The absence of systematic nonperturbative models to describe intense atomic laser field ionization has stimulated the development of purely numerical methods [1,2,3]. For example, stabilization of atoms against ionization by an intense high-frequency laser beam has been widely studied over the past decade [4,5]. All the approaches to stabilization with a linearly poralized laser field (Floquet theory in the Kramers Henneberger frame, direct numerical integration of the wave equation, classic methods, etc.) have in common the presence of a highly distorted wave function that is spread along the polarization direction over a region whose width is given by the classic motion of the electron. This effect, sometimes referred as dichotomy when the wave function shows mostly a two-peaked structure, is the starting point for our understanding of the physical origin of the atomic stabilization. The existence of this effect is surprising because the electron is not ionized during this motion close to the nucleus. In elliptically polarized fields stabilization is also expected, but the shape of the stabilized wave packet changes significantly [6].

To our knowledge, no clear experiments to observe atomic stabilization have been reported. One of the main difficulties in designing such an experiment is the high frequency of the laser photons needed. To get proper Kramers Henneberger stabilization it is necessary to use a short-wavelength, intense laser field. It is widely believed that this effect will appear as soon as convenient light sources become available.

Because such intense fields are employed, several studies of the relativistic dynamics related to stabilization have been developed [7] that show how changes in the electron’s mass influence the classic excursion of the electron. To our knowledge, there are no detailed quantum studies of the influence of the magnetic field on the electron dynamics, though some classic relativistic analyses have been carried out [8]. Relativistic mass change is at least a second-order effect in υ/c, whereas the drift that is due to the magnetic field is a first-order effect in υ/c and therefore it would appear first, as the increase in the field strength drives the electron to higher velocities.

The problem with magnetic effects is that they are absent from the standard one-dimensional model, that has been widely used for study of atomic stabilization. Fortunately today it is feasible to solve two-dimensional models that are able to account for magnetic-field effects. Three-dimensional models are still a bit too impractical for use in ionization situations, for which large numerical grids have to be employed.

There is a range of laser parameters (frequency and intensity) at which the velocity is small enough to permit us to assume that the electron dynamics is given by the Schrödinger equation and large enough to include the first-order relativistic correction: The influence of the magnetic field. To study this region, we present numerical solutions of the time-dependent Schrödinger equation

which describes an atom interacting with an electromagnetic field. Atomic units (au; e=m=1au, c=137 au) are used throughout this paper. Because of computer limitations we work with a two-dimensional model, considering a flat atom in the xy plane with a soft-core Coulomb potential:

where a is a parameter that avoids singularity at the origin and was chosen to be a=1 au in our calculations. The spatial grid where the Schrödinger equation is solved has 1300×3000 equally separated points, with a step size of Δx=Δy=0.2 au. In this grid and for the chosen value of a, the ground-state energy of the model is E_B =-0.43 au, as determined by the standard imaginary-time propagation method.

To have the electron dynamics restricted to the xy plane we assume that the electric field is linearly polarized along the y axis and that the pulse propagates along the x axis. Then the magnetic field is parallel to the z axis. Therefore the classic Lorentz force will drive the electron (initially at rest) inside the z=0 plane. The electric field is

where f(x, t) is the laser turn-on, which is chosen to be linear and to last four cycles of the field. The vector potential is defined by

Therefore the two-dimensional time-dependent Schrödinger equation for such an atomic model interacting with a laser field is

Observe that the space dependence of the fields has been retained. We are not working in the dipole approximation or, in other words, we are retaining the magnetic-field effect in the nonrelativistic Lorentz force.

The numerical method employed in this paper to solve the Schrödinger equation out of the dipole approximation, is a generalization of those methods that were used to solve the dipole Schrödinger equation [5,9] in the case of spatially dependent fields. The wave function at a given time t+Δt is calculated from the wave function in the previous time step t [10]:

where we have defined

Now we express the exponentials in the Cayley unitary form (which preserves the norm of the wave function), and we solve the equations by using the Crank Nicholson finite-difference scheme. It is not possible to use Fourier-transform methods directly because the term A_y (x, t)(∂/∂y) mixes momentum and space coordinates.

Laser parameters at which stabilization is expected are chosen for our simulations. Figure 1 shows the probability density of the electron after 10 cycles of the field, comparing these kinds of (nondipolar) simulation with the corresponding two-dimensional calculation in the dipole approximation [E_y (x, t)≃E_y (t)=E ₀ f(t) sin(ω_Lt)] with the same laser parameters. Figure 1 corresponds to a laser field of amplitude E ₀=15 au (intensity, 7.9×10¹⁸ W=cm ²) and frequency ω_L =1 au (photon energy, 27 eV). This is a set of parameters for which a nonrelativistic study is reasonable but the first order correction υ/c is not extremely small. We selected these parameters because they were also selected for a recent paper in this journal [6]; we use a similar two-dimensional model, but we take into account the space dependence of the fields.

Working in the dipole approximation, we find that the wave function shows the expected dichotomy along the classic excursion, that is showed in the contour plot of Figure 1(a). In this case the dipole approximation could seem to be reasonable because the laser wavelength is close to 45 nm (or 861 au), which is still much bigger than the atomic size. But the dipole approximation is not valid because of the influence of the magnetic field. Figure 1(b) shows the computation for the same laser parameters with the space dependence of the fields retained. The ionized population that reaches the boundaries of the integration surface in both cases is negligible (less than 10^-6 anytime).

 

Figure 1. Probability density |Ψ(x, y, t)|² of the electron after 10 cycles of the field for E ₀=15 au (intensity, 7.9×10¹⁸ W/cm ²) and ω_L =1 au (photon energy, 27 eV). A linear envelope four-cycles turn-on has been employed. (a) results obtained in the dipole approximation, (b) simulation including the space dependence of the fields [Eq. (5)]. In both cases, contour plot lines are set to the same linear scale.

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The effect of the Lorentz force is clear: The spread of the wave function is observed not only along the polarization axis but also along the propagation axis. The ionized population is affected by the momentum transferred by incident photons, which push it toward the propagation direction and break the symmetry along the polarization axis. The dichotomy is not so relevant as in the dipole case. Because of the intense electric field, the electron reaches a high velocity (in fact, we are not too far from a breakdown of the validity of the nonrelativisitic wave equation, as has been shown in the context of atomic stabilization). At such high velocities, a small magnetic field is enough to explain a drift of the electron along the propagation axis. A trivial integration of the Lorentz force equations for a free electron in a squared pulse laser field (no turn-on time, for simplicity) gives a magnetic drift (or drift along the propagation direction) that is proportional to $E_0^2$=(${c\omega }_L^2$ ) (electron initially at rest) in addition to the well-known oscillation at 2ω_L ; this is the origin of the drift shown in Fig. 1(b). Obviously, in the case of the numerical simulation presented here, the laser turn-on and the atomic potential reduce the magnitude of the drift.

To conclude, we have presented a two-dimensional atomic model, with a standard soft-core Coulomb potential, that shows the effects of the magnetic field in the non-relativistic domain. In our opinion, this effect is relevant to the expected phenomenology related to stabilization. A magnetic field can induce a small drift that prevents the electron from oscillating steadily and indicates that Kramers Henneberger treatments have to be reconsidered at intense fields and high frequencies. To observe the expected stabilization with a dichotomy of the wave function it would be necessary to consider the superposition of laser fields (standing-wave or similars fields) where electric fields are added and magnetic fields could interfere destructively.