On-chip high-order parametric downconversion in the excitonic Mott insulator Nb$_3$Cl$_8$ for programmable multiphoton entangled states

Spontaneous parametric downconversion (SPDC) and four-wave mixing in $χ^{(2)}$ and $χ^{(3)}$ media underpin most entangled-photon sources, but direct generation of higher-order entangled multiphoton states by $n$-th order parametric downconversion remains extremely challenging because conventional materials exhibit tiny high-order nonlinearities. Here we show that single-layer Nb$_3$Cl$_8$, an excitonic Mott insulator on a breathing Kagome lattice, supports exceptionally large nonlinear susceptibilities up to seventh order. Many-body GW--Bethe--Salpeter and time-dependent BSE / Kadanoff--Baym simulations yield resonant $χ^{(2)}$--$χ^{(7)}$ for monolayer Nb$_3$Cl$_8$, with $|χ^{(4)}|$ and $|χ^{(5)}|$ surpassing values in prototypical transition metal dichalcogenides by 5--9 orders of magnitude. We trace this enhancement to flat bands and strongly bound Frenkel excitons with ferroelectrically aligned out-of-plane dipoles. Building on experimentally demonstrated 1$\times N$ integrated beam splitters with arbitrary power ratios, we propose an on-chip architecture where each output arm hosts an Nb$_3$Cl$_8$ patch, optionally gated by graphene to tune the complex $n$-photon amplitudes. Using the ab-initio $χ^{(3)}$ and $χ^{(4)}$ values, we predict that three-photon GHZ$_3$ and four-photon cluster-state sources in this platform can achieve $n$-photon generation rates up to $\sim 10^8$ and $\sim 10^6$ times larger, respectively, than silica-fiber- and MoS$_2$-based implementations with comparable geometry. We derive the quantum Hamiltonian and explicit $n$-photon generation rates for this platform, and show how suitable interferometric networks enable electrically and spectrally tunable GHZ, $W$, and cluster states based on genuine high-order nonlinear processes in a 2D excitonic Mott insulator.