Nonlocality with ultracold atoms in a lattice

We study the creation of nonlocal states with ultracold atoms trapped in an optical lattice. We show that these states violate Bell inequality by measuring one- and two-body correlations. Our scheme only requires beam-splitting operations and global phase shifts, and can be realized within the current technology, employing single-site addressing. This proposal paves the way to study multipartite nonlocality and entanglement in ultracold-atomic systems.

Figure 1

Quantum nonlocality of neutral atoms in an optical lattice (here limited, for illustration sake, to few lattice sites). The protocol consists of four operations involving two internal (hyperfine) states and is implemented by a state-selective optical lattice. Here, $a$ and $b$ indicate hyperfine levels, and $l$ is the lattice site. (a) Preparation of a Mott state with one atom per lattice site in level $a$, with each atom being schematically represented by a dot. (b) Balance pulse coupling neighboring wells: this is made of a $\pi /2$ Rabi pulse, schematically represented by a green waving line (b.I), followed by a relative shift of the state-selective lattice (b.II). (c) Collective phase shift $\theta$ or $\phi$; see text. (d) Balance pulse coupling the internal levels of each atom. The protocol ends with the measurement of the number of particles in the internal ground level.

Figure 2

Maximal relative violation ${\xi }_N$ of Eq. (1), optimized over different local phases, as a function of the number of parties $N$ (crosses). The orange line is ${\xi }_N^{\mathrm{glob}}$, where all local phases are equal, ${\theta }_k=\theta$ and ${\phi }_k=\phi$. The dashed gray line is a power-law fit $1.5/N$.

Figure 3

Relative quantum violation ${\xi }_N^{\mathrm{glob}}$ taking the same local phase shift $\theta$ and $\phi$ expressed in $\pi$ rad, at each lattice site. Different panels refer to $N=2,4,10,30$ (from left to right). The color scale is cut at ${\xi }_N^{\mathrm{glob}}=1$ and violation of Bell's inequality (${\xi }_N^{\mathrm{glob}}<1$) is obtained in the colored regions.