Entanglement of a Laguerre-Gaussian cavity mode with a rotating mirror

It has previously been shown theoretically that the exchange of linear momentum between the light field in an optical cavity and a vibrating end mirror can entangle the electromagnetic field with the vibrational motion of that mirror. In this paper we consider the rotational analog of this situation and show that radiation torque can similarly entangle a Laguerre-Gaussian cavity mode with a rotating end mirror. We examine the mirror-field entanglement as a function of ambient temperature, radiation detuning, and orbital angular momentum carried by the cavity mode.

Figure 1

(Color online) Arrangement for entangling the rotation of a mirror with a Laguerre-Gaussian (LG) mode using angular momentum exchange. A Gaussian $\left(G\right)$ field of optical charge 0 enters the cavity through a fixed partially transparent input coupler (IC) that does not affect its charge. A perfectly reflecting rear mirror (RM) rotating about the cavity axis on a support $S$ adds a charge $2l$ to the beam on reflection. Both the IC and the RM are spiral phase elements. The angular deflection of the RM from equilibrium $({\varphi }_0=0)$ is indicated by the angle $\varphi$. The charge on the beams at various points is also indicated.

Figure 2

Logarithmic negativity $E_N$ as a function of ambient temperature, for the parameters of Table .

Figure 3

(Color online) Solid line, deviation $\Delta E_N$ of $E_N$ from its maximum value $E_0\sim 0.5$ as a function of $T$. The graph shows the low temperature end of the curve. Dashed line, mean number of thermal phonons $\overline{n}$ [Eq. (7)] at frequency ${\omega }_{\mathrm{eff}}$ as a function of $T$. The vertical dotted line corresponds to the temperature $T_c$ at which thermal phonons of energy larger than $\hslash {\omega }_{\mathrm{eff}}$ start exciting the rotor, degrading the entanglement. Parameters of Table .

Figure 4

The logarithmic negativity $E_N$ plotted as a function of the detuning $\Delta$ (normalized by the angular frequency ${\omega }_{\varphi }$ of the rotating mirror) of the laser radiation from the cavity resonance. The three different curves correspond to different masses of the rotating mirror. The maximum of the entanglement occurs practically at $\Delta ={\omega }_{\mathrm{eff}}\sim 0.8{\omega }_{\varphi }$ (see the discussion in Sec. 4C) for all three curves. For this plot we used the parameters from Table except for the following changes: $L=0.1\mathrm{mm}$, $R=250\mu \mathrm{m}$, $Q_{\varphi }={10}^7$, $F={10}^4$, and $T=400\mathrm{mK}$.

Figure 5

Logarithmic negativity $E_N$ as a function of the orbital angular momentum carried by the Laguerre-Gaussian mode. The threshold angular momentum $l_c\sim 21$ at which entanglement begins to appear is indicated. Parameters of Table , temperature $T=10\mathrm{K}$.