Semiclassical wave packet dynamics in Schrodinger equations with periodic potentials

TL;DR

This paper develops semiclassical wave packet solutions for Schrödinger equations with periodic potentials, revealing how they evolve under effective mass dynamics and validating adiabatic decoupling over Ehrenfest times.

Contribution

It introduces a method to construct asymptotic wave packet solutions in periodic potentials and analyzes their validity over Ehrenfest time scales.

Findings

Wave packets are concentrated around the effective phase-space flow.

The envelope dynamics follow a homogenized Schrödinger equation with time-dependent effective mass.

Adiabatic decoupling holds up to Ehrenfest time scales.

Abstract

We consider semiclassically scaled Schrodinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow obtained by Peierl's substitution, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrodinger equation with time-dependent effective mass. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid up to Ehrenfest time scales.