We consider the dynamics of a vibrating and rotating end-mirror of an optical Fabry-P{erot} cavity that can sustain Laguerre-Gaussian modes. We demonstrate theoretically that since the intra-cavity field carries linear as well as angular momentum, radiation pressure can create bipartite entanglement between a vibrational and a rotational mode of the mirror. Further we show that the ratio of vibrational and rotational couplings with the radiation field can easily be adjusted experimentally, which makes the generation and detection of entanglement robust to uncertainties in the cavity manufacture. This constitutes the first proposal to demonstrate entanglement between two qualitatively different degrees of freedom of the same macroscopic object.
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