  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
$\phi_{NL}$ produced by the space-time curvature and achieve a scaling of the
standard 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 the
intensity of the probe field. In a regime of high number of photons, this
effect can dominate over the linear phase shift.
