which the second electron tunnels, and thus cause an overall suppression in the yield. The cutoff energy, however, would remain unchanged. The saddle-point equation (22), on the other hand, provides constraints on the momentum of the first electron. This equation can be viewed as a rescattered ATI-like process, in which one electron rescatters with its parent ion and subsequently reaches the detector. The difference is that, in Eq. (22), part of the kinetic energy of the first electron is transferred to the core upon rescattering. For high-order ATI, elastic rescattering takes place. Explicitly, the momentum component of the first electron parallel to the laser-field polarization is given by −A(t′′ ) −

√ √ 2Ediff ≤ p1∥ ≤ −A(t′′ ) + 2Ediff ,

(25)

where Ediff = Eret (t′′ ) − (E2g − E2e ) − p21⊥ /2 and Eret (t′′ ) denotes the kinetic energy of the first electron upon return. The electron is more likely √ to return near a crossing of the laser field. Hence, one may use the approximation A(t) ≃ 2 Up . Furthermore, the kinetic energy is bounded by Eret (t′′ ) ≤ 3.17Up . Thus, Ediff

(max)

≤ 3.17Up − (E2g − E2e ) − p21⊥ /2 and

√ √ √ √ (max) (max) −2 Up − 2Ediff ≤ p1∥ ≤ −2 Up + 2Ediff .

(26)

√ Eq. (26) shows that the momentum-space region for p1∥ is centered around −2 Up and bounded (max)

by 2Ediff . If the parameters inside the square root are positive, i.e., if 3.17Up ≥ (E2g − E2e ) + p21⊥ /2, the rescattering process described by Eq. (22) allows the existence of a classically allowed region for p1∥ . For increasing perpendicular momentum and/or bound-state energy difference, √ this region will become more and more localized around −2 Up until it collapses. This allows one to derive a threshold for the RESI mechanism, which is below that for electron-impact ionization. If (E2g − E2e ) ≪ 3.17Up the well-known cutoff of 10Up for rescattered above-threshold ionization is recovered. For details we refer to (31, 72). In Fig. 19, we provide a schematic representation of the momentum-space regions populated by the RESI mechanism. In the figure, we assumed that the driving-field intensity was far√ above the threshold, so that the largest momenta of the first electron may extend far beyond ±2 Up . The momentum constraints defining such regions leads us to expect electron-momentum distributions with the following characteristics: √ (1) Maxima at the points (p1|| , p2|| ) = (±2 Up , 0) and, since both electrons are indistinguish√ able, also at (p1|| , p2|| ) = (0, ±2 Up ). √ (2) Along the axis, the distributions may extend up to momenta pn|| = 2 5Up . This limit is given by the rescattered ATI cutoff. This maximal momentum, however, will vary depending on the driving-field intensity.


p2||/[Up]

35

1/2

4

2

-4

-2

2

4 p /[U ]1/2 1|| p

-2

-4

Figure 19. Schematic representation of the regions of the parallel momentum plane populated by the recollision-excitationtunneling ionization mechanism is shown. The black circles indicate the expected maxima of the electron momentum distributions, and the shadowed region indicates the region populated by the RESI mechanism. We are considering different sets of trajectories, whose start and recollision times are separated by half a cycle of the field, and the symmetrization p1 ←→ p2 with respect to the indistinguishability of the two electrons.

√ (3) The width of the distributions will never be larger than the bounds pn∥ = ±2 Up . These bounds correspond to the direct-ATI cutoff energy. If, however, the prefactor Vp2 e exhibits nodes, the distributions may be narrower. (4) The electron-momentum distributions occupy the four quadrants of the parallelmomentum plane. For long pulses, which can be approximated by a monochromatic wave, they are expected to occupy these four quadrants equally. For few-cycle pulses, there may be some distortions and the symmetry pn∥ → −pn∥ may be broken, depending on the carrier-envelope phase. Our constraints, however, do not yield distributions occupying only the second and the forth quadrants of the parallel momentum plane. In the literature, due to the time delay between the rescattering of the first electron and the ionization of the second electron, it has been suggested that this would be the case. This has been backed, however, employing classical models, for which excitation is difficult to model, and for which tunneling does not exist. Apart from that, in classical computations, electronimpact ionization is also present, and may obscure RESI in the first and third quadrants of the parallel momentum plane. In our model, however, other physical mechanisms are absent from the start. Hence, RESI can be singled out and studied in more detail. Finally, it is worth mentioning that RESI electron-momentum distributions occupying the four quadrants have also been observed in classical-trajectory computations, for which the binding potential has not been neglected in the continuum (57, 69, 71). This is evidence that our results are not an artifact of the strong-field approximation1 . (5) In comparison to the constraints defined by electron-impact ionization, the lower momentum regions are much more populated. This is in agreement with experimental findings (48, 49), and other theoretical computations (56). These constraints agree with the solutions of the saddle-point equations for this mechanism. These solutions have been presented and discussed in detail in (72). Another interesting issue is that the momenta in the saddle-points (22) and (23) are disentangled. This facilitates the computations to a great extent. 1 One should note, however, that, due to the presence of the Coulomb potential the symmetries (p , t) → (−p , t ± T /2) and 2 2 (p1 , t′ , t′′ ) → (−p1 , t′ ± T /2, t′′ ± T /2) over half a cycle are no longer present, so that there are distortions in the patterns encountered with regard to those found employing the SFA.

36


In this context, one should note that the above-stated constraints are very different from those delimited by electron-impact ionization with time delay of at least a quarter of a cycle between the recollision of the first electron and the ionization of the second electron. This would imply that the first electron leaves near a field crossing and the second electron near a field maximum, without, however, explicitly considering the recollision-excitation with subsequent tunneling mechanism. This time delay would lead to the population of the second and fourth quadrants of the parallel-momentum plane. These constraints can be directly inferred from Eq. (9), if we consider

[p1|| + A(t)]2 + [p2|| + A(t + ∆t)]2 +

2 ∑

[ ]2 p2n⊥ = k + A(t′ ) − 2Eg2 ,

(27)

n=1

where ∆t ≥ T /4. This time delay would move the center of the hypersphere in momentum space. Specifically in the momentum the center of the electron√ momentum distributions would √ plane,√ move from (p1∥ , p2∥ ) ≃ (±2 Up , ±2 Up ) to (p1∥ , p2∥ ) ≃ (±2 Up , ∓ϵ), where ϵ ≪ 1. Hence, the distributions would be at least covering the axis p2∥ = 0. If ϵ > 0 this would imply that the second and fourth quadrants would be populated. Even though Eq. (27) is a rough estimate, in which the time delay ∆t is incorporated in an ad-hoc manner and which does not account for the presence of the binding potential in the continuum, it already shows that time delayed electron-impact ionization would lead to rather different constraints. 2.4.3.

Statistical thermalization model

In many physical situations, such as nonsequential triple and quadruple ionization, more than two active electrons must be considered. An ab initio modeling of such processes is not tractable, as only recently this has been achieved for a two-electron system in an intense field. Furthermore, the implementation of an S-matrix model is probably a hopeless task. In fact, even for the simplest case of two active electrons, the computation of NSDI transition amplitudes from each of the two Feynman diagrams discussed above, which only give the two main rescattering mechanisms of electron-impact ionization and RESI, is technically a non-trivial matter. Apart from the technical difficulties encountered, this approach is not practical as each process is specific to the Feynman diagram in question. Hence, it cannot be generalized to an arbitrary number of electrons. This renders the problem intractable. For this reason, instead, we turn to the classical model which constitutes the limit of the Smatrix approach for high enough driving field intensities. We generalize this model to an arbitrary number of electrons. We assume that, at a time t′ , an electron leaves its parent ion by tunneling ionization. The probability per unit time according to which this electron tunnels is taken to be the quasi-static rate R(t′ ). This electron propagates in the continuum under the influence of the external laser field only, gaining kinetic energy from the field, and , at a subsequent time t, recollides with its parent ion. Thereby, it shares part of its kinetic energy upon return with the remaining N −1 active electrons, which are bound. Within a thermalization time ∆t, this energy is redistributed among the bound electrons. At a time t+∆t, all active electrons leave and reach the detector with momenta p1 , . . . , pN . In this model, we do not make any assumption on how this energy is redistributed, i.e., all the dynamics between the recollision of the first electron and the ionization of the remaining electrons are neglected. In this sense, it is a statistical model like those used in many areas of physics such as molecular physics (130, 131) and high-energy collisions (132). The differential electron-momentum distribution is then given by the expression ∫ F (p1 , p2 , . . . , pN ) =

( ) (N ) (N ) dt′ R(t′ )δ E0 + Ekin − Eret (t) |Vp1 k |2 ,

(28)


37

(N )

where E0 > 0 denotes the total ionization potential of the N − 1 (up to the recollision time (N ) t inactive) electrons, Eret (t) is the energy of the first electron upon return and Ekin the final kinetic energy of the whole N -electron system. The δ function expresses the fact that the total kinetic energy of the N participating electrons at the time t + ∆t is fixed by the first-ionized electron at its recollision time t. The time delay ∆t is the only free parameter in our model, which will be adjusted according to the available experimental data. For the above-stated equation, one may consider two possible scenarios. If 1∑ [pn + A(t + ∆t)]2 =+ 2 N

(N ) Ekin

(29)

n=1

and |Vp1 k |2 = const., this implies that the N −1 bound electrons have been dislodged by the first in a hard collision. Hence, the first electron is trapped in the core and leaves together with the remaining electrons after the energy has been redistributed. The constant form factor is related to the fact that we are neglecting the dynamics of the problem during and after recollision, and only incorporating effects related to the momentum-space integration1 . It could also be, however, that the returning electron still continues on its path and transfers only some fraction of its kinetic energy to the remaining N − 1 electrons in a glancing collision. In this latter case, we assume that this energy is transferred by a long-range, Coulomb-type interaction and that the first electron does not get involved in the thermalization process taking place at the core. In this case, 1∑ [pn + A(t + ∆t)]2 , 2 N

(N )

Ekin = [p1 + A(t)]2 +

(30)

n=2

i.e., the first electron leaves instantaneously and the remaining electrons after a time interval ∆t, and Vp 1 k =

1 , (p1 − k)2

(31)

which is a signature of the Coulomb-type interaction if the core is assumed to be spatially localized (see discussion in Sec. 2.5.1). The Feynman diagrams corresponding to both scenarios are given in Fig. 20. One should note that the argument of the δ function in Eq. (28) gives a hypersphere in the (N ) 3N −dimensional momentum space, whose radius is determined by Eret (t) − E0 and whose center is at (p1 , . . . , pN ) = (−A(t + ∆t), . . . , −A(t + ∆t)) for the hard-collision scenario and (p1 , p2 . . . , pN ) = (−A(t), −A(t + ∆t), . . . , −A(t + ∆t)) for the glancing collision. The time lag between the recollision of the first electron and the ionization of the remaining electron moves the peaks of the electron-momentum distributions from the region in momentum space in which all momenta are equal and exhibit the same sign towards the region in which the momenta are unequal and, in the case of a delay of a quarter of a cycle, may even exhibit different signs. For comparison with the experiments (16, 19), however, one must consider the distribution as a function of the momentum component P∥ of the ion parallel to the laser-polarization direction. ∑ If the momentum of the absorbed laser photons can be neglected, P = − N n=1 pn . This yields

should note, that, even though |Vp1 k |2 = const. is formally what one would obtain by including a contact-type interaction, the idea behind the present model is different, namely that a hard collision took place, and the energy is being redistributed without making further assumptions on the system dynamics apart from the time delay ∆t 1 One


38

Figure 20. Feynman diagrams depicting the two thermalization processes in Ref. (99). In the diagram shown in panel (a), the first electron transfers part of its kinetic energy upon return to the core in a hard collision and leaves together with the remaining electrons after a thermalization time ∆t, while in panel (b) it transfers energy to its parent ion through a glancing collision, leaving immediately. This figure has been published previously in (99)

F (P∥ ) =

∫ ∏ N

(

∫ 3

d pi

2

d p1⊥ δ P∥ +

i=2

N ∑

) pi∥

F (p1 , p2 , . . . , pN ).

