Rapid cooling of a strain-coupled oscillator by an optical phase-shift measurement

We consider an optical probe that interacts with an ensemble of rare-earth ions doping a material in the shape of a cantilever. By optical spectral hole burning, the inhomogeneously broadened transition in the ions is prepared to transmit the probe field within a narrow window, but bending of the cantilever causes strain in the material which shifts the ion resonances. The motion of the cantilever may thus be registered by the phase shift of the probe. By continuously measuring the optical field we induce a rapid reduction of the position and momentum uncertainty of the cantilever. Doing so, the probing extracts entropy and thus effectively cools the thermal state of motion towards a known, conditional oscillatory motion with strongly reduced thermal fluctuations. Moreover, as the optical probe provides a force on the resonator proportional to its intensity, it is possible to exploit the phase-shift measurements in order to create an active feedback loop, which eliminates the thermal fluctuations of the resonator. We describe this system theoretically, and provide numerical simulations which demonstrate the rapid reduction in resonator position and momentum uncertainty, as well as the implementation of the active cooling protocol.