Spin-orbit effect on strong-field ionization of krypton

A recent pump-probe experiment employing tunable, linearly polarized x rays demonstrated that ${\mathrm{Kr}}^{+}$ ions produced via strong-field ionization in a linearly polarized laser field are aligned, but that the degree of alignment is greatly overestimated by nonrelativistic strong-field ionization models. An effective one-electron model of strong-field ionization is presented that includes the effect of spin-orbit interaction. The method makes use of a flexible finite-element basis set and determines ionization rates in this square-integrable basis using a complex absorbing potential. It is found that even at the electric-field strength corresponding to the saturation intensity for the ionization of Kr, there is very little mixing between the $4p_{3∕2}$ and $4p_{1∕2}$ outer-valence orbitals. This shows that the uncoupled $m_l,m_s$ projection quantum numbers are inappropriate to describe the ${\mathrm{Kr}}^{+}$ states that are populated by strong-field ionization of krypton. For the x-ray probe step, a description is developed, within a density-matrix formalism. It is demonstrated that the inclusion of spin-orbit interaction in the ionization process provides satisfactory agreement with the experimental observation. Possibilities for time-resolved studies utilizing fs and sub-fs laser pulses are indicated.

Figure 1

(Color online) Cycle-averaged ionization rates of the $4p$ orbitals of krypton as a function of laser intensity, calculated for linearly polarized light.

Figure 2

Probability, as a function of peak intensity, of finding either neutral Kr or the $4p$-type states of ${\mathrm{Kr}}^{+}$ after exposure of neutral krypton to a laser pulse of Gaussian temporal shape (FWHM $40\mathrm{fs}$). $4p_j^{-1}$ in the figure legend means that a hole is present in the $4p_j$ orbital.

Figure 3

$R^{\left[2\right]}$ [Eq. (35)] as a function of the probe delay $t_1$ in units of the spin-orbit period $T^{\left(\mathrm{SO}\right)}$. In the experiment suggested by the figure, the resonant x-ray absorption by a tunnel-ionized noble gas atom (Ne, Ar, Kr, Xe) would be measured for two different angles ${\vartheta }_p$ between the (pump) laser and (probe) x-ray polarizations. $R^{\left[2\right]}$ is the ratio of the absorption probabilities at ${\vartheta }_p=0°$ and ${\vartheta }_p=90°$.