Induced transparency in optomechanically coupled resonators

In this work we theoretically investigate a hybrid system of two optomechanically coupled resonators, which exhibits induced transparency. This is realized by coupling an optical ring resonator to a toroid. In the semiclassical analyses, the system displays bistabilities, isolated branches (isolas), and self-sustained oscillation dynamics. Furthermore, we find that the induced transparency window sensitively relies on the mechanical motion. Based on this fact, we show that the described system can be used as a weak force detector and the optimal sensitivity can beat the standard quantum limit without using feedback control or squeezing under available experimental conditions.

Figure 1

Schematic for the coupled-resonator system with optomechanical coupling. An optical ring resonator (the gray ring) is coupled to a toroid (the red ring) evanescently. In the toroid, the optical mode and mechanical mode interact through radiation pressure.

Figure 2

Left panel: The bifurcation diagram for the position of the mechanical resonator in the ($\mathrm{\Delta },I_{\text{in}}$) plane in an unresolved sideband regime. The red solid curve labels the boundary of saddle-node bifurcations and the black dashed curve labels the boundary of Hopf bifurcations. The stable region is denoted by region I, which is the area outside of the red and black curves. The area within the red curve is region II, in which multiple steady states exist. Region III, enclosed with a black curve, is the parametric unstable region, with self-sustained oscillations created in the Hopf bifurcation. In the right panel, (a) is a cross section of the left panel labeled by the white dashed line. The blue, red, and green curves denote stable, unstable, and parametric unstable solutions. (b) shows the corresponding light transmission of (a). The parameters are $k_a/{\omega }_m=0.6, k_b/{\omega }_m=4, k_{\text{ex}}/{\omega }_m=4, \delta /{\omega }_m=20, g_1/{\omega }_m=0.03, g_2/{\omega }_m=4$, and ${\gamma }_m/{\omega }_m=0.1$. (c) and (d) present the dynamics and its corresponding phase-space picture for one point in region III $(\mathrm{\Delta }/{\omega }_m,I_{\text{in}}/{\omega }_m)=(-0.25,1.1\times {10}^5)$, as labeled using a white star in the left panel.

Figure 3

Left panel: The bifurcation diagram for the position of the mechanical resonator in the ($\mathrm{\Delta },I_{\text{in}}$) plane in a resolved sideband regime. Right panel: The transmission as a function of detuning $\mathrm{\Delta }$ with different input light intensities. (a) $I_{\text{in}}/{\omega }_m=1$, (b) $I_{\text{in}}/{\omega }_m=200$, (c) $I_{\text{in}}/{\omega }_m=230$, and (d) $I_{\text{in}}/{\omega }_m=250$, indicated by the white dashed lines in the left panel. The parameters are ${\kappa }_a/{\omega }_m=0.1, {\kappa }_b/{\omega }_m=0.0002, {\kappa }_{\text{ex}}/{\omega }_m=0.1, \delta /{\omega }_m=0, g_1/{\omega }_m=0.001, g_2/{\omega }_m=0.02$, and ${\gamma }_m/{\omega }_m=0.001$.

Figure 4

(a) The scheme for weak force detection by measuring phase shifts when a light passes through the hybrid CRIT-optomechanical system. (b) The dispersion of the system as a function of detuning $\mathrm{\Delta }$ with (red dashed line) and without (blue solid blue line) external forces. (d) is the zooming in of the labeled region of (b). (c) The phase shift as a function of the external force $f$. The blue solid line presents the numerical result and the red dashed line is the approximate analytical result based on Eq. (13). $f_0=\sqrt{\hslash m{\omega }_m^2}$. The parameters are ${\kappa }_a/{\omega }_m=0.1, {\kappa }_b/{\omega }_m=0.0002, {\kappa }_{\text{ex}}/{\omega }_m=0.1, \delta /{\omega }_m=0, g_1/{\omega }_m=0.001, g_2/{\omega }_m=0.01, I_{\text{in}}/{\omega }_m=0.1$, and ${\gamma }_m/{\omega }_m=0.1$.