Relativistic Dynamics of highly charged hydrogen-like systems exposed to intense high-frequency electromagnetic fields

We solve the time dependent Dirac equation in a basis consisting of the eigenstates of the field free Hamiltonian $H_0=c \mbox{\boldmath $\alpha$} \cdot {\bf p} + V(r)+ m c^2 \beta$. The dynamics induced by the time dependent perturbation, $H'_v=c \mbox{\boldmath $\alpha$} \cdot {\bf A}({\bf r},t)$ in the velocity gauge, or equivalently $H'_l={\bf r} \cdot {\bf E}({\bf r},t) (1-\mbox{\boldmath $\alpha$} \cdot \hat{\bf k})$ in the length gauge, is resolved by solving the first order differential equation arising from the expansion. The number of continuum states needed in order to get converged results is reduced by complex scaling of the coordinates, $r \rightarrow r e^{i \theta}$. We investigate the importance of relativistic effects for various field strengths, $E_0$, and nuclear charges by comparing the solutions of the Dirac equation with those of the Schr{\"o}dinger equation. The dynamics is described both within and beyond the dipole approximation (${\bf A}({\bf r},t) \approx {\bf A}(t)$). For high $E_0$ and high photon energy, $\hbar \omega$, it is clear that this approximation breaks down. However, for increasing $Z$, the electron is increasingly tightly bound, which to some extent reduces the importance of non-dipole effects.