Entanglement analysis and experimental proposal for measuring Bell-type correlations in an injected optical parametric oscillator with structured light

We study a type-II optical parametric oscillator under injection of laser beams with first-order Hermite-Gauss modes. We develop the equations describing the dynamics of this system in the Wigner representation of the density matrix. These equations are solved numerically and compared with the analytical solution in the nondepletion regime. We show that, when the injected beams are much less intense than the pumping beam, the system still exhibits quantum entanglement, which can be witnessed via Duan-Simon criterion and Bell inequalities. We also propose an experimental apparatus for measuring Bell inequalities. This system can generate spin-orbit hyperentangled states, and is thus a useful resource to quantum technology applications and fundamental physics.