Frozen Gaussian Approximation-Based Two-level Methods for Multi-frequency Time Dependent Schrödinger Equation

In this paper, we develop two-level numerical methods for the time-dependent Schrödinger equation (TDSE) in multi-frequency regimes. This work is motivated by attosecond science [P.B. Corkum and F. Krausz, Nature Physics, 3 (2007), 381–387], which refers to the interaction of short and intense laser pulses with quantum particles generating wide frequency spectrum light and allowing for the coherent emission of attosecond pulses (1 attosecond = 10 second). The principle of the proposed methods consists in decomposing a wavefunction into a low/moderate frequency (quantum) contribution, and a high frequency contribution exhibiting a semi-classical behavior. Low/moderate frequencies are computed through the direct solution of the quantum TDSE on a coarse mesh, and the high frequency contribution is computed by frozen Gaussian approximation [M.F. Herman and E. Kluk, Chem. Phys., 91 (1984), 27–34]. This paper is devoted to the derivation of consistent, accurate and efficient algorithms performing such a decomposition and the time evolution of the wavefunction in the multi-frequency regime. Numerical simulations are provided to illustrate the accuracy and efficiency of derived algorithms.