  We report on multi-photon statistics of correlated twin beams produced in a
periodic poled micro-ring resonator on thin-film lithium niobate. Owing to high
cavity confinement and near perfect quasi-phase matching, the photons pairs are
produced efficiently in single modes at rates reaching 27 MHz per $\mu$W pump
power. By using a pump laser whose pulse width impedance matches with the
cavity, those photons are further created in single longitudinal modes with
purity reaching 99\%, without relying on later-on filtering. With a
dual-channel photon-number resolving detection system, we obtain directly the
joint detection probabilities of multi-photon states up to three photons, with
high coincidence to accidental contrast for each. Used as a single photon
source, it gives heralded $g_H^{(2)}(0)$ around 0.04 at a single photon rate of
650 kHz on chip. The findings of our research highlight the potential of this
nanophotonic platform as a promising platform for generating non-classical,
few-photon states with ideal indistinguishability, for fundamental quantum
optics studies and information applications.
