  We demonstrate strong dispersive coupling between a fluxonium superconducting
qubit and a 690 megahertz mechanical oscillator, extending the reach of circuit
quantum acousto-dynamics (cQAD) experiments into a new range of frequencies. We
have engineered a qubit-phonon coupling rate of
$g\approx2\pi\times14~\text{MHz}$, and achieved a dispersive interaction that
exceeds the decoherence rates of both systems while the qubit and mechanics are
highly nonresonant ($\Delta/g\gtrsim10$). Leveraging this strong coupling, we
perform phonon number-resolved measurements of the mechanical resonator and
investigate its dissipation and dephasing properties. Our results demonstrate
the potential for fluxonium-based hybrid quantum systems, and a path for
developing new quantum sensing and information processing schemes with phonons
at frequencies below 700 MHz to significantly expand the toolbox of cQAD.
