Momentum distributions in time-dependent density-functional theory: Product-phase approximation for nonsequential double ionization in strong laser fields

We investigate the possibility to deduce momentum space properties from time-dependent density functional calculations. Electron and ion momentum distributions after double ionization of a model helium atom in a strong few-cycle laser pulse are studied. We show that, in this case, the choice of suitable functionals for the observables is considerably more important than the choice of the correlation potential in the time-dependent Kohn-Sham equations. By comparison with the solution of the time-dependent Schrödinger equation, the insufficiency of functionals neglecting electron correlation is demonstrated. We construct a functional of the Kohn-Sham orbitals, which in principle yields the exact momentum distributions of the electrons and the ion. The product-phase approximation is introduced, which reduces the problem of approximating this functional significantly.

Figure 1

Contour plots of the momentum pair density ${\rho }^{2+}(k_1,k_2,T)$ of the electrons freed in double ionization. Results calculated from the uncorrelated functional (7) using the EKSO (right-hand side) are compared to the TDSE (left-hand side) solution. Momentum pair densities for $\lambda =780\mathrm{nm}$, $(N=3)\text{−}\text{cycle}$ laser pulses with different effective peak intensities are shown.

Figure 2

Ion momentum density of the model ${\mathrm{He}}^{2+}$ ion after interaction with $\lambda =780\mathrm{nm}$, $(N=3)\text{−}\text{cycle}$ laser pulses with different effective peak intensities. The density calculated using the EKSO in the uncorrelated functional (8) is compared to results from the TDSE.

Figure 3

Ion momentum density of the model ${\mathrm{He}}^{2+}$ ion after interaction with $\lambda =780\mathrm{nm}$, $(N=3)\text{−}\text{cycle}$ laser pulses with different effective peak intensities. The densities are calculated from the uncorrelated functional (8) using the EKSO and the orbitals obtained with $v_c=0$ (TDHF) and $v_c^{\mathrm{LK}}$ (LK).

Figure 4

Ion momentum density of the model ${\mathrm{He}}^{2+}$ ion calculated from the correlated functionals in the PP approximation using the EKSO. Results for $\lambda =780\mathrm{nm}$, $(N=3)\text{−}\text{cycle}$ laser pulses with different effective peak intensities are compared to the ion momentum density obtained from the TDSE.

Figure 5

Contour plots of the exchange-correlation function $g_{\mathrm{xc}}(x_1,x_2,t)$ for two effective peak intensities of $\lambda =780\mathrm{nm}$, $(N=3)\text{−}\text{cyckle}$ laser pulses as acquired from the solution of the TDSE. For clarity values larger than 10 are shown as 10.