A Fractional Variational Approach for Modelling Dissipative Mechanical Systems: Continuous and Discrete Settings

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linear change of variables. This principle relies on a particular restriction upon the admissible variation of the curves. In the case of the half-derivative and mechanical Lagrangians, i.e. kinetic minus potential energy, the restricted fractional Euler-Lagrange equations model a dissipative system in both directions of time, summing up to a set of equations that is invariant under time reversal. Finally, we show that the discrete equations are a meaningful discretisation of the continuous ones.Comment: Key words: Variational analysis, Mechanical systems, Lagrangian mechanics, Damping, Fractional derivatives, Discretisation, Variational integrators. 13 pages, no figures. Contributed paper to 6th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Contro

Two-photon finite-pulse model for resonant transitions in attosecond experiments

We present an analytical model capable of describing two-photon ionization of atoms with attosecond pulses in the presence of intermediate and final isolated autoionizing states. The model is based on the finite-pulse formulation of second-order time-dependent perturbation theory. It approximates the intermediate and final states with Fano's theory for resonant continua, and it depends on a small set of atomic parameters that can either be obtained from separate \emph{ab initio} calculations, or be extracted from few selected experiments. We use the model to compute the two-photon resonant photoelectron spectrum of helium below the N=2 threshold for the RABITT (Reconstruction of Attosecond Beating by Interference of Two-photon Transitions) pump-probe scheme, in which an XUV attosecond pulse train is used in association to a weak IR probe, obtaining results in quantitative agreement with those from accurate \emph{ab initio} simulations. In particular, we show that: i) Use of finite pulses results in a homogeneous red shift of the RABITT beating frequency, as well as a resonant modulation of the beating frequency in proximity of intermediate autoionizing states; ii) The phase of resonant two-photon amplitudes generally experiences a continuous excursion as a function of the intermediate detuning, with either zero or $2\pi$ overall variation.Comment: 23 pages, 13 figure