† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 11274219), the Science and Technology Planning Project of Guangdong Province of China (Grant No. 180917124960522), and the Program for Promotion of Science at Universities in Guangdong Province of China (Grant No. 2018KTSCX062).
We review the recently improved quantitative rescattering theory for nonsequential double ionization, in which the lowering of threshold due to the presence of electric field at the time of recollision has been taken into account. First, we present the basic theoretical tools which are used in the numerical simulations, especially the quantum theories for elastic scattering of electron as well as the processes of electron impact excitation and electron impact ionization. Then, after a brief discussion about the properties of the returning electron wave packet, we provide the numerical procedures for the simulations of the total double ionization yield, the double-to-single ionization ratio, and the correlated two-electron momentum distribution.
Nonsequential double ionization (NSDI) of atoms in strong laser fields has attracted considerable interest both experimentally and theoretically for more than three decades, and still remains one of the most interesting and challenging topics in strong-field physics. The interest on NSDI was first stimulated by the early measurements of the number of doubly charged ions of rare gas atoms exposed to intense laser pulses versus the peak laser intensity.[1–5] The characteristic “knee” structure observed in experiment provides the distinct evidence for NSDI since double-ionization yields could be several orders of magnitude greater than the prediction by the standard theory based on the assumption that electrons are emitted one by one, with the second ionization independent of the previous one. However, with the only available experimental data for the total yield of doubly charged ion, the mechanisms responsible for the two-electron transition were still a matter of controversy. For example, it was suggested that the second electron could be shaken-off by a nonadiabatic change of the potential caused by the emission of the first electron.[3] Alternatively, it was argued[6] that the first electron, released by tunneling through the potential barrier of the combined atomic and field potential, is driven back to its parent ion by the laser field and ionizes the second electron, in a process analogous to an (e,2e) collision. The debate on the mechanisms responsible for NSDI was further settled by the differential ionization measurements, which were first carried out at the turn of this century, for momentum distributions of the doubly charged ion along the laser polarization direction[7] and the momentum correlations between the two outgoing electrons.[8] Those differential ionization measurements provide more detailed insight into the dynamics of the NSDI process and provide a much improved testing ground for the different theoretical models in the dispute. Nowadays, it has already been widely accepted that the main mechanisms leading to NSDI are the laser-induced recollisional (e,2e) and the recollisional excitation with subsequent ionization of the second electron from the excited state of the parent ion.[9]
Over the past three decades, there have been no shortage of theoretical efforts devoted to NSDI (for a review, see Refs. [10] and [11]). It has been well recognized that five interactions are involved in NSDI: the interactions of the laser field with the two electrons, the Coulomb interactions between the doubly charged ion and the two outgoing electrons, and the electron–electron interaction. All the five interactions are strong and roughly equal in importance, so ideally they must be treated on an equal basis. However, in actual numerical calculations, one has to resort to simplifying approximations to make the calculations tractable. In general, the numerical methods developed for NSDI fall into two categories. The first approach takes the set of fundamental interactions of the electrons and solves the time dependent Schrödinger equation (TDSE) or the time dependent Newton equations (TDNE) for the motion of the two electrons. Indeed, a complete description of NSDI could be obtained by an exact solution of the TDSE. However, it is still a formidable computational challenge because of computer speed and memory requirements, and the results alone would not offer much insight into the basic mechanisms for the double ionization processes. So far, only a very few TDSE calculations, with more or less approximations, have been carried out for NSDI.