  Recently observed quantum emitters in hexagonal boron nitride (hBN) membranes
have a potential for achieving high accessibility and controllability thanks to
the lower spatial dimension. Moreover, these objects naturally have a high
sensitivity to vibrations of the hosting membrane due to its low mass density
and high elasticity modulus. Here, we propose and analyze a spin-mechanical
system based on color centers in a suspended hBN mechanical resonator. Through
group theoretical analyses and ab-initio calculation of the electronic and spin
properties of such a system, we identify a spin doublet ground state and
demonstrate that a spin-motion interaction can be engineered which enables
ground state cooling of the mechanical resonator. We also present a toolbox for
initialization, rotation, and readout of the defect spin qubit. As a result the
proposed setup presents the possibility for studying a wide range of physics.
To illustrate its assets, we show that a fast and noise resilient preparation
of a multicomponent cat state and a squeezed state of the mechanical resonator
is possible; the latter is achieved by realizing the extremely detuned,
ultrastrong coupling regime of the Rabi model, where a phonon superradiant
phase transition is expected to occur.

Quantum Physics;Mesoscale and Nanoscale Physics