  A quartz Bulk Acoustic Wave resonator is designed to coherently trap phonons
in a way that they are well confined and immune to suspension losses so they
exhibit extremely high acoustic $Q$-factors at low temperature, with $Q\times
f$ products of order $10^{18}$ Hz. In this work we couple such a resonator to a
SQUID amplifier and investigate effects in the strong signal regime. Both
parallel and series connection topologies of the system are investigated. The
study reveals significant non-Duffing response that is associated with the
nonlinear characteristics of Josephson junctions. The nonlinearity provides
quasi-periodic structure of the spectrum in both incident power and frequency.
The result gives an insight into the open loop behaviour of a future Cryogenic
Quartz Oscillator in the strong signal regime.
