Dynamical preparation of Einstein-Podolsky-Rosen entanglement in two-well Bose-Einstein condensates

We propose to generate Einstein-Podolsky-Rosen (EPR) entanglement between groups of atoms in a two-well Bose-Einstein condensate using a dynamical process similar to that employed in quantum optics. A local nonlinear $S$-wave scattering interaction has the effect of creating spin squeezing at each well, while a tunneling coupling, analogous to a beam splitter in optics, introduces an interference between these fields that causes interwell entanglement. We consider two internal modes at each well so that the entanglement can be detected by measuring a reduction in the variances of the sums of local Schwinger spin observables. As is typical of continuous variable (CV) entanglement, the entanglement is predicted to increase with atom number. It becomes sufficiently strong at higher numbers of atoms so that the EPR paradox and steering nonlocality can be realized. The entanglement is predicted using an analytical approach and, for larger atom numbers, using stochastic simulations based on a truncated Wigner function approximation. We find generally that strong tunneling is favorable, and that entanglement persists and is even enhanced in the presence of realistic nonlinear losses.

Figure 1

Creation of spatial entanglement between modes $a_i$ and $b_i$ at different sites. The optical scheme is depicted in (a) and the BEC two-well proposal in (b). Local spin squeezing can be produced when the two modes at each well evolve for a time $\tau$ under the nonlinear Hamiltonian (11). Depending on the exact configuration, there can be coupling terms between the two modes at each site. The well height is controlled to allow a tunneling cross interaction between the modes. This is equivalent to the modes interfering using a beam splitter, and the effect is to generate an interwell spatial entanglement.

Figure 2

Squeezing of local Schwinger spin operators versus interaction time $\tau$. The plot shows the squeezing ${\Delta }^2{\widehat{J}}^{\theta }/n_0$ (solid lines), and ${\Delta }^2{\widehat{J}}^{\theta +\pi /2}/n_0$ (dashed lines), where the shot-noise level is $n_0=\left|\langle {\widehat{J}}^Y\rangle \right|/2$. Squeezing is obtained when $S={\Delta }^2{\widehat{J}}^{\theta ,\theta +\pi /2}/\left(\right|\langle {\widehat{J}}^Y\rangle |/2)<1$. Inset shows the optimal phase choice $\theta$ for squeezing. The dimensionless coupling parameters correspond to ${}^{87}$Rb atoms at magnetic field $B=9.116\mathrm{G}$ (blue lines), with corresponding scattering lengths, as explained in the text. We also give results for the case without local cross couplings (red lines), that is, $g_{12}=0,g_{22}=g_{11}$. Here $N_A=200$.

Figure 3

Inference squeezing $S_{-}=[{\Delta }^2({\widehat{J}}_C^{\theta }-{\widehat{J}}_D^{\theta })/n_0]$ (solid lines) and $S_{+}=[{\Delta }^2({\widehat{J}}_C^{\theta +\pi /2}+{\widehat{J}}_D^{\theta +\pi /2})/n_0]$ (dashed lines) after the beam-splitter interaction, where the shot-noise level is $n_0=\left(\right|\langle {\widehat{J}}_C^Y\rangle |+|\langle {\widehat{J}}_D^Y\rangle \left|\right)/2$. Here $N=N_A=N_B=200$. The parameters are the same as Fig. 2. The red lines show the result without local cross couplings (i.e., $g_{12}=0,g_{22}=g_{11}$).

Figure 4

Entanglement ($E_{\mathrm{product}}<1$) based on the criterion in product form (25). EPR paradox entanglement is obtained when $E_{\mathrm{product}}<0.5$. The solid and dotted lines stand for $N=N_A=N_B=200$ and for $N=2000$ with couplings corresponding to $B=9.116\mathrm{G}$; while the dash-dotted and dashed lines assume no local cross couplings (i.e., $g_{12}=0$, $g_{11}=g_{22}$, for $N=200$ and for $N=2000$).

Figure 5

EPR paradox entanglement, shown by the simultaneous inference squeezing of $S_{-}=[{\Delta }^2({\widehat{J}}_C^{\theta }-g{\widehat{J}}_D^{\theta })/n_0]$ (solid lines) and $S_{+}=[{\Delta }^2({\widehat{J}}_C^{\theta +\pi /2}+g^{\prime }{\widehat{J}}_D^{\theta +\pi /2})/n_0]$ (dashed lines) with optimal $g$ and $g^{\prime }$, where the shot noise level is $n_0=\left(\right|\langle {\widehat{J}}_C^Y\rangle \left|\right)/2$. Plots show different local cross couplings: (a) $B=9.116\mathrm{G}$, and (b) $B=9.086\mathrm{G}$. Insets show the optimal value of factors $g$ (solid lines) and $g^{\prime }$ (dashed lines) with time. Here $N_A=N_B=200$, and other parameters are as for Fig. 2.

