Atomic entanglement generation and detection via degenerate four-wave mixing of a Bose-Einstein condensate in an optical lattice

The unequivocal detection of entanglement between two distinct matter-wave pulses is a significant challenge that has yet to be experimentally demonstrated. We describe a realistic scheme to generate and detect continuous-variable entanglement between two atomic matter-wave pulses produced via degenerate four-wave mixing from an initially trapped Bose-Einstein condensate loaded into a one-dimensional optical lattice. We perform a comprehensive numerical investigation for fixed condensate parameters to determine the maximum violation of separability and Einstein-Podolsky-Rosen inequalities for field quadrature entanglement, and describe and simulate an experimental scheme for measuring the necessary quadratures.

Figure 1

(Color online) Band structure of an optical lattice with $s=1$. Two atoms in the (moving) condensate mode 0 can collide into modes 1 and 2, conserving energy and quasimomentum.

Figure 2

(Color online) (a) The intensity of the various optical lattices as a function of time. The solid (red) line indicates the intensity of the optical lattice generating the entanglement, which is ramped on and off with a continuous first derivative using parabolic curves. This process adiabatically transfers momentum to quasimomentum states and vice versa. The dashed (green) line indicates the Bragg pulse used to split the phase reference into two. The dotted (blue) line indicates the simultaneous Bragg pulses used to beam split each of modes 1 and 2 with local oscillators in modes 0 and 3. (b) A momentum space schematic of each stage of the process.

Figure 3

(Color online) Momentum density and relative number fluctuations versus the time held in the optical lattice, $t_1$. Populations of each discrete momentum mode for (a) the periodic model and (b) the trapped model. The initial condensate is barely visible in the periodic model as it occupies just a single momentum mode. We see strong population growth around momenta $k_1$ and $k_2$, as well as low-energy scattering about $k_0$. In (c) and (d), the solid line is the number in the entangled modes, $⟨{\widehat{N}}_1+{\widehat{N}}_2⟩$, and the dashed line is the variance of the number difference, $\text{Var}\left[{\widehat{N}}_1-{\widehat{N}}_2\right]$. Relative number squeezing is observed when this variance is less than the total number, and a strong correlation is exhibited at early times in (c), the data from the homogenous model, and (d), the data from the trapped model. At later times the nonclassical correlation is lost.

Figure 4

(Color online) Entanglement demonstration as a function of the hold time in the optical lattice, $t_1$. The solid line represents the result for the homogenous calculation, while the dashed line is the result for the trapped calculation. (a) The separability criterion [Eq. (15)] is maximally violated when the entangled populations $N_1+N_2\approx 9$. The presence of the trap reduces the level of violation, but the system still demonstrates the EPR paradox as given by Eq. (18). (b) The two systems display similar behavior as that in (a) with respect to the EPR criterion, Eq. (19). Ensembles of 1000 stochastic trajectories were used to generate these results.

Figure 5

(Color online) Minimal values of (a) the separability parameter $\mathcal{S}$ and (b) the EPR parameter $\mathcal{E}$ with different initial seed populations in mode 1. Crosses (blue) represent results for periodic systems and circles (red) represent those for trapped systems. Larger seeds appear to degrade the amount of entanglement violation. Each point corresponds to a different time $t_1$ that minimizes the separability parameter for each system and seed size.

Figure 6

(Color online) The measured dependences of the (a) separability and (b) EPR parameters on the phase of the Bragg lasers, ${\theta }_1-{\theta }_2={\varphi }_1+{\varphi }_2$. Dots (black) indicate data from stochastic simulations for each phase angle. Solid (red) lines represent fits to the data of (a) sinusoidal form with minimum value $\mathcal{S}=1.52$ and (b) two-frequency sinusoidal form with minima of $\mathcal{E}=0.43$. These results are directly calculated from number correlations at time $t_3$.