The solution to Maxwell-Bloch systems using an integral-equation-based framework has proven effective at capturing collective features of laser-driven and radiation-coupled quantum dots, such as light localization and modifications of Rabi oscillations. Importantly, it enables observation of the dynamics of each quantum dot in large ensembles in a rigorous, error-controlled, and self-consistent way without resorting to spatial averaging. Indeed, this approach has demonstrated convergence in ensembles containing up to $10^4$ interacting quantum dots. Scaling beyond $10^4$ quantum dots tests the limit of computational horsepower, however, due to the $\mathcal{O}(N_t N_s^2)$ scaling (where $N_t$ and $N_s$ denote the number of temporal and spatial degrees of freedom). In this work, we present an algorithm that reduces the cost of analysis to $\mathcal{O}(N_t N_s \log^2 N_s)$. While the foundations of this approach rely on well-known particle-particle/particle-mesh and adaptive integral methods, we add refinements specific to transient systems and systems with multiple spatial and temporal derivatives. Accordingly, we offer numerical results that validate the accuracy, effectiveness and utility of this approach in analyzing the dynamics of large ensembles of quantum dots.

中文翻译：

---

半经典麦克斯韦-布洛赫系统的加速技术：离散量子点系综的应用

Maxwell-Bloch 系统的解决方案使用基于积分方程的框架已被证明在捕获激光驱动和辐射耦合量子点的集体特征方面是有效的，例如光定位和 Rabi 振荡的修改。重要的是，它能够以严格、误差控制和自洽的方式观察大型集合中每个量子点的动力学，而无需求助于空间平均。事实上，这种方法已经证明了包含高达 $10^4$ 相互作用量子点的集合的收敛性。超过 $10^4$ 量子点的扩展测试了计算能力的极限，然而，由于 $\mathcal{O}(N_t N_s^2)$ 缩放（其中 $N_t$ 和 $N_s$ 表示时间和空间的数量自由程度）。在这项工作中，我们提出了一种将分析成本降低到 $\mathcal{O}(N_t N_s \log^2 N_s)$ 的算法。虽然这种方法的基础依赖于众所周知的粒子-粒子/粒子-网格和自适应积分方法，但我们添加了特定于瞬态系统和具有多个空间和时间导数的系统的改进。因此，我们提供的数值结果验证了这种方法在分析大型量子点集合动力学方面的准确性、有效性和实用性。