Optomechanical tests of a Schr\"odinger-Newton equation for gravitational quantum mechanics

We show that optomechanical systems can test the Schr"{o}dinger-Newton equation of gravitational quantum mechanics due to Yang et al. This equation is motivated by semiclassical gravity, a widely used theory of interacting gravitational and quantum fields. From the many-body Schr"{o}dinger-Newton equation follows an approximate equation for the center-of-mass dynamics of macroscopic objects. This predicts a distinctive double-peaked signature in the output optical quadrature power spectral density of certain optomechanical systems. Since the Schr"{o}dinger-Newton equation lacks free parameters, these will allow its experimental confirmation or refutation.