Correlation-based entanglement criterion in bipartite multiboson systems

We describe a criterion for the detection of entanglement between two multiboson systems. The criterion is based on calculating correlations of Gell-Mann matrices with a fixed boson number on each subsystem. This applies naturally to systems such as two entangled spinor Bose-Einstein condensates. We apply our criterion to several experimentally motivated examples, such as $S^z{S}^z$ entangled BECs, ac-Stark-shift–induced two-mode squeezed BECs, and photons under parametric down-conversion. We find that entanglement can be detected for all parameter regions for the most general criterion. Alternative criteria based on a similar formalism are also discussed together with their merits.

Figure 1

Various entanglement identifiers (a) ${\mathcal{T}}^{\left(N\right)}$ (b) ${\varepsilon }^{\left(N\right)}$ (c) the von Neumann entropy $S$ for two BECs entangled via a $S^z{S}^z$ interaction as a function of $\tau$.

Figure 2

The values of ${\varepsilon }^{\left(N\right)}$ for the macroscopic entanglement state (32) as a function of $t$ for $N$ odd and even, separately.

Figure 3

The values of ${\mathcal{T}}^{\left(N\right)}, T_{\max }$ and ${\varepsilon }^{\left(N\right)}$ for the macroscopic entanglement state (32) with $t=2.5/N$. For each case, ${\varepsilon }^{\left(N\right)}>1$, which proves entanglement.

Figure 4

The values of ${\varepsilon }^{{}^{\prime }\left(N\right)}$ parameter for $N\le 20$. If ${\varepsilon }^{{}^{\prime }\left(N\right)}>1$, entanglement of the BEC state can be revealed by means of the condition based on the spin operators.