Two-electron ionization and stabilization beyond the dipole approximation

A two-dimensional model atom is employed to study the ionization behavior of helium subjected to strong laser fields in the high-frequency regime. The evolution of the system is studied by means of numerical integration of the Schrödinger equation beyond the dipole approximation. Ionization probabilities of the two-electron atom in highly intense laser fields have been calculated for different pulse shapes. It is confirmed that the mutual repulsion between the two electrons as well as the length of the laser pulses significantly alter the ionization probabilities of the system. Nondipole effects are shown to lead to a considerable increase of the ionization probabilities. For certain laser pulse shapes, a regime of ionization suppression is investigated which exists in addition to two-electron stabilization. The applicability of our model scheme to the case of heliumlike systems is discussed, and it is shown that stabilization may also occur in these systems, and how it is altered in dependence of the nuclear charge.

Figure 1

Ionization probabilities of the inner and the outer electron as a function of the peak electric field strength $E_0$. The frequency of the laser is $\omega =3\mathrm{a.u.}$, while the pulse consists of each four optical cycles linear turn-on and turn-off, and eight optical cycles at fixed amplitude. In (a), the ionization probabilities are displayed for the case where the repulsion between the two electrons has been taken into account, while it has been neglected in (b).

Figure 2

Contour plots of the inner (a) and outer (b) electron probability density after $10\text{cycles}$ interaction with the same laser pulse as in Fig. 1, at peak field strength $E_0=45\mathrm{a.u.}$ The $x$ axis is the polarization direction of the laser field, while the laser propagates along the $y$ axis. Here, the dipole approximation has not been carried out in the computations, such that the retardation of the laser pulse has been taken into account. In the lower row, subfigures (c) and (d), the electron densities are displayed for calculations which have been performed within the dipole approximation. For better visibility, 15 contour lines are shown for a density range between 0 and $1\times {10}^{-3}$.

Figure 3

Displacement of the center of masses of the electrons along the laser polarization axis (a), and in the propagation direction (b) as a function of time. As in Fig. 1, the pulse consists of each four optical cycles linear turn-on and turn-off, and eight cycles at maximal field strength $E_0=45\mathrm{a.u.}$ and frequency $\omega =3\mathrm{a.u.}$ For comparison, the displacement of a classical electron subjected to the same laser pulse in the propagation direction is also shown in (b).

Figure 4

Ionization probabilities of the outer electron as a function of the peak electric field strength $E_0$, where either the retardation of the laser pulse has been taken into account or has been neglected (dipole approximation). The laser pulses have been chosen to be the same as in Fig. 1, while for better visibility the scale has been changed.

Figure 5

Ionization probabilities of the inner and the outer electron as a function of the peak electric field strength $E_0$. The frequency of the laser is $\omega =3\mathrm{a.u.}$, while the pulse consists of each two optical cycles linear turn-on and turn-off, and eight optical cycles at fixed amplitude. In (a), the ionization probabilities are displayed for the case where the repulsion between the two electrons has been taken into account, while it has been neglected in (b).

Figure 6

Ionization probabilities of the inner and the outer electron of helium for varying pulse lengths. The frequency of the laser is $\omega =3\mathrm{a.u.}$, with the pulse consisting of each four optical cycles linear turn-on and turn-off, and a center part of constant amplitude $E_0$ with a length of 4, 6, 10, and $12\text{cycles}$, respectively. In (a), the ionization probabilities are displayed for the inner electron, while in (b) those of the outer electron are shown. The numbers in the caption refer to the length (in optical cycles) of the turn-off, intermediate, and turn-off stages of the laser pulses employed throughout the respective calculations.

Figure 7

Ionization probabilities of the inner and the outer electron of ${\mathrm{Li}}^{+}$ as a function of the peak electric field strength $E_0$. The frequency of the laser is $\omega =5\mathrm{a.u.}$, while the pulse consists of each two optical cycles linear turn-on and turn-off, and eight optical cycles at constant amplitude. In (a), the ionization probabilities are depicted with the repulsion between both electrons being taken into account, while it has been neglected in (b).

Figure 8

Ionization probabilities of the inner and the outer electron of the doubly charged beryllium ion. The frequency of the pulse is $\omega =7.5\mathrm{a.u.}$, while its shape consists of each two optical cycles linear turn-on and turn-off, and eight optical cycles at constant peak electric field strength $E_0$. Depicted in (a) are the ionization probabilities when the repulsion between both electrons is taken into account, while it has been neglected in (b).

Figure 9

Same as Fig. 8a, but here the dipole approximation has been employed throughout the calculations.

Figure 10

Ratio of double to single ionization probabilities for the three two-electron systems after interaction with a laser pulse of $12\text{cycles}$ total duration, where in each case the frequency $\omega$ has been chosen to slightly exceed the binding energy of the respective inner electron. To allow for comparison of the curves, the peak field strength $E_0$ has been scaled with $1∕Z^3$.