Acceleration techniques for semiclassical Maxwell-Bloch systems: An application to discrete quantum dot ensembles

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.