Einstein-Podolsky-Rosen entanglement and steering in two-well Bose-Einstein-condensate ground states

We consider how to generate and detect Einstein-Podolsky-Rosen (EPR) entanglement and the steering paradox between groups of atoms in two separated potential wells in a Bose-Einstein condensate. We present experimental criteria for this form of entanglement and propose experimental strategies for detecting entanglement using two- or four-mode ground states. These approaches use spatial and/or internal modes. We also present higher-order criteria that act as signatures to detect the multiparticle entanglement present in this system. We point out the difference between spatial entanglement using separated detectors and other types of entanglement that do not require spatial separation. The four-mode approach with two spatial and two internal modes results in an entanglement signature with spatially separated detectors, conceptually similar to the original EPR paradox.

Figure 1

Two-mode case: a double-well one-spin-orientation BEC. The $a$ and $b$ are operators for two modes at $A$ and $B$. The $a$ and $b$ are prepared with a two-mode number difference squeezing and an entanglement by adiabatic cooling to the ground state. We develop signatures to detect the interwell entanglement using interwell spin operators.

Figure 2

Four-mode case: a double-well two-spin-orientation BEC. We suppose the modes $a_i$ and $b_i$ are spatially separated. Modes $a_1$ and $a_2$ could be different spatial modes or different spin components of the same well. The pair $a_1,b_1$ (and $a_2,b_2$) can become entangled due to the interwell couplings. We allow for the asymmetric case where the pair $a_2,b_2$ has much greater numbers than $a_1,b_1$ ($N_2\gg N_1$) and also consider a case where modes $a_2$ and $b_2$ need not be entangled (${\kappa }_2=0$).

Figure 3

Fock number state $|N\rangle$ incident on a beam splitter (BS) produces an entangled state (2).

Figure 4

Entanglement of Fig. 3 detected by the HZ entanglement criterion ($E_{\text{HZ}}<1$) [Eq. (7)]. Higher-order entanglement is indicated by the purple doted, cyan dash-dotted, and red short dashed curves. The correlation does not confirm EPR steering entanglement from Eq. (14), which requires $E_{\text{HZ}}<0.5$.

Figure 5

Double Fock number state incident on a beam splitter (BS) produces a number-conserving entangled state.

Figure 6

Hillery-Zubairy entanglement criterion using a double Fock number state and beam splitter. The graph shows the criterion (7) for $m=n=2$ (solid blue line) and $m=n=4$ (red dashed line). Einstein-Podolsky-Rosen, which requires $E_{\mathrm{HZ}}<0.5$ steering is observable with $m=n=2$ and $N<5$.

Figure 7

Entanglement in the ground state of the BEC Hamiltonian (31), using the HZ criterion (16), plotted against the coupling constant for both positive and negative couplings for $N=100$ atoms. Plots show the first-order HZ entanglement as a function of $Ng/\kappa$, with $\kappa >0$ held fixed and $g$ varied. The mean spin is in the direction defined by $J_{AB}^X$. The HZ entanglement criterion $E_{\text{HZ}}<1$ indicates a two-mode entanglement and $E_{\text{HZ}}<0.5$ indicates EPR steering. The dashed red line gives the HZ criterion $E_{\text{HZ}}^{\prime }$ for the rotated modes $a^{\prime }$ and $b^{\prime }$. The predictions for the respective second-order entanglement criterion $E_{\text{HZ}}^{\left(2\right)}$ [Eq. (9)] are given by the dotted and starred curves.

Figure 8

Same as Fig. 7 but for much lower particle number with $N=6$. First-order entanglement $E_{\text{HZ}}^{\left(1\right)}$ in ($a,b$) is shown by the solid blue line; second-order entanglement $E_{\text{HZ}}^{\left(2\right)}$ is shown by the purple line with dots. First-order entanglement in ($a^{\prime },b^{\prime }$) is shown by the dashed red line; second-order entanglement is shown by the black line with stars. The second-order entanglement criterion becomes more sensitive where the nonlinearity is higher.

