Density-based one-dimensional model potentials for strong-field simulations in He, ${\mathrm{H}}_2{}^{+}$, and ${\mathrm{H}}_2$

We present results on the accurate one-dimensional (1D) modeling of simple atomic and molecular systems excited by strong laser fields. We use atomic model potentials that we derive from the corrections proposed earlier using the reduced ground-state density of a three-dimensional (3D) single-active electron atom. The correction involves a change of the asymptotics of the 1D Coulomb model potentials while maintaining the correct ground-state energy. We present three different applications of this method: we construct correct 1D models of the hydrogen molecular ion, the helium atom, and the hydrogen molecule using improved parameters of existing soft-core Coulomb potential forms. We test these 1D models by comparing the corresponding numerical simulation results with their 3D counterparts in typical strong-field physics scenarios with near- and mid-infrared laser pulses, having peak intensities in the ${10}^{14}-{10}^{15}\mathrm{W}/{\mathrm{cm}}^2$ range, and we find an impressively increased accuracy in the dynamics of the most important atomic quantities on the time scale of the excitation. We also present the high-order harmonic spectra of the He atom, computed using our 1D atomic model potentials. They show a very good match with the structure and phase obtained from the 3D simulations in an experimentally important range of excitation amplitudes.

Figure 1

Upper panel: the plot of the square root of reduced density of the hydrogen molecular ion with $d=2.0$ (in blue). Lower panel: the plot of the density-based model potential (14) (in purple) and the improved soft-core Coulomb model (15) with the indicated parameter values (in gold). The ground-state energy is $E_0=-1.1026$ for both of the potentials, using $\mathrm{\Delta }z=0.2$ in (14).

Figure 2

Pulse shape of the near-infrared (in blue) and mid-infrared (in red) laser pulse, with the indicated parameters corresponding to Eq. (20).

Figure 3

Results for a static hydrogen molecular ion with $d=2.0$, driven by the near-infrared laser pulse with $F=0.1$ (left panels) and $F=0.15$ (right panels). We plot the time dependence of the mean values $〈z〉\left(t\right)$ in (a),(b) and the standard deviations ${\sigma }_z\left(t\right)$ in (c),(d) and the ground-state population losses $g\left(t\right)$ in (e),(f). Results of the corresponding 3D simulations are plotted in blue.

Figure 4

Results for two static hydrogen molecular ions with intermolecular distances of $d=1.4$ (left panels) and $d=2.6$ (right panels), driven by the near-infrared laser pulse with $F=0.15$. We show the time dependence of the mean values $〈z〉\left(t\right)$ in (a),(b) and the standard deviations ${\sigma }_z\left(t\right)$ in (c),(d) and the ground-state population losses $g\left(t\right)$ in (e),(f). Results of the corresponding 3D simulations are plotted in blue.

Figure 5

Time dependence of the mean values $〈z〉\left(t\right)$ in (a),(b) and the standard deviations ${\sigma }_z\left(t\right)$ in (c),(d) and the ground-state population losses $g\left(t\right)$ in (e),(f) for a helium atom driven by the near-infrared laser pulse with $F=0.15$ (left panels) and $F=0.20$ (right panels). Results of the corresponding 3D simulations are plotted in blue.

Figure 6

Time dependence of the mean values $〈z〉\left(t\right)$ in (a) and the ground-state population losses $g\left(t\right)$ in (b) for a helium atom driven by the near-infrared laser pulse with $F=0.25$.

Figure 7

Scaled power spectra versus the harmonic order of the emitted radiation for a helium atom driven by the near-infrared laser pulse with $F=0.15$ in (a), $F=0.20$ in (b), and $F=0.25$ in (c). Results of the corresponding 3D simulations are plotted in blue. The scaling function of Eq. (21) was applied to all of the 1D results of this figure.

Figure 8

Results for a static hydrogen molecule with $d=1.4$, driven by the near-infrared laser pulse with $F=0.07$ (left panels) and $F=0.1$ (right panels). We plot the time dependence of the mean values $〈z〉\left(t\right)$ in (a),(b) and the standard deviations ${\sigma }_z\left(t\right)$ in (c),(d) and the ground-state population losses $g\left(t\right)$ in (e),(f). Results of the corresponding 3D simulations are plotted in blue.

Figure 9

Results for a static hydrogen molecular ion with $d=2.0$ (left panels), and for a helium atom (right panels), driven by the mid-infrared laser pulse with $F=0.15$ for ${\text{H}}_2{}^{+}$ and $F=0.2$ for He. We plot the time dependence of the mean value $〈z〉\left(t\right)$ in (a),(b), the standard deviation ${\sigma }_z\left(t\right)$ in (c),(d), and the ground-state population loss $g\left(t\right)$ in (e),(f). The color coding of the curves in the left and right panels corresponds to those in Fig. 3 and Fig. 5, respectively.