Generalized space-translated Dirac and Pauli equations for superintense laser-atom interactions

Abstract

We obtain a generalization of the nonrelativistic space-translation transformation to the Dirac equation in the case of a unidirectional laser pulse. This is achieved in a quantum-mechanical representation connected to the standard Dirac representation by a unitary operator $T$ transforming the Foldy-Wouthuysen free-particle basis into the Volkov spinor basis. We show that a solution of the transformed Dirac equation containing initially low momenta $p$ ($p/mc\ensuremath{\ll}1)$ will maintain this property at all times, no matter how intense the field or how rapidly it varies (within present experimental capabilities). As a consequence, the transformed four-component equation propagates independently electron and positron wave packets, and in fact the latter are propagated via two two-component Pauli equations, one for the electron, the other for the positron. These we shall denote as the Pauli low-momentum regime (LMR) equations, equivalent to the Dirac equation for the laser field. Successive levels of dynamical accuracy appear depending on how accurately the operator $T$ is approximated. At the level of accuracy considered in this paper, the Pauli LMR equations contain no spin matrices and are in fact two-component Schr\"odinger equations containing generalized time-dependent potentials. The effects of spin are nevertheless included in the theory because, in the calculation of observables which are formulated in the laboratory frame, use is made of the spin-dependent transformation operator $T$. In addition, the nonrelativistic limit of our results reproduces known results for the laboratory frame with spin included. We show that in intense laser pulses the generalized potentials can undergo extreme distortion from their unperturbed form. The Pauli LMR equation for the electron is applicable to one-electron atoms of small nuclear charge$\phantom{\rule{0.28em}{0ex}}(\ensuremath{\alpha}Z\ensuremath{\ll}1)$ interacting with lasers of all intensities and frequencies $\ensuremath{\omega}\ensuremath{\ll}m{c}^{2}.$