We present the results of numerical experiments on a two-dimensional model atom driven by a high-intense laser pulse. The electron wave-packet behavior is studied in a range of laser parameters corresponding to the dynamic stabilization regime. Wave packet localization in this regime with arbitrary laser polarizations is shown to manifest itself macroscopically by high-order harmonic production in the form of long trains of attosecond pulses. Calculations for the sub-relativistic regime of laser-atom interaction are carried out without making the dipole approximation in order to take into account the Lorentz force effect in wave packet evolution. The transition from polychotomy to the magnetic-field-induced drifting at very high laser intensities is documented which results in the electron delocalization. As a consequence, the intensity dependence of the atomic survival probability as well as that of the efficiency of high-order harmonic production possess a wide “stabilization window” followed by an abrupt drop because of the magnetic field effect.
© Optical Society of America
A number of interesting phenomena arising from the atomic electron interaction with an intense laser field has been discovered during the last two decades (see, e.g., review  and references therein). Most of these phenomena are connected with ionization, and the analysis of classical electron trajectories in a continuum or the examination of electron wave-packet dynamics proves to be useful for better insight into the physics of these processes.
Among the most surprising results in the theory of strong-field laser-atom interactions has been the prediction of atomic stabilization against ionization in a superintense high-frequency laser field. The possibility of stabilization was discovered initially for the case of constant-amplitude laser field (adiabatic stabilization, see  and references therein) and was confirmed later for the more realistic case of the pulsed field (dynamic stabilization) in numerical experiments on both 1D  and 3D  model atoms. Stabilization was predicted to manifest itself by the formation of localized wave packets which in linearly polarized field exhibit two-peak  or, more generally, multi-peak  structure (the so-called dichotomy and polychotomy, respectively). The dynamics of transition from the ground-state wave function to a fully developed localized wave packet was investigated in Ref.  both analytically and numerically by an example of the electron bound in a 1D model short-range potential.
All above-mentioned predictions concerning stabilization were based essentially on using the dipole approximation for laser-atom interaction. Till recently this approximation was widely used in theoretical studies of strong-field phenomena. In this approximation, the spatial dependence of the vector potential of the electromagnetic field is ignored and, hence, magnetic-field effects in a single-atom response are neglected. Such approximation is justified for the laser intensities not high enough to accelerate the electrons to relativistic velocities, but for higher intensities its validity becomes questionable [7–18]. This is relevant mostly to those processes in which the special types of electronic trajectories or wave-packet structures play a key role and that are sensitive to the factors leading to their distortions, the magnetic field being one of these factors. The typical example is high-order harmonic generation (HHG). This process is known to be very sensitive to the laser ellipticity that strongly affects the efficiency of the underlying “recollision” mechanism. The other factor that can be crucial for HHG is the magnetic field of the laser pulse, as has been confirmed by both classical  and quantum-mechanical  simulations. The reason is that the Lorentz force exerted by the magnetic component of the laser field deflects electrons along the wave propagation direction thus diminishing their interaction with the parent ions. This effect can be significant for attosecond pulse production in the few-optical-cycle regime, since electrons detached from atoms during the fast turn-on of the intense laser pulse can be accelerated to very high velocities prior to coming back to the parent ions . At the same time, it’s worth notice that the electron velocities sufficient for the Lorentz-force-induced suppression of attosecond burst production are still substantially smaller than the speed of light and, hence, nonrelativistic quantum mechanics remains valid in this case.
As pointed out in Ref. , the predictions concerning atomic stabilization should be also revised from the viewpoint of the validity of the dipole approximation. Indeed, since at very strong laser fields the Lorentz force that pushes the electron away from the parent ion becomes appreciable, the question arises whether this magnetic-field-induced electronic drift prevents atomic stabilization. This issue was addressed in a number of works from both classical and quantum-mechanical points of view [7–11,14–18]. The results of Monte-Carlo classical simulations  indicate that the stabilization breakdown does may occur at very high laser intensities. On the other hand, in a range of parameters explored in a number of quantum-mechanical calculations [9,10,15], no considerable suppression of atomic stabilization has been found. Eventually, the evidence of the breakdown of the dynamic stabilization has been reported very recently [16–18] that was obtained in numerical experiments on a 2D model atom. The dipole approximation has been found to fail at rather moderate laser intensities where the nonrelativistic Schrödinger equation still remains valid.
Atomic stabilization, on the one hand, is interesting in its own right as a rather counterintuitive effect of decreasing ionization probability with increasing laser intensity. On the other hand, an atom interacting with superintense laser field in the stabilization regime is of considerable interest as a long-lived highly nonlinear system that possesses a number of uncommon features. For example, as shown below, in the dynamic stabilization regime high-order harmonics are produced with unusual time- and polarization dependence. These characteristic features are the consequences of the peculiarities of electron dynamics and can be used, e. g., as a probe of stabilization.
