Calculations of rates for strong-field ionization of alkali-metal atoms in the quasistatic regime

Tunneling and over-the-barrier ionization of alkali-metal atoms in strong electromagnetic fields are studied using the single-electron model (valence electron plus atomic core) and the frozen-core approximation. The lowest-state energies and widths (ionization rates) at different values of applied field, obtained using the Stark shift expansion and the Ammosov-Delone-Krainov formula, respectively, are compared with the corresponding values determined numerically by the complex rotation method. Good agreement for the energies is obtained at the field strengths corresponding to the tunneling regime. In contrast, the rates obtained by the Ammosov-Delone-Krainov formula significantly overestimate numerical results. After introducing a correction in the formula that accounts for the dependence of the binding energy on the field strength, good agreement in the tunneling regime is obtained for the rates too. A disagreement that still remains in the over-the-barrier ionization regime indicates that at stronger fields further corrections of the rate formula, such as those related to the form of the bound-state wave function, are required. Finally, it is demonstrated that numerically determined ionization rates are not too sensitive to the choice of model for the effective core potential and good results can be obtained using a simple local pseudopotential.

Figure 1

Potential energy $V=V_{\mathrm{core}}\left(r\right)-Fz$, i.e., its $x=y=0$ cut (red lines), and the lowest energy level $E$ (blue lines) of the valence electron of sodium atom for three different strengths of applied electric field: (a) $F=0$, (b) $F=0.008$, and (c) $F=0.015$ a.u. The effective core potential $V_{\mathrm{core}}\left(r\right)$ is represented by the Hellmann pseudopotential (8). Cases (b) and (c) correspond to the tunneling and OBI regimes, respectively. For comparison, the sums (gray lines) of the Coulomb potential $V_C=-1/r$ and the corresponding field contributions $-Fz$ (dashed lines) are shown in the same graphs.

Figure 2

Lowest-energy of alkali-metal atoms versus the electric-field strength $F$, determined numerically by solving the eigenvalue problem of the Hamiltonian (2) (bullets) and by the use of the expansion (4) up to the second (gray lines) and fourth (red lines) powers of $F$. For comparison the lowest energy of the hydrogen atom is given at the bottom. Vertical gray lines mark the field strengths $F_s$ dividing the tunneling and OBI areas for each atom.

Figure 3

Lowest-state widths and ionization rates of alkali-metal atoms in the static electric field $F$ determined numerically (diamonds), by the ADK formula (dashed lines), by the ADK formula with second- (fourth-) order Stark shift correction [solid gray (red) lines], and by the ADK formula corrected using numerically calculated energy (black dots). The corresponding energies [obtained numerically and by the use of Eq. (4)] are shown in the insets (dots and solid lines, respectively). Vertical gray lines mark the field strengths $F_s$ dividing the tunneling and OBI areas for each atom.

Figure 4

Ionization rates $w_{\mathrm{alt}}$ of alkali-metal atoms versus the laser field intensity $I$. Vertical gray lines mark the field intensities dividing the tunneling and OBI areas for each atom.