Entangling the rovibrational modes of a macroscopic mirror using radiation pressure

We consider the dynamics of a vibrating and rotating end mirror of an optical Fabry-Pérot cavity that can sustain Laguerre-Gaussian modes. We demonstrate theoretically that since the intracavity 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 is a proposal to demonstrate entanglement between two qualitatively different degrees of freedom of the same macroscopic object.

Figure 1

(Color online) The arrangement proposed in this work for entangling the vibrational and rotational modes of a mirror. A Laguerre-Gaussian beam is incident on the resonant cavity formed by two spiral phase elements, the transmissive but fixed input coupler and the perfectly reflective moving rear mirror. The rear mirror is mounted on a helical spring $S$, which provides a vibrating restoring force along the $z$ axis as well as torsion opposite the direction $\varphi$. The deflection of the rear mirror from its angular equilibrium position $({\varphi }_0=0)$ is indicated by the angle $\varphi$; the $z$ deflection of the mirror has not been shown for clarity. The charge on the Laguerre-Gaussian beams at various points has been indicated.

Figure 2

(Color online) Radiation pressure-induced entanglement $\mathcal{E}\left(\omega \right)$ between a vibrational and a rotational mode of the moving mirror. The parameters are $M=1\mathrm{\mu }\mathrm{g}$, $R=15\mathrm{\mu }\mathrm{m}$, vibrational and rotational quality factors $Q_z=Q_{\varphi }={10}^6$, ${\omega }_z\simeq {\omega }_{\varphi }=1\mathrm{MHz}$, $l=82$, $L\simeq 4\mathrm{mm}$, cavity finesse $F=2.5\times {10}^4$, $\lambda =812.7\mathrm{nm}$, $\Delta ={\omega }_{\varphi }$, and $P_{\mathrm{in}}=1\mathrm{mW}$.

Figure 3

Maximum entanglement ${\mathcal{E}}_{\max }$ at $T=1\mathrm{K}$ between a vibration and a rotational mode of the rear mirror of Fig. 1 as a function of the percent fractional imbalance in the couplings with the radiation field. The entanglement decreases with increasing asymmetry in the couplings. In the figure for a $200\mathrm{Hz}$ difference between rotational and vibrational frequencies centered at $1\mathrm{MHz}$, the coupling imbalance, about 0.02%, destroys the entanglement completely. The remaining parameters are the same as in Fig. 2.