Fig. 1: Schematic representation of the experimental setup and control strategy.

a Shown is a clamped nano/micromechanical cantilever carrying a single defect spin, which in this study is exemplified by an NV electronic spin. The interaction between the NV center spin and the cantilever motion is mediated by strain, while single qubit gates are achieved by microwave (MW) pulses. Initialization and readout processes are facilitated by optical pulses, with a photon detector used for readout. A magnetic tip at the edge of the cantilever creates a gradient field that allows interaction with a nearby spin ensemble, coupling it to the mechanical motion of the cantilever. In the inset, we schematically represent a network of interacting spins as a quantum bus. When the number of spins in the network is large, it can be modeled as a bosonic system (e.g., magnons) by the Holstein–Primakoff transformation. Note that it is simplified here as the fundamental mode of the cantilever’s vibration. b The control sequence of the probe spin to perform the cooling protocol. First, the probe spin is reset to the \(\left\vert 0\right\rangle\) state by optical pulses, then a π/2 MW pulse brings it to a state of equal superposition, which is followed by n inversion MW pulses applied periodically with a carefully chosen time interval τ. To ensure that the effective probe-oscillator coupling strength under pulses is comparable to the oscillator frequency ω, enabling the oscillator to be driven, n should be on the order of ω/g₀. Another π/2 MW pulse is applied before the probe is optically read into the computational base. The above process is repeated M times, bringing the oscillator and the spin ensemble progressively closer to their thermal ground states. Note that the probe to be read must be post-selected in \(\left\vert 0\right\rangle\) each time, otherwise the whole process starts over.