Optomechanical Bell Test

Over the past few decades, experimental tests of Bell-type inequalities have been at the forefront of understanding quantum mechanics and its implications. These strong bounds on specific measurements on a physical system originate from some of the most fundamental concepts of classical physics—in particular that properties of an object are well-defined independent of measurements (realism) and only affected by local interactions (locality). The violation of these bounds unambiguously shows that the measured system does not behave classically, void of any assumption on the validity of quantum theory. It has also found applications in quantum technologies for certifying the suitability of devices for generating quantum randomness, distributing secret keys and for quantum computing. Here we report on the violation of a Bell inequality involving a massive, macroscopic mechanical system. We create light-matter entanglement between the vibrational motion of two silicon optomechanical oscillators, each comprising approx. $10^{10}$ atoms, and two optical modes. This state allows us to violate a Bell inequality by more than 4 standard deviations, directly confirming the nonclassical behavior of our optomechanical system under the fair sampling assumption.

Figure 1

(a) Schematic of the setup: blue detuned drive pulses interact with the mechanical resonators (devices $A$ and $B$) producing entangled photon-phonon pairs. The light-matter entanglement is in the path basis ($A$ or $B$), corresponding to the device in which the Stokes scattering event took place. The generated photons are detected in single-photon detectors giving the measurement results $a_{b1}$ and $a_{b2}$. The detection of the phonons is done by transferring their states to another optical mode by using a red drive after some time $\mathrm{\Delta }\tau$ and, subsequently, obtaining the results $a_{r1}$ and $a_{r2}$. Note that for technical reasons the photons created by the blue and red drives are detected on the same pair of detectors, but with a time delay $\mathrm{\Delta }\tau =200\mathrm{ns}$. Therefore we have time separation of the two parties of the Bell test instead of space separation (as commonly done). $\mathrm{BS}1/2$ represent beam splitter $1/2$. (b) Scanning electron microscope image of one of the optomechanical devices, represented with a star symbol in (a) and (c), next to the coupling waveguide (top). (c) Illustration of our experimental sequence: one party of the Bell test measures in which detector path the Stokes photon is found at time $t=0$, while the other performs the same measurement for the anti-Stokes photon after a time $t=\mathrm{\Delta }\tau$. We probe their correlations in order to violate the CHSH inequality. Since the two photons never interacted directly (only through the mechanics), the observed correlations are a direct consequence of the correlations between the Stokes photons and phonons.

Figure 2

Correlation coefficients for various phase settings. We set the blue phase parameter ${\varphi }_b$ to 0 (orange) and $0.5\pi$ (green), while we scan the red pulse’s phase setting ${\varphi }_r$ over more than $2\pi$. The optimal angles to test the CHSH inequality are shown with different symbols. The associated measured values can be found in Table 1.

Figure 3

Visibility as a function of generation rate and state transfer probability. We sweep the power of the blue pulse while keeping the red state transfer probability fixed, inducing absorption of the optical field in the silicon structure, and see that for excitation probabilities up to around 3% the measured visibility exceeds the threshold to violate the CHSH inequality (left). When increasing only the red pump power (right) a similar behavior can be observed, allowing us to increase the state transfer probability beyond 14%, while still being able to overcome the classical bound (orange shaded region). The visibilities $V=|E({\varphi }_b,{\varphi }_r){|}_{\max }$ are measured in a single phase setting at the optimal angles ${\varphi }_b=0$ and ${\varphi }_r=0.337\pi$.