(32)

i=1

Specifically, for the hard-collision model, (2π)(3N −1)/2 F (P∥ ) = √ N Γ[(3N − 1)/2]

∫

( )3/2(N −1) dt′ R(t′ ) ∆E (h) , +

(33)

(N )

where ∆E (h) = Eret (t) − E0 − [P∥ − N A(t + ∆t)]2 /2N and the function x+ is defined as x+ = xΘ(x), where Θ is the unit-step function. For the glancing collision model, ) ∫ ∫ ( )3N/2−2 ( )2 −2 (2π)3N/2−1 1( ′ ′ (gl) (gl) F (P∥ ) = √ dt R(t ) dp1∥ ∆E ∆E + p −k 2 1∥ + 4 N − 1Γ[3N/2 − 1] ( ) 3N 3N ∆E (gl) ×2 F1 − 2; 2; − 1; (34) ( )2 , 2 2 ∆E (gl) + 1 p1∥ − k 2

where the energy difference ∆E (gl) in the glancing collision is ∆E

(gl)

= Eret (t) −

(N ) E0

[ ] ]2 (N − 1) ) 2 1[ 1 ( − p + A(t) − A(t + ∆t − P + p1∥ 2 1∥ 2 N −1 ∥

(35)


39

and 2 F1 (a; b; c; d) denotes the Hypergeometric function. In the original papers (99), we provide more details on such derivations, and also expressions in case the ion transverse momentum has just been partly integrated over. Finally, it is also worth mentioning that time lags between the rescattering of the first electron and the ionization of the second or remaining active electrons have been identified in many regimes of laser-induced nonsequential double or multiple ionization. For instance, first investigations of NSDI in the multiphoton regime (81) reported a time lag between the ionization of the first electron and the second. Furthermore, in recollision followed by excitation and subsequent tunnel ionization, there is also a time delay between the former and the latter. One has no guarantee, however, that the kinematic constraints determined by electron momentum distributions in such processes will lead to a hypersphere in momentum space whose center is displaced. Hence, the present model is to be understood as a first approximation for the actual processes. A concrete example is the RESI mechanism occurring in NSDI, which populates the four quadrants of the parallel momentum plane. There is however, some evidence for the validity of present model and the constraints presented in this section, in form of the measured electron momentum distributions in (16) for laser-induced nonsequential triple and fourfold ionization. 2.5.

Electron momentum distributions

We will now briefly discuss the outcome of our work on laser-induced nonsequential double and multiple ionization, which has been developed employing the models in the previous section. In the following results, in order to perform a direct comparison with existing experiments, the electron momentum distributions are either partially or fully integrated over the transverse electron momenta. Explicitly, these distributions read ∫∫ F (p1∥ , p2∥ ) = d2 p1⊥ d2 p2⊥ |MR (p1 , p2 ) (36) + ML (p1 , p2 ) + p1 ↔ p2 |2 , where MR (p1 , p2 ) and ML (p1 , p2 ) give the transition amplitudes related to the right or left peak, respectively. As indicated, unless stated otherwise, the distributions are symmetrized with respect to electron exchange. For monchromatic fields, we employ the symmetry A(t ± T /2) = −A(t), where T = 2π/ω to compute ML (p1 , p2 ) from the right-peak amplitude. For few-cycle pulses, this property does not hold and the amplitudes have to be computed independently. For more than two active electrons, we employ the expressions in Sec. 2.4.3. In Secs. 2.5.1, 2.5.2, 2.5.4 and 2.5.5, we explore different aspects of electron-impact ionization. For more details, we refer to the original papers. Specifically, in Sec. 2.5.1 we provide a summary of how this physical mechanism is influenced by the type of electron-electron interaction or finalstate electron-electron repulsion when integrating over different transverse momentum ranges (see (47, 88)). We also discuss the results (55) on the influence of different excited bound states, or of the residual ionic potential, in the electron-momentum distributions. Subsequently, in Sec. 2.5.2, we focus on our work on NSDI with few-cycle laser pulses (107, 108), with particular emphasis on how asymmetries in the electron-momentum distributions can be used for diagnosing the carrier-envelope phase. We also comment on the fact that this effect has been measured experimentally and could be traced back to a change in the dominant set of electron trajectories (128). Furthermore, in Sec. 2.5.4, we compare our calculated results of ion momentum distributions from nonsequential multiple ionization, in the context of statistical thermalization model, with the available experimental data. In the model, we consider both types of electron-electron interaction, i.e., contact and Coulomb interaction, through which the the first ionized electron interacts with and kicks off the other N − 1 bound electrons. Finally, in Sec. 2.5.5, we bring our results on diatomic molecules (30), in which the electron-impact ionization transition amplitude has been extended in order to account for two-center quantum interference effects.

40


Finally, in Sec. 2.5.6 we perform a rigorous analysis of the RESI mechanism, placing particular emphasis on how the shapes of the electron momentum distributions are influenced by the initial state in which the second electron was bound, the state to which it is excited and the type of interaction by which it is excited. We also show characteristic below- and above-threshold behaviors for this mechanism. Unless otherwise stated, we will employ a linearly polarized monochromatic field, for which A(t) = 2 2.5.1.

√

Up cos (ωt) eˆz .

(37)

Electron-electron dynamics: the type of interaction and the influence of the initial bound state

The type of interaction by which the second electron is dislodged, as well as the states in which it is initially bound, have a very strong influence on the shapes of the electron-momentum distributions. In our framework, this is clear, as different kinds of electron-electron interaction and/or bound-state wavefunctions will lead to different prefactors (5), which will introduce a different bias in the electron-momentum distributions. In the context of the strong-field approximation, this has been first reported in (105, 106), for neon and argon. If a contact-type interaction (δ,0)

V12

(r1 , r2 ) = δ(r1 − r2 )δ(r2 )

(38)

placed at the position of the ion was taken, circular-shaped distributions peaked at p1∥ = p2∥ = √ ± Up have been found. Physically, this suggested that the electrons left most likely with equal momenta and around the field crossing. In contrast, if a more realistic, long-range interaction (C)

V12 (r1 , r2 ) =

1 |r1 − r2 |

(39)

was considered in the model, the distributions were peaked at unequal momenta, and exhibited a broadening. In our framework, the interaction given by Eq. (38) leads to Vpn k = const. In contrast, a contact-type interaction (δ)

V12 (r1 , r2 ) = δ(r1 − r2 )

(40)

outside the origin of our coordinate system or a long-range, Coulomb-type interaction will lead to (1s)

Vpn ,k ∼ η(p1 , k)

1 [2|E02 | + p ˜ 2 ]2

+ p1 ↔ p2

(41)

for an initial 1s state. In the above-stated equations, the function η(p1 , k) depends on the ˜ interaction V12 and p ˜=p ˜ 1 (t) + p ˜ 2 (t) − k(t). For a contact-type interaction, η(p1 , k) = const., while for a Coulomb type interaction η(p1 , k) = 1/[p1 − k]2 . In the length gauge, p ˜ n (t) = ˜ ˜ pn + A(t) and k(t) = k + A(t), while in the velocity gauge p ˜ n (t) = pn and k(t) = k. In these early computations, the length gauge has been employed. We have also used the length gauge in our computations unless otherwise stated. Apart from the form of the electron-electron interaction, it may also be that both electrons repel each other when propagating in the continuum. In the SFA, this can be accounted for by replacing the product Volkov stated in the matrix element (5) by the exact correlated two-


41

electron Volkov wavefunction (V,C)

(V )

(V )

Ψp1 ,p2 = ψp1 (r1 )ψp2 (r2 )1 F1 (−iζ; 1; i(pr − p · r))C(ζ),

(42)

where r = r1 − r2 , p = (p1 − p2 )/2 are the relative electron coordinates, ζ = |p1 − p2 |−1 , 1 F1 (a; b; z) denotes the confluent hypergeometric function and the normalization factor reads C(ζ) = exp[−πζ/2]Γ(1 + iζ).

(43)

Hence, |C(ζ)|2 =

2πζ . exp[2πζ] − 1

(44)

An inspection of Eqs. (43) and (44) shows that the prefactor C(ζ) favors unequal momenta. This is physically expected due to the presence of final-state repulsion. This effect reflects itself as a broadening in the electron-momentum distributions, and is present both for the contact and Coulomb-type interactions. For details we refer to (88). Both the influence of the long range of the Coulomb interaction and the final-state repulsion can be seen in Fig. 21. In the absence of final-state repulsion, √ the contact type interaction (38) leads to circular-shaped distributions peaked at p1∥ = p2∥ = ±2 Up (Fig. 21.(a)). In this specific case, the prefactor Vpn k = const. Hence, there is no momentum bias due to the type of interaction and the electron-momentum distribution is mainly determined by phase-space effects. One should note that, for the driving-field intensity considered in the figure, the probability √ density may extend far beyond the momenta ±2 Up . This can be directly seen in the saddlepoint equation (17), which delimits the classical region for electron-impact ionization. In fact, this equation shows that this is the most probable, but not the maximal drift momentum the first electron may have upon return. If, on the other hand, the interaction V12 is taken to be long-range, there is a dramatic change in the shapes of the electron-momentum distributions. In this case, instead of circular-shaped distributions, one observes a V-shaped structure. This structure is shown int the right columns of Fig. 21. It is a signature of the long-range interaction, and has been observed relatively recently, both experimentally (50, 51), and by means of other methods, such as classical-trajectory computations (46), or the fully numerical solution of the time-dependent Schrödinger equation (44). In this context, it is worth mentioning that, in the early 2000s, the experiments did not have enough resolution to identify this structure. Hence, it was believed that the contact-type interaction led to a better agreement with the experimental data. As the final-state repulsion is incorporated, a broadening in the direction of the antidiagonal p1∥ +p2∥ = const. is observed if the contact-type interaction is taken (Fig. 21.(c)), and a suppression along the diagonal p1∥ = p2∥ occurs for the distributions computed with the Coulomb-type interaction (Fig. 21.(d)). This is a consequence of the fact that electron-electron repulsion favors unequal momenta. Finally, for this intensity regime, the outcome of the SFA computation and its classical limit yield very similar results (see Fig. 21.(e) and (f)). There are only minor differences in the boundary of the classical region, due to the fact that classical models underestimate the yield at this boundary. Significant discrepancies between both models only occur in the so-called threshold regime, when the maximal kinetic energy acquired by the electron from the field is just enough to enable it to overcome the second ionization potential (133). The influence of the final-state electron-electron repulsion and the type of interaction on the distributions can be seen even more clearly if the transverse momenta are relatively small. In Fig. 22, computations in which such momenta have only been partly integrated over are presented for the contact-type interaction. If at least one transverse momentum is taken to be small, there is a splitting in the maxima for the contact-type interaction. This suppression is shown in Fig. 22 (panels (d) and (f), second column from the left). For larger transverse momentum

