Integral boundary conditions for the time-dependent Schrödinger equation: Atom in a laser field

Abstract

We formulate exact integral boundary conditions for a solution of the time-dependent Schrödinger equation that describes an atom interacting, in the dipole approximation, with a laser pulse. These conditions are imposed on a surface ͑boundary͒ which is usually chosen at a finite ͑but sufficiently remote͒ distance from the atom where the motion of electrons can be assumed to be semiclassical. For the numerical integration of the Schrödinger equation, these boundary conditions may be used to replace mask functions and diffuse absorbing potentials applied at the edge of the integration grid. These latter are usually introduced in order to ͑approxi-mately͒ compensate for unphysical reflection which occurs at the boundary of a finite region if a zero-value condition is imposed there on the solution. The present method allows one to reduce significantly the size of the space domain needed for numerical integration. Considering the numerical solution for a one-dimensional model, we demonstrate the effectiveness of our approach in comparison with some other numerical methods.

Hybrid surface-flux method for extraction of the ionization amplitude from the calculated wave function

“…The mathematically consistent approach to deal with this problem is the utilizing of the integral boundary conditions [4,7,20]. However, their technical implementation in the many-dimensional case becomes quite cumbersome.…”

“…From the complete Volkov functions set one may obtain the Green's function in an analytic form [7,10] and thus have an opportunity to completely determine the wavefunction evolution far from the center. Hence the wavefunction evolution appears to be non-trivial only in the vicinity of the center, so the Eq.…”

“…Among them one should mention those rest upon the evaluation of the projection to the approximate continuum wave function [1,2], the space Fourier transform of the wavefunction asymptotic part [3], the temporal Fourier transform (tFT) of the wavefunction [4,5] or the autocorrelation function [6]. The latter approach has also the refined version, namely the technique built upon the tFT of the probability amplitude flux through a certain closed surface [7][8][9][10][11][12] (following [10], hereinafter we will refer to this method as the t-SURFF). Finally, McCurdy et al have suggested an approach rest upon the evaluation of the flux of the probability amplitude for the scattering function derived by the Green's function operator action to the wavefunction after the end of the pulse [13,14] (in what follows we will refer to this method as the E-SURFF).…”

“…In the present work we propose a method for the amplitudes extraction combining the advantages of both the E-SURFF [13] and the t-SURFF [7] approaches. The paper structure is as follows.…”

“…Next, the E-SURFF [13,14] method implementation requires the enormous space region in order not to allow the ejected electron wave packet to reach its boundaries before the end of the laser pulse action. At the same time, the t-SURFF approach [7][8][9][10] though allowing to utilize rather compact space grid (under the condition of the employing of any methods for the supposition of the electron wavefunction unphysical reflection from the grid boundary, see below), demands the TDSE solving for a time period large enough to provide the probability to "flow" outwards the outer closed surface.…”