  This work presents some peculiarities of the near quantum ground state
behaviour of curved (phonon trapping) Bulk Acoustic Wave (BAW) cavities when
compared to a conventional mechanical resonator. The curved cavity system
resolves the quandary of the conventional mechanical system where the
Bose-Einstein distribution requires higher frequencies for lower quantum
occupation factors contrary to the constraint of an inverse frequency
dependence of the quantum fluctuations of displacement. We demonstrate how the
non-trivial cavity geometry can lead to better phonon trapping, enhancing the
variance of zero-point-fluctuations of displacement. This variance becomes
independent of overtone number (or BAW resonance frequency) overcoming the
constraint and allowing better observation of quantum effects in a mechanical
system. The piezoelectric electro-mechanical coupling approach is qualitatively
compared to the parametric optomechanical technique for the curved BAW
cavities. In both cases the detectible quantity grows proportional to the
square root of the overtone number, and thus the resonance frequency. Also, the
phonon trapping improves with higher overtone numbers, which allows the
electrode size to be reduced such that in the optimal case the parasitic
capacitive impedance becomes independent of the overtone number, allowing
effective coupling to very high frequency overtones.
