  We demonstrate non-classical correlations between phonons and photons created
using opto-mechanical spontaneous parametric down-conversion in a system based
on a soft-clamped ultracoherent membrane oscillator inside of a Fabry-P\'erot
optical resonator. We show that phonons stored in the mechanical oscillator,
when subsequently read out, display strong signs of quantum coherence, which we
demonstrate by single-photon counting enabled by our state-of-the-art optical
filtering system. We observe a violation of the classical Cauchy-Schwarz
inequality with a confidence of >92%. The presented system demonstrates the
potential for studies of low-frequency quantum effects in sub-millimeter size
nanogram-scale mechanical oscillators.
