Clauser-Horne-Shimony-Holt Bell inequality test in an optomechanical device

We propose here a scheme based on the measurement of quadrature phase coherence that is aimed at testing the Clauser-Horne-Shimony-Holt Bell inequality in an optomechanical setting. Our setup is constituted by two optical cavities dispersively coupled to a common mechanical resonator. We show that it is possible to generate Einstein-Podolsky-Rosen–like correlations between the quadratures of the output fields of the two cavities and, depending on the system parameters, to observe the violation of the Clauser-Horne-Shimony-Holt inequality.

Figure 1

Schematic of the detection scheme. Outputs of the cavities are directed to different beam splitters, where they are mixed with local oscillator (LO) fields. The mixed signals are sent to photodetectors D1, E1, D2, and E2, characterized by fields $d_1, e_1, d_2, e_2$, respectively. Unlike the case of (balanced) homodyne detection schemes, where the signals emerging from the two branches of each beam splitter—in our case directed towards detectors (i) D1 and E1 and (ii) D2 and E2—are combined, we keep track of all four signals and their intensity correlations described by Eqs. (9a)–(9d).

Figure 2

(a) Value of $\mathcal{F}$ as a function of ${\alpha }_{\mathrm{i}}$ and $r=G_{+}/G_{-}$. Parameters: ${\kappa }_{\mathrm{a}}={\kappa }_{\mathrm{c}}=\kappa =0.1, {\kappa }_{\mathrm{e}}=0.9\kappa , \gamma =1\times {10}^{-5}$—all energies are expressed in units of ${\omega }_{\mathrm{m}}$ ($\hslash =1$) throughout the manuscript. The dashed curve corresponds to the exact boundary region $\mathcal{F}=1/2$, as determined from the solution of Eqs. (2a)–(2c) (b) Boundary $\mathcal{F}=1/2$ for different values of $r_{\mathrm{e}}={\kappa }_e/\kappa$. Smaller regions are associated with smaller values of $r_{\mathrm{e}}$. For ${\alpha }_{\mathrm{i}}\simeq 0.2$ it is possible to observe a crossing between boundary regions for different values of $r_{\mathrm{e}}$, hinting at a nontrivial relation between entanglement and violation of the CHSH BI (see text). The solid line corresponds to the exact boundary as in panel (a), and the dashed line corresponds to the expression given in Eq. (16).

Figure 3

Dependence of the value of $\mathcal{F}$ (for $r\to r_{\mathrm{opt}}$ and ${\alpha }_{\mathrm{i}}, {\chi }_{\mathrm{i}}\to 0$) on the thermal population baths associated with the mechanical noise (${\overline{n}}_{\mathrm{m}}$, blue—flattest—curve), internal noise (${\overline{n}}_{\mathrm{i}}$, red—intermediate—curve), and external noise (${\overline{n}}_{\mathrm{e}}$, black—steepest—curve). Solid lines correspond to the exact solution from the equations of motion with each noise source considered independently. Dashed lines are the approximations given in Eq. (7a) and Eqs. (18a)–(18d). Values of $\mathcal{F}$ above the horizontal dashed line at $\mathcal{F}=1/2$ correspond to the violation of the BI. Parameters: (a) $\kappa =0.01, {\kappa }_e=0.9\kappa , G_{-}=0.2, \gamma =1\times {10}^{-5}$; (b) ${\kappa }_e=0.99\kappa$, all other parameters are the same as in panel (a).

Figure 4

Noise dependence of the $\mathcal{F}=1/2$ boundary in the presence of a finite coherent input. Smaller regions correspond to a large value of the noise. All parameters are the same as in Fig. 2.

Figure 5

Comparison between the value of $\mathcal{F}$ calculated from the full solution of the equations of motion (full lines, $\kappa =0.01$, 0.02, 0.1; larger values correspond to smaller regions for which $\mathcal{F}>1/2$), with the solution obtained in the rotating-wave approximation (dashed line).