[12–15] On the contrary, it is easier to solve the TDNE for NSDI,[16] and a number of predictions with at least semiquantitative agreement with a relatively wide variety of experimental data have been achieved.[17,18] The second approach is based on the rescattering model from which conceptual clarity derives more easily, even though this might come at the expense of possible loss of physical reality. Within this framework, the S-matrix theory was first developed by extending the strong field approximation (SFA) for high order above threshold ionization (HATI)[19,20] to double ionization.[21] While the TDSE results contain all possible outcomes, which are mixed together, from a given initial state, the S-matrix theory allows one to select any particular item, and hence is more powerful in identifying the specific mechanisms of the NSDI process. Similarly, a recently developed full quantum theory for NSDI, called quantitative rescattering (QRS) model, is also based on the rescattering model. While laser-induced rescattering processes can be qualitatively interpreted by the classical rescattering model,[6,22] the QRS model provides the quantitative description. In addition, the semiclassical theory including the tunneling effect, indeed, has also been widely recognized as a useful approach to simulate the NSDI.[23–27]
The QRS model was first proposed for HATI,[28,29] in which the momentum distribution of the HATI photoelectron is factorized as a product of the returning electron wave packet (RWP) and the differential cross sections (DCS) for elastic scattering of the returning electron with the parent ion. The only difference between HATI and NSDI is that the returning electron experiences different types of collisions, one is elastic scattering and the other is inelastic scattering including electron impact ionization and electron impact excitation. Based on this philosophy, the QRS model was extended to NSDI.[30–32] The past decade has witnessed great success of the QRS model in dealing with various laser-induced rescattering processes.[33,34] Recently, the QRS model for NSDI has been further improved[35–39] by taking into account the lowering of threshold due to the presence of electric field at the instant of recollision.[40]
In this review, we will limit ourselves to the presentation of the improved QRS model for NSDI and its applications to some of the major experimental observations. This article is organized as follows. In Section
Unless indicated otherwise, atomic units (a.u.) are used throughout this paper.
Ionization of atoms is the first step of the laser-induced rescattering processes such as HATI,[41] high-order harmonic generation (HHG),[42] and NSDI.[5,8]
In general, two mechanisms have been classified for ionization of atoms exposed to an intense laser field by using the Keldysh parameter[43]
The most commonly used theory for ionization of atoms by a linearly polarized laser field is the Ammosov–Delone–Kainov (ADK) approach,[44] which is generally valid for tunneling ionization when the adiabaticity parameter γ is lower than 1.
Suppose an atom is exposed to a linearly polarized laser pulse (along the z axis) with carrier frequency ω and the carrier envelope phase (CEP) φ, the field can be expressed as
The ADK model has also been generalized to molecular systems.[45] However, Tong and Lin[46] found that the ADK rates significantly overestimate the numerically calculated static ionization rates for atoms in the barrier-suppression regime, hence they modified the original ADK model to extend the ADK rates from the tunneling ionization region to the near- and over-the-barrier regimes by introducing an empirical correction factor. The modified ADK rate w[|F(t)|] is given by[46]
The ADK model represents the limit when γ → 0 of the theory proposed by Perelomov, Popov, and Terent’ev (PPT),[47] which is valid for arbitrary γ values. In the PPT model, the ionization rate reads
With the calculated ionization rate, one can obtain the total ionization probability by a laser pulse as
The strong field approximation is a very effective tool to deal with the interaction of atoms with strong laser field. While the SFA can also be used to obtained the total ionization probability of an atom by a laser pulse, it further provides a fairly good quantitative description of the photoenergy spectra as well as the momentum distributions. In general, the SFA model is valid when Ip ≤ 2Up. This condition implies that when the electron appears in the continuum it is only under the influence of a very strong laser field. In addition, when the electron comes back to the parent ion it has a large kinetic energy, so that the Coulomb force of the atomic potential can be neglected.
