Nonlinear generation of quantum-entangled photons from high-Q states in dielectric nanoparticles

We develop multipolar theory of nonlinear generation of entangled photons from subwavelength dielectric particles due to the spontaneous parametric downconversion. We demonstrate that optical excitation in resonance with the high-quality supercavity mode of the aluminum gallium arsenide (AlGaAs) nanodisk leads to a strong enhancement of generation of entangled photon pairs associated with electric and magnetic dipole modes. Our rigorous numerical results are corroborated by an analytical model, universally describing formation of high-$Q$ resonant states and dark states due to the interference and interplay of the parent multipoles, namely, magnetic dipoles and magnetic octupole. Our findings and description can be instructive for quantum and nonlinear nanophotonics applications.