Asymmetry-Based Quantum Backaction Suppression in Quadratic Optomechanics

As the field of optomechanics advances, quadratic dispersive coupling (QDC) represents an increasingly feasible path toward qualitatively new functionality. However, the leading QDC geometries generate linear dissipative coupling and an associated quantum radiation force noise that is detrimental to QDC applications. Here, we propose a simple geometry that dramatically reduces this noise without altering the QDC strength. We identify optimal regimes of operation, and discuss advantages within the examples of optical levitation and nondestructive phonon measurement.

Figure 1

Membrane-cavity system. (a) Fabry-Perot cavity asymmetrically split by a membrane (blue) of transmission $|t_m|$ into two subcavities of lengths $L_j$. The subcavities are coupled to each other through the membrane at rate $J$. Drive and detection are conducted through external coupling at rates ${\kappa }_j^{\mathrm{ext}}$, while photons are lost at rates ${\kappa }_j^{\mathrm{int}}$. (b) Cavity transmission versus membrane position $\mathrm{\Delta }x$ and drive detuning $\mathrm{\Delta }\equiv {\omega }_{\mathrm{in}}-{\omega }_0$. In this example, $|t_m{|}^2=7\times {10}^4\mathrm{ppm}$, $|t_1{|}^2=6\times {10}^4\mathrm{ppm}$, $|t_2{|}^2=4\times {10}^4\mathrm{ppm}$, and $L_1/L_2={10}^3$. Grey dashed lines show the uncoupled ($J=0$) subcavity frequencies ${\omega }_{\mathrm{in}}={\omega }_j\equiv {\omega }_0[1+(-1{)}^j\mathrm{\Delta }x/L_j]$; orange dashed lines show eigenfrequencies ${\omega }_{\pm }$.

Figure 2

$S_{FF}({\mathrm{\Omega }}_m)$ relative to MIM, for a “typical” canonical setup: cavity length $L=10\mathrm{cm}$, transmissions $|t_m{|}^2={10}^4\mathrm{ppm}$ [54] and $|t_2{|}^2=0$, losses ${\mathcal{L}}_j=1\mathrm{ppm}$ [55, 56], and mechanical frequency ${\mathrm{\Omega }}_m=2\pi \times 240\mathrm{kHz}$, with a wide range of input transmissions $|t_1{|}^2$. The solid (dashed) curves are in the resolved (unresolved) sideband regime, and markers indicate optimal membrane position [Eq. (12)]. For $|t_1{|}^2={10}^3\mathrm{ppm}$, the force noise can be reduced by a factor of 250. Inset: optimal reduction of force noise. The shading indicates the unresolved sideband regime, and the dotted line corresponds to the resolved-sideband approximation [Eq. (16)].

Figure 3

Resolvable fluctuation $x_{\mathrm{res}}^2$ for QND measurement of the mechanical ground state ($n=0$). Our scheme exhibits clear advantage for a wide range of front mirror transmission $|t_1{|}^2$ (black and purple), achieving a factor of 25 improvement over the MIM configuration in a near-single-port system with $|t_1{|}^2={10}^2\mathrm{ppm}$ (black), and provides additional improvement when optimizing $|t_1{|}^2$ [red; see Eq. (19)]. Further reducing $|t_1{|}^2$ provides no benefit due to reduced collection efficiency. The system parameters are ${\mathcal{L}}_1={\mathcal{L}}_2=1\mathrm{ppm}$, $|t_2{|}^2=0\mathrm{ppm}$ (loss-limited Bragg stack), $|t_m{|}^2={10}^4\mathrm{ppm}$, and $L=10\mathrm{cm}$, ${\omega }_{\mathrm{in}}={\omega }_{+}$ in the resolved-sideband regime. The dotted vertical line indicates $L_1=L/2$. The $y$ axis indicates the minimal zero-point variance $x_{\mathrm{ZPF}}^2$ required to resolve the ground state.