Silicon optomechanical crystal resonator at millikelvin temperatures

Optical measurements of a nanoscale silicon optomechanical crystal cavity with a mechanical resonance frequency of $3.6$ GHz are performed at subkelvin temperatures. We infer optical-absorption-induced heating and damping of the mechanical resonator from measurements of phonon occupancy and motional sideband asymmetry. At the lowest probe power and lowest fridge temperature ($T_{\mathrm{f}}=10$ mK), the localized mechanical resonance is found to couple at a rate of ${\gamma }_{\mathrm{i}}/2\pi =400$ Hz ($Q_{\mathrm{m}}=9\times {10}^6$) to a thermal bath of temperature $T_{\mathrm{b}}\approx 270$ mK. These measurements indicate that silicon optomechanical crystals cooled to millikelvin temperatures should be suitable for a variety of experiments involving coherent coupling between photons and phonons at the single quanta level.

Figure 1

(a) SEM image of the OMC device. (b) Zoomed in SEM image showing details of the waveguide-cavity coupling region. (c) Normalized optical cavity reflection spectrum ($R$). Detunings from resonance of $\Delta =\pm {\omega }_{\mathrm{m}}/2\pi =\pm 3.6$ GHz are denoted by the red and blue arrows, respectively. (d) Schematic of the fiber-based heterodyne receiver used to perform optical and mechanical spectroscopy of the OMC cavity. $\lambda$ meter: wave meter, $\phi$-m: electro-optic phase modulator, VOA: variable optical attenuator, VC: variable coupler, RSA: real-time spectrum analyzer.

Figure 2

(a) Measured mechanical linewidth $\gamma$ for $\Delta ={\omega }_{\mathrm{m}}$ (red) and $-{\omega }_{\mathrm{m}}$ (blue) at a fridge temperature of $T_{\mathrm{f}}=4$ K. The vertical blue dashed line indicates the threshold $n_{\mathrm{c}}$ beyond which the mechanical resonance self-oscillates for $\Delta =-{\omega }_{\mathrm{m}}$, resulting in a 40 dB increase in the mechanical signal level. Black circles indicate the values of ${\gamma }_{\mathrm{i}}$ obtained by taking the average of the detuned data. The inset shows ${\gamma }_{\mathrm{OM}}$ determined by subtracting ${\gamma }_{\mathrm{i}}$ from the red-detuned $\gamma$ (circles), and from cooperativity $C$ using the calibrated $\langle n\rangle$ (squares). A linear fit (red line) yields $g_0/2\pi =840$ kHz. (b) Calibrated mechanical mode occupancy $\langle n\rangle$ versus intracavity photon number $n_{\mathrm{c}}$. Blue and red circles are measured with probe laser detunings $\Delta =\pm {\omega }_{\mathrm{m}}$, respectively. The mode occupancy $n_{\mathrm{f}}$ corresponding to $T_{\mathrm{f}}=4$ K is indicated by a black solid line. (c) Series of red-detuned NPSD for range of $n_{\mathrm{c}}$. Here the NPSD is plotted as $S_{\mathrm{xx}}=x_{\mathrm{zpf}}^2{S}_{\mathrm{bb}}$, where $x_{\mathrm{zpf}}=4.1$ fm is the zero-point amplitude of the breathing mode.

Figure 3

(a) Mechanical linewidth versus $n_{\mathrm{c}}$ for red- (red circles, $\Delta ={\omega }_{\mathrm{m}}$) and blue-detuned (blue circles, $\Delta =-{\omega }_{\mathrm{m}}$) probes at $T_{\mathrm{f}}=185$ mK. The blue dashed line indicates the self-oscillation threshold. (b) Measured resonant signal power (circles) versus optical detuning, $\Delta$, for $n_{\mathrm{c}}\sim 1.5$. The red curve shows a best fit to the data, yielding $C={\gamma }_{\mathrm{OM}}/{\gamma }_{\mathrm{i}}=3.9$ at $\Delta ={\omega }_{\mathrm{m}}$. Here we have assumed, to good approximation when ${\omega }_{\mathrm{m}}/\kappa \gg 1$ and ${(\Delta -{\omega }_{\mathrm{m}})}^2\gg {(\Delta +{\omega }_{\mathrm{m}})}^2$, a back-action damping ${\gamma }_{\mathrm{OM}}\left(\Delta \right)=(4g_0^2{n}_{\mathrm{c}}/\kappa ){[1+4{(\Delta -{\omega }_{\mathrm{m}})}^2/{\kappa }^2]}^{-1}$, cooled mode occupancy $\langle n\rangle \propto {[1+{\gamma }_{\mathrm{OM}}\left(\Delta \right)/{\gamma }_{\mathrm{i}}]}^{-1}$, and heterodyne resonant signal power $\propto \langle n\rangle {\gamma }_{\mathrm{OM}}$. The black dashed curve shows the expected signal in the absence of back-action cooling ($C\to 0$), consistent with assuming ${\gamma }_{\mathrm{i}}$ is equal to the measured time-averaged linewidth. (c) Corresponding measured (circles) mechanical linewidth versus $\Delta$. The red curve is a fit with $C(\Delta ={\omega }_{\mathrm{m}})$ constrained ($=3.9$), but assuming a Voigt line shape with additional Gaussian frequency jitter term. The dashed black curve is the best fit for $C$ constrained ($=3.9$) but with no additional frequency jitter term.

Figure 4

(a) Measured phonon occupancy versus intracavity photon number for a resonant probe at $T_{\mathrm{f}}=10$ mK. The red line shows a power-law fit to the occupancy. The right axis shows the equivalent bath temperature, $T_{\mathrm{p}}$. (b) Measured mechanical linewidth versus $n_{\mathrm{c}}$ for resonant (gray circles) and red-detuned (red circles) probes. The blue circles show the best-fit values of ${\gamma }_{\mathrm{i}}={\gamma }_0+{\gamma }_{\mathrm{p}}$ to both $T_{\mathrm{f}}=10$ and 635 mK data sets. The best-fit value for ${\gamma }_0$ and a smooth curve fit to the inferred ${\gamma }_{\mathrm{p}}$ are shown as a black dashed line and a solid red line, respectively. (c) Diagram illustrating the proposed model of optical-absorption-driven heating of the mechanics and (d) schematic showing the various baths coupled to the localized mechanical mode (see text for details). (e) Calibrated mode occupancy versus $n_{\mathrm{c}}$ for $T_{\mathrm{f}}=10$ mK and 635 mK. Best fits to both temperature data sets using the proposed heating model are shown as solid curves, with shaded regions representing the variation in the fit for ${\gamma }_0/2\pi =306\pm 28$ Hz. The inset shows the measured asymmetry $\xi$ as a function of $n_{\mathrm{c}}$, with the prediction of the best-fit model shown as solid curves. $T_{\mathrm{f}}=10$ mK data (fits) are shown as purple circles (solid curves). $T_{\mathrm{f}}=635$ mK data (fits) are shown as green circles (solid curves).