  Ultra-high sensitivity detection of quantum-scale displacements in cavity
optomechanics optimises the combined errors from measurement back-action and
imprecisions from incoming quantum noises. This sets the well-known Standard
Quantum Limit (SQL). Normal quantum cavity optomechanics allows cooling and
detection of a single degree of freedom, along the cavity axis. However, a
recent breakthrough that allows quantum ground-state cooling of levitated
nanoparticles [Delic et al, arxiv:1911.04406], is uniquely 3D in character,
with coupling along the $x$, $y$ and $z$ axes. We investigate current
experiments and show that the underlying behaviour is far from the addition of
independent 1D components and that ground-state cooling and sensing analysis
must consider- to date neglected- 3D hybridisation effects. We characterise the
additional 3D spectral contributions and find direct and indirect hybridising
pathways can destructively interfere suppressing of 3D effects at certain
parameters in order to approach, and possibly surpass, the SQL. We identify a
sympathetic cooling mechanism that can enhance cooling of weaker coupled modes,
arising from optomechanically induced correlations.
