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 (OT) 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 OT number, and thus the resonance frequency. Also, the phonon trapping improves with higher OT numbers, which allows the electrode size to be reduced such that in the optimal case the parasitic capacitive impedance becomes independent of the OT number, allowing effective coupling to very high frequency OTs. © 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.