Lattice topology and spontaneous parametric down-conversion in quadratic nonlinear waveguide arrays

We analyze spontaneous parametric down-conversion in various experimentally feasible one-dimensional quadratic nonlinear waveguide arrays, with emphasis on the relationship between the lattice's topological invariants and the biphoton correlations. Nontrivial topology results in a nontrivial "winding" of the array's Bloch waves, which introduces additional selection rules for the generation of biphotons, independent of existing control using the pump beam's spatial profile and phase-matching conditions. In finite lattices, nontrivial topology produces single-photon edge modes, resulting in "hybrid" biphoton edge modes, with one photon localized at the edge and the other propagating into the bulk. When the single-photon band gap is sufficiently large, these hybrid biphoton modes reside in a band gap of the bulk biphoton Bloch wave spectrum. Numerical simulations support our analytical results.