Figure 1

Schematic diagram of the system. (a) Side view of the toroid. (b) Top view. We show the coupling of the cavity fields to the field in the waveguide.Reuse & Permissions

Figure 2

Regions of (a) 1-1-mode (“reduction”) entanglement, or (b) of 1-2-mode (“bipartition”) entanglement. In (a) the dashed curve encompasses the (red) region where the two-mode reduction consisting of the second-harmonic optical mode (system $\mathrm{S}$) and the mechanical mode $\mathrm{M}$ is not separable, the dotted curve encompasses the (green) region where the fundamental optical mode (system $\mathrm{F}$) and the mechanical mode are not separable, and the dashed-dotted curve delineates the (blue) region where the two optical modes are entangled. A similar explanation holds for subfigure (b), but with the dashed curve corresponding to entanglement between the fundamental optical mode and the two-mode system formed by the second-harmonic mode and the mechanical mode. The dotted curve delineates inseparability of the second-harmonic from the two-mode system formed by the other two modes, and likewise the dashed-dotted curve bounds the region where the mechanical mode is entangled with the two-mode system consisting of the optical modes. The solid curve encloses the region of instability. (c) and (d) are similar to (a) and (b), respectively, but with a larger mechanical $Q$ factor. Because of the smaller mechanical decay rate, the region of instability starts at significantly lower powers, especially on the blue-detuned side of the figure.Reuse & Permissions

Figure 3

Tripartite logarithmic negativity, $E_{\mathcal{N}}^{\mathrm{tri}}$, for five different input powers and both low- and high-$Q_{\mathrm{m}}$ cases.Reuse & Permissions

Figure 4

Similar to Fig. 2, but varying the second-harmonic generation rate on the vertical axis. (a) and (b) have $Q_{\mathrm{m}}=5970$, whereas (c) and (d) have $Q_{\mathrm{m}}=597000$. (${\chi }_0=2\pi \times 700$Hz, $P_{\mathrm{in}}={10}^{-3}$W.)Reuse & Permissions

Figure 5

Schematic of detection system. Part of the input field is used as the local oscillator of two homodyne detectors (this necessitates adding the second harmonic onto the local oscillator), one recording the field quadratures before the toroid, and one after.Reuse & Permissions