Four-point correlators with BPS bound states in AdS${}_3$ and AdS${}_5$

We consider heavy-heavy-light-light (HHLL) correlators in AdS/CFT, focussing on the D1D5 CFT${}_2$ and the $\mathcal{N}=4$ super Yang-Mills theory. Out of the lightest $1\mathrm{/}2$-BPS operator in the spectrum, $O$, we construct a particular heavy operator $O_H$ given by a coherent superposition of multi-particle operators $O^n$, and study the HHLL correlator. When $n$ is of order of the central charge, we show that the bulk equation that computes our boundary HHLL correlators is always a Heun equation. By assuming that the form of the correlator can be continued to the regime where $n$ is $\mathcal{O}(1)$, we first reproduce the known single-particle four-point correlators for $n=1$ and then predict new results for the multi-particle correlators $⟨O^n{O}^{n}OO⟩$. Explicit expressions can be written entirely in terms of $n$-loop ladder integrals and their derivatives, and we provide them for $n=2$ and $n=3$ both in position and in Mellin space. Focussing on the AdS${}_5$ case, we study the OPE expansion of these multi-particle correlators and show that several consistency relations with known CFT data are non-trivially satisfied. Finally, we extract new CFT data for double and triple-particle long operators.