  The nonclassical signature of a photon-phonon pair can be tested effectively
by violating Cauchy-Schwarz and Bell's inequality, which can arise due to
antibunching phenomena in coupled bosonic systems. In this paper, we analyze
the measurement criteria imposed on the second-order coherence functions and
investigate the quantum correlations leading to the suppression of
multi-photon-phonon excitation in a single optomechanical cavity upon driving
it with two pumping fields. It is also shown that the Cauchy-Schwarz violation
can serve as an ideal precursor to demonstrate stronger tests of locality
related to Bell's inequality. We consider weak driving and optomechanical
coupling coefficient parameters in the system that enables the unconventional
nature of photon (phonon) blockades while operating in the resonance of cavity
detuning and mechanical frequency. These findings are valuable for generating
sub-Poissonian signals in optimal conditions and have potential applications in
hybrid systems for on-demand single photon (phonon) detection.