Figure 6

EPR paradox is predicted ($E_{\mathrm{EPR}-\mathrm{product}}<1$), for a variety of local cross couplings. Here $N=N_A=N_B=2000$;$B=9.116\mathrm{G}$ (solid line),$B=9.086\mathrm{G}$ (dotted line), and with no cross-couplings, that is, $g_{12}=0,g_{22}=g_{11}$ (dashed line). We use the optimal value of factors $g$ and $g^{\prime }$ for each case. Other parameters are as for the last figures.

Figure 7

Effect of atom number: (a) $N=N_A=N_B=200$ (dashed lines) and $N=2000$ (solid lines) on the EPR paradox entanglement with optimal gain factors $g$ and $g^{\prime }$, for fixed couplings $B=9.116\mathrm{G}$. Inset shows the individual squeezing inferences $S_{-}=[{\Delta }^2({\widehat{J}}_C^{\theta }-g{\widehat{J}}_D^{\theta })/n_0]$ (solid lines), $S_{+}=[{\Delta }^2({\widehat{J}}_C^{\theta +\pi /2}+g^{\prime }{\widehat{J}}_D^{\theta +\pi /2})/n_0]$ (dashed lines) with optimal $g$ and $g^{\prime }$, for $N=2000$. (b) The optimal $E_{\mathrm{EPR}-\mathrm{product}}$ versus different number of atoms. Inset shows the corresponding $g$, $g^{\prime }$ versus $N$.

Figure 8

Entanglement (a) $E_{\mathrm{product}}$ and (b) $E_{\mathrm{EPR}-\mathrm{product}}$ after evolution of the full Hamiltonian (9) including tunneling for $N=200$ and $B=9.116\mathrm{G}$, without the application of the beam splitter. The results are shown for values $\tilde{\kappa }=0.1$ (blue solid lines), $\tilde{\kappa }=0.8$ (red dashed lines), $\tilde{\kappa }=0.9$ (green dotted lines), and $\tilde{\kappa }=1$ (cyan dash-dotted lines).

Figure 9

Entanglement (a) $E_{\mathrm{product}}$ and (b) $E_{\mathrm{EPR}-\mathrm{product}}$ after evolution of the full Hamiltonian (9) including tunneling for $N=2000$ and $B=9.116\mathrm{G}$, without the application of the beam splitter. Meaning of the lines is the same as in Fig. 8.

Figure 10

Entanglement $E_{\mathrm{product}}$ after evolution of the full Hamiltonian (9) including tunneling for $N=200$ and $B=9.116\mathrm{G}$, (a) without and (b) with the application of the beam splitter. The results are shown for values $\tilde{\kappa }=0.01$ (blue solid lines), $\tilde{\kappa }=0.5$ (red dashed lines), and $\tilde{\kappa }=1$ (green dotted lines).

Figure 11

EPR paradox entanglement as measured by $E_{\mathrm{EPR}-\mathrm{product}}$ without tunneling and with the application of the beam splitter, for $B=9.116\mathrm{G}$ and $N_A=N_B=2000$. (a) Only linear losses are enabled; no losses (solid blue lines), ${\tilde{\gamma }}_1=5\times {10}^{-3}$ (dashed red lines), ${\tilde{\gamma }}_1={10}^{-2}$ (dotted green lines). (b) Only two-body intraspecies losses are enabled; no losses (solid blue lines), ${\tilde{\gamma }}_{22}=5\times {10}^{-6}$ (dashed red lines), ${\tilde{\gamma }}_{22}={10}^{-5}$ (dotted green lines).

Figure 12

EPR paradox entanglement as measured by $E_{\mathrm{EPR}-\mathrm{product}}$ without tunneling and with the application of the beam splitter, for $B=9.116\mathrm{G}$ and $N_A=N_B=2000$. Only two-body interspecies losses are enabled; no losses (solid blue lines), ${\tilde{\gamma }}_{12}={10}^{-4}$ (dashed red lines), ${\tilde{\gamma }}_{12}={10}^{-3}$ (dotted green lines).