Stable and efficient numerical method for solving the Schr�dinger equation to determine the response of tunneling electrons to a laser pulse

Abstract

The form Ψ(x, t)=[F0(x)+F1(x, t)e−iωt+F−1(x, t)eiωt]e−iEt/ℏ is used for the wave function in the transient solutions. This expression is similar to the three dominant terms in the steady-state solution from the Floquet theory, except that now F1 and F−1 depend on t as well as x. The function F0 is the static solution, and separate partial differential equations are given for F1 and F−1. Polynomial extrapolation is used to satisfy boundary conditions at the ends of the grid. The numerical solutions are shown to converge and to be numerically stable even for simulated times exceeding 2000 cycles of the radiation field. The examples show delays corresponding to the semiclassical tunneling transit time, the classical time for traversing the inverted barrier. A resonance is seen when electrons promoted above the barrier by absorbing quanta from the radiation field have the closed line integral of momentum between the turning points equal to an integral multiple of Planck's constant. A second resonance occurs when the period of oscillation for the radiation equals the semiclassical tunneling transit time for electrons that absorb one photon from the radiation but are still below the barrier. This resonance decays at a rate corresponding to the tunneling dwell time, and, thus, it is not present in the steady state. These observations suggest a semiclassical picture of the tunneling process. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 703–710, 1998