Fig. 4: Anti-crossing in the normal mode spectrum.

a Output spectrum of the coupled upper-branch polariton and the mechanical mode versus the drive power. Due to the magnon self-Kerr effect, the frequency of the polariton shifts by increasing the drive power (black dashed line). The scattered mechanical sideband is fixed at 7.216313 GHz (white dashed line). During the process of varying the drive power, the cross-Kerr effect causes a 1-kHz mechanical frequency shift, which has a negligible impact. b Polaromechanical cooperativity C_+,b versus the frequency of the upper-branch polariton ω₊. The cooperativity gets significantly improved when the polariton resonates with the mechanical sideband at the CPA frequency.