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4 -4 -3 -2 -1 0 1 2 3 4

p2ÈÈ @UP D12

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4

4 3 2 1 0 -1 -2 -3 -4

p2ÈÈ @UP D12

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4

4 3 2 1 0 -1 -2 -3 -4

p2ÈÈ @UP D12


42

4 3 2 1 0 -1 -2 -3 -4 -4 -3 -2 -1 0 1 2 3 4

HaL

HcL

HeL

p1ÈÈ @UP D12

HbL

HdL

HfL

p1ÈÈ @UP D12

Figure 21. Fig. 4 in Ref. (88), depicting electron-impact ionization NSDI probability densities for neon (E01 = 0.9 a.u., E02 = 1.51 a.u.), in a driving field of intensity I = 1015 W/cm2 , integrated over all transverse momenta. The left and right panels have been computed employing the contact-type and the Coulomb-type interaction at the position of the ion, respectively. The upper four are computed from the quantum-mechanical amplitude, in the absence (panels (a) and (b)) or presence (panels (c) and (d)) of final-state electron-electron repulsion. Panels (e) and (f) have been calculated employing the classical limit of the electron-impact ionization transition probability without final-state repulsion.

ranges, the final-state repulsion leads only to a broadening, as shown in panel (b), second column from the left. These features may be interpreted in terms of the available classically allowed momentum-space regions according to the saddle-point equation (11). For large transverse momenta, this region is relatively small, so that the presence of final-state repulsion does not alter the distributions considerably. For a Coulomb-type interaction, final-state Coulomb repulsion also causes a suppression along p1∥ = p2∥ . This suppression is superimposed to the above-mentioned V-shaped structure, and is more extreme if at least one of the transverse momenta is small. The outcome of our SFA computations for this interaction in the absence and presence of final-state repulsion is shown in the second right and far right panel of Fig. 22, respectively. In particular, one observes that, in the presence of this repulsion, the peaks of the electron-momentum distributions extend far √ beyond 2 Up . Furthermore, the peaks in the distributions are not symmetric with respect to the diagonal p1∥ = p2∥ if one considers unequal ranges for the transverse momenta p1⊥ and p2⊥ when integrating over such variables. This is due to the fact that the Coulomb prefactor is only symmetric upon interchange of all momentum components p1 ↔ p1 , while for Vpn k = const. the transverse and parallel momenta can be interchanged independently. For more details on resolved transverse momenta we refer to our previous publications (47, 88). In the results discussed above, we have assumed that the active electrons were either bound in 1s states or, for a three-body contact interaction (38) in the bound states √ ψ (n) (rn ) ∼ exp[− 2|E0n |rn ]/rn ,

(45)

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4 -4 -3 -2 -1 0 1 2 3 4

HdL

4 3 2 1 0 -1 -2 -3 -4

p2ÈÈ @UP D12

p1ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4

HbL

43

4 3 2 1 0 -1 -2 -3 -4

p2ÈÈ @UP D12

HeL

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4 -4 -3 -2 -1 0 1 2 3 4

HcL

4 3 2 1 0 -1 -2 -3 -4

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4

HaL

p2ÈÈ @UP D12

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4 -4 -3 -2 -1 0 1 2 3 4

p2ÈÈ @UP D12

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4

4 3 2 1 0 -1 -2 -3 -4

p2ÈÈ @UP D12

p2ÈÈ @UP D12

4 3 2 1 0 -1 -2 -3 -4

p2ÈÈ @UP D12


4 3 2 1 0 -1 -2 -3 -4 -4 -3 -2 -1 0 1 2 3 4

HaL

HcL

HeL

HfL

p1ÈÈ @UP D12

p1ÈÈ @UP D12

HbL

HdL

HfL

p1ÈÈ @UP D12

Figure 22. Differential electron momentum distributions without and with electron-electron repulsion in the final state, as a function of the electron momenta parallel to the laser field, for a three-body contact interaction (38)(far left and second left columns) and a Coulomb-type interaction (far right and second right panels). The first and third column display the results in the absence of final-state repulsion, while in the second and fourth column this feature has been incorporated. We consider argon (E01 = 0.58 a.u., E02 = 1.015 a.u. in a laser field of frequency ω = 0.057 a.u. and intensity I = 2.5 × 1014 W/cm2 (panels (a)-(d)) and I = 4.7 × 1014 W/cm2 ( panels (e) and (f)). In panels (a) and (b), |p1⊥ | ≥ 0.5 a.u, and in panels (c) and (d), |p1⊥ | ≤ 0.5 a.u. In the remaining panels, |p1⊥ | or |p2⊥ | ≤ 0.1 a.u. These results have been first published as Figs. 2 and 3 in (88).

with n = 1, 2, of a zero-range potential. Furthermore, by placing the contact-type interaction at the position of the ion, we imposed that the initial bound state of the second electron is very localized. In real life, however, common species used in NSDI experiments are neon and argon. The mostly loosely bound electrons in the former and the latter species are released in the continuum from a 2p or a 3p state, respectively. Hence, a legitimate question is whether the shapes of the electronmomentum distributions are affected by the bound states from which the active electrons are released. In (55), we considered that the first and second electrons were bound in 1s, 2p, 3p and localized states, for different types of electron-electron interaction and in the absence of final-state repulsion. Some of these results will be discussed in this work. The form factors obtained for 2p and 3p states read (2p)

Vpn ,k ∼ η(p1 , k)

p˜ [2|E02 | + p ˜ 2 ]3

+ p1 ↔ p2

(46)

+ p1 ↔ p2 ,

(47)

and (3p)

Vpn ,k ∼ η(p1 , k)

p˜(˜ p2 − 2|E02 |) [2|E02 | + p ˜ 2 ]4

where p˜ and η(p1 , k) are defined as above. In the two left columns in Fig. 23, we present the results obtained with the contact-type interaction (40) and different initial electronic bound states. As an overall feature, circularshaped distributions are only obtained if we place this interaction at the position of the ion. Physically, this implies that the initial wavefunction of the second electron is localized at r2 = 0. These results are shown in Fig. 23 (d). For all other types of initial bound-states, the distributions


44

4

4

/[U ]

0

0

p

-2

1/2

2

0

0

p

2

p

p

-2

0

2

4

-4 4

0

2

4

2

0

0

-4

-2

0

p

-2

3

-4 4 -4 4

2

-2

0

2

4

(d)

(b) 2

2

0

0

2||

p

2||

p

2

4 1/2

(d)

(b) 1/2

-2

p

-4

1

-4

-4

/[U ]

4

s

-2

3

-4

/[U ]

(c)

2

2||

p 2||

-2

s

1

p

4

(a) /[U ]

2

1/2

4

(c)

(a)

-2

0

p

2 1/2

/[U ]

1||

p

4

-4

localized -4

-4

-4 -2

p

2

localized

-4 -4

-2

-2

-2

p

2

-2

0

p

2 1/2

/[U ]

1||

p

4

-4

-2

0

p

2 1/2

/[U ]

1||

p

4

-4

-2

p

0

1/2

2

4

/[U ]

1||

p

Figure 23. Electron-momentum distributions computed using a contact-type (far left and second left columns) and Coulomb interaction (far right and second right columns), as functions of the electron-momentum components parallel to the laserfield polarization. We considered neon (|E01 | = 0.79 a.u. and |E02 | = 1.51 a.u.) in a monochromatic linearly polarized field of frequency ω = 0.057 a.u. and intensity I = 8 × 1014 W/cm2 . In panels (a), (b) and (c), both electrons are initially bound in 1s, 2p and 3p states (equations (15), (13) and (14)), respectively, whereas in part (d) the first electron is initially in a 1s state and the wavefunction of the second electron is localized at r2 = 0. The transverse momenta have been integrated over. This figure has been published as Figs. 1 and 2 in (55)

exhibit a broadening along the anti-diagonal p1∥ + p2∥ = const and peaks far below p1∥ = p2∥ = √ ±2 Up . Interestingly, there is not much difference in the shapes of the distributions, regardless of whether the second electron was released from a 1s, 2p or 3p state (Fig. 23.(a), (b) and (c), respectively). This led us to conclude that different initial states do not influence electron-impact ionization considerably, as long as their spatial extension remains the same. In contrast, more recent results obtained for RESI show that, if excitation is involved, the shapes of the electronic bound states are important (see (31) and Sec. 2.5.6 for discussions). Furthermore, the finger-like structure is absent throughout. This is in very good agreement with recent results obtained using a classical-trajectory method (46), which observed similar distributions taking V12 as a short-range, Yukawa-type potential. On the other hand, if V12 is of Coulomb type, a finger-like structure is observed throughout. In our framework, this is related to the functional form of η(p1 , k). If the electronic √ bound states are spatially extended, this structure does not extend beyond the momenta 2 Up (see panels (a) to (c)). If, on the other hand, the initial state of the second electron is localized at r2 = 0, the V-shaped structure extends far beyond such momenta. In this case, the prefactor Vpn k is proportional to η(p1 , k) + (p1 ↔ p2 ). Physically, a localized wavefunction mimics the role of a residual binding potential, which is absent in our SFA treatment. The above-mentioned results lead us to conclude that the long-range nature of the Coulomb interaction√is responsible for the V-shaped structure, while the fact that this structure extends beyond 2 Up is due to the influence of the residual ionic potential. These statements are in full agreement with the results in (46), obtained using classical trajectories and the pertaining analysis (see also discussion in Sec. 2.2). 2.5.2.

