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.

翻译：本文研究布洛赫模型衍生过渡模型的数值求解。布洛赫方程描述了受电磁波驱动的量子系统密度矩阵的时间演化。在高频低振幅条件下，该方程渐近简化为非刚性能级速率方程。作为中间状态，过渡模型控制密度矩阵的对角部分，属于含高频准周期强迫项与指数衰减强迫项的线性问题统一框架。对此类问题的数值求解具有挑战性。通过将高阶平均技术适配到该场景，我们分离慢（速率）动力学与快（振荡与衰减）动力学，从而推导出新型微宏观问题。我们建立了分解中微观部分及其时间导数量级的估计式，证明该新问题为非刚性系统。因此，可采用标准数值格式以一致精度求解该微宏观问题。为验证该方法，我们先后在玩具问题与过渡布洛赫模型上展示了数值结果。