A coherent semiclassical transport model for pure-state quantum scattering

Abstract

Abstract. We present a time-dependent semiclassical transport model for coherent pure-state scattering with quantum barriers. The model is based on a complex-valued Liouville equation, with interface conditions at quantum barriers computed from the steady-state Schrödinger equation. By retaining the phase information at the barrier, this coherent model adequately describes quantum scattering and interference at quantum barriers, with a computational cost comparable to that of classical mechanics. We construct both Eulerian and Lagrangian numerical methods for this model, and validate it using several numerical examples, including multiple quantum barriers.Key words. multiscale method, semiclassical limit, Liouville, coherent, quantum barrier.AMS subject classifications. 65M06, 65Z05, 81Q20, 81S30, 81T80. IntroductionThe motion of electrons in a plasma or a semiconductor can be modeled with classical mechanics when the change in the applied potential is moderate. But in a region where the potential changes rapidly over a length on the order of a de Broglie wavelength, quantum mechanics is required to accurately capture wave phenomena such as tunneling, resonance, and partial transmission and reflections. Because quantumscale parameters often control the accuracy and consistency of the solution, one often must resolve the dynamics entirely at the quantum scale. But for large-scale problems, such an approach is numerically infeasible. If the quantum region is sufficiently localized, a viable approach is to solve the problem using a multiscale method that combines the large-scale classical model with the small-scale quantum model.In [7,8] the authors presented a multiscale approach which accurately models the interaction of a quantum wave packet with a thin barrier in the semiclassical regime as the scaled Planck constant ε vanishes. This thin-barrier model accurately describes the weak limit of the moments of solutions to the pure-state Schrödinger and mixedstate von Neumann equation for an isolated thin quantum barrier (a barrier of width on the order of a de Broglie wavelength). This model assumes that the dwell time of the particle in the barrier is sufficiently short so that the behavior of the wave packet may be adequately described by the steady-state Schrödinger equation. Such an assumption is realistic for O(ε) thin barriers in the semiclassical limit as ε → 0, but it is inadequate when either ε is finite or the width of the barrier is significant in comparison to the width of the wave packet.Another shortcoming of the thin-barrier model is that it can only generate a decoherent solution. Quantum mechanics is in essence wave mechanics and the Schrö-

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