Few cycle pulses

With the advance of femtosecond laser technology, laser pulses consisting of only a few laser cycles have become available in strong field physics [44]. One may describe a few-cycle (e.g., ncycle) laser pulse by the vector potential A(t) = A0 F (t) sin(ωt + ϕ)ex , with the positive-definite envelope F(t) defined as F (t) = exp[−4(ωt − nπ)2 /(nπ)2 ] for 0 ≤ t ≤ nT (T = 2π/ω). For such few-cycle pulses, the specific pulse shape depends strongly on the phase of the carrier wave (having frequency ω) with respect to the pulse envelope, the so-called carrier-envelope (CE)


2

4

4

(b)

2

(c)

2

(d)

-2

-2

-2

-2

-4 -4

-4 -4

-4 -4

2

4

1/2

2

4

(e)

2

-2

0

2

4

2

-2

0

2

4

4

4

(f)

-4 -4

(g)

2

(h)

0

0

-2

-2

0

2

4

1/2

-2

0

2

4

4

(i)

2

-4 -4

-2

0

2

4

4

(j)

2

-4 -4

-2

0

2

(k)

2

(l)

0

0

-2

-2

-4 -4

-2

0

p

2 1/2

/[U ]

1||

p

4

-4 -4

-2

0

p

2

/[U ]

1||

4

-4 -4

-2

0

2

1/2

p

p

1/2

/[U ]

1||

p

4

-4 -4

2||

0 -2

p

2||

0 -2

-2

0

p

2 1/2

/[U ]

1||

p

4

4

(a)

2

0 -2

2

-2

0

2

4

(e)

2

-2

0

2

4

(f)

-2

-4 -4 4 2

-2

0

2

4

(g)

-4 -4 4 2

0

0

0

0

-2

-2

-2

-2

2

-2

0

2

4

(i)

-4 -4 4 2

-2

0

2

4

(j)

-4 -4 4 2

-2

0

2

4

(k)

2

0

0

0

0

-2

-2

-2

-2

p

1||

0

2

/ [U ] p

1/2

4

-4 -4

-2

p

1||

0

2

/ [U ] p

1/2

4

-4 -4

-2

p

1||

0

2

/ [U ] p

4 1/2

-2

0

2

4

0

2

4

2

4

(h)

-4 -4 4

-2 -4 -4

(d)

0

-2

-4 -4 4 2

4

(c)

0

-2

-4 -4 4 2

4

(b)

0

-4 -4 4

4

4

p

p /[U ]

2

-4 -4

1/2

-2

4

p

-4 -4

2||

0 -2

p

0 -2

/ [U ]

2||

p

p /[U ]

0

4

4 2

-2

1/2

0

p

-2

p

0

/ [U ]

0

2||

0

2

/ [U ]

2||

0

4

1.25E-5 3.75E-5 6.25E-5 8.75E-5 2.5E-5 7.5E-5 5E-5 1E-4 0

1/2

4

(a)

p

p /[U ]

1/2

2

p

4

45

-2

(l)

-4 -4

-2

p

1||

0

/ [U ]

1/2

p

Figure 24. Electron momentum distributions computed for neon subject to a four-cycle pulse (n = 4) of frequency ω = 0.057 a.u., for various intensities and CE phases. The left-hand and the right-hand part of the figure correspond to the quantummechanical and to the classical computation, respectively. The upper, middle, and lower rows are for I = 4 × 1014 W/cm2 (Up = 0.879 a.u.), I = 5.5 × 1014 W/cm2 (Up = 1.2 a.u.), and I = 8 × 1014 W/cm2 (Up = 1.758 a.u.), respectively. The CE phases are for panels (a), (e) and (i): ϕ = 0.8 π; for panels (b), (f) and (j): ϕ = 0.9 π; for panels (c), (g) and (k): ϕ = 1.0 π; and for panels (d), (h) and (l): ϕ = 1.1 π. This figure has been published as Figs. 1 and 9 in (107).

phase ϕ. When subject to such a few-cycle laser pulse, the electron-momentum distribution of NSDI will lose the symmetry with respect to the antidiagonal [(p1∥ , p2∥ ) → −(p1∥ , p2∥ )], in contrast to what we have in the preceding section for a long pulse. In addition, its pattern will strongly depend on the value of the CE phase and the distribution can exhibit dramatic changes upon a small variation of the CE phase. All these features can be seen in Fig. 24, which exhibits NSDI electron-momentum distributions calculated for Ne subject to a 4-cycle pulse, obtained from both the quantum-mechanical S-matrix amplitude Eq. (36) (left-hand panels), and from its classical limit Eq. (11) (right-hand panels) (107, 108). Upon a critical value of the CE phase (e.g., ϕ ≈ 1.1π at 4 × 1014 W/cm2 , as shown in Fig. 24), the distribution shifts from the first into the third quadrant. For increasing intensity, this critical value of the CE phase moves to smaller values. Minor differences are only observed at the boundary of the classically allowed region, or around the critical CE phases, for which the momenta start to change sign. The dependence of the electron-momentum distribution on the CE phase can be explained by a change in the dominant set of orbits of the tunnel-ionized electron rescattering inelastically off its parent ion, in the context of the rescattering scenario. For an orbit to make an important contribution, two conditions must be satisfied: first, the probability that the electron tunnel out at a time t′ must not be too small and, second, the subsequent acceleration must be strong enough for the first electron return to its parent ion (at the time t) with sufficient kinetic energy Eret (t). These two conditions reduce the significance of start times within the trailing part of the pulse (t′ ≥ nT /2). Both in the quantum-mechanical and in the classical calculations, only the first return of the electron to the ion has been considered. Due to wave-function spreading, the contributions of the longer orbits are suppressed. The most remarkable result of these investigations is the surprisingly high sensitivity of the (p1∥ , p2∥ )-momentum distribution to variations of the CE phase. This provides a highly potential for a precise determination and control of the CE phase. It is noteworthy that the features discovered in our model have been observed by a later experiment on NSDI of argon irradiated by few-cycle laser pulses (128). 2.5.3.

Polarization-gated pulses

Another example for which the shape of the driving field influences the electron momentum distributions considerably is NSDI with polarization-gated pulses. In this case, an adequate superposition of circular or elliptically polarized pulses leads to a time-dependent ellipticity, and may allow even for a higher degree of control. NSDI by elliptically polarized pulses has been recently studied in (95) Below we briefly recall the results from our previous publication (134), in which we extend

46


our classical model to the elliptically polarized case. We consider the superposition of two time delayed few-cycle laser pulses with counter-rotating circular polarizations. Without less of generality, the dynamics can be restricted to the xy plane. Explicitly, the electric fields El (t) and Er (t) of the left- and right-circularly polarized pulses read El (t) = E0 e−2ln(2)((t−Td /2)/τp )/2 [ˆ x cos(ω(t − Td /2) + ϕ) + yˆ sin(ω(t − Td /2) + ϕ)]

(48)

Er (t) = E0 e−2ln(2)((t+Td /2)/τp )/2 [ˆ x cos(ω(t + Td /2) + ϕ) − yˆ sin(ω(t + Td /2) + ϕ)]

(49)

and

respectively. In the above-stated equations, E0 is the peak-field amplitude, ω is the carrier frequency, τp is the pulse duration, Td is the time delay between the two circularly polarized pulses and ϕ is the CE phase. The unit vectors in the x and y directions are denoted by x ˆ and yˆ. The resulting pulse will then change abruptly with the time delay. For long delays, there will be a very short time interval for which the polarization of the resulting pulse is approximately linear, and for which rescattering of the first electron may occur. Outside this interval, the polarization of the resulting pulse is nearly circular and rescattering is negligible. Hence, the yield decreases. As this time delay decreases, the gate becomes less efficient and there will be a large time region for which the polarization is almost linear and rescattering may occur. We compute electron-momentum distributions employing our classical model, which has been generalized in order to account for elliptical polarization. The main difference, with respect to its linearly polarized counterpart, is the initial distribution of electrons. Apart from the quasi-static tunneling rate Wt (t′ ) ∼ |Esum (t′ )| exp[

−2(2|E01 |)3/2 ], 3|Esum (t′ )|

(50)

where Esum = Ex + Ey is the vector sum of the components of the resulting pulse, we now include a gaussian transverse velocity distribution Wl (vl ) =

1 vl exp[−( )2 ], 2 (πδvl ) δvl

where the initial width is given by δvl = (Esum / trajectory in the ensemble then reads

√

(51)

2|E01 )1/2 . The resulting weight for each

W (t′ , vl ) = Wt (t′ ) × Wl (vl ) × (t − t′ )−3 ,

(52)

where the purely time dependent term mimics the quantum mechanic wave-packet spreading. In Fig. 25, we depict electron-momentum distributions obtained with this model. For long delays (see, e.g., the most left column in Fig. 25), the electron momentum distributions resemble very much those obtained from a linearly polarized, single few-cycle pulse, i.e., they are strongly localized in either the positive or negative momentum region and change considerably with the CE phase. This is due to the fact that a single set of electron trajectories is close to the center of the gate and dominates the yield. As the delay decreases, the gate becomes less and less efficient and other sets of trajectories contribute. This renders the distributions more and more symmetric, until they resemble those of a monochromatic, linearly polarized wave(see, e.g., the most right column in Fig. 25).


47

Figure 25. NSDI electron momentum distributions computed for neon in a polarization-gated pulse described by Eqs. (48) and ( 49). The peak intensity of the two circularly polarized pulses is 2 × 1014 W/cm2 , their length is four cycles (n = 4) and their frequency is ω = 0.057 a.u. The distributions are plotted as functions of the momentum components pn∥ (n = 1, 2) parallel to the nearly linearly polarized part of the combined pulse. The CE phase is varied from the top to the bottom of the figure, and the delay ωTd between the two pulses from its left to its right. In the first, second, third and fourth rows from the top [panels (a) to (d), panels (e) to (h), panels (i) to (l), and (m) to (p), respectively], the CE phases ϕ are 0.5π, 0.8π, 1.0π, and 1.2π respectively. In the first [panels (a), (e), (i) and (m)], second [panels (b), (f), (j) and (n)], third [panels (b), (f), (k) and (o)], and fourth [panels (c), (g), (l) and (p)] columns from the left, we depict distributions for the delay phases ωTd = 8π, 6π, 4π, and 2π. This figure has been published as Fig. 1 in (134)

2.5.4.

Multiply charged ion momentum distributions

In this section, we will present the results of our statistical thermalization model (see Sec. 2.4.3) on ion momentum distributions from nonsequential multiple ionization. In the model we consider either that energy is transferred to the N-1 bound electrons by a hard or glancing collision. In the former case, the first electron is trapped, and all electrons leave after a time delay ∆t, while in the latter case, the first electron leaves immediately, transferring energy to the bound N − 1 electrons by a soft collision. By comparing the calculated ion momentum distribution, in which an appropriate delay time is employed, with the available experimental data, we deduce an upper limit of the electron thermalization time in laser-induced nonsequential multiple ionization process. In Fig. 26, we display the calculated longitudinal ion-momentum distributions for nonsequential quadruple and triple ionization of neon, under different recollision scenarios considered in Sec. 2.4.3. For both scenarios, the time delay ∆t between the recollision of the first electron and the time when all N electrons or N-1 electron leave the ion influences both the width and the peaks of the momentum distributions. For the case of small time delay (e.g., ∆t ≤ 0.2T ), the peak positions of the two humps will move towards zero momentum with the increase of ∆t. This is expected since, with increasing the time delay, the N (or N-1) electrons become free at a later time and the corresponding vector potential of the laser field is small. Hence, the electrons’ drift momenta will be smaller and in consequence, the ion’s momentum becomes small. Furthermore,


48

t = 0

Hard-collision

Counts

t = 0.1T

t = 0

Hard-collision

t = 0.1T

t = 0.2T

t = 0.2T

t = 0.3T

t = 0.3T

Glancing-collision

Counts

Glancing-collision

-10 -8 -6 -4 -2

0

2

P / (U ) ||

p

4 1/2

6

8 10

-10 -8 -6 -4 -2

0

2

P / (U ) ||

4

6

8 10

1/2

p

Figure 26. Calculated distribution of the longitudinal ion momentum for nonsequential quadruple ionization of neon at 2.0 PW/cm2 (left panel) and for nonsequential triple ionization of neon at 1.5 PW/cm2 (right panel), in the context of the statistical thermalization model. The curves in the upper panels are calculated from Eq. (33) for the hard-collision scenario, while the curves in the lower panels are from Eq. ( 34) for the glancing-collision scenario. The various delay times employed in the calculations are indicated in the upper panels. This figure is taken from Fig. 2 in (99).