In the length gauge and the dipole approximation, the laser–electron interaction can be expressed by
Introducing the strong field approximation, i.e., replacing 〈Ψ
The form of fSFA(
It should be noted that, in the numerical calculations, one needs to use a short-range potential
The angular and momentum distribution of an electron emitted in the direction of
For a linearly polarized laser field, the system has cylindrical symmetry. As a result, the two-dimensional momentum distribution is defined as
It should be noted that the numerical results depend on the screening factor α. It has been found that the photoelectron energy spectra of SFA2 at low energies increase very rapidly as α decreases. Nevertheless, in the high energy regime, when E > 4Up, the dependence becomes relatively weak, and the shape of the energy spectra almost remains the same. Usually, in the numerical calculations, the value of α is chosen in such a way that direct ionization (SFA1) dominates the photoelectron spectra for the energies bellow 2Up.[52,53]
While the SFA allows for intuitive physical interpretation of the intense laser–atom interaction, the numerical solutions of the time-dependent Schrödinger equation, in principle, provide a full quantum description of various phenomena, including the ATI spectra, for an atom in a strong laser field. Within the single-active-electron approximation, the TDSE for single ionization of an atom in the presence of a linearly polarized laser field can be written, in the length gauge, as
The probability amplitude is obtained by projecting the total final wave function at the end of the laser pulse onto eigenstates of a continuum electron with momentum
Indeed, by solving the TDSE one can obtain the most accurate numerical results provided that the calculations are practical. Unfortunately, the TDSE calculations require a large amount of computing time and are impractical for long pulses at high intensities, so that results can only be obtained for a restricted regime of laser parameters. In addition, the insight into physics is hidden even though the numerical solutions of TDSE can be obtained.
The QRS model was first proposed for HATI[28] in which laser-induced elastic scattering of electron with the parent ion plays a decisive role.[29,59,60] In this section, we briefly summarize the standard potential scattering theory which has been well documented in the text books; see Refs. [61] and [62], for example.
The scattered wavefunction of an electron by a potential V(
For a short-range potential which falls faster than r−2 as r → ∞, the wavefunction of the scattered electron in the asymptotic region is given by
To get the scattering amplitude, we solve Eq. (
For a plane wave when V(r) = 0, the radial component ul(k,r)/kr in Eq. (
When r → ∞, the boundary condition satisfied by ul(k,r) for V(r) = 0 is
By matching the coefficients of the outgoing spherical waves in Eqs. (
For the scattering by a pure Coulomb potential
However, for a real situation frequently encountered in electron scattering in which a short-range potential V(r) is added to a pure Coulomb potential Vc(r) due to decreasing screening of the nucleus of the target ion by its electrons, one still needs to resort to the partial wave decomposition. Similarly, we expand the wavefunction as
The boundary condition satisfied by
Laser-induced electron impact excitation is one of the main mechanisms in NSDI. To simulate the correlated two-electron momentum distributions based on the QRS model, one needs to calculate the DCSs for electron impact excitation of parent ions.
The distorted-wave Born approximation (DWBA) is one of most efficient methods dealing with inelastic collision processes. In this section, we briefly review the DWBA method for electron impact excitation of singly charged ion.
Suppose we have an electron with momentum
Both the exact initial state wavefunction Ψi(
The wavefunctions for the electron in both initial ground state Φi(
The distorting potentials Ui and Uf used in Eq. (
In DWBA, the direct transition amplitude for excitation from an initial state Φi to a final state Φf is given by
To evaluate the scattering amplitude, we perform the standard partial wave expansions. The details of numerical calculations for the scattering amplitude have been presented in Ref. [63]. Finally, the spin-averaged differential cross section for electron impact excitation from (Ni,Li) to (Nf,Lf) is given by
The DWBA is a relatively simple model with respect to other sophisticated theoretical models, such as R-matrix method[64] and convergent close-coupling (CCC) approach.[65] For the calculations of correlated two-electron momentum distributions of NSDI at high laser intensities, the DWBA model is a good candidate since it can be calculated very quickly. Furthermore, the main important physics has been considered in DWBA such that the shape of the DCS is typically in fairly good agreement with the experimental measurements. However, it is well known that, at low energies, the total cross sections (TCS) predicted by the DWBA significantly exceed the experimental values, especially at energies near the threshold, which implies that the DCSs of DWBA are also larger than the absolute experimental data. This suggests that the DCSs of DWBA at low energies require a special treatment before they are used to simulate the parallel momentum distributions for the returning electron after recollisional excitation of the parent ion. This special treatment has been accomplished by introducing a calibration procedure for DWBA.[66]
Now we consider electron impact ionization which is another dominant mechanism in NSDI. Different from the process of electron impact excitation, in (e, 2e) process, there are two outgoing electrons. Suppose we have an electron with momentum
Similar to electron impact excitation, the transition amplitude for the (e, 2e) process can also be evaluated by using the DWBA model. The initial state wavefunction is exactly the same as that for electron impact excitation, i.e.,
The standard partial wave expansions are also made to evaluate the scattering amplitude for (e, 2e). For details, see Ref. [67].
Again, the DWBA model for (e, 2e) is also only valid at high impact energies since the final-state electron–electron interaction, which is important in the collision process at intermediate and low energies, has not been taken into account. The group of Madison has improved the DWBA model by including the final-state Coulomb interaction directly in the final-state wavefunction.[68,69] It has been found that, for intermediate energies, the post-collision interaction between the two outgoing electrons has an important effect and leads to an overall improvement in the agreement between experiment and theory.
For electron impact ionization of He+ ion, the final-state electron–electron interaction has been taken into account in the Brauner, Briggs, and Klar (BBK) model[70] in which an approximate three-body scattering wavefunction for the final state that satisfies the asymptotic three-body Schrödinger equation (
In the BBK model which was originally proposed for the process of electron impact ionization of hydrogen, a plane wave is used to describe the incoming electron since the target is neutral. However, for electron impact ionization of He+ ion, a Coulomb wave should be employed to account for the Coulomb attraction between the incident electron and the target ion. The Coulomb wavefunction for the incident electron takes the form
It should be mentioned that some deficiencies still exist in the BBK wavefunction although it satisfies the asymptotic three-body Schrödinger equation exactly. The major limitation of the 3C wave function lies in the fact that the influence on the strength of the interaction of any two particles by the presence of a third one has not been taken into account. This deficiency was first corrected for the case in which the two outgoing electrons have equal energies by introducing effective Sommerfeld parameters,[72] and was later generalized for any energy sharing in all geometries.[73] Such a modification reflects the dynamic screening of the three-body Coulomb interactions.
Numerical simulations for spectra are usually performed for a single laser intensity. However, when atoms are exposed to a laser field, they experience different intensities across the beam profile. For a Gaussian beam, the intensity distribution is given by
Since experiment collects electrons originating from atoms located anywhere in the interaction volume, direct comparison of the experimental data with the theoretical spectra is acceptable only when the focal volume effect has been taken into account. Therefore, integration over the focus volume should be performed by finding the volume of each isointensity shell. For example, the focal volume averaged total yield of single ionization at a peak intensity I0 can be obtained by
Laser-induced rescattering processes have been well understood based on the three-step classical rescattering model.[6,22] In the first step, the electron tunnels out from an atom. In the second step, the electron moves in the laser field and returns to the parent core. Finally, in the third step, the electron collides with the parent ion. During the collision, both elastic and inelastic scatterings could take place. The elastic scattering of the returning electron with the parent ion leads to HTAI,[75] and the recombination of the returning electron with the parent ion produces HHG.[76] On the other hand, NSDI[77] is due to inelastic scattering of the returning electron by the parent ion, in which two mechanisms could be involved, one is the direct ionization of the parent and the other one is recollision excitation with subsequent ionization. In the simulation of all the laser-induced rescattering processes, the RWP which describes the momentum distribution of the returning electron plays a significantly important role.
According to the QRS model, the momentum distribution of the HATI photoelectron with momentum
The detected photoelectron momentum
The RWP is then obtained by
It has been established[28] that, equation (
Figure
Another important property of the RWP is that it is almost independent of the target. Figure
The QRS model for NSDI was first applied to the calculations of the total yield of doubly charged ion.[30] The total yield of doubly ionization due to NSDI in a linearly polarized laser pulse at a single peak intensity I can be expressed by
As mentioned in Subsection
In Fig.
