Two-photon double ionization of atoms in attosecond x-ray radiation fields

We consider two-photon double ionization of helium with 100, 200, and 400 eV excess energy for the two ejected electrons, corresponding to photon energies of 89.5, 139.5, and 239.5 eV, respectively. We focus on the case of ultrashort pulses (two oscillations of the field) and develop an approach to calculate the two-photon transition matrix elements within the lowest order of the time-dependent perturbation theory. One of the major difficulties in calculating such amplitudes is the infinite summation over a complete set of intermediate states. In the subfemtosecond regime, however, this summation can be performed accurately by means of the closure approximation. This results in a simple expression for the two-photon amplitude that contains a dipole term and a quadrupole term. The dipole term can be clearly associated to a process in which each electron absorbs a photon whereas the quadrupole term is associated to a process in which one electron absorbs two photons and ejects the second one by collision. We analyze in detail how the relative weight of both processes influences the behavior of the electron energy and angular distributions. In particular we study how the shape of these distributions changes with the amount of electron correlations taken into account in both initial and final states. For 100 eV excess energy, our results for the electron energy distribution are compared with those obtained by solving the time-dependent Schroedinger equation. All these results unveil the crucial role of electron correlations in this transient regime of ionization which is neither sequential nor direct.