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.
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