Article Dans Une Revue Communications in Mathematical Sciences Année : 2024

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

Résumé

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.
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Dates et versions

hal-04171818 , version 1 (31-07-2023)
hal-04171818 , version 2 (04-11-2024)

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  • HAL Id : hal-04171818 , version 2

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Brigitte Bidégaray-Fesquet, Clément Jourdana, Léopold Trémant. Multi-frequency averaging and uniform accuracy towards numerical approximations for a Bloch model. Communications in Mathematical Sciences, 2024. ⟨hal-04171818v2⟩
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