  In this study, we observe the nonlinear behavior of the two-photon geometric
phase for polarization states using time-correlated photons pairs. This phase
manifests as a shift of two-photon interference fringes. Under certain
arrangements, the geometric phase can vary nonlinearly and become very
sensitive to a change in the polarization state. Moreover, it is known that the
geometric phase for $N$ identically polarized photons is $N$ times larger than
that for one photon. Thus, the geometric phase for two photons can become two
times more sensitive to a state change. This high sensitivity to a change in
the polarization can be exploited for precision measurement of small
polarization variation. We evaluate the signal-to-noise ratio of the
measurement scheme using the nonlinear behavior of the geometric phase under
technical noise and highlight the practical advantages of this scheme.
