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

Abstract

We consider the hydrogen atom $\mathrm{H}(1s)$ exposed to an ultrashort laser pulse with a central frequency ${\ensuremath{\omega}}_{0}$ ranging from several hundreds of eV to 1.5 keV ($\ensuremath{\approx}55$ a.u.) and a peak intensity of $3.51\ifmmode\times\else\texttimes\fi{}{10}^{16}\phantom{\rule{4pt}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$. We study the excitation of the atom by stimulated Raman scattering, a process involving pairs of frequencies (${\ensuremath{\omega}}_{1},{\ensuremath{\omega}}_{2}$). These frequencies are non-negligible components of the pulse Fourier transform and they satisfy the condition ${E}_{g}+\ensuremath{\hbar}{\ensuremath{\omega}}_{1}={E}_{b}+\ensuremath{\hbar}{\ensuremath{\omega}}_{2},{E}_{g}$ and ${E}_{b}\ensuremath{\equiv}{E}_{n}$ being the ground-state and the excited-state energy, respectively. The numerical results obtained by integrating the time-dependent Schr\odinger 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 $\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}$ in the dipole approximation and, in first order of PT, the expression of ${\mathbf{A}}^{2}$ taken for first-order retardation effects. ($\mathbf{A}$ denotes the vector potential of the field and $\mathbf{P}$ 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 $\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}$ contribution to $1s\ensuremath{-}ns$ and $1s\ensuremath{-}nd$ transition probabilities uses analytic expressions of the corresponding Kramers--Heisenberg matrix elements. At fixed pulse duration $\ensuremath{\tau}=6\ensuremath{\pi}$ a.u. ($\ensuremath{\approx}0.48$ 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 $\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}$ dominates over the ${\mathbf{A}}^{2}$ contribution for ${\ensuremath{\omega}}_{0}$ values of a few a.u. We also study the dependence of the Raman process on the pulse duration for several values of ${\ensuremath{\omega}}_{0}$. In the case ${\ensuremath{\omega}}_{0}=13\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{u}.\phantom{\rule{4pt}{0ex}}(\ensuremath{\approx}354\phantom{\rule{4pt}{0ex}}\mathrm{eV})$ 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 $ns,\phantom{\rule{4pt}{0ex}}np$, and $nd$ states.