Characterizing twin-particle entanglement in double-well potentials

We consider a pair of twin atoms trapped in double-well potentials. For each atom, two orthogonal spatial modes are accessible: the states $|L\rangle$ and $|R\rangle$ spatially localized in the left and right wells respectively. Furthermore the twin atoms are distinguishable thanks to an additional degree of freedom. We propose a method for experimentally quantifying the particle entanglement between these atoms which allows us to probe a violation of Bell's inequality. It is based on measuring the correlations in the atoms' momentum distribution. If the tunneling and the energy difference between the wells are tunable, then full state tomography is achievable.

Figure 1

Measuring the momentum distribution of atoms trapped in a double-well potential is similar to measuring the far-field intensity distribution in a double-slit experiment. For atoms in the single-particle pure state ${\rho }_1$ (left), interferences in the momentum distribution $n\left(p\right)$ (b) reflect the coherences of the density matrix and the position distribution (a) the populations. For atoms in a nonseparable two-particle state (c), the mean momentum distributions $n\left(p_1\right), n\left(p_2\right)$ of each particle separately (d) are not enough to characterize the state, while interferences in the momentum second-order correlation (e) relate to the coherences of the two-particle density matrix.

Figure 2

Different terms ${\rho }_{ijkl}$ of a two-qubit density matrix $\rho$ contribute to the pattern of $G^{\left(2\right)}(p_{+},p_{-})$ with different patterns ${\varphi }_{ijkl}(p_{+},p_{-})$, whose real part is plotted here.

Figure 3

Examples of correlation functions in momentum space $G^{\left(2\right)}(p_1,p_2)$ (bottom row) for two-atom states (top row) of different purities and concurrences (middle row). Spatially nonseparable patterns can be obtained not only for entangled pure states like ${\rho }_a$ and ${\rho }_c$, but also for mixed states like ${\rho }_d$ and the Werner state ${\rho }_e$. For the product state ${\rho }_b, G^{\left(2\right)}(p_1,p_2)$ is simply the product of the oscillating mean densities of the two atoms. $G^{\left(2\right)}(p_1,p_2)$ provides information on some terms only of the two-atom density matrix $\rho$ (see text). Therefore the fully entangled state ${\rho }_c$ and the fully mixed state ${\rho }_d$ exhibit the same checkerboard pattern. However, in the particular case of the state ${\rho }_a$, the maximal contrast of the fringe pattern allows full reconstruction of the state.

Figure 4

Left: We consider here a maximally entangled state of the form $\left|\mathrm{\Psi }\right(\alpha )\rangle$ defined in Eq. (14). Without assuming initial knowledge on this state, the experimentally measurable correlations allow setting a lower bound on its concurrence (top), characterizing its entanglement, and computing the parameter $S_{\mathrm{Bell}}$ which describes a Bell inequality violation in the $\left\{\right|L\rangle ,|R\rangle \}$ basis (bottom). Without experimental limitation on the fringes contrast (yellow/light-gray line) one can in principle measure a maximal entanglement (the concurrence being 1) and a maximal violation of a Bell inequality ($S_{\mathrm{Bell}}=2\sqrt{2}$) for the Bell states $|\mathrm{\Psi }\left(0\right)\rangle =1/\sqrt{2}\left(|LL\rangle +|RR\rangle \right)$ and $|\mathrm{\Psi }(\pi /2)\rangle =1/\sqrt{2}\left(|LR\rangle +|RL\rangle \right)$. The effect of a finite experimental contrast reduces the lower bound on the concurrence and the region of the Bell violation [dash-dotted red and (lower) blue lines]. Right: In the case of a Werner state ${\rho }_W$ [defined in Eq. (15)], the correlation measurement in the $\left\{\right|L\rangle ,|R\rangle \}$ basis captures well the state entanglement: the lower bound on the concurrence for a perfect contrast coincides with the concurrence itself (top, yellow/light-gray and dashed black lines).

Figure 5

By transforming the double-well potential one can perform a full tomography of the two-atom state. Two different transformations come into play: (a) In coupled double wells the atoms can tunnel and the state undergo rotation in the $\left\{\right|L\rangle ,|R\rangle \}$ basis [see Eq. (18)]. (b) In decoupled, asymmetric double wells a phase difference accumulates between the left and right wells. For a $\pi /2$ phase difference, the imaginary parts of the $c, d, e, f$ elements of the initial density matrix $\rho$ are projected onto the real part of the new density matrix ${\rho }^{\prime }$ [see Eq. (19)]. (c) and (d): After the rotation illustrated in (a), decomposition of the diagonal coherences of the rotated density matrix $\tilde{\rho }$ in function of the elements of the nonrotated density matrix $\rho$. Only the nonzero coefficients of the decomposition are plotted here. For $\theta =\pi /2, Im(\tilde{a}-\tilde{b})=1/2Re(d-e)$ and $Im(\tilde{a}+\tilde{b})=1/2Re(c-f)$. Since $d+e$ and $c+f$ are known, on can reconstruct the real parts of $c, d, e$, and $f$. Applying the same operation to ${\rho }^{\prime }$, one can reconstruct their imaginary parts.

Figure 6

Entangled two-atom state in double wells (bottom) is obtained, for example, through four-wave mixing, starting from a BEC in an excited transverse state of the potential (top left). The two atoms have opposite momenta $\pm p_0$ along the longitudinal axis $x$ (top right).