Entanglement generation between two spinor Bose-Einstein condensates with cavity QED

We analyze a scheme for generating entanglement between two spinor Bose-Einstein condensates (BECs). The BECs are off-resonantly coupled to a common photon mode and are controlled by external lasers to induce a $S^z{S}^z$ interaction, where $S^z$ is the total spin of the BEC. We directly simulate the scheme numerically for small systems and show the performance of the scheme. The scaling of the entanglement to large-scale systems under realistic conditions of spontaneous emission and cavity photon loss is analyzed. It is shown that both entanglement in the beyond-continuous-variable regime can be generated, where the entanglement is of the order of the maximal entanglement between the systems.

Figure 1

Scheme for entanglement generation between two spinor Bose-Einstein condensates (BECs). The BECs are placed in an optical cavity, in the strong coupling regime allowing for coherent transfer with coupling $G$ between a cavity photon and an optical transition between two internal states ($b_{1,2}\leftrightarrow e_{1,2}$) of the atoms. The BEC exists initially in a spin coherent state between the ground states of the atoms $a_{1,2},b_{1,2}$. The two BEC entanglement is initiated with an laser transition $g$ detuned from the resonance by energy ${\Delta }_l$. The cavities are also detuned from the excitation by an energy ${\Delta }_c$. The cavities are connected by an optical fiber forming a common mode $c$.

Figure 2

The entanglement generated by the proposed scheme for [(a),(b)] $N=1$ and [(c),(d)] $N=8$. Plots (a) and (c) show the off-resonant case with ${\Delta }_c/g=10$ and ${\Delta }_l/g=20$, while plots (b) and (d) show the resonant case with ${\Delta }_c/g={\Delta }_l/g=10$. In each plot the entanglement due to a pure $S^z{S}^z$ interaction (solid lines), the proposed scheme with ${\Gamma }_s={\Gamma }_c=0$ (dashed lines), and the proposed scheme with cavity decay $\hslash {\Gamma }_c/g=0.1$ and spontaneous emission $\hslash {\Gamma }_s/g=0.01$ are shown (dotted lines). $G/g=1$ is used for all cases and the time scale is measured in units of $\Omega =\frac{G^2{g}^2}{2\hslash {\Delta }_c{\Delta }_l^2}$. The time scale of the ideal $S^z{S}^z$ interaction curves (solid lines) have been adjusted to fit the shape of the simulated curves.

Figure 3

Entanglement at the time $\Omega t=\frac{1}{\sqrt{2N}}$ for various boson numbers $N$. $\delta E$ is the difference between the ideal entanglement generated by a pure $S^z{S}^z$ interaction and the proposed scheme. Once again, each plot shows the scaling of three different possible detunings: ${\Delta }_l/g=15$ (diamonds), ${\Delta }_l/g=15\sqrt{N}$ (squares), and ${\Delta }_l/g=5N$ (triangles). The proposed scheme with ${\Gamma }_s={\Gamma }_c=0$ (a) and the proposed scheme with cavity decay $\hslash {\Gamma }_c/g=0.1$ and spontaneous emission $\hslash {\Gamma }_s/g=0.01$ are shown (b). $G/g=1$, ${\Delta }_c/g=2$ is used for all cases.

Figure 4

Entanglement at the time $\Omega t=\frac{\pi }{4N}$ for various boson numbers $N$. $\delta E$ is the difference between the ideal entanglement generated by a pure $S^z{S}^z$ interaction and the proposed scheme. Each plot shows the scaling of three different possible detunings: ${\Delta }_l=15$ (diamonds), ${\Delta }_l=15\sqrt{N}$ (squares), and ${\Delta }_l=5N$ (triangles). The proposed scheme with ${\Gamma }_s={\Gamma }_c=0$ (a) and the proposed scheme with cavity decay $\hslash {\Gamma }_c/g=0.1$ and spontaneous emission $\hslash {\Gamma }_s/g=0.01$ are shown (b). $G/g=1$, ${\Delta }_c/g=2$ is used for all cases.

Figure 5

A plot of the partial $Q$ distribution of BEC 1 at time $\Omega t=1/\sqrt{2N}$. Figures show (20) with projections on BEC 2 for the $k_2$ values shown. Parameters used are ${\Delta }_c/g=2$, ${\Delta }_l/g=20$, $G/g=1$, and $N=8$.

Figure 6

A plot of the partial $Q$ distribution of BEC 1 at time $\Omega t=1/\sqrt{2N}$. Figures show (20) with projections on BEC 2 for the $k_2$ values shown. Parameters used are ${\Delta }_c/g={\Delta }_l/g=10$, $G/g=1$, and $N=8$.