the vector potential at the release time t + ∆t will spread over a larger range with the increase of ∆t, and in consequence, the widths of the humps increase. With ∆t increasing further (see, e.g., ∆t = 0.3T), the two humps will merge into one peak. In Fig. 27, we compare the results of the statistical thermalization model, under two different recollision scenarios, with the experimental data for nonsequential quadruple and triple ionization of neon (16). The curves in the upper panels are calculated from Eq. (33) for the hard-collision scenario, while in the lower panels the curves are calculated from Eq. (34) for the glancingcollision scenario. The red and the green curves correspond to the time delay of ∆t = 0 and 0.17T, respectively. The black curves are taken from the data of figure 2 of (16). For the hardcollision scenario, it is found that the best agreement with the data, for both quadruple and triple ionization, is achieved when one value of the time delay ∆t of 0.17T is chosen. In contrast, for the glancing-collision scenario, there is no optimal value of the time delay that yields equally good agreement with the data for both quadruple and triple ionization. For example, the choice of a time delay of ∆t = 0.17T yields better agreement with the data for quadruple ionization (left lower panel in Fig. (27)), while for triple ionization, a better fit is obtained with ∆t = 0. Therefore, we conclude that the thermalization model in which the first electron interacts with the core in a hard collision, is trapped and takes active part in the thermalization process gives the best description of nonsequential multiple ionization of neon, considering the fact that, with this scenario, a good agreement with the data can be obtained for both triple and quadruple ionization with one value of time delay. Inspired by the fact that the thermalization model with the hard-collision scenario describes well the multiple ionization of neon, we further apply this model to the case of argon. In Fig. 28, the model calculations, performed for quadruple ionization of argon at two laser intensities, are compared with the experimental data (16). A good agreement is found if a larger time delay ∆t = 0.265T is chosen in the model. The comparison suggests that more time is needed for the redistribution of the excess energy between the recolliding electron and the bound electrons in argon than in neon. It should be noted that, within our statistical model, all of the dynamics of the physical system between the time of recollision and the time when the electrons become free are hidden in the time delay ∆t, which is the sum of the thermalization time and a possible additional ’dwell’ time. The same value of the ∆t may reflect different dynamical mechanisms: the returning electron


Exp. Data

Hard-collision

49

Exp. Data

Hard-collision

t = 0.0T

t = 0.0T t = 0.17T

Counts (arb. units)

Counts (arb. units)

t = 0.17T

Glancing-collision

-10 -8 -6 -4 -2 P

||

0

2

/ (U ) p

4 1/2

6

8

10

Glancing-collision

-10 -8 -6 -4 -2 0 P

||

2

/ (U )

4

6

8 10

1/2

p

Figure 27. Comparison of the ion-momentum distributions of the statistical thermalization model calculations with the experimental data for nonsequential quadruple (left panel) and triple (right panel) ionization of neon at laser intensities of 2.0 and 1.5 PW/cm2 , respectively. The curves in the upper panels are calculated from Eq. (32) for the hard-collision scenario, while in the lower panels the glancing-collision scenario is employed according to Eq. (33). For the red and green curves, the thermalization time △t are taken as 0 and 0.17 T, respectively. The black curves are from the experimental data of figure 2 of (16). This figure is taken from Figs. 3 and 4 in (99).

may form an excited complex with N −1 bound electrons (135) or, from another perspective, the decay of the complex may be affected by the laser field (136). Both of these effects are, however, entangled and indistinguishable within our thermalization model. For example, in the case of argon, the optimal time delay of ∆t = 0.265T is very close to a quarter of the field period. Upon this delay, the escape of the electron complex is easiest. Therefore, one may also suspect that the thermalization process in argon is fast but the electrons do not leave before the field is near its maximum. Nevertheless, the compatibility of our simple model with the data available so far, for both neon and argon, allows one to infer a tighter limit on the thermalization time. All that is necessary for that purpose is to restrict the electron recollision time range ∆trec . This can be done by, for example, using an additional perpendicular polarized driving field, or by simply decreasing the laser intensity, to limit the temporal range of return times for the first electron. 2.5.5.

Molecules

In this section, we will present the results of our model for NSDI of molecules. As molecules possess extra degrees of freedom, the electron correlation in strong field molecular ionization becomes more complicated. On the one hand, due to the multiple center nature, structural interference from the different ionization pathways (i.e., different ionic core) may show up in the electron-momentum distribution. On the other hand, molecular ionization dynamics depends on their ground state orbital symmetry and their relative alignment with respect to the excitation laser polarization axis. Indeed, experimental studies have shown a qualitatively different correlated electron momentum pattern for O2 as compared to that of N2 (28). This difference has been ascribed to the different symmetry of the ground states of molecules, i.e., antibonding for O2 and bonding symmetry for N2 . Another study has shown that the electron correlation pattern of N2 depends strongly on the alignment angle between the molecule and the laser polarization axis (29). Here we will extend our S-matrix treatment and its classical analog to molecular systems and investigate how the molecular orbital symmetry and the molecular alignment with respect to the laser polarization influence the electron-momentum distributions. For simplicity, we will

50


Exp. Data

1.2PW

t = 0

Counts (arb. units)

t = 0.265T

Exp. Data

1.5PW

t = 0 t = 0.265T

-10

-8

-6

-4

-2

P

||

0

2

/ (U )

4

6

8

10

1/2

p

Figure 28. Comparison of the ion-momentum distributions of the statistical thermalization model calculations with the experimental data for nonsequential quadruple ionization of argon at 1.2 (upper panel) and 1.5 PW/cm2 (lower panel). The red and green curves are calculated from Eq. (32) for the contact scenario, with △t = 0 and 0.265T, respectively. The black curves are from the experimental data of figure 3 of (16). This figure is taken from Fig. 6 in (99).

consider the specific case of diatomic homonuclear molecules and assume frozen nuclei, i.e., the linear combination of atomic orbitals (LCAO) approximation. For details, we refer to (30). Explicitly, the molecular bound-state wave function for each electron reads [ ] (n) (n) (n) ψ0 (rn ) = Cψ ϕ0 (rn − R/2) + ϵϕ0 (rn + R/2)

(53)

√ where n = 1, 2, ϵ = ±1, and Cψ = 1/ 2(1 + ϵS(R), with ∫ [ S(R) =

(n)

ϕ0 (rn − R/2)

]∗

(n)

ϕ0 (rn + R/2)d3 r.

(54)

The positive and negative signs for ϵ correspond to a symmetric and an antisymmetric combination of 1s orbitals, respectively. The binding potential of this molecule, as seen by each electron, is given by V (rn ) = V0 (rn − R/2) + V0 (rn + R/2),

(55)

where V0 corresponds to the binding potential of each center in the molecule. The above-stated assumptions lead to (s)

2Cψ ˜ ′ ) · R/2]I(k(t ˜ ′ )) cos[k(t (2π)3/2

(56)

(a)

2iCψ ˜ ′ ) · R/2]I(k(t ˜ ′ )), sin[k(t (2π)3/2

(57)

Vk0 = − or Vk0 = −


51

for the symmetric and antisymmetric cases, respectively, with ˜ ′ )) = I(k(t

∫

˜ ′ ) · r1 ]V0 (r1 )ϕ(1) (r1 ). d3 r1 exp[ik(t 0

(58)

Thereby, we have neglected the integrals for which the binding potential V0 (r) and the bound(1) state wave function ϕ0 (r) are localized at different centers in the molecule. Assuming that the electron-electron interaction depends only on the difference between the coordinates of both electrons, i.e., V12 = V12 (r1 − r2 ), one may write the prefactor Vpn k as (s)

2Cψ (2) V12 (p1 − k) cos[P(t) · R/2]φ0 (P(t)) (2π)9/2

(59)

(a)

2iCψ (2) V12 (p1 − k) sin[P(t) · R/2]φ0 (P(t)), 9/2 (2π)

(60)

Vpn k = or Vpn k =

˜ with P(t) = p ˜ 1 (t)+˜ p2 (t)− k(t), for symmetric and antisymmetric orbitals, respectively. Thereby, (2) φ0 (P(t))

∫

(2)

=

d3 r2 exp[iP(t) · r2 ]ϕ0 (r2 ),

(61)

d3 rV12 (r) exp[i(p1 − k) · r],

(62)

and ∫ V12 (p1 − k) =

with r = r1 − r2 . Specifically, in the velocity and length gauges, the argument in Eqs. (59), (60) is given by P(t) = p1 + p2 − k and P(t) = p1 + p2 − k + A(t), respectively. The structure of the highest occupied molecular orbital is embedded in Eqs. (56)-(60). The simplest way to proceed is to consider these prefactors and the single-center action (3). The multiple integral in (2) will be solved using saddle-point methods. Similar to the atomic case, the transition amplitude is computed by means of a uniform saddle-point approximation. A more rigorous approach would be to incorporate the prefactors (59) or (60) in the action. This would lead to modified saddle-point equations, in which the structure of the molecule, in particular scattering processes involving one or two centers, are taken into account. For simplicity, we will restrict our investigation to single-atom saddle-point equations, together with the two-center prefactors (59) or (60). Before showing our calculated results within the S-matrix formalism and the interference patterns therein, we will first present a detailed analysis of the interference conditions. In terms of the momentum components pi∥ , or pi⊥ (i = 1, 2), parallel or perpendicular to the laser-field polarization, this condition may be written as cos [ζR/2] or sin [ζR/2] , in terms of the argument ζ. Explicitly, this argument is given by ζ = ζ∥ + ζ⊥ ,

(63)

with [ ζ∥ =

2 ∑ i=1

] pi∥ − k(t) cos θ

(64)

52


and ζ⊥ = p1⊥ sin θ cos φ + p2⊥ sin θ cos(φ + α).

(65)

In the above-stated equations, θ gives the alignment angle of the molecule, φ corresponds to the angle between the perpendicular momentum p1⊥ and the polarization plane, and α yields the angle between both perpendicular momentum components. Since we are dealing with nonresolved transverse momenta, we integrate over the latter two angles. In the velocity and in the length gauge, k(t) = k and k(t) = k − A(t), respectively. Interference extrema will then be given by the condition (ζ⊥ + ζ∥ )R = nπ.

(66)

For a symmetric linear combination of atomic orbitals, even and odd n correspond to interference maxima and minima, respectively, whereas, in the antisymmetric case, this condition is reversed. An inspection of Eqs. (64) and (65), together with the above-stated condition, provides an intuitive picture of how the interference patterns change with the alignment angle θ. For parallel alignment, the only contributions to such patterns will be due to ζ∥ . In this particular case, the interference condition may be written as p1∥ + p2∥ =

nπ + k(t), R cos θ

(67)

where cos θ = 1. Eq. (67) implies the existence of well-defined interference maxima or minima, which, to a first approximation, are parallel to the anti-diagonal p1∥ = −p2∥ . This is only an approximate picture, as k, according to the saddle-point equation (21), is dependent on the start time t′ and on the return time t. Furthermore, since t′ and t depend on the transverse momenta of the electrons, Eq. (67) is also influenced by such momenta. Finally, in the length gauge, there is an additional time dependence via the vector potential A(t) at the instant of rescattering. As the alignment angle increases, the contributions from the term ζ⊥ related to the transverse momenta start to play an increasingly important role in determining the interference conditions. The main effect such contributions have is to weaken the fringes defined by Eq. (67), until, for perpendicular alignment, the fringes completely vanish and the electron-momentum distributions resemble those obtained for a single atom. This can be readily seen if we consider the interference condition for θ = π/2, which is p1⊥ cos φ + p2⊥ cos(φ + α) =

nπ . R

(68)

Eq. (68) gives interference conditions which do not depend on k(t), and which vary with the angles φ and α. As one integrates over the latter parameters, which is the procedure adopted for distributions with non-resolved transverse momentum, any structure which may exist in Eq. (68) is washed out. In Fig. 29, we display electron momentum distributions computed in the velocity gauge for a highest occupied molecular orbital modeled by the symmetric combination of two 1s orbitals and various alignment angles, employing the S-matrix model (left-hand panels) and its classical limit (right-hand panels). In order to avoid addtional momentum bias, we consider V12 as a three-body contact-type interaction. The figure shows that, for small alignment angles (see, e.g., Fig. 29(a) and (b)), the patterns caused by the two-center interference survive the integration over the transverse momentum components. Furthermore, the fringes exhibit a very good qualitative agreement with the interference conditions derived above. There is also a very good agreement between the outcome of the classical and the quantum-mechanical models.


