Quantifying the Mesoscopic Nature of Einstein-Podolsky-Rosen Nonlocality

Evidence for Bell’s nonlocality is so far mainly restricted to microscopic systems, where the elements of reality that are negated predetermine results of measurements to within one spin unit. Any observed nonlocal effect (or lack of classical predetermination) is then limited to no more than the difference of a single photon or electron being detected or not (at a given detector). In this paper, we analyze experiments that report the Einstein-Podolsky-Rosen (EPR) steering form of nonlocality for mesoscopic photonic or Bose-Einstein condensate systems. Using an EPR steering parameter, we show how the EPR nonlocalities involved can be quantified for four-mode states, to give evidence of EPR-nonlocal effects corresponding to a two-mode number difference of $10^5$ photons, or of several tens of atoms (at a given site). Applying to experiments, we also show how the variance criterion of Duan, Giedke, Cirac and Zoller for EPR entanglement can be used to determine a lower bound on the number of particles in a pure two-mode EPR-entangled or steerable state.

Figure 1

The $\delta$-scopic EPR nonlocality is realized (${\epsilon }_{\delta }<1$) when $\epsilon$ is below the line shown, for the given $\delta$. Data $i-xxv$ are for the experiments referenced in Fig. 9 of Ref. [13], while data $a$ [15], $b$ [16], $c$ [22], $d$ [23], $e$ [19], $f$ [20], $g$ [24], $aa$ [25], $bb$ [26], $cc$ [27], $dd$ [32] are later experiments. All results determine EPR correlations between spatially-separated optical fields, except those given by blue stars, which determine correlations between mesoscopic groups of cold atoms ($aa$, $bb$, $cc$) or between hybrid systems ($dd$). The cold-atom groups have negligible ($aa$) or small separations $\sim 10\mu \mathrm{m}$ ($bb$, $cc$).

Figure 2

The entanglement parameter $D$ is plotted versus the date for a sample of experiments. Data ($ii-xxv$) are the values reported for atomic (stars) and optical (triangles) experiments, as listed in Fig. 9 of Ref. [13]. Data $c$ [22], $d$ [23], $e$ [19], $f$ [20], $g$ [24], $h$ [18] are for optical experiments (diamonds); data $aa$ [25], $dd$ [32], $ee$ [29] are for atomic or hybrid experiments (stars); and $ff$ [35], $gg$ [38] are for mechanical-oscillator experiments (squares). Note that $D<D_{\overline{n}}^{(l)}$ implies the EPR state $|\psi ⟩$ has a mean number of bosons greater than $\overline{n}$. The $D_{\overline{n}}^{(l)}$ are plotted for $\overline{n}=1$, 2, 3, 4. Here, $D<D_{n_0}$ requires states of more than $n_0$ bosons. We plot $D_{n_0}$ for $n_0=2$, 3, 4. For $n_0=10$, $D_{n_0}\simeq 0.2228$.