  Hong-Ou-Mandel (HOM) interferometry has emerged as a valuable tool for
quantum sensing applications, particularly in measuring physical parameters
that influence the relative optical delay between pair photons. Unlike
classical techniques, HOM-based quantum sensors offer higher resolution due to
their intrinsic dispersion cancellation property. Despite this advantage,
achieving precise measurements of optical delay crucial for practical
applications often involves time-consuming integration and post-processing with
traditional statistical methods. To address this challenge, our recent work
focused on optimizing optical delay measurements in a time-efficient manner. By
carefully selecting the length of a 1 mm periodically-poled KTP (PPKTP) crystal
for pair photon generation, we achieved a remarkable group index measurement
precision of $\sim 6.75\times 10^{-6}$ per centimeter of sample length,
surpassing the previous maximum precision by over 400$\%$. These current
measurements maintain fast detection and high photon counts, which are
essential for practical quantum sensing applications. The HOM-based method,
while limiting the measurement range, can be extended by compensating for
photon delay using an optical delay stage. As a proof-of-principle, we measured
the group index variation of PPKTP over a temperature range up to
200$^{\circ}$C with a precision in the range of one part per million
($\sim$10$^{-6}$). This advancement not only contributes to quantum sensing but
also holds promising implications for high-precision and long-range
measurements in quantum optical coherence tomography.
