Giant microwave-optical Kerr nonlinearity via Rydberg excitons in cuprous oxide

Microwave-optical conversion is key to future networks of quantum devices, such as those based on superconducting technology. Conversion at the single quantum level requires strong nonlinearity, high bandwidth, and compatibility with a millikelvin environment. A large nonlinearity is observed in Rydberg atoms, but combining atomic gases with dilution refrigerators is technically challenging. Here we demonstrate that a strong microwave-optical nonlinearity in a cryogenic, solid-state system by exploiting Rydberg states of excitons in \cuprite. We measure a microwave-optical cross-Kerr coefficient of $B_0 = 0.022 \pm 0.008 $ m V$^{-2}$ at 4~K, which is several orders of magnitude larger than other solid-state systems. Our results highlight the potential of Rydberg excitons for nonlinear optics, and form the basis for a microwave-optical frequency converter based on Cu$_2$O.