  Enhancing and tailoring light-matter interactions offer remarkable nonlinear
resources with wide-ranging applications in various scientific disciplines. In
this study, we investigate the construction of strong and deterministic
tripartite `beamsplitter' (`squeeze') interactions by utilizing cavity-enhanced
nonlinear anti-Stokes (Stokes) scattering within the spin-photon-phonon degrees
of freedom. We explore the exotic dynamical and steady-state properties
associated with the confined motion of a single atom within a high-finesse
optical cavity. Notably, we demonstrate the direct extraction of vacuum
fluctuations of photons and phonons, which are inherent in Heisenberg's
uncertainty principle, without requiring any free parameters. Moreover, our
approach enables the realization of high-quality single-quanta sources with
large average photon (phonon) occupancies. The underlying physical mechanisms
responsible for generating nonclassical quantum emitters are attributed to
decay-enhanced single-quanta blockade and the utilization of long-lived
motional phonons, resulting in strong nonlinearity. This work unveils
significant opportunities for studying hitherto unexplored physical phenomena
and provides novel perspectives on fundamental physics dominated by strong
tripartite interactions.
