Multi-frequency averaging and uniform accuracy towards numerical approximations for a Bloch model

TL;DR

This paper develops a high-order averaging method to efficiently and accurately solve a transitional model derived from the Bloch equation, enabling uniform accuracy in numerical approximations of quantum systems under high-frequency forcing.

Contribution

It introduces a novel micro-macro decomposition and averaging technique to handle the transitional Bloch model with non-stiff properties, improving numerical solution accuracy.

Findings

The micro-macro problem is non-stiff, allowing stable numerical solutions.

Numerical results confirm the method's uniform accuracy for the transitional Bloch model.

The approach effectively separates slow and fast dynamics in high-frequency regimes.

Abstract

We are interested in numerically solving a transitional model derived from the Bloch model. The Bloch equation describes the time evolution of the density matrix of a quantum system forced by an electromagnetic wave. In a high frequency and low amplitude regime, it asymptotically reduces to a non-stiff rate equation. As a middle ground, the transitional model governs the diagonal part of the density matrix. It fits in a general setting of linear problems with a high-frequency quasi-periodic forcing and an exponentially decaying forcing. The numerical resolution of such problems is challenging. Adapting high-order averaging techniques to this setting, we separate the slow (rate) dynamics from the fast (oscillatory and decay) dynamics to derive a new micro-macro problem. We derive estimates for the size of the micro part of the decomposition, and of its time derivatives, showing that this new problem is non-stiff. As such, we may solve this micro-macro problem with uniform accuracy using standard numerical schemes. To validate this approach, we present numerical results first on a toy problem and then on the transitional Bloch model.