In the present work the dynamics of atomic electron driven by a high-intense laser pulse is investigated in numerical experiments on a two-dimensional model atom. The electron wave-packet behavior and its macroscopic manifestations for both linear and circular laser polarizations are studied in a range of laser parameters corresponding to the dynamic stabilization regime. Calculations for the sub-relativistic regime of laser-atom interaction are carried out beyond the dipole approximation in order to take into account the magnetic component of the laser field. The changes of wave-packet time evolution and their effect in HHG are studied in the laser intensity domain where the Lorentz-force-induced transition from stabilization to complete ionization occurs.
2. Two-dimensional model
In this paper we deal with the situations when the classical electron motion is confined within at most two spatial dimensions. In view of reduced dimensionality of a classical problem, we used in our quantum-mechanical calculations a two-dimensional model similar to that utilized earlier in studies of laser polarization effects in atomic ionization and HHG [19–21]. In the calculations with arbitrary laser polarizations carried out in the dipole approximation we considered the atomic electron to be initially in the ground state of a smoothed 2D Coulomb potential V (r)=-(a 2+r 2)-1/2 with a=0.8, the binding energy being equal to 0.5. Here r=ix+jy, x and y being the coordinates in the laser polarization plane. In computations for linearly polarized laser field carried out without making the dipole approximation we used the same model potential except that r=ix+kz, where x and z are the coordinates along the laser polarization vector and the field propagation direction, respectively. Almost all results presented here were obtained for the electric field taken as a trapezoidal pulse with two-cycle linear ramps and ten-cycle flat top of amplitude Eo , the frequency ωo being equal to 1 (in atomic units). The Schrödinger equation written as
When the dipole approximation is employed, the well-known split-operator fast Fourier transform technique  can be readily used. Otherwise, since the spatial dependence of the vector potential leads to the impossibility of separating momentum and space coordinates in (1) because of the coupling term proportional to pA, Fourier methods can’t be applied directly to this equation if we work out of the dipole approximation. But, so far as atomic sizes are much less than the laser wavelength, we may expand the vector potential on z and retain only the terms up to the linear one. Making the unitary transformation
where vx =A(0, t)/c (A(0,t) is the vector potential at z=0) and vz =/(2c), we obtain the equation solvable numerically by means of the split-operator FFT technique:
We integrated equations (1) or (3) on a square grid of side L≈200 a. u. centered at the atomic core, the absorbing boundaries being used in order to avoid the wave function reflections from the edges.
3. Results (in the dipole approximation)
3.1 Electron localization
Movie in Fig. 1 shows the wave packet time evolution in linearly polarized laser pulse with Eo =15. After the pulse turn-on the electron probability distribution corresponding initially to the atomic ground state begins to move forth and back along the polarization axis following the electric field oscillations while its spatial width is increasing due to the quantum-mechanical spreading. Since the amplitude of classical electron oscillations αo =Eo / is much larger than the atomic size, and the velocity of the electron is very high when it passes the vicinity of the ion, at early stage the ionic potential has no visible effect on the wave packet evolution, except that the transverse spreading is somewhat slower because of the Coulomb focusing. The other regime begins to set in when the electron wave packet has spread along the polarization axis over the whole oscillation radius αo . Since then the wings of the wave packet begin to overlap with the ionic potential every time the center of the packet approaches one of the turning points. The ionic attraction impedes further wave packet spreading and leads to the electron probability redistribution. Ultimately, the wave packet becomes localized between the two classical turning points and gains a multi-peak structure. This dynamics is in agreement with the scenario described in Ref. .
As an example of the localized wave packet, the snapshot of the electron probability distribution taken at time t=12T (i.e., just before the pulse turn-off) for Eo =15 is shown in Fig. 2 where a three-peak structure is seen. The two side peaks corresponding to the minima of the time-averaged Kramers-Henneberger potential are mainly due to the ground dressed state population while the central peak originates from excited bound dressed states populated due to the fast turn-on of the laser field.
Below we present the results for circularly polarized laser pulse with Eox =Eoy =15.
After the laser pulse turn-on the initially isotropic electron probability distribution starts to revolve as a classical particle passing periodically the vicinity of the ion while quantum-mechanically spreading. As soon as the electron wave packet has spread over the size comparable with αo , it begins to change its form getting localized around the circular arc coincident with the locus of its overlaps with the nucleus. The snapshot of the electron probability distribution taken after 12 cycles of a laser pulse is shown in Fig. 4. Sickle-shape wave packet localization [21,23] is clearly seen that can be attributed to the population of the manifold of Kramers-Henneberger bound states because of the fast laser pulse turn-on.
This behavior differs fundamentally from that inherent in the normal regime, i.e. when there is no stabilization. Movie in Fig. 5 shows the wave packet evolution in this latter case for circularly polarized laser pulse with Eox =Eoy =0.1 and ωo =0.2. The ionized electron wave packet in this case has the appearance of a spiral unwinding from the nucleus [19,20] as seen in Fig. 6 where the snapshot taken after 9 cycles of a laser pulse is shown.
3.2 High-order harmonic generation
The above-mentioned sharp features of stabilized wave packets give rise to the harmonic production [4,21,24]. As an example, the square of the Fourier transform of the dipole acceleration for the case of linearly polarized laser pulse with Eo =15 is shown in Fig. 7. A comb of harmonics up to the 79-th one is seen in the stabilized atom photoemission spectrum.