4

(a)

4

q= 0 2

0

0

-2

-2

q = 30

4

0

(c)

-2

0

2

(d)

0

2

0

0

-2

-2

-2

0

2

0

0

-2

-2

(c) 1/2

-4 -4

-2

0

p1||/[Up]

2 1/2

-4 4 -4

-2

0

p1||/[Up]

2 1/2

4

-2

0

θ = 60

2 0

-4 4 -4 4

θ = 30

0

2

0

0

-2

-2

-4 -4

-2

0

p1|| / [Up]

-2

0

2

4

2

4

(d) θ = 900

2

p2|| / [Up]

p2||/[Up]

-4 -4 4

4

0

θ = 90

1/2

2

(b)

1/2

-4 4 -4 4

θ = 60

θ=0

2

p2|| / [Up]

p2||/[Up]

-4 -4 4

4

(a)

(a)

2

1/2

2

(b)

53

2 1/2

-4 4 -4

-2

0

p1|| / [Up]

1/2

Figure 29. Figs. 2 and 6 in (30) depicting electron momentum distributions as functions of the parallel momenta (p1∥ , p2∥ ), for several alignment angles of model molecule. The left-hand and the right-hand panels of the figure correspond to the quantum-mechanical and to the classical computation, respectively. We consider the velocity gauge, symmetric orbitals. The field intensity and frequency have been taken as I = 5 × 1014 W/cm2 , and ω = 0.057 a.u., respectively, and the ionization potentials E01 = 0.573 a.u. and E02 = 0.997 a.u. correspond to N2 at the equilibrium internuclear distance R = 2.068 a.u. The position of the interference minima, estimated by assuming that the first electron returns at a field crossing, are indicated by the lines in the figure. Panel (a), (b), (c) and (d) correspond to alignment angles θ = 0, θ = 300 , θ = 600 and θ = 900 , respectively.

For parallel alignment, sharp interference fringes at p1∥ +p2∥ = const, according to Eq. (67), can be clearly seen in Fig. 29(a). For small alignment angles, such as that in Fig. 29(b), the maxima and minima start to move towards larger parallel momenta. Furthermore, there exists an increase in the momentum difference between consecutive maxima or minima, and the interference fringes become less defined. This is due to the fact that the term ζ⊥ , which washes out the interference patterns, is getting increasingly prominent. For large alignment angles, such as that in Fig. 29(c), the contributions from this term are very prominent and have practically washed out the twocenter interference. Finally, for perpendicular alignment, the distributions resemble very √ much those obtained for the single-atom case, i.e., circular distributions peaked at p1∥ = p2∥ = ±2 Up . This is expected, since the term responsible for the two-center interference fringes is vanishing for θ = 900 . Interference fringes parallel to p1∥ + p2∥ = const are also present in the length gauge, and for antisymmetric orbitals. This is shown in the upper panels of Fig. 30, for parallel alignment angle. In fact, the main difference as compared to the symmetric, velocity-gauge case, is the position of such patterns, in agreement with Eq. (67). There is also some blurring in the patterns, in the length gauge, possibly caused by the fact that the vector potential A(t) depends on the return time t. This latter quantity is different for different transverse momenta. The patterns, however, can be also clearly identified in this gauge. In all cases, however, there is no evidence of a straightforward connection between an enhancement or suppression of the yield in the lowmomentum region and the symmetry of the orbital. For instance, in the velocity gauge, the yield is enhanced if the orbital is antisymmetric. The length-gauge distributions, on the other hand, exhibit a suppression in that region regardless of the orbital symmetry. In the lower panels of Fig. 30, we display the distributions along p1∥ = p2∥ = p∥ . Similarly to the velocity-gauge, symmetric case, the minima and maxima of the distributions roughly agree with the analysis from Eq. (67), which is indicated by the lines in the figure. However, it is noteworthy that, for the √ length-gauge distributions [Figs. 30(d) and (e)], there is an overall displacement of roughly 2 Up in the position of the patterns. This is consistent with the modified interference conditions in the length gauge, considering the fact that k(t) = k − A(t) in this case.


54

4

4

4

(b)

(a)

(c)

2

2

0

0

0

p2||/[Up]

1/2

2

-2

-2

-2

length gauge symmetric

-4 -4

-2

0

1/2 2

p1||/[Up]

4

-2

0

1/2

p1||/[Up]

2

4

-4 -4

(d)

-2

0.08

0.6

0.06

0.4

0.04

0.4

0.2

0.02

0.2

-2

0

2 1/2

p||/[Up]

4

0.00 -4

0

1/2 2

4

2

4

p1||/[Up]

(f)

1.0

(e)

0.8

0.0 -4

velocity gauge antisymmetric

0.10

1.0

Yield (arb. units)

-4 -4

length gauge antisymmetric

0.8 0.6

-2

0

2 1/2

p||/[Up]

4

0.0 -4

-2

0

1/2

p||/[Up]

Figure 30. Fig. 5 in (30) depicting electron momentum distributions for a parallel-aligned molecule (θ = 0), different orbital symmetries and gauges. The upper and lower panels give the contour plots as functions of the parallel momenta, and the distributions along p1∥ = p2∥ = p∥ , respectively. We integrate over the transverse momenta, and employ the same molecule and field parameters as in Fig. 29. The interference minima according to Eq. (67) are indicated by the vertical lines in the figure. Panel (a) and (d), (b) and (e), and (c) and (f) correspond to symmetric orbitals in the length gauge, antisymmetric orbitals in the length gauge and antisymmetric orbitals in the velocity gauge, respectively. For panels (d), (e) and (f), the units in the vertical axis have been chosen so that their upper values are unity (the original values have been divided by 0.016, 0.01 and 0.04, respectively).

2.5.6.

Excitation

In this section, we will present the outcome of our model for the RESI mechanism. An assessment of how the distributions populate the momentum-space regions, with increasing drivingfield intensity, can be made if all the prefactors in Eq. (13) are taken to be constant. Momentumdependent form factors will introduce an additional bias. Fig. 31 depicts such distributions, for an intensities varying from just enough for the first electron to excite the second upon return, i.e, such that 3.17Up ≃ (E2g − E2e ) (panel (a)), to just above the electron-impact ionization threshold, i.e., for which 3.17Up ≃ E2g (panel (c)). The intermediate intensity (panel (b)) allows RESI to occur, but not electron-impact ionization. The behavior displayed in the figure is in very good agreement with the constraints defined in Sec. 2.4.2, and with we observe that, ( Fig. 19.) In fact, ( √ ) for the lowest intensity, the distributions exhibit peaks at pn∥ , pm∥ = ±2 Up , 0 , n ̸= m, and are very localized around such points. This is due√to the fact that the classically allowed region for the first electron consists of the points ±2 Up only. Their widths are determined √ √ by the direct-ATI constraints and, as predicted, extend from −2 Up to 2 Up . As the intensity increases, so does the momentum region for which the first electron is allowed to rescatter, classically. As a direct consequence, the distributions become more and more elongated. Their widths, however, do not change as the tunneling process for the second electron has no classical counterpart. In particular Fig. 31(c) shows a very similar shape, compared to the estimated constraints in Fig. 19. If, on the other hand, we consider that the second electron is excited to a specific state, such state will leave an imprint in the electron momentum distributions. We assume that the second electron, initially in 1s, is either excited to a 2s or 2p state. In the latter case we consider the coherent superposition ⟩ 1 ( (2) ⟩ (2) ⟩ (2) ⟩) (2) ψ2p = √ ψ2px + ψ2py + ψ2pz , 3

(69)


4

(b)

(a)

55

(c)

p2||/[Up]

1/2

2 0 -2 -4 -4

-2

0

2

p1||/[Up]

1/2

4

-4

-2

0

p1||/[Up]

2

-4

4

1/2

-2

0

p1||/[Up]

2

4

1/2

Figure 31. (Color online) Fig. 2 in (31) depicting electron momentum distributions for Helium (E1g = 0.97 a.u., E2g = 2 a.u. and E2e = 0.5 a.u.) in a linearly polarized, monochromatic field of frequency ω = 0.057 a.u.. Panels (a), (b) and (c) correspond to a driving-field intensity I = 2.2 × 1014 W/cm2 , I = 2.5 × 1014 W/cm2 and I = 3.0 × 1014 W/cm2 , respectively.

which includes the three degenerate p states. Explicitly, for the 1s → 2s excitation, these prefactors read (2s) Vp2 e

(1s→2s)

Vp1 e,kg where κ =

√

[ ]2 p˜2 (t)∥ + p22⊥ − 2E2e ∼ [ ]2 [ p˜2 (t)∥ + p22⊥ + 2E2e ]2

(70)

[( ] )2 η1 k − p1∥ + p21⊥ , E2g , E2e ∼ V12 (p1 − k) ( , )2 [ k − p1∥ + p21⊥ + ζ 2 (E2g , E2e )]3

(71)

(p1 − k)2 , √ √ η1 (κ2 , E2g , E2e ) = κ2 ( 2E2g + 2 2E2e ) + (2E2g )3/2 √ −2(2E2e )3/2 − 6E2e 2E2g .

(72)

and ζ(E2g , E2e ) =

√

2E2e +

√

2E2g .

(73)

For the 1s → 2p excitation, we find. (2p)

Vp 2 e

√[ ]2 p˜2 (t)∥ + p22⊥ ∼( p2 (t)) )2 β(˜ [ ]2 2 2E2e + p˜2 (t)∥ + p2⊥

(74)

and (1s→2p)

Vp1 e,kg

∼ V12 (p1 − k)η2 (κ2 , E2g , E2e )β(κ),

(75)

where 2

η2 (κ , E2g , E2e ) =

√ ζ(E2g , E2e ) κ2 (ζ 2 (E2g , E2e ) + κ2 )3

.

(76)

and the angular dependence is given by β(q) = (sin θq cos φq + sin θq sin φq + cos θq ).

(77)


56 4

(a)

4 2

0

0

p2||/[Up]

1/2

2

(b)

-2

-2

Contact

-4 -4 4

-2

0

2

4

(c)

Contact

-4 -4 4 2

0

0

0

2

4

(d)

p2||/[Up]

1/2

2

-2

-2

-2

Coulomb

-4 -4

-2

0

p1||/[Up]

2 1/2

4

Coulomb

-4 -4

-2

0

p1||/[Up]

2

4

1/2

Figure 32. (Color online) Fig. 3 in (31) showing the velocity-gauge electron momentum distributions for Helium in a linearly polarized, monochromatic field of frequency ω = 0.057 a.u. and intensity I = 2.16 × 1014 W/cm2 . In panels (a) and (c), the first electron has been excited to 2s, while in panels (b), and (d) it has been excited to 2p. The interaction employed is indicated in the figure.

