Cavity Optomechanics with Stoichiometric SiN Films

We study high-stress SiN films for reaching the quantum regime with mesoscopic oscillators connected to a room-temperature thermal bath, for which there are stringent requirements on the oscillators’ quality factors and frequencies. Our SiN films support mechanical modes with unprecedented products of mechanical quality factor $Q_m$ and frequency ${\nu }_m$ reaching $Q_m{\nu }_m\simeq 2\times {10}^{13}\mathrm{Hz}$. The SiN membranes exhibit a low optical absorption characterized by $\mathrm{Im}(n)\lesssim {10}^{-5}$ at 935 nm, representing a 15 times reduction for SiN membranes. We have developed an apparatus to simultaneously cool the motion of multiple mechanical modes based on a short, high-finesse Fabry-Perot cavity and present initial cooling results along with future possibilities.

Figure 1

(a) Experiment schematic: Membrane window in a short cavity. (b) Illustration of $(j,k)=(3,3)$ membrane mode amplitude $a_{jk}$. (c) Optical mode $\psi (x,y)$ [red (medium gray) solid line] compared to membrane modes $(j,k)=(2,2)$ (top) and (6, 6) (bottom) for a $0.5\mathrm{mm}\times 0.5\mathrm{mm}$ square membrane [blue (dark gray) solid line].

Figure 2

Mechanical quality factors of ${\mathrm{Si}}_3{\mathrm{N}}_4$ films measured as a function of resonant frequency for diagonal ($j=k$) modes. Three different data sets are represented: Membrane frame corners resting on a curved surface (open circles), membrane frame glued at three corners (closed circles), membrane frame glued over the full frame (green triangles). The solid black line is a guide to the eye of the maximum $Q_m$ realized. The gray-shaded area represents a $Q$-frequency product less than $k_b{T}_{\mathrm{room}}/h=0.6\times {10}^{13}\mathrm{Hz}$. The dashed blue line is an estimate of the thermoelastic limit based upon an extension of Zener theory for a thin plate at room temperature [7, 11, 17]. (inset) Representative mechanical ringdown for the (8, 8) mode at 6.6 MHz.

Figure 3

Measured cavity finesse $F$ as a function of membrane position $\Delta z$ about the center of the cavity (black circles). Absent the membrane the finesse is $F_0=9160$. The shaded regions represent the envelope of the calculated finesse variation for $\mathrm{Im}(n)=0.6\times {10}^{-5}$ [red (dark gray)] and $\mathrm{Im}(n)=0$ [green (light gray)] for comparison. (Insets) Examples of the local dependence as a function of $\Delta z$ compared to theoretical curves for $\mathrm{Im}(n)=0.6\times {10}^{-5}$.

Figure 4

Demonstration of cavity cooling of the (6, 6) (red circles) and (1, 1) (gray squares) modes with frequencies and initial quality factors of ${\nu }_m=4.82\mathrm{MHz}$; $Q_m=1.5\times {10}^6$ and ${\nu }_m=0.80\mathrm{MHz}$; $Q_m=5.8\times {10}^4$, respectively, where $Q_m$ is measured via ringdown. For these data the cavity output power at resonance is fixed at $P_{\mathrm{out}}^0=10\mu \mathrm{W}$. The representative error bars show the uncertainty due to our knowledge of $g$. (a) Measured mode temperature (points) and theoretically expected cooling (lines) as a function of detuning $\Delta$ from the cavity resonance. (b) Corresponding measured effective mechanical linewidth ${\gamma }_{\mathrm{eff}}$ and theoretical expectation.

Figure 5

(a) Calculated achievable phonon number as a function of coupling strength $g_{\mathrm{eff}}$ for parameters described in the text. The solid line is for a cavity with $\gamma =2\pi \times 5\mathrm{MHz}$ and the dashed line for $\gamma =2\pi \times 1\mathrm{MHz}$. The green (dark gray) line would result for $\gamma =2\pi \times 5\mathrm{MHz}$ upon increasing $Q_m$ to $4\times {10}^7$, which may be possible for a thinner SiN film. (b) Emergence of normal-mode splitting as the cavity linewidth is varied for fixed mechanical frequency ${\omega }_m=2\pi \times 5\mathrm{MHz}$ and coupling $g_{\mathrm{eff}}=2\pi \times 1\mathrm{MHz}$. The exact spectrum $S_{bb}(\omega )$ is displayed in this density plot where zero weight corresponds to the blue (dark gray) shading and white to maximum mechanical response. The lines mark the position of the parameters used in (a).