Proposal to Test Bell’s Inequality in Electromechanics

Optomechanical and electromechanical systems offer an effective platform to test quantum theory and its predictions at macroscopic scales. To date, all experiments presuppose the validity of quantum mechanics, but could in principle be described by a hypothetical local statistical theory. Here we suggest a Bell test using the electromechanical Einstein-Podolski-Rosen entangled state recently generated by Palomaki et al., Science 342, 710 (2013), which would rule out any local and realistic explanation of the measured data without assuming the validity of quantum mechanics at macroscopic scales. It additionally provides a device-independent way to verify electromechanical entanglement. The parameter regime required for our scheme has been demonstrated or is within reach of current experiments.

Figure 1

(a) Electromechanical circuit (formed by the $LC$ resonator and the mechanical oscillator) and cascaded microwave cavities with integrated qubits, used as single-photon detectors. (b) Level scheme of superconducting qubit coupled to the intracavity microwave field. The solid green line represents the resonant transition that we seek to drive, while the dashed line indicates an off-resonant transition.

Figure 2

Bell correlations against cooperativity optimized with respect to ${\tau }_1$, ${\tau }_2$, and $\upsilon$ for different transmissivities ${\lambda }_t=1$, 0.98, 0.95, 0.92 (red, yellow, green, blue lines), and initial mechanical occupation numbers $n_0=0.1$, 0.25 (solid, dashed lines). Other parameters are $\overline{n}=40$ and ${\kappa }_{LC}/{\omega }_m\approx 1/8$.

Figure 3

Optimal values of ${\tau }_1$ (top), ${\tau }_2$ (middle), and $\upsilon$ (bottom) for the corresponding values of $S$ in Fig. 2.