Collective dynamics Using Truncated Equations (CUT-E): simulating the collective strong coupling regime with few-molecule models

The study of molecular polaritons beyond simple quantum emitter ensemble models (e.g., Tavis-Cummings) is challenging due to the large dimensionality of these systems (the number of molecular emitters is $N\approx 10^{6}-10^{10}$) and the complex interplay of molecular electronic and nuclear degrees of freedom. This complexity constraints existing models to either coarse-grain the rich physics and chemistry of the molecular degrees of freedom or artificially limit the description to a small number of molecules. In this work, we exploit permutational symmetries to drastically reduce the computational cost of \textit{ab-initio} quantum dynamics simulations for large $N$. Furthermore, we discover an emergent hierarchy of timescales present in these systems, that justifies the use of an \textit{effective} single molecule to approximately capture the dynamics of the entire ensemble, an approximation that becomes exact as $N\rightarrow \infty$. We also systematically derive finite $N$ corrections to the dynamics, and show that addition of $k$ extra effective molecules is enough to account for phenomena whose rates scale as $\mathcal{O}(N^{-k})$. Based on this result, we discuss how to seamlessly modify existing single-molecule strong coupling models to describe the dynamics of the corresponding ensemble, as well as the crucial differences in phenomena predicted by each model. We call this approach Collective dynamics Using Truncated Equations (CUT-E), benchmark it against well-known results of polariton relaxation rates, and apply it to describe a universal cavity-assisted energy funneling mechanism between different molecular species. Beyond being a computationally efficient tool, this formalism provides an intuitive picture for understanding the role of bright and dark states in chemical reactivity, necessary to generate robust strategies for polariton chemistry.