According to the classical rescattering model, the maximum kinetic energy that a returning electron can gain from the field is 3.17Up, which is proportional to the laser intensity. This indicates that a minimum intensity, referred to as the threshold intensity, is required for the returning electron to have enough energy to ionize an electron of the residual ion or to promote this electron to an excited state. Nevertheless, this threshold intensity has never been observed in experiment.[82] van der Hart and Burnett[40] argued that the incorrect threshold intensity appears from the assumption in the recollision model that scattering can be treated as if it takes place in a free atom. As a matter of fact, this assumption is not true. In a free atom or ion, an incoming electron can promote the ground-state electron to an excited state and remain a continuum electron only when its kinetic energy is larger than the energy difference between the ground state and the excited state, since the total energy of a continuum electron must be positive. In contrast, in the case when the collision takes place in an electric field, the combined atomic and electric potentials form a barrier below zero. As a result, the projectile electron can escape from the atom or ion even with negative energy as long as its energy is higher than the potential barrier.[40] Therefore, in the laser-induced recollision process, electron impact excitation of the parent ion can still take place even when the returning electron has an energy less than the energy difference between the ground state and the excited state. This phenomenon is usually referred to as “lowering of threshold”. This does not mean, however, that the energy level of an excited state is lowered.
The maximum barrier height in the combined Coulomb and electric field potentials is given by[40]
Unfortunately, it is still a formidable task to perform actual numerical calculations with the lowering of threshold included. Alternatively, van der Hart and Burnett[40] suggested that, to account for the lowering of threshold, one should adjust the collision energy so that the returning electron energy Er (=Ei) corresponds to the incoming energy Er + |Vb| for electron impact excitation and ionization in the field-free case. Based on these considerations, the QRS model could be improved, and thus equation (
As demonstrated in Fig.
As indicated in Subsection
It is found that as α is increased, the model calculations converge to the experimental data, as demonstrated in Fig.
It should be noted that, for He, singlet cross sections for both electron impact excitation and electron impact ionization of the parent ion should be used in the numerical simulations for NSDI, since the two electrons involved in the process start in the singlet ground state, and their singlet coupling is preserved during recollision. This effect is virtually absent in other noble gases with many valence electrons and hence spin-averaged cross sections should be used.
With the improved QRS model, Chen et al.[36] have also simulated the total yields of doubly charged ions for NSDI of He and Ne in intense laser pulses at wavelengths of 390 nm and 400 nm, respectively, and the obtained results are in very good agreement with the experimental data over a broad range of laser intensities.
While double ionization yields versus laser intensity provide the first evidence that strong-field NSDI occurs in favor of the classical recollision model, the measured experimental data are relative. In contrast, the absolute values of the double-to-single ionization ratio can be determined in experiments and hence offer a more stringent test of the theoretical models. Theoretically, to simulate the double-to-single ionization ratio, one needs to obtain the absolute values for both double and single ionization yields.
In the previous section, the RWPs used in the simulations of the double ionization yield are evaluated by using SFA2. Nevertheless, it has been demonstrated[28] that the RWP of SFA2 only has the correct momentum or energy dependence but not the absolute value with respect to the RWP calculated by solving the TDSE. This might indicate that, in order to obtain the absolute total yield of doubly charged ion, one should use the RWP from TDSE. Unfortunately, the TDSE calculations are very time-consuming and very computationally demanding for long pulses at high intensities. Therefore, it is almost impossible to solve the TDSE for the simulations of real experiments, as for the situations considered here. However, since the only difference between the RWPs from SFA2 and TDSE, for a given laser pulse at a peak intensity I with a wavelength λ and a pulse duration Γ, is a normalization factor, it is still possible for one to obtain the correct absolute RWPs by renormalizing the RWP from SFA2 according to[38]
To obtain the double-to-single ionization ratio, one also needs to evaluate the total single ionization yield. Several theoretical models, such as ADK, PPT, SFA, and TDSE, can be used to calculate the single ionization yield. All of these theoretical models, except TDSE, are not able to reproduce the correct absolute ionization yields despite that the general trend of the ionization yield as a function of laser intensity can be well predicted by PPT and SFA. Again, this indicates that one also needs to calibrate the PPT or SFA results. For single ionization, a similar procedure could be applied by using the ionization probability from solving the TDSE for a short pulse at low intensity to renormalize the ionization probability from PPT or SFA. Details of the TDSE calculations and the normalization procedure can be found in Refs. [38] and [55], respectively.
