Stimulated Raman scattering of an ultrashort XUV radiation pulse by a hydrogen atom

We consider the hydrogen atom H(1s) exposed to an ultrashort laser pulse with a central frequency ?0 ranging from several hundreds of eV to 1.5 keV (approximate to 55 a.u.) and a peak intensity of 3.51x10(16)W/cm(2). We study the excitation of the atom by stimulated Raman scattering, a process involving pairs of frequencies (omega(1),omega(2)). These frequencies are non-negligible components of the pulse Fourier transform and they satisfy the condition E-g+h omega(1)=Eb+h omega(2), E-g and E-b equivalent to E-n being the ground-state and the excited-state energy, respectively. The numerical results obtained by integrating the time-dependent Schrodinger equation (TDSE) are compared with calculations in lowest order perturbation theory (LOPT). In LOPT we consider, in the second order of PT, the contribution of the term A center dot P in the dipole approximation and, in first order of PT, the expression of A(2) taken for first-order retardation effects. (A denotes the vector potential of the field and P is the momentum operator.) We focus on the Raman excitation of bound states with principal quantum numbers n up to n=13. The evaluation in perturbation theory of the A center dot P contribution to 1s-ns and 1s-nd transition probabilities uses analytic expressions of the corresponding KramersHeisenberg matrix elements. At fixed pulse duration tau=6 pi a.u. (approximate to 0.48 fs), we find that the retardation effects play an important role at high frequencies: they progressively diminish as the frequency decreases until the contribution of A center dot P dominates over the A(2) contribution for omega(0) values of a few a.u. We also study the dependence of the Raman process on the pulse duration for several values of omega(0). In the case omega(0)=13a.u.(approximate to 354eV) where dipole and nondipole contributions are of the same order of magnitude, we present the Raman excitation probability as a function of the pulse duration for excited ns, np, and nd states.