Qprop with faster calculation of photoelectron spectra☆,☆☆
Introduction
Intense-laser–matter experiments brought forward many surprising results that were inaccessible to conventional perturbative theoretical approaches (see, e.g., [1]). As a consequence, new but less rigorous or semi-classical methods have been developed [2], [3], [4], [5] that, however, need to be tested against numerical ab initio solutions. Already the solution of the time-dependent Schrödinger equation (TDSE) for a single active electron in an effective atomic potential and in the presence of a classical, strong laser field can be a demanding computational task [6], [7], [8], [9].
The present paper is devoted to a revised version of Qprop — a position-space TDSE-solver for a single active electron bound in a spherically symmetric potential and subject to an external, time-dependent, space-homogeneous electric field (representing the laser field in dipole approximation). Qprop was introduced in Ref. [7]. A revised version of Qprop employing the time-dependent surface flux method (t-SURFF) for the calculation of PES as proposed in Ref. [10] was published in Ref. [9]. Using t-SURFF, the PES are calculated from the probability flux through a surface located sufficiently far away from the effective range of the binding potential. The time interval over which the flux is captured is limited by the simulation time. Hence, those components of the electronic wavefunction that represent the slowest electrons of interest should reach the t-SURFF surface during the simulation time. In practice, that means that the simulation time might be many times the actual laser pulse duration, in particular for the simulation of ultra-short pulse experiments. In this paper, we introduce Qprop 3.0, where this post-pulse propagation just to capture the slow electrons is avoided using the “trick” proposed in [8] called i-SURFV: once the laser field is off, the evolution of the system is described by a time-independent Hamiltonian, and the contribution to the surface flux after the pulse up to infinity can be calculated in a single step. Refining the formulas used in Qprop 2.0 (()), it is possible to reduce this evaluation to an action of a non-local operator.
The paper is organized as follows. Section 2 contains the mathematical formulation of the upgraded version of t-SURFF (i.e., i-SURFV) that is implemented in Qprop 3.0. In Section 3, the most important functions and data structures are described. Section 4 contains examples. Two examples were already in the Qprop 2.0 paper [9], thus demonstrating nicely the improvement in performance using i-SURFV. A few more demo configurations that may serve as useful templates for a user have been added.
Atomic units are used throughout the paper unless other units are explicitly given.
Section snippets
Hamiltonian and wavefunction
We consider a single active electron, initially bound by the atomic potential, under the influence of a laser field. This system is described by the TDSE with the Hamiltonian in velocity gauge Here, is the binding potential of the atom, is the vector potential in dipole approximation (i.e., the electric field is ), and is the imaginary potential which plays the role of an absorber to exclude unphysical reflections
Parameters and flags
In the current version, all parameters defining coordinate, momentum, and time grids, the potentials, and the laser are moved to the *param files. Most of the flags that allow to switch between different methods or to turn on and off the generation or the storage of specific output are also put into those files. Thus, it is no longer necessary to touch *.cc files for a wide range of problems. All parameters and flags are commented so that their function should become very clear while going
Examples
The quickest way to run an example is to go to its folder and launch the do_all bash script by typing ./do_all.sh in the terminal. Alternatively, one may make and execute the programs for imaginary time propagation (imag_prop), real-time propagation (real_prop) and t-SURFF (tsurff or tsurff_mpi for an MPI-parallelized version) manually. These programs have been slightly revised and renamed in Qprop 3.0. Table 1 shows the old program names in Qprop 2.0, the new names in Qprop 3.0, together with
Plotting guide
For the user’s convenience, scripts written in Python that were used to visualize the Qprop 3.0 generated data are added to the package. They are located in scr/plots.
Photoelectron distributions can be plotted using plot_pes. py. Leave the desired filename uncommented in the upper section of it, choose the number of angles and the polarization as =‘xz’ for linear or =‘xy’ otherwise.
Fig. 1(a) was produced with polar_canvas=1, Fig. 1(c) with plot_type=‘1D’, Fig. 2(a) was based on the same script
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work was supported by the project BA 2190/10 of the German Research Foundation (DFG) .
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The review of this paper was arranged by Prof. Stephan Fritzsche.
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This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).