A semi-relativistic formulation of light-matter interaction is derived using the so called propagation gauge and the relativistic mass shift. We show that relativistic effects induced by a super-intense laser field can, to a surprisingly large extent, be accounted for by the Schrödinger equation, provided that we replace the rest mass in the propagation gauge Hamiltonian by the corresponding time-dependent field-dressed mass. The validity of the semi-relativistic approach is tested numerically on a hydrogen atom exposed to an intense XUV laser pulse strong enough to accelerate the electron towards relativistic velocities. It is found that while the results obtained from the ordinary (non-relativistic) Schrödinger equation generally differ from those of the Dirac equation, demonstrating that relativistic effects are significant, the semi-relativistic formulation provides results in quantitative agreement with a fully relativistic treatment.Triggered by rapid technological advances [1][2][3][4] and new infrastructure projects [5] there is an increased interest in the description of quantum systems exposed to super-intense laser fields. In spite of the importance to address the relativistic regime [6] there are comparatively few such studies reported in the literature, probably due to the very nature of the timedependent Dirac equation which is notoriously hard to solve.Several issues make the Dirac equation tougher to solve numerically than its non-relativistic counterpart, the Schrödinger equation. The fact that the numerical space is increased by a factor four owing to the four components of the Dirac wave function, as opposed to a scalar wave function in the nonrelativistic case, is but the least of problems. The existence of a negative energy continuum is harder to tackle -for several reasons [7,8]. Firstly, many numerical time integration techniques require a numerical time step restricted by the inverse of the rest mass energy of the particle at hand, thus rendering calculations for realistic laser pulses infeasible. Secondly, spurious states may contaminate the spectrum of the numerical representation of the Hamiltonian.Another, more subtle complication is the fact that inclusion of the spatial dependence of the external field, which is imperative for ultrastrong fields [9], is harder to achieve in a consistent manner for the Dirac equation than is the case for the Schrödinger equation [10]. While this particular challenge to a large extent has been lifted by introducing the so-called propagation gauge to the Dirac equation [11][12][13], a formulation of the Schrödinger equation which allows us to include relativistic effects is still desirable, albeit seemingly too much to hope for. However, as it turns out, the apparently naive approach of simply substituting the rest mass with the relativistic mass of the electron, does in fact, to a surprisingly large degree, accommodate for relativistic effects induced by external electromagnetic fields.In the following we outline the theoretical framework. The semi-rel...

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We study the photoionization process of a hydrogen atom initially prepared in a circular Rydberg state. The atom is exposed to a two-cycle laser pulse with a central wavelength of 800 nm. Before the atom approaches saturation, at field intensities of the order of 10 17 W/cm 2 , relativistic corrections to the ionization probability are clearly seen. The ionization is predominantly driven by the radiation pressure in the propagation direction of the laser field, not by the electric field. Direct comparisons with the full numerical solution of the time-dependent Dirac equation demonstrate quantitative agreement with a semirelativistic approximation, which is considerably easier to implement.