Analytical description of quantum emission in optical analogs to gravity

We consider a moving refractive index perturbation in an optical medium as an optical analog to waves under the influence of gravity. We describe the dielectric medium by the Lagrangian of the Hopfield model. We supplement the field theory in curved spacetime for this model to solve the scattering problem for all modes and frequencies analytically. Because of dispersion, the kinematic scenario of the field modes may contain optical event horizons for some frequencies. We calculate the spectra of spontaneous emission in the frame comoving with the perturbation and in the laboratory frame. We also calculate the spectrally resolved photon number correlations in either frame. The emitted multimode field comes in different types depending on the presence of horizons. We show that these types are robust against changes in the system parameters and thus are genuine features of optical and nonoptical analogues. These methods and findings pave the way to new observations of analog gravity in dispersive systems.