Abstract
We study the problem of estimating the phase shift due to the general relativistic time dilation in the interference of photons using a non-linear Mach-Zender interferometer setup. By introducing two non-linear Kerr materials, one in the bottom and one in the top arm, we can measure the non-linear phase φ_NL produced by the space-time curvature and achieve a scaling of the standard deviation with photon number (N) of 1/N^β where β > 1, which exceeds the conventional Heisenberg limit of a linear interferometer (1/N). The non-linear phase shift is an effect that is amplified by the intensity of the probe field. In a regime of high number of photons, this effect can dominate over the linear phase shift.