Passive Cooling of a Micromechanical Oscillator with a Resonant Electric Circuit

We cool the fundamental mode of a miniature cantilever by capacitively coupling it to a driven rf resonant circuit. Cooling results from the rf capacitive force, which is phase shifted relative to the cantilever motion. We demonstrate the technique by cooling a 7 kHz cantilever from room temperature to 45 K, obtaining reasonable agreement with a model for the cooling, damping, and frequency shift. Extending the method to higher frequencies in a cryogenic system could enable ground state cooling and may prove simpler than related optical experiments in a low temperature apparatus.

Figure 1

Schematics of the cantilever cooling and detection electronics. (a) Cantilever and associated rf circuitry. (b) Motional detection electronics. Near ${\omega }_c$ the rf circuit looks like a short to ground as shown. (c) Equivalent circuit for the cantilever and detection electronics near $\omega \approx {\omega }_c$.

Figure 2

Cantilever thermal spectra for four values of rf power. The $x$ axis for each spectrum has been shifted to align the maxima of the three data sets. $S_v$ is the measured noise referred to the input of the amplifier. Solid lines are fits to the model in Fig. 1c, giving the temperatures indicated by arrows. (Inset) Cantilever frequency shift $\Delta f$ vs rf power.

Figure 3

$T_{\mathrm{eff}}$ as a function of rf power. The solid line is the temperature predicted by Eq. (6) using $\Gamma$ from the fit in the inset and $e_n=e_n(R_{\mathrm{eq}})$, while the dashed line takes into account additional noise due to the rf source. (Inset) Damping rate vs rf power. The solid line is a linear fit to the first ten points.

Figure 4

Variation of the cantilever resonance with respect to rf frequency. (a) Cantilever frequency $f_c$ and (b) damping rate $\Gamma$ vs normalized rf detuning $\Delta \Omega /\gamma$ for several values of $P_{\mathrm{rf}}$. The missing points between $-0.7\mathrm{MHz}$ and 0 MHz correspond to a region of instability where $\Gamma$ becomes negative. Solid lines are fits to Eqs. (4, 5).