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We consider the dynamics of a vibrating and rotating end-mirror of an opticalFabry-P{\'erot} cavity that can sustain Laguerre-Gaussian modes. We demonstratetheoretically that since the intra-cavity field carries linear as well asangular momentum, radiation pressure can create bipartite entanglement betweena vibrational and a rotational mode of the mirror. Further we show that theratio of vibrational and rotational couplings with the radiation field caneasily be adjusted experimentally, which makes the generation and detection ofentanglement robust to uncertainties in the cavity manufacture. Thisconstitutes the first proposal to demonstrate entanglement between twoqualitatively different degrees of freedom of the same macroscopic object.

Entangling the ro-vibrational modes of a macroscopic mirror using radiation pressure