  We study a setup where a single negatively-charged silicon-vacancy center in
diamond is magnetically coupled to a low-frequency mechanical bending mode and
via strain to the high-frequency phonon continuum of a semi-clamped diamond
beam. We show that under appropriate microwave driving conditions, this setup
can be used to induce a laser cooling like effect for the low-frequency
mechanical vibrations, where the high-frequency longitudinal compression modes
of the beam serve as an intrinsic low-temperature reservoir. We evaluate the
experimental conditions under which cooling close to the quantum ground state
can be achieved and describe an extended scheme for the preparation of a
stationary entangled state between two mechanical modes. By relying on
intrinsic properties of the mechanical beam only, this approach offers an
interesting alternative for quantum manipulation schemes of mechanical systems,
where otherwise efficient optomechanical interactions are not available.
