Two-photon above-threshold ionization of hydrogen over the photon energy range from 15 eV to 50 keV

We investigate the absorption of two identical photons from the ground state of hydrogen-like atoms over an energy range that extends beyond that explored up to now. Our approach is based on a hybrid formula, valid in second-order perturbation theory, in which the ${\mathbf{A}}^2$ contribution from the nonrelativistic Hamiltonian is treated exactly, while the $\mathbf{A}·\mathbf{P}$ contribution is calculated in dipole approximation. We find that, at least up to 50 keV, the nonrelativistic dipole approximation, based only on the $\mathbf{A}·\mathbf{P}$ contribution, determines the values of the total cross section. Our numerical results, covering photon energies from 90 nm (13.7 eV) to 0.0248 nm (50 keV) are in very good agreement with most previous theoretical works. Differences with recent results are discussed.

Figure 1

The contribution ${\sigma }_{\mathbf{A}·\mathbf{P}}^{\mathrm{DA}}$ to the total cross section [see Eq. (11)] for linearly (solid line) and circularly (dashed line) polarized photons vs the photon energy in the range 13.7–800 eV. Other calculations are also presented; see the legend. The much smaller contribution ${\sigma }_{{\mathbf{A}}^2}$ is also shown (magnified by a factor of 2 $\times {10}^3$).

Figure 2

The term ${\sigma }_{\mathbf{A}·\mathbf{P}}^{\mathrm{DA}}$ in the total cross section [see Eq. (11)] for linearly (solid line) and circularly (dashed line) polarized photons vs the photon energy in the range 0.8–50 keV. The quantity ${\sigma }_{{\mathbf{A}}^2}$ is also shown (dotted line).

Figure 3

The term ${\sigma }_{{\mathbf{A}}^2}$ of the total cross section [see Eq. (11)] vs the photon energy in the range 5–50 keV and linear polarization. Solid line: calculation with the nonrelativistic energy conservation law [see Eq. (3)]; dashed line: calculation with the relativistic law [see Eq. (B16)]. The dotted line is based on the plane-wave approximation; see Eq. (B19).

Figure 4

The total cross section ${\sigma }^{\mathrm{hyb}}$ and the term ${\sigma }_{{\mathbf{A}}^2}$ [see Eq. (11)] using the relativistic and nonrelativistic conservation laws expressed in Eqs. (B16) and (3), respectively. The photon energies vary from 50 to 200 keV, and linear polarization is considered.