This dependence gets washed out as the transverse momenta are integrated over. We also assume that the electron is excited by either a contact or a Coulomb-type interaction. For the (δ) (C) former, V12 (p1 − k) = const., while for the latter V12 (p1 − k) ∼ 1/(p1 − k)2 . As previously discussed, different types of interaction influence the electron-momentum distributions considerably for electron-impact ionization. Hence, they are expected to affect the shapes of the electron momentum distributions for RESI as well. In Fig. 32, we show electron-momentum distributions computed for the lowest intensity in Fig. 31, under the above-stated assumptions. The shapes of these distributions depend very strongly on the excited state of the second electron. If this state is√ 2s, the distributions are very localized along the axis pn∥ = 0 and drop very steeply around ± Up . This is due to the fact that the prefactor (70) exhibits a node near these momenta. This prefactor also decays very steeply as the transverse momenta increase, and exhibits maxima for pn∥ = 0. If, on the other hand, the electron gets excited from 1s to 2p, the distributions are much broader and exhibit minima at the (2p) axis pn∥ = 0. This behavior can also be traced back to the prefactor Vp2 e related to the tunnel ionization of the second electron. This prefactor, apart from decaying much more slowly with the transverse momenta pn⊥ , exhibit nodes at vanishing parallel momenta. For all panels in the figure, the Coulomb-type interaction introduces a bias towards lower momenta. Specifically for the 1s to 2p case (panel (d)), we observe the absence of the nodes at p1∥ = p2∥ and p1∥ = −p2∥ , which are present if a contact-type interaction is taken (panel (c)). These are not real nodes, (1s→2p) but stem from the fact that the Vp1 e,kg decays faster for the contact-type case. Similar patterns can be observed if the highest intensity in Fig. 31 is taken. The main difference is that the regions close to vanishing momenta are much more populated, and that the type of interaction V12 plays a more critical role. Both features are a consequence of the fact that now there is an extensive momentum region for which rescattering of the first electron is classically allowed. This means that momentum regions near p1∥ = p2∥ = 0 are within the constraints defined in Sec. 2.4.2 and that, physically, it will also matter whether the second electron was excited by a Coulomb or contact-type interaction. Apart from that, we have computed RESI electron momentum distributions for argon. This species has been employed in recent experiments in which the driving-field intensity was around or below the electron-impact ionization threshold (22), and have been displayed in Fig. 15. Studies for NSDI below the threshold have also been reported in (136). Our SFA results for the parameters in this latter reference (see Fig. 2 therein) are displayed in Fig. 34. We have assumed


4

4

(a)

2

0

0

-2

-2

p2||/[Up]

1/2

2

Contact

-4 -4

-2

0

2

(b)

-4

4

Contact -4

-2

X Axis Title

4 1/2

0

2

4

X Axis Title

4

(c)

2

p2||/[Up]

57

(d)

2

0

0

-2

-2

-4

Coulomb -4

-2

0

p1||/[Up]

2 1/2

4

-4

Coulomb -4

-2

0

p1||/[Up]

2

1/2

4

Figure 33. (Color online) Fig. 4 in (31) for velocity-gauge electron momentum distributions for the same parameters as in the previous figure, but driving-field intensity I = 3 × 1014 W/cm2 . In panels (a) and (c), the first electron has been excited to 2s, while in panels (b), and (d) it has been excited to 2p. The interaction employed is indicated in the figure.

that the second electron, initially in a 3p state, was excited either to the 4s or the 4p state, from which it subsequently tunnels. In all cases, we found that the RESI probability densities are symmetric with respect to the origin p1∥ = p2∥ = 0 of the parallel momentum plane and equally distributed in its four quadrants. This specific feature resembles very much that obtained in (136), and also in the more recent work (22). One should note, however, that the distributions obtained for a 3p → 4s excitation are much more localized along the axes pn∥ = 0 than those observed in the experiments. The distributions obtained by us considering that the electron was excited to the 4p state, on the other hand, are considerably broader and exhibit shallow nodes along such axes. For all cases, there exist secondary sets of maxima along p1∥ = p2∥ and p1∥ = −p2∥ . The behaviors observed in the figure can be traced back to specific characteristics of the wave function of the state to which the second electron is excited, and from which it subsequently tunnels (for details see our recent publication (31)). The discrepancies between our results and the experimental findings may be due to several issues. Firstly, our SFA model does not account for the Stark shifts introduced by the field in the atomic bound-state energies. This is expected to be particularly important near the field maximum, which is close to the ionization time of the second electron. Furthermore, we have no guarantee that excitation is happening to a specific state, either 4s or 4p, and the experimental findings may be related to a coherent superposition of both. Finally, the presence of the Coulomb potential in the continuum means that the field-dressed momenta of both electrons are no longer conserved (137–139), and that, due to being accelerated or decelerated by the binding potential, these momenta may change sign. In fact,it has been recently shown, in the context of direct above-threshold ionization, that the orbits found in the SFA are not topologically identical to those obtained employing coulomb-corrected S-Matrix models. In fact, the presence of the Coulomb potential in the continuum leads to other types of orbits, whose interference may lead to striking quantum-interference effects (140). Since, however, we are integrating over transverse degrees of freedom, we do not expect interference to be important (for more details see Appendix B in (31)). Nevertheless, the presence of the Coulomb potential in the electron propagation is expected to cause distortions in the cross-shaped or ring-shaped structures shown in this work. This is due to the fact that the action related to the RESI mechanism discussed in this work will no longer be symmetric upon half a cycle. In fact, within the SFA, for a long enough pulse


58

(a)

4

0

0

-2

-2

Contact

-4 -4

4

-2

0

2

4

(c)

2

1/2

-2

0

2

0

0 -2

Coulomb

-4

0

0

-2

-2

4

Contact -4

(d)

4

2

-2

-2

0

-2

0

2

p1||/[Up]

1/2

4

2

-4

4

(c)

Contact -4

4

2

2

0

0

-2

-2

-2

0

-2

0

2

4

(d)

Coulomb

-4

-4 -4

(b)

2

-4 -4

4

4

(a)

2

Contact

-4

1/2

p2||/[Up]

1/2

4

(b)

2

p2||/[Up]

2

p2||/[Up]

p2||/[Up]

1/2

4

-4

-2

0

p1||/[Up]

2 1/2

Coulomb

4

-4

-2

0

2

p1||/[Up]

1/2

4

-4

Coulomb -4

p1||/[Up]

2

4

1/2

Figure 34. (Color online) Figs. 7 and 8 in (31) depicting velocity-gauge electron momentum distributions for argon in a linearly polarized, monochromatic field of frequency ω = 0.057 a.u. The far left and second left columns correspond to the process in which the second electron is excited from 3p to 4s, while in the far right and second right columns, excitation from 3p to 4p occurs. The laser-field intensity in panels (a) and (c), and panels (b) and (d) is I = 9 × 1013 W/cm2 and I = 1.5 × 1014 W/cm2 , respectively. The type of interaction V12 taken is indicated in the figure. The contour plots have been normalized with respect to the highest yield in each panels. The highest yields on panels (a) and (c) are between one and a half and two orders of magnitude smaller than those on panels (b) and (d).

one can assume the field is approximately monochromatic so that A(t) = ±A(t ± T /2), where T = 2π/ω is the driving-field period. This implies that the action related to the left and right peaks in the distributions, in which all times are displaced by half a cycle, will be symmetric upon pn∥ → −pn∥ . This will no longer hold if the Coulomb potential is accounted for. Physically, these distortions can be traced back to the fact that, if the Coulomb potential is present, the electron will no longer tunnel most probably with vanishing momentum at peak field. This issue is, however, still under investigation. The features presented in this section, together with the analysis in Sec. 2.4, lead us to conclude that a time lag between the rescattering of the first electron and the ionization of the second electron does not imply that only the second and the fourth quadrants of the parallel momentum plane will be populated. In fact, a rigorous analysis of the RESI mechanism resulted in electron-momentum distributions equally populating the four quadrants of the momentum plane. Qualitatively similar results have been obtained employing a classical model, by restricting the rescattering time of the first electron around the field crossing and the ionization of the second electron around the maximum of the field (57, 69, 71). Since the model in (57, 69, 71) includes the full classical dynamics of the two-electron problem, and thus considers the residual binding potentials in the continuum, this is clearly not an artifact of our SFA model. An interesting feature, for instance in (57, 69) is that, as the intensity of the driving field increases, the regions around the origin p1∥ = p2∥ = 0 of the parallel momentum plane starts to fill in. In other words, one moves from a ring-shaped to a cross-shaped distribution, in agreement with the constraints in Sec. 2.4.2 and the electron-momentum distributions in this section. In such models, however, these crosses or rings are slightly distorted, i.e., the above-mentioned symmetry is broken. This is very likely due to the presence of the residual Coulomb potential, as discussed above. Possibly, the fact that in the outcome of many computations other rescattering mechanisms such as electron-impact ionization are present masks the RESI contributions in the first and third quadrant of the parallel momentum plane. This may suggest that the RESI mechanism only populates the second and the fourth quadrants of the p1∥ p2∥ plane. Another possibility is that, instead of looking at RESI, most classical models mainly address electron-impact ionization with a time delay. This would lead to different momentum constraints similar to those obtained


59

in our thermalization model. If a time delay of a quarter of a cycle is taken, these constraints populate only the second and fourth quadrants of the parallel momentum plane (29, 58–60). Apart from that, the above-stated results show that RESI in nonsequential double ionization has a strong potential for ultra-fast imaging. In fact, the shapes of the electron momentum distributions are considerably more influenced by the state in which the second electron is bound and in particular the state to which it is excited than electron-impact ionization. If this is already the case for atoms, this dependence could even be more extreme for molecules. Indeed, this double ionization mechanism is becoming increasingly important for such systems (29) and also for driving-field intensities below the threshold.

3.

Complex systems in intense fields

In this section, we address the electron-electron correlation in the interaction of more complex systems, such as molecules, clusters and solids, with intense laser field. It has been shown in the preceding sections that, for atomic systems, the electron-electron correlation shows up clearly in nonsequential double or multiple ionization. Due to the large number of electronic and nuclear degrees of freedom, the strong field dynamics of complex systems becomes more complicated. While SAE models can account for HHG and single/sequential ionization of atoms to a great extent, they are no longer valid for complex systems and multielectron effects have to be considered. The study of the interaction of complex systems with intense laser pulses has been promoted essentially by two considerations. First, it is well known that the production of coherent XUV radiation with unprecedented attosecond pulse duration, from HHG of the gas targets, has suffered from a severe problem, namely its extremely low conversion efficiency. This has inhibited its widespread applications. In contrast, complex systems, such as clustered gases or solids, seem to be a suitable nonlinear medium that may overcome this shortage and provide an attosecond pulse with considerably improved pulse energy. Secondly, it has been demonstrated that the structural information of the atoms and small molecules can be encoded by recording the photoelectron or harmonics spectra in the interaction of the system with intense laser pulses. However, it is not clear at all if this ultrafast structure imaging can be extended into the larger complex systems, considering the fact that the collective effects become pronounced and may deteriorate the imaging in larger systems.

