Noise-resistant optomechanical entanglement via cross-Kerr effect

We propose a scheme to achieve entanglement between cavity mode and mechanical mode by breaking dark mode (DM) effect via cross-Kerr effect. This DM induced by the coupling of multiple degenerate mechanical modes to a common cavity mode, is decoupled from the cavity mode and bright mode, and can be broken by controlling cross-Kerr nonlinearity. We find that at nonzero temperature, light and vibrations are separable in DM-unbreaking regime but entangled in DM-breaking regime. The introduction of cross-Kerr effect significantly enhances optomechanical entanglement. Especially, the threshold of thermal phonon number increases by three orders of magnitude compared to the DM-unbreaking regime. Our scheme may be extended to optomechanical system with $N$ degenerate mechanical modes, where each mechanical mode can be entangled with optical mode only when $N-1$ different cross-Kerr strengths are introduced. This study offers a method for protecting and enhancing delicate quantum resources from DM and thermal noise to construct entangled optomechanical networks.