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We study the problem of estimating the phase shift due to the generalrelativistic time dilation in the interference of photons using a non-linearMach-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$\phi_{NL}$ produced by the space-time curvature and achieve a scaling of thestandard deviation with photon number ($N$) of $1/N^{\beta}$ where $\beta > 1$,which exceeds the conventional Heisenberg limit of a linear interferometer($1/N$). The non-linear phase shift is an effect that is amplified by theintensity of the probe field. In a regime of high number of photons, thiseffect can dominate over the linear phase shift.

Quantum limited measurement of space-time curvature with scaling beyond the conventional Heisenberg limit