On-chip high-order parametric downconversion in the excitonic Mott insulator Nb₃Cl₈ 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 Nb₃Cl₈, 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 Nb₃Cl₈, 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 1times N integrated beam splitters with arbitrary power ratios, we propose an on-chip architecture where each output arm hosts an Nb₃Cl₈ 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 GHZ₃ and four-photon cluster-state sources in this platform can achieve n-photon generation rates up to sim 10⁸ and sim 10⁶ times larger, respectively, than silica-fiber- and MoS₂-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.