As an example, in Fig.
With the carefully prepared absolute double and single ionization yields, it is straightforward to obtain the double-to-single ionization ratio.
Using the improved QRS model, Chen et al.[35,36,38] have recently calculated the double-to-single ionization ratios for He and Ne in intense fields with various different laser parameters, and the numerical results were found to be in good accordance with the experimentally measured data.[5,82–85] Figure
Very recently, the QRS model has also been employed to investigate the pulse-duration dependence of the double-to-single ionization ratio of Ne by intense 780-nm and 800-nm laser fields.[38]
While the early measurements of the total ionization yields for doubly charged ions greatly stimulated the investigation of NSDI, the experimental data could only give little insight into the ionization mechanism compared to the correlated two-electron momentum distributions (CMD). Since the first measurement performed by Weber et al.[77] at the turn of this century, the CMD have been shown to provide the most detailed information of NSDI, and hence yield direct and intuitive insight into the dynamics of laser–atom interaction.
In 2010, just two years after it was first development for HATI and HHG,[86] the QRS model was quickly extended to the simulations of CMD for NSDI of He, Ne, and Ar atoms.[31,32,87] Different from the calculations of total ionization yields for doubly charged ions in which only the TCSs for electron impact excitation and electron impact ionization of the parent ion are needed, in the simulations of CMD for NSDI, it is necessary to evaluate the DCSs for the corresponding inelastic collision processes. Consequently, the numerical procedures become much more complicated. Very recently, the QRS model was further improved and the NSDI processes of He were revisited.[37,39]
Within the framework of the present QRS model, the CMD for recollisional (e,2e) process and recollisional excitation with subsequent tunneling of the second electron are simulated separately.
According to the QRS model, the CMD for laser-induced (e,2e) collision in NSDI can be factorized as a product of the RWP and the field-free differential cross section for ionization of the parent ion by the impact of the laser-induced returning electron. In experiment, the CMD for NSDI are measured only for the momentum components along the laser polarization axis for the two outgoing electrons. Thus, to compare with experiment, the calculated TDCS for the laser-free (e,2e) process needs to be projected on the polarization axis. For a given incident energy, the two-electron momentum spectra along the incident direction for the laser-free (e,2e) process can be obtained by integrating the TDCS over ϕ2 and E2
For laser-induced recollisional (e,2e) process, the two electrons are still under the influence of the laser field after the collision, and thus at the end of the laser pulse each electron will gain an additional momentum −Ar, where Ar is the vector potential at the time when the recollision takes place. Therefore, by using the relation in Eq. (
Based on the classical simulation[28] for HATI in which laser-induced elastic scattering takes place, the vector potential Ar at the collision time is approximately determined by the relation in Eq. (
To obtain the CMD for a given intensity, one has to consider the contributions from all collisions at different incident energies weighted by the RWP. This gives
Finally, to compare with experimental measurements, the integration over the focus volume should be performed
We comment that the above numerical procedures for recollisional (e,2e) in NSDI are directly applicable to all rare gas atoms except that for He the spin conservation should be considered such that in Eq. (
Apart from recollisional (e,2e), another main mechanism in NSDI is recollisional excitation-tunneling. At a single peak intensity, the CMD for recollisional excitation-tunneling of atoms in a linearly polarized laser pulse can be expressed as a product of the parallel momentum distribution for the scattered electron after recollisional excitation and the parallel momentum distribution for tunneling-ionized electron from the excited state.
