Entanglement properties of degenerate four-wave mixing of matter waves in a periodic potential

In a recent experiment, Campbell et al. [Phys. Rev. Lett. 96, 020406 (2006)] observed degenerate four-wave mixing of matter waves in a one-dimensional optical lattice. We analyze the essential quantum features of the experiment to show that entanglement is created between the quadratures of the two scattered atomic clouds and is a true many-body (rather than two-body) phenomenon. We demonstrate that well-known bipartite entanglement inequalities are significantly violated and that this is robust to a moderate level of coherent seeding. The system is thus a promising candidate for generating spatially separated macroscopically entangled atomic samples.

Figure 1

(Color online) The mode occupations as a function of time. The solid line and the lower dashed line are the averages of $4.34\times {10}^6$ trajectories of the positive-$P$ representation equations for $N_0$, $N_1$, and $N_2$ in the spontaneous case. Note that $N_1=N_2$ and that the Wigner results are indistinguishable. The dotted line and the upper dashed lines are the Wigner results ($6.6\times {10}^5$ trajectories) for $N_0$, $N_1$, and $N_2$ with an initial seed, $N_1\left(0\right)=100$. All quantities plotted in this and subsequent figures are dimensionless. The horizontal axis is a parametrized time, $\xi =\chi {\mid \alpha \left(0\right)\mid }^{2}t$.

Figure 2

(Color online) The Duan correlation, $V\left({\widehat{X}}_{-}\right)+V\left({\widehat{Y}}_{+}\right)$, with and without an injected seed. The lower three lines are for the spontaneous case, with the dotted line being analytical, the dash-dotted line being the positive-$P$ prediction, and the full line being the Wigner prediction. The dashed line is the Wigner prediction with $N_1\left(0\right)=100$. Note that the analytical solution is almost indistinguishable from the positive-$P$ solution until the latter turns and begins to increase.

Figure 3

(Color online) The EPR correlation, $V^{\inf }\left({\widehat{X}}_1\right)V^{\inf }\left({\widehat{Y}}_1\right)$. The dash-dotted line is the spontaneous positive-$P$ result, the solid line is the spontaneous Wigner result, and the dashed line is the stimulated Wigner result with $N_1\left(0\right)=100$. The analytical solution is again indistinguishable from the numerical solutions up until they begin to turn.