Nonlinear spin-orbit interaction of light in a cold Rydberg atom medium

We demonstrate that a nonlinear spin-orbit interaction (SOI) emerges in cold Rydberg atomic gases, where the orbital characteristics (spatial structure) of light fields become coupled to their polarization (spin). This SOI originates from the giant, spatically inhomogeneous nonlinear optical response arising from the strong, long-range interactions between Rydberg atoms. By extending the theoretical framework beyond the conventional paraxial approximation, we derive a light propagation equation that rigorously incorporates such inhomogeneous nonlinearity. Our study reveals that the SOI is driven by the gradient of inhomogeneous nonlinear response and therefore vanishes in linear or homogeneous media. When two orthogonal light fields co-propagate, the SOI generates novel nonlinear energy transfer and pronounced structure deformation, which are not captured without considering the SOI and in the paraxial approximation. Furthermore, owing to its direct dependence on the gradient of the nonlocal nonlinearity, the SOI is inherently mode-dependent. As a consequence, the structural deformation becomes particularly significant for light field with odd parity symmetry. Our investigation establishes novel pathways for exploring the nonlinear SOI and examining inhomogeneous nonlinear optical responses beyond the paraxial approximation in cold Rydberg atomic gases.