Violating Bell’s inequalities in vacuum

We employ an approach wherein the ground state entanglement of a relativistic free scalar field is directly probed in a controlled manner. The approach consists of having a pair of initially nonentangled detectors locally interact with the vacuum for a finite duration $T$, such that the two detectors remain causally disconnected, and then analyzing the resulting detector mixed state. We show that the correlations between arbitrarily far-apart regions of the vacuum cannot be reproduced by a local hidden-variable model, and that as a function of the distance $L$ between the regions, the entanglement decreases at a slower rate than $\sim \exp [-{(L∕cT)}^3]$.

Figure 1

The world lines of detectors $A$ and $B$ are shown for the duration of the interaction. The horizontal and vertical axes are space and time, respectively. The arrows denote the emitted radiation. Notice that the radiation emitted by detector $A\left(B\right)$ does not affect detector $B\left(A\right)$, since for $t>T$ the interaction is switched off.

Figure 2

Emission and exchange processes. The exchange amplitude, portrayed on the right, is dominated by a single off-shell emission followed by an on-shell absorption, while the double emission amplitude, portrayed on the left, consists of two off-resonance processes. Thus, as ${\Omega }_{A,B}$ increases, the emission term $\parallel E_A\parallel \parallel E_B\parallel$ decreases more rapidly than the exchange term $\mid ⟨0\mid X_{AB}⟩\mid$.

Figure 3

Violation of the CHSH inequality. The dashed line represents the negativity of the detectors before passing through the filter, while the solid line represents the quantity $M\left(\rho \right)-1$, which is calculated after the detectors have passed through the filter. Since the CHSH inequality can be written in the form $M\left(\rho \right)<1$ [25], we see that for $\Omega \to \infty$, the CHSH inequality is maximally violated.