  Shaping single-mode operation in high-power fibres requires a precise
knowledge of the gain-medium optical properties. This requires accurate
measurements of the refractive index differences ($\Delta$n) between the core
and the cladding of the fiber. We exploit a quantum optical method based on
low-coherence Hong-Ou-Mandel interferometry to perform practical measurements
of the refractive index difference using broadband energy-time entangled
photons. The precision enhancement reached with this method is benchmarked with
a classical method based on single photon interferometry. We show in classical
regime an improvement by an order of magnitude of the precision compared to
already reported classical methods. Strikingly, in the quantum regime, we
demonstrate an extra factor of 4 on the accuracy enhancement, exhibiting a
state-of-the-art $\Delta$n precision of $6.10^{-7}$. This work sets the quantum
photonics metrology as a powerful characterization tool that should enable a
faster and reliable design of materials dedicated to light amplification.
