Lightcone fluctuations in a nonlinear medium due to thermal fluctuations

We study the flight time fluctuations of a probe light propagating in a slab of nonlinear optical material with an effective fluctuating refractive index caused by thermal fluctuations of background photons at a temperature $T$, which are analogous to the lightcone fluctuations due to fluctuating spacetime geometry when gravity is quantized. A smoothly varying second order susceptibility is introduced, which results in that background field modes whose wavelengths are of the order of the thickness of the slab give the main contribution. We show that, in the low-temperature limit, the contribution of thermal fluctuations to the flight time fluctuations is proportional to $T^4$, which is a small correction compared with the contributions from vacuum fluctuations, while in the high-temperature limit, the contribution of thermal fluctuations increases linearly with $T$, which dominates over that of vacuum fluctuations. Numerical estimation shows that, in realistic situations, the contributions from thermal fluctuations are still small compared with that from vacuum fluctuations even at room temperature.