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

Inducing Strong Non-Linearities in a Phonon Trapping Quartz Bulk Acoustic Wave Resonator Coupled to a Superconducting Quantum Interference Device