Nonclassicality of optomechanical devices in experimentally realistic operating regimes

Enforcing a nonclassical behavior in mesoscopic systems is important for the study of the boundaries between the quantum and the classical world. Recent experiments have shown that optomechanical devices are promising candidates to pursue such investigations. Here we consider two different setups where the indirect coupling between a three-level atom and the movable mirrors of a cavity is achieved. The resulting dynamics is able to conditionally prepare a nonclassical state of the mirrors by means of projective measurements operated over a pure state of the atomic system. The nonclassical features are persistent against incoherent thermal preparation of the mechanical systems and their dissipative dynamics.

Figure 1

(a) Scheme of the system. (b) Energy levels of the atom driven by an off-resonant two-photon Raman transition.

Figure 2

Maximum violation of the Bell-CHSH inequality against the displacement $d$. From top to bottom, the curves correspond to $V=1$, 3, and 5, with $\eta t=2d$ and ${\theta }_1\simeq 3\pi /2$, and are optimized with respect to $\theta$. The inset shows, from top to bottom, the logarithmic negativity $E$ against $V$ for projected states with $p=0$, 1, and 2, for $d=2$.

Figure 3

Wigner function of the conditional mirror state against ${\xi }_r=\text{Re}\left(\xi \right)$ and ${\xi }_i=\text{Im}\left(\xi \right)$, for $V=3$ and $d=0$. Panels (a), (b), and (c) correspond to $\eta t=2$, 3, and 4, respectively.

Figure 4

Density plot of fidelity against $V$ and $\eta$. Darker regions correspond to smaller values of $F_W$.

Figure 5

Wigner function of the mirror under dissipation, for $\gamma \sim 0.1\eta$ and $V=5$.

Figure 6

Negative volume of $W({\mu }_1,{\mu }_2)$ against $V$ for $\eta t=5$. Inset: Wigner function $W_1\left({\mu }_1\right)$ at ${\mu }_2=-(1+i)$, $\eta t=2$, and $T=0$.

Figure 7

Numerically optimized violation of the Bell-CHSH inequality for the two-mirror state against $\eta t$ and $V$.

Figure 8

Wigner function for a mechanical system open to dissipation. (a) Wigner function of a single mirror for ${\mu }_2=1+i$, $\eta /\gamma =2$, and $\gamma t=V=1$. (b) ${\mathcal{V}}_{-}$ against $V$ and $\eta /\gamma$ for $\gamma t=1$ (we assume that all the relevant parameters are the same for both mirrors).

Figure 9

Violation of the CHSH inequality as a function of $\gamma t$ for four values of $\eta /\gamma$.