Steady State Entanglement in the Mechanical Vibrations of Two Dielectric Membranes

We consider two dielectric membranes suspended inside a Fabry-Perot cavity, which are cooled to a steady state via a drive by suitable classical lasers. We show that the vibrations of the membranes can be entangled in this steady state. They thus form two mechanical, macroscopic degrees of freedom that share steady state entanglement.

Figure 1

The setup: Two mechanically vibrating membranes (brown) are suspended inside a Fabry-Perot cavity, which is driven by external lasers (green). Dissipation occurs via mechanical damping (brown) and cavity decay (blue).

Figure 2

The entanglement of the two mechanical vibrations in the steady state measured by the logarithmic negativity $E_N$ as a function of $c_{bn}$ and $c_{cm}$. Left plot: ${\Delta }_{bn}=4.07\mathrm{MHz}$, ${\Delta }_{cm}=20.84\mathrm{MHz}$, and $\mathcal{T}=8\mathrm{mK}$ ($n_{{\omega }_m}=1000$). Right plot: ${\Delta }_{bn}=6.12\mathrm{MHz}$, ${\Delta }_{cm}=33.18\mathrm{MHz}$, and $\mathcal{T}=100\mathrm{mK}$ ($n_{{\omega }_m}=13085$). The remaining parameters are ${\omega }_m=1\mathrm{MHz}$, $m={10}^{-9}\mathrm{g}$, $T=0.2$, $L=1\mathrm{mm}$, $n=2\times {10}^3$, $=3n=6\times {10}^3$, $q_{01}=-L$, $q_{02}=2L$, ${\Gamma }_{bn}={\Gamma }_{cm}={\omega }_m/10$, and $Q={10}^7$ for both plots. ${\Delta }_{bn}$, ${\Delta }_{cm}$, $c_{bn}$, and $c_{cm}$ have been optimized numerically for each case.

Figure 3

The steady state entanglement of the two mechanical vibrations for $\mathcal{T}=100\mathrm{mK}$, $c_{bn}=60$, and $c_{cm}=486$. (a) $E_N$ as a function of ${\Delta }_{bn}$ and ${\Delta }_{cm}$, (b) $E_N$ as a function of ${\Gamma }_{bn}$ and ${\Gamma }_{cm}$, (c) $E_N$ as a function of $m$ and ${\omega }_m$, and (d) $E_N$ as a function of $Q$ and $T$. All other parameters as in Fig. 2.