First Benchmark of Relativistic Photoionization Theories against 3D ab initio Simulation

Photoelectron spectra and ionization rates encompassing relativistic intensities and hydrogenlike ions with relativistic binding energies are obtained using a quasiclassical $S$-matrix approach. These results, along with those based on the imaginary time method, are compared with three-dimensional, $½$-period ab initio simulations for a wide range of ionization potentials and electric field amplitudes. Significant differences between the three results are demonstrated. Time-dependent simulations indicate that the peak ionization current can occur before the peak of the electric field.

Figure 1

Snapshots of charge density $|\rho |=|e{\mathrm{\Psi }}^{†}(\mathbf{r},t){\widehat{\tau }}_3\mathrm{\Psi }(\mathbf{r},t)|$ at (a) $ct/\lambda =¼$ and (b) $ct/\lambda =⅜$, respectively, obtained from simulations. Ionization potential $I_p/m{c}^2=0.259$ and peak laser electric field normalized to the characteristic electric field $E_y/E_c=-0.0684$. Panel (a) is at the peak of the electric field while for panel (b) the field is reduced by the factor $\sqrt{2}$.

Figure 2

Comparison of ionization rates (averaged over an optical period, normalized by $c/\lambda$) based on the $S$-matrix method (dotted), ITM (dashed), two-dimensional simulations (triangles), and three-dimensional simulations (circles) of the KG equation. The $\times$ marks the result of a high resolution, axisymmetric TDSE, in the dipole approximation. The lower and upper abscissae show the ionization potential and the normalized vector potential, respectively. [Abscissa values yield the relativistic Keldysh parameter ${\gamma }_R=\sqrt{3{\mathrm{\Xi }}^2/(1+{\mathrm{\Xi }}^2)}/a$, where $\mathrm{\Xi }$ is a function of $I_p/m{c}^2$ and is defined following Eq. (1).] The peak laser electric field, normalized to the characteristic electric field, is nearly constant across the plot and listed in the text prior to description of Fig. 1.

Figure 3

Instantaneous ionization rate (normalized by $c/\lambda$) versus the phase of laser pulse for ionization potential $I_p/m{c}^2=0.259$ and peak laser electric field normalized to characteristic electric field $E_y/E_c=-0.0684$. With the half-cycle pulse format, in either two dimensions or three dimensions, the peak photoelectron current occurs before the peak of the electric field (at $\phi \equiv 0$). A case with finite pulse envelope is shown for comparison. Positive phases can be explained as a delay due to finite radius of the diagnostic sphere, but negative phases must be interpreted as a decrease in ionization current with an increase in the field. The ionization fraction (integrated curve) is too small for depletion of the bound state wave function to play a role.