Abstract
We present an adaptable, fast, and robust method for integrating the time-dependent Schrödinger equation. We apply the method to calculations of High Harmonic (HHG) and Above Threshold Ionisation (ATI) spectra for a single atomic electron in an intense laser field. Our approach implements the stabilized bi-conjugate gradient method (BiCG-STAB) for solving a sparse linear system to evolve the electronic wavefunction in time. The use of this established method makes the propagation scheme less restrictive compared to other schemes which may have particular requirements for the form of the equation, such as use of a three-point finite-difference approximation for spatial derivatives. Our method produces converged solutions significantly faster than existing methods, particularly if high accuracy is required. We demonstrate that this approach is suitable for a range of different parameters and show that in many circumstances significant gains can be made with the use of a fourth-order time propagator as opposed to the more common second-order Crank-Nicolson (CN) method.
Highlights
There is substantial interest in using the interaction of atoms in intense laser fields as a source of coherent short-wavelength light[1,2,3,4,5]
Numerical integration of the Time-Dependent Schrödinger Equation (TDSE) has been an essential tool in understanding intense laser-atom interactions since practical applications of this field of research became apparent with the discoveries of Above Threshold Ionization (ATI)[10] and High Harmonic Generation (HHG)[11]
As remarked earlier the splitting of the workload of the two schemes is very similar with both involving direct inversion of the part of the matrix that is diagonal in the orbital quantum number l — in the bi-conjugate gradient (BiCG)-STAB scheme this is done through the preconditioner — while parts off-diagonal in l are handled by the iterative scheme
Summary
There is substantial interest in using the interaction of atoms in intense laser fields as a source of coherent short-wavelength light[1,2,3,4,5]. Despite the centrality of the TDSE to this field, high accuracy computations are rarely performed This is due in part to the substantial computational requirements of such a task, and due to the difficulty in directly comparing the results of numerical simulation with experimental observables. This is a result of both uncertainty in the precise form of the laser field and the necessity of making approximations in treating many electron atoms.
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