It should be stressed that high-order harmonics in the dynamic stabilization regime are emitted as a long train of attosecond pulses. Interestingly, pulse amplitudes in this train increase, in spite of ionization. It can be ascertained by means of the wavelet analysis [25–27] that proves to be very useful mathematical tools for probing the temporal behavior of HHG [28,29]. A typical time profile of high-order harmonics in the dynamic stabilization regime obtained from the wavelet analysis of the polarization signal is shown in Fig. 8 (the time dependence of the Morlet wavelet [25–27] coefficient for 57-th harmonic of linearly polarized laser field is plotted). The temporal behavior seen in Fig. 8 can be understood from the close examination of the wave packet time evolution (see Fig. 1). High-order harmonics appear when the electron wave packet gets structured. The side peaks of oscillating wave packet come to the origin one time per cycle with nearly zero velocity and, hence, don’t participate in high-order harmonic production. On the contrary, the central peak encounters the ion twice per cycle at high speed, thus producing short bursts of high-order harmonics. Sharpening of features during wave packet localization leads to an increase of the amplitude of these bursts. The alternation of higher- and lower-intense pulse is because of the packet asymmetry.
Fourier- and wavelet analysis of the polarization signal for different values of the laser ellipticity give the results similar to those of Figs. 7 and 8 that can be also explained from the structure arguments (these results will be published elsewhere). Most essentially, dynamic stabilization allows rather efficient high-order harmonic production even in the case of circularly polarized laser field, in contrast to the normal regime. This fact can be readily understood from comparing pictures of wave packet evolution presented in Figs. 3 and 5.
Thus, the above-stated unusual properties of high-order harmonics prove to be very characteristic manifestations of the dynamic stabilization. This regime of attosecond pulse production using HHG differs essentially from those explored by now [13,30–33].
4. Results (beyond the dipole approximation)
The results obtained without making the dipole approximation are presented below.
4.1 Electron delocalization
The comparison of wave packet evolution obtained in numerical calculations with and without the dipole approximation for different values of Eo shows that the magnetic-field effect is negligible at Eo <10. As the value of Eo increases further, the changes in the wave packet evolution become significant. Movies in Figs. 9–11 present three different scenarios of this evolution corresponding to three different values of Eo .
At Eo =15 the electron localization observed is still rather similar to that obtained with the dipole approximation, with the difference that a portion of the electronic density leaks from the atom in the propagation direction (Fig. 9). At Eo =20 the wave packet splits into the two distinct parts (Fig. 10). The first part remains strongly localized in the vicinity of the ion and exhibits a polychotomous structure as a signature of remaining stabilization that occurs due to the trapping of some portion of the electronic density by the Kramers-Henneberger effective potential. The second part gets delocalized, spreading and drifting away from the atom in z direction because of the Lorentz force. At Eo =22.5 (Fig. 11) no evidence remains of any localization that signifies full ionization.
The snapshots of the electron probability distribution taken just before the laser pulse turn-off (Fig. 12) for these three cases illustrate the transition from the polychotomy to the magnetic-field-induced drifting regime.
As a consequence of electron delocalization at E o=22.5, no high-order harmonic production is observed in this case (Fig. 13, lower curve).
4.2 “Stabilization window”
The probability Ps (τ) of finding the electron inside the box -xm ≤x≤xm , -zm ≤z≤zm just after the laser pulse turn-off can be used as a measure of an atom survival. At xm =zm =50 a. u. and τ=14T we obtain the field dependence of the atom survival probability shown in Fig. 14. The abrupt drop beginning from Eo ≈10 is seen that means the stabilization breakdown [16–18]. The high-order harmonic intensities depend on the field strength in a similar way (see Fig. 15). Nevertheless, it should be stressed that, despite the detrimental effect of the magnetic field, a rather wide range of laser intensities (“stabilization window”) persists in which stabilization and attosecond pulse train production take place as the theory based on the dipole approximation predicts.
In conclusion, an atom interacting with high-intense laser pulse in the dynamic stabilization regime provides a long-lived source of high-order harmonics emitted in the form of long trains of attosecond pulses. The peculiarities of electron localization dynamics cause a number of unusual properties of HHG process in this regime among which are an increase of harmonic peak intensity with time despite ionization and the possibility of HHG with arbitrary laser polarizations including circular one. The magnetic field of the laser pulse leads to the loss of electron localization through splitting the wave function into the localized and delocalized parts. The localized fraction begins to decrease as the laser field exceeds some critical value and it drops dramatically at higher fields. It is important to emphasize that the net effect of the electric and magnetic components of the laser field in atomic ionization is by no means the prevention of strong-field stabilization but the existence of a “stabilization window” extending over a rather wide interval of laser intensities. The similar “window” for efficient high-order harmonic generation in the dynamic stabilization regime exists as well.
The authors wish to acknowledge M. V. Fedorov for the interest to this work and helpful discussions. We also thank M. D. Chernobrovtseva for help in preparing the manuscript. This work was supported in part by the State Scientific and Technical Programs “Physics of Quantum and Wave Processes” and “Fundamental Metrology”, and the Russian Foundation for Fundamental Research.
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