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.

Figure 1

A schematics of the setup allowing one to detect the vibrations of a cantilever using homodyne detection. The dimensions of the cantilever are $100\times 10\times 10\mu {\text{m}}^3$ (for more details, see Sec. 4).

Figure 2

Mechanical resonator subject to optical probing. The physical properties are described in the text ($\omega =2\pi \times 1$ MHz, $\gamma =2\pi \times 10$ Hz, laser full power 1 mW, initial temperature 400 mK). The sequence used here is the following: (1) we start at thermal equilibrium with no probing laser; (2) after 1 $\mu \mathrm{s}$, we apply the probe laser at half power (feedforward); (3) at 1.5 $\mu \mathrm{s}$ we apply the probe laser at full power; (4) at 2.5 $\mu \mathrm{s}$ we detect the probe laser phase continuously, which progressively localizes the resonator on a sine-wave oscillation with random amplitude and phase; (5) after 10 $\mu \mathrm{s}$, we apply an active feedback process (which includes a 1-$\mu \mathrm{s}$ delay time) that keeps the resonator near its rest position by acting on the probe laser power. Top panel: Mean position of the resonator (the gray-shaded area of the curve indicates the uncertainty). Middle panel: Two times the standard deviation (root mean square) of the position and momentum variables. Bottom panel: Variation in the probe laser flux normalized to its final constant value, showing the excitation at half power and the oscillatory feedback modulation, conditioned on the measurement outcome.