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.