Enhanced optomechanical levitation of minimally supported dielectrics

Optically levitated mechanical sensors promise isolation from thermal noise far beyond what is possible using flexible materials alone. One way to access this potential is to apply a strong optical trap to a minimally supported mechanical element, thereby increasing its quality factor $Q_m$. Current schemes, however, require prohibitively high laser power ($\sim 10$ W), and the $Q_m$ enhancement is ultimately limited to a factor of $\sim 50$ by hybridization between the trapped mode and the dissipative modes of the supporting structure. Here we propose a levitation scheme taking full advantage of an optical resonator to reduce the circulating power requirements by many orders of magnitude. Applying this scheme to the case of a dielectric disk in a Fabry-Perot cavity, we find a tilt-based tuning mechanism for optimizing both center-of-mass and torsional-mode traps. Notably, the two modes are trapped with comparable efficiency, and we estimate that a $10-\mu$m-diameter, 100-nm-thick Si disk could be trapped to a frequency of $\sim 10$ MHz with only 30 mW circulating in a cavity of (modest) finesse 1500. Finally, we simulate the effect that such a strong trap would have on a realistic doubly tethered disc. Of central importance, we find torsional motion is comparatively immune to $Q_m$-limiting hybridization, allowing a $Q_m$-enhancement factor of $\sim 1500$. This opens the possibility of realizing a laser-tuned 10 MHz mechanical system with a quality factor on the order of ${10}^9$.

Figure 1

A proposed levitation geometry, comprised of a doubly tethered dielectric disk immersed in a cavity optical field. Inset: The intensity profile of the cavity's ${\text{TEM}}_{00}$ and ${\text{TEM}}_{10}$ modes relative to the structure.

Figure 2

Quadratic optomechanical coupling. When two optical modes of linear coupling constants $G_1$ (red) and $G_2$ (green) are degenerate (at $\xi =0$ and ${\omega }_{\gamma }={\omega }_0$), an intermode scattering rate $\mathrm{\Gamma }$ leads to adiabatic frequencies ${\omega }_{\pm }$ (black dashed curves). The underlying blue gradient represents the optical (amplitude) susceptibility of the mixed modes, with linewidth $\kappa$ set by the decay rate of the cavity. The blue arrow indicates an optical mode with positive quadratic coupling ${\partial }_{\xi }^2{\omega }_{+}>0$, which can be used to generate a stable optical trap.

Figure 3

Single-mode and enhanced cavity traps. The cavity comprises two mirrors with radius of curvature 2.5 cm separated by a length $L=4.9$ cm, and a 50-nm-thick ${\text{Si}}_3{\text{N}}_4$ membrane ($n=2$) positioned near the waist ($x_0=0$). Blue arrows indicate stable cavity traps, solid curves show the analytical theory, and diamonds show a numerical solution with no approximations. (a) Detuning of ${\text{TEM}}_{00}$ and ${\text{TEM}}_{10}$ cavity modes vs displacement $x_0$ for an aligned (${\theta }_z={\theta }_y=0$) membrane. (b) Detuning of the same modes vs tilt about the $\widehat{y}$ (tether) axis for $x_0=0$. (c) and (d) show refined plots of the regions indicated by dotted boxes in (a) and (b) for fixed tilts ${\theta }_z=0,0.1,0.2,0.3$ mrad about $\widehat{z}$. In both cases, the gap is tuned linearly with ${\theta }_z$.

Figure 4

Response of a doubly tethered silicon disk to a strong optical trap. Structure has thickness $t=110$ nm, disk radius $r=5\mu \text{m}$, tether length $l=45\mu \text{m}$, and tether width $d=100$ nm. (a) Relevant low-frequency (untrapped) mechanical modes that are symmetric (“s”) and antisymmetric (“a”) about the $y=0$ plane, or torsional (“t”). (b) Eigenfrequencies vs optical trap strength $\propto \sqrt{P}$. The units of trap strength are scaled to remove the dependence on the trapping mechanism (see text). Gray curves correspond to noninteracting in-plane modes. (c) Enhancement $U_{\text{opt}}/U_{\text{mat}}$ for the modes in (b). For CM motion, $U_{\text{opt}}/U_{\text{mat}}$ plummets from hybridization with “violin” modes such as $s_2$. (d) Enhancement for the CM-like (blue) and torsional (red) modes at higher trap strength. For each group, the lightest, intermediate, and darkest shades correspond to $\sigma /r=0.5, 0.75$, and 1, respectively. The CM mode $Q_m$ enhancement is continuously limited by hybridization with tether modes [e.g., inset (i)], whereas the torsional hybridization is suppressed [inset (ii)] until the finite spot size of the trap causes coupling with the “flappy” mode of the disk [inset (iii)]. Shown for comparison is the torsional $Q_m$ enhancement for an infinitely stiff structure (black curve). The values of $\sigma /r$ shown here are compatible with high-finesse cavities: even $\sigma /r=1$ could achieve $F>{10}^4$ for a properly manufactured structure [28].

Figure 5

Purely quartic CM and TM coupling. (a) Detuning of the ${\text{TEM}}_{00}$ and ${\text{TEM}}_{10}$ cavity modes vs displacement of a 50-nm-thick ${\text{Si}}_3{\text{N}}_4$ disk for a cavity of length $L=4.7$ cm and ${\theta }_z$ between 0 and 6 mrad. (b) Detuning vs tilt ${\theta }_y$ under the same conditions. Both (a) and (b) exhibit a transition from quadratic (6 mrad) to double-well (2 mrad) potentials. In between (near 4 mrad), the optomechanical coupling is purely quartic.

Figure 6

Limits of perturbation theory. (a) Detuning of the ${\text{TEM}}_{00}$ cavity resonances (near wavelength $\lambda =1.55\mu \text{m}$) for a cavity of length $L=3.5$ cm and mirror radius of curvature 2.5 cm, as a function of disk displacement $x_0$ from the waist. The disk is 110 nm thick and made of single-crystal silicon (index $n=3.48$), and is aligned with the cavity mode (${\theta }_y={\theta }_z=0$). Curves show the results of first-order degenerate perturbation theory, including (i) a single ${\text{TEM}}_{00}$ mode, (ii) $\pm 2 {\text{TEM}}_{00}$ modes (i.e., 5 in total), and (iii) $\pm 100$ (201 total) ${\text{TEM}}_{00}$ modes. The gradient scale shows transmission through the cavity calculated using transfer matrices in 1D for comparison (end mirror amplitude transmission coefficients set to 0.3 for visibility). The qualitative behavior of perturbation theory approaches that of the transfer matrices; however, including $\pm 1000$ modes does not noticeably change the result from the solid black curve, hinting that this perturbation is too large to be quantitatively captured by a first-order theory. (b) Displacement-mediated avoided crossings between the ${\text{TEM}}_{00}$ and ${\text{TEM}}_{10}$ modes, including $\pm 100$ of each type (402 in total), for ${\theta }_z=0.0,0.1,0.2$, and 0.3 mrad. The gap tuning mechanism is roughly linear in ${\theta }_z$, though a shift in the crossing point is introduced via interactions with adjacent longitudinal modes. (c) Tilt-mediated avoided crossings under the same conditions as (b). The gap tuning is again linear in ${\theta }_z$, but the crossing modes are no longer symmetrically tuned by ${\theta }_y$, leading to a skewed crossing. Using this numerically determined quadratic coupling, a disk of radius $r=75\mu \text{m}$ (i.e., equal to the ${\text{TEM}}_{00}$ spot size) experiences a trap frequency ratio ${\omega }_{\text{TM}}/{\omega }_{\text{CM}}=0.43$.