The parallel momentum distribution for the scattered electron after excitation at a given incident energy Ei can be obtained by projecting the DCS dσexc/dΩ for laser-free electron impact excitation of parent ion onto the polarization direction
For tunneling ionization of electron from excited state of the parent ion, several theoretical models, such as ADK, PPT, SFA, and TDSE, can be used to evaluate the parallel momentum distribution. Among all of these theoretical methods, the ADK model is the simplest one. Using the ADK model, the ionization rate, with the depletion effect taken into account, can be expressed as
With the parallel momentum distributions
Finally, to compare with experiment, an integration over the focus volume should be performed according to
As indicated in Eqs. (
In a previous work, the CMD for recollisional excitation-tunneling in NSDI of He was simulated by Chen et al.[32] who used the DWBA model to calculate the DCSs for electron impact excitation of He+ while the parallel momentum distribution for tunneling-ionization was evaluated by using the modified ADK model. However, it is well known that, the total cross sections predicted by the DWBA often exceed the experimental values significantly at low energies, even though the angular distribution of the DCS from DWBA calculations is typically in fairly good agreement with experiment. On the other hand, a damping factor was introduced empirically in the modified ADK model. Unfortunately, the damping factor is not available for ionization from the excited states although it has been well estimated for ionization from the ground state of a few atoms and ions.[46]
Very recently, the recollisional excitation-tunneling process in NSDI of helium in 800 nm laser pulse was revisited and the CMD measured by Staudte et al.[14] was re-simulated by Chen et al.[39] using the improved QRS model. In those
CMD simulations, the DCSs for electron impact excitation of He+ were calculated using the state-of-the-art many-electron R-matrix theory, and the tunneling ionization rates for electrons in the excited states were evaluated by solving the TDSE. The calculated CMD is displayed in Fig.
We have reviewed the quantitative rescattering (QRS) model for nonsequential double ionization (NSDI) of atoms in strong laser field. With the recent improvement, i.e., by taking into account the lowering of threshold energy due to the presence of electric field, the QRS model has been employed to simulate the total yield of the doubly charged ion as a function of laser intensity, and the correlated two-electron momentum distributions (CMD) for recollisional (e,2e) and excitation-tunneling as well. Since the obtained results highly depend on the scattering cross sections involved in the simulations, the QRS model provides a stringent test of the quantum collision theories for the traditional laser-free scattering processes. It has been found that with the carefully prepared accurate total and differential cross sections for both electron impact ionization and electron impact excitation, the total yields of doubly charged ion and the CMD for recollisional (e,2e) and excitation-tunneling measured in experiments can be well predicted by the QRS model. However, to simulate the double-to-single ionization ratio, both the returning electron wave packet calculated by using the SFA2 and the total yield for single ionization evaluated by using the simple models, such as ADK, PPT and SFA, need to be calibrated according to the corresponding results by solving the TDSE for short laser pulses. This calibration reflects the consistency of the QRS model for different types of recollision processes.
Despite the great success of the QRS model in making quantitative predictions on recollision phenomena including NSDI, there is still a lot of space left for improvement. For example, within the framework of the QRS model, it is still hard to predict the momentum distributions for capture-tunneling, although the mechanism can be interpreted qualitatively.[32] Furthermore, the deficiency of the QRS model remains that the quantum interference effects in NSDI at low intensities[88–91] are absent in the model results. This is mainly due to the fact that the transition amplitudes are not available when the ADK model is used to evaluate the parallel momentum distributions of the tunneling electron. In a very recent work,[39] the parallel momentum distributions of the tunneling electron are obtained by projecting the two-dimensional momentum spectra, evaluated by solving the TDSE, onto the polarization direction. Since the TDSE not only provides the probabilities but also the necessary amplitudes, a way to study the interference effect in NSDI based on the QRS model has been paved.
So far, the QRS model has only been used for treating NSDI in linearly polarized laser field. In the recent years, many efforts have shifted to NSDI in elliptically polarized laser field,[92] two-color laser field,[93] and especially counter-rotating circularly polarized two-color field[94–96] for which the semiclassical model has been widely employed. With the established validity of the QRS model for HATI in two-color strong laser fields,[97,98] it is reasonable to expect that the QRS model could also be applied to NSDI in two-color fields. This is, of course, a part of our work in the near future.
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