Detection of non-local spin-entanglement by light emission from a superconducting pn-junction

We model a superconducting pn-junction in which the n- and the p-site are contacted through two optical quantum dots, each embedded into a photonic nanocavity. Whenever a Cooper pair is transported from the n-site to the p-site, two photons are emitted. When the two electrons of a Cooper pair are transported through different quantum dots, polarization entangled photons are created, provided that the Cooper pairs retain their spin-singlet character although being spatially separated on the two quantum dots. We show that a CHSH Bell-type measurement is able to detect the entanglement of the photons over a broad range of microscopic parameters, even in the presence of parasitic processes and imperfections. Observing this signature is a direct proof of crossed Andreev reflection, or, equivalently, Cooper pair splitting, retaining the spin-singlet wave function.