A Semiclassical Transport Model for Thin Quantum Barriers

Abstract

'ublic reporting burden tor this collection ot intormation is estimated to average 1 hour per response, including the time tar reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Abstract. We present a one-dimensional time-dependent semiclassical transport model for mixed state scattering with thin quantum barriers. The idea is to solve a stationary Schr6dinger equation in the thin quantum barrier to obtain the scattering coefficients, and then use them to supply the interface condition that connects the two classical domains. We then build in the interface condition to the numerical flux, in the spirit of the Hamiltonian-preserving scheme introduced by Jin and Wen for a classical barrier. The overall cost is roughly the same as solving a classical barrier. We construct a numerical method based on this semiclassical approach and validate the model using various numerical examples.

A semiclassical transport model for two-dimensional thin quantum barriers

“…Since we also require an additional swap array, we find that 100 4 is a practical limit for brute calculation. Because a problem often requires at least 100 grid-points over a unit interval to resolve details, the finite-volume method developed in [11] is ineffective for general multi-dimensional semiclassical problems. A sparse matrix algorithm may alleviate some of the difficulty [10]; however, such an approach is viable only when the density information is sufficiently local (such as a front), which is typically an exception for von Neumann solutions.…”

“…One may also solve the boundary value problem (27) by using the transfer matrix method [11,15,7]. Since the solution is constant in the y-direction, the semiclassical impulse force is normal to the barrier curve.…”

“…The choice between the two equivalent interface conditions is an issue of implementation. An Eulerian method, such as the finite-volume method developed in [11], combines information by pulling information from the appropriate bicharacteristics upwind of the barrier. A Lagrangian method, such as the particle method, pushes the information to the appropriate bicharacteristics located downwind of the barrier.…”

“…The interested reader is encouraged to see [11,20] and references therein for a more complete discussion of the background of the semiclassical limit and von Neumann equation.…”

“…In [11,20], we developed a semiclassical model for one-dimensional thin quantum barriers for mixed-state dynamics. In the present work we extend the results from the first paper to higher dimensions, discuss the numerical implementation of the model, and demonstrate convergence to the semiclassical limit.…”