3.1.

Molecules

Similar to the well studied atomic systems, the first evidence of electron-electron correlation in the interaction between small molecules and intense laser field was found in total ionization yield measurement, from which the experimental data with linear polarized laser pulses exhibited a much higher double ionization yield than the prediction of the SAE model by many orders of magnitude (11, 12, 120, 141). The comparison between the data with linear and circular polarization gave a hint that the electron recollision mechanism, as it is the case for NSDI of atomic systems, may also be responsible for the NSDI in molecules. Soon after these ion yield measurements, the recolliding electron has been found to play a key role in the production of the charged ionic fragments emitted from the double ionization of the most simplest molecule, i.e., molecular hydrogen (142, 143). Clearer evidence can be found in the differential ion momentum and electron-electron momentum correlation distribution (28), with the help of COLTRIMS technique. It has been shown that, for NSDI of molecular N2 , a very similar correlated electron momentum pattern to that of atomic systems, i.e., a momentum distribution peaked at nonvanishing parallel momenta, was present, giving a strong support that electron recollision is the essential mechanism behind NSDI of N2 . Compared to atomic systems, molecules possess extra nuclear degrees of freedom and therefore, some novel aspects specific to molecular strong field dynamics, in the context of the electron


60

recollision scenario, may be envisioned. Indeed, as was shown in (28), a qualitatively different correlated electron momentum pattern was observed for O2 in comparison with that of N2 . For O2 , the emitted electrons from NSDI were accumulated around the area close to the origin. The different features between N2 and O2 were ascribed to the different symmetry of the molecular ground states of molecules, i.e., antibonding for O2 and bonding symmetry for N2 . Further study has unveiled that the electron correlation pattern also depends strongly on the alignment angle between molecular axis and the laser polarization axis (29). On the other side, due to the multiple center nature of molecular systems, the ionized electron may be emitted from different atomic center and, upon acceleration in the laser field, revisit more than one center. The different pathways may result in structural interference that shows up in the electron-electron correlation distribution (30). While it is not surprising that, similar to atomic systems, electron-electron correlation can be clearly demonstrated in nonsequential double ionization of small molecules, an increasing number of evidence has been accumulated that electron correlation may also occur in HHG in molecular systems. It is well known that the single active electron approximation is sufficient to account for the essential physics and that, to first approximation, multielectron effects can be safely ignored for HHG in atomic systems. Within the strong-field community, consensus has been achieved that the valence electron from the outmost atomic orbital mainly determines the essential strong field atomic dynamics 1 . However this consensus seems to be invalid for molecular systems and sub-outmost orbitals, in addition to the highest occupied molecular orbital (HOMO), has been demonstrated to play a role for strong-field molecular dynamics )(see, e.g., (33, 35, 36) and references therein). The contribution from the sub-outmost orbitals shows up as a consequence of the specific molecular structure: there may exist lower-lying electronic states with binding energy close to HOMO and furthermore, the preferential field ionization of more deeply bound states becomes possible due to the different geometries of the corresponding molecular orbitals. Recently, multielectron effects in HHG of molecules have attracted much attention, as HHG serves as a useful tool for ultrafast imaging of molecular structure and dynamics (35, 39). Multielectron effects may affect ultrafast imaging in two ways: on one hand, they may pose a serious obstacle towards ultrafast molecular imaging. On the other hand, they have the potential to unveil the underlying attosecond multielectron dynamics whose fingerprint is encoded in the amplitude, phases and polarization of the emitted harmonic light. For example, recent studies (36, 144) have shown that the sub-outmost molecular orbital (the so-called HOMO-1), contributes to the harmonic emission in N2 molecules. It was found that the HHG cutoff from the HOMO - 1 contribution extends beyond that of the HOMO. Further studies on CO2 molecules has identified that at least three orbitals, and three states of the ion contribute to the harmonic emission (35). In this work, the authors have found the fingerprint of multiple molecular orbitals participating in the HHG process and suggested that, by measuring the phases and amplitudes of the harmonic emission, it is possible not only to follow the nuclear vibrational motion, but also to decode the underlying attosecond multielectron dynamics. 3.2.

Clusters and solids

In this section, we address the intense field dynamics of the most complicated systems, i.e., clusters and solids. Compared to atoms and small molecules, collective processes are expected to dominate for both systems due to the fact that a large number of delocalized electrons are present. Donnelly et al (145) have for the first time observed the enhanced efficiency of high-harmonics from rare-gas clusters irradiated by infrared pulses. Later experiments (146, 147) have con1 Note,

however, that multielectron effects do introduce quantitative changes in the high-order harmonic spectra of multielectron atoms; for a discussion of this issue see, e.g., (161)


61

firmed this HHG efficiency enhancement and ascribed it to collective oscillation of electrons. Shim et al (148) have further found √ a resonant enhancement of the third harmonic yield when ωM ie → 3ωLaser , where ωM ie = Q/R3 , with Q the total ionic charge and R the cluster radius. This behavior has been studied theoretically by means of particle-in-cell simulations (149) and a pronounced resonant enhancement of the low-order harmonic yields is found when the Mie plasma frequency of the ionizing and expanding clusters resonates with the harmonic frequency. However it was also shown that higher order harmonics are suppressed by the fact that the coherent electron motion is inhibited by the strong, nonlinear resonant coupling of the cluster electrons with the laser field. In addition to the studies on the noble-gas clusters, Ganeev et al (150) have demonstrated that HHG in fullerenes also exhibited an extended cutoff and enhancement of harmonic intensity when comparing with those from monomer carbon particles. It is interesting to note that, for clusters, rescattering may also be responsible for their strong field dynamics. The multielectron response and the collective effects, however, become much more pronounced in complex systems and is unavoidably involved in the strong field processes. One example has been found recently in the study in multiple ionization of C60 cluster (151), in which the fragments were found to be induced by the electron recollision through the ellipticity dependence measurements. The observation of recollision in such larger systems is important as it may be a promise to using the ionized electron to image the structure of the larger systems, similar to what has been shown for small molecular systems (32). However, for larger complex systems such as C60 , multielectron response becomes important and affects the recolliding electron dynamics. It was shown in (40) that the recolliding electron can excite collective modes which may contribute predominantly to the harmonic generation yield. Therefore an efficient tomographic imaging of complex systems, which involves retrieving the structural information of targets encoded in the high harmonic spectrum or photoelectron spectra, will be contaminated by multielectron effects. Compared to the clusters, the interaction of intense laser field with solids becomes even more complicated due to their very complex structure. The interaction dynamics is different on the surface as compared to the bulk of a solid. For the latter, the propagation effect of the intense laser pulses plays an important role. For the former, ionization leads to the formation of a plasma that is not bound and free to expand as in clusters and the physics of the interaction is to some extent similar to that in clusters. In the context of short laser pulses, e.g., less than picosecond, in principle two competing microscopic processes are responsible for the plasma formation on solid surfaces: strong field ionization, which removes the electrons from the valence band to the conduction band, and impact ionization, in which the free electron in the conduction band imparts part of its energy to those in the valence band, resulting in the ionization of the latter (152). The interplay between strong field ionization and impact ionization depends on the pulse duration. With the increase in the laser pulse duration, the contribution of impact ionization becomes significant. It becomes comparable to strong field ionization at about 200 fs (153). On the other hand, for very short pulses where direct impact ionization is negligible, a different mechanism (154), in which the ionization process is materialized by using the energy from both the photon and the electron in the conduction band whose energy alone is not enough to overcome the bandgap, starts to play a major role in the production of the free electrons. This process is very similar to RESI in atomic case. It should be noted that, while numerous studies have been performed to understand the physical mechanism behind intense laser field-solid interaction, electron collision has not been recognized for solids.

4.

Summary and perspectives

Nonsequential double and multiple ionization of atoms, reviewed in the first part of this paper, demonstrated the important role of electron recollision in understanding the electron-electron

62


correlation in intense laser fields. However, this understanding is only valid for a very limited dynamical range, i.e., in the high-intensity, long wavelength regime. On one hand, if one enters the very low intensity regime, it is still an open question to which extent the electron recollision scenario can still apply. In this case, the role of atomic excitation has to be included and the detailed structure of the target will become an important issue, which has seldom been discussed before and poses a challenge for further theoretical studies. On the other hand, a great majority of processes investigated so far falls in long-wavelength limit, for which the electron motion is considerably fast with respect to the oscillation of the laser field such that a quasi-static treatment can be justified. Recent advances in free-electron lasers (155, 156) as well as in high-order harmonic sources (157), both of which can deliver short VUV and/or XUV laser pulses with unprecedented intensities, opens up the possibility of investigating electron-electron correlation in strong field multiple ionization in the short wavelength regime. Nonsequential mechanisms for the simultaneous removal of valence electrons have been found recently both experimentally and theoretically (26, 27, 158, 159). With the further increase of the laser photon energy, one is confronted with a totally different situation: the ionization may begin from inner shell electrons, with concomitant Auger transitions. It can be envisioned that, in this case, the electron from the inner shells of the target, and therefore the specific target structure, will play an increasingly important role in strong field processes. For example, a very recent experimental study on photoionization of different rare gases, performed at the wavelength of 13.7 nm, i.e., 90.5 eV photon energy, has revealed that the degree of the nonlinear photoionization of xenon is significantly higher than for other noble gas species (160). This peculiar feature observed for xenon gives an indication that the collective giant 4d resonance of xenon plays a key role in this short wavelength regime. A comprehension of this process involves the interplay of inner-shell electron structure, electron-electron correlation and the resonance and thus poses a great challenge for theoretical investigations. Even if, in the past few years, considerable progress has been made towards the understanding of how complex systems, such as molecules and clusters, interact with intense laser fields, an accurate treatment of such targets, and in particular the understanding of electron-electron correlation in this context is still an open question to a very large extent. This poses an unprecedent challenge to theorists, as, traditionally, theoretical strong-field laser physics places far more emphasis on the external laser field than on an accurate treatment of the targets. In fact, in many approaches, such as the strong-field approximation, the latter is reduced to a source term. For extended systems, however, this is no longer a reasonable assumption. In fact, this discussion moves beyond the current boundaries of intense-field laser physics and moves into the realms of quantum chemistry, molecular physics and condensed-matter theory. At the present moment, one may identify two particular challenges in this context. The first of them is an appropriate treatment of the residual binding potentials in these complex systems and how they affect strong-field phenomena. For small systems, such as atoms or diatomic molecules, these potentials may be neglected to a first approximation, and the continuum may be approximated by field-dressed plane waves. This is the underlying assumption behind most semianalytical strong-field approaches, such as the strong-field approximation. For large molecules, clusters or solids, however, the distinction between bound states and continuum states is less clear. Hence, it will be necessary to incorporate the binding potentials in the electron propagation. Apart from that, an appropriate treatment of electron-electron correlation in complex systems is not yet available. If even for relatively small molecules multielectron effects may be important, and mask structural interference, for more complex targets they are expected to play a major role. As discussed above, already the interaction of a two-electron system with an intense field has posed a great challenge to theorists in the past two decades. This leads us to believe that many-body systems in strong laser fields constitutes one of the most rich and challenging topics within this area in the years to come.

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