Homogenization of the Schrodinger equation with a time oscillating potential

TL;DR

This paper investigates the homogenization of the Schrödinger equation with a time-oscillating potential in a periodic medium, demonstrating electron transitions between Bloch bands and justifying the Fermi golden rule in semiconductors.

Contribution

It introduces a novel homogenization approach combining classical techniques and Bloch wave theory to model electron transitions induced by oscillating potentials.

Findings

Partial transfer of electrons between Bloch bands.

Validation of the Fermi golden rule for transition probabilities.

A new homogenization framework for time-dependent potentials.

Abstract

We study the homogenization of a Schrodinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory.