Cooling a mechanical resonator to the quantum regime by heating it

We consider a mechanical resonator made of diamond, which contains a nitrogen-vacancy center (NV center) locating at the end of the oscillator. A second-order magnetic gradient is applied and induces coupling between mechanical modes and the NV center. By applying a proper external magnetic field, the energy difference between NV center electron spin levels can be tuned to match the energy difference between two mechanical modes $a$ and $b$. A laser is used for continuously initializing the NV center electron spin. The mode $a$ with lower frequency is driven by a thermal bath. We find that the temperature of the mode $b$ is significantly cooled when the heating bath temperature is increased. We discuss the conditions that quantum regime cooling requires and confirm the results by numerical simulation. Finally we give the intuitive physical explanation of this unusual effect.