Figure 9

Repulsive interaction case for $N=100$, showing individual spin variances and mean spin $|\langle J_{AB}^X\rangle |$ for the ground-state solution of the Hamiltonian (31) in the regime where there is a strong repulsive self-interaction $g/\kappa$ for $N=100$. Here $\kappa >0$ is fixed and $g$ is varied. Entanglement is obtained when ${\Delta }^2{J}_{AB}^Z+{\Delta }^2{J}_{AB}^X<N/2$ and squeezing of the individual spin $J_{AB}^{\theta }$ is obtained when ${\Delta }^2{J}_{AB}^{\theta }<N/4$. For simplicity we have dropped the subscripts $AB$ in the labeling of the variances in the figure. We note that the sum of the spin variances has a minimum value with a critical value of the coupling $g/\kappa$.

Figure 10

Three-dimensional variance ellipsoid corresponding to $N=100$ and a repulsive interaction at the optimum coupling of $Ng/\kappa =40$. Spin variances are reduced on both axes parallel to the $X\text{−}Z$ plane to show strong but not perfect planar quantum squeezing. The variance increases perpendicular to the squeezing plane along the $Y$ axis.

Figure 11

Higher-order entanglement for the case of $N=100$. Other parameters are as in Fig. 7. Entanglement is obtained when $E_{\text{HZ}}<1$. The red dashed line shows a reducing $E_{\text{HZ}}^{\left(1\right)}$ entanglement as $g/\kappa$ is increased above a certain level. The solid black line (starred) shows the second-order entanglement measure $E_{\text{HZ}}^{\left(2\right)}$. The second-order entanglement $E_{\text{HZ}}^{\left(2\right)}<1$ is achievable at higher $g/\kappa$ ratios.

Figure 12

Pairs of Fock states transmitted through a beam splitter. The pair $a_1,b_1$ is coupled and becomes entangled, as does $a_2,b_2$.

Figure 13

The entanglement of pairs of Fock states transmitted through a beam splitter can be detected via the spin Hillery-Zubairy criterion (23) for the asymmetric case where the pair $a_2,b_2$ has much greater numbers $N_2\gg N_1$ ($N_2=100$). Entanglement is confirmed if $E_{\text{HZ}}^{\text{spin}\left(1\right)}<1$.

Figure 14

Detecting interwell entanglement using a second local oscillator mode pair: entanglement of the ground state for a two-well potential, at $T=0\text{K}$, for the four-mode model of Fig. 2, with a variety of local cross couplings $g_{ij}$, for $N_1=5$ and $N_2=100$. Here $\kappa ={\kappa }_1={\kappa }_2>0$ and $g_{11}$ is varied with the other values of $g_{ij}$ held in a fixed ratio; $E_{\text{HZ}}^{\text{spin}\left(1\right)}<1$ indicates entanglement and $E_{\text{HZ}}^{\text{spin}\left(1\right)}<0.5$ indicates EPR steering. Curves are labeled in order of nesting as follows: magenta dash dotted curve, equal couplings; blue dotted curve, nonzero cross couplings corresponding to ${}^{87}\text{Rb}$ Feshbach resonance with $a_{11}=100.4a_0,a_{12}=80.8a_0,\text{and}{a}_{22}=95.5a_0$; black dashed curve, without cross correlations $g_{12}=0$ and $g_{22}=g_{11}$; and green solid curve, negative relative cross coupling $g_{11},g_{12}<0$. The inset shows the effect of increasingly symmetric atom numbers.

Figure 15

Effect of optimally entangling the second local oscillator mode pair: entanglement of the ground state for a two-well potential, at $T=0\text{K}$, for the four-mode model of Fig. 2. Here $\kappa ={\kappa }_1={\kappa }_2$, $g_{12}=0$, and both $g_{11}$ and $g_{22}$ are varied so that $N_1{g}_{11}/{\kappa }_1=N_2{g}_{22}/{\kappa }_2$; $E_{\text{HZ}}^{\text{spin}\left(1\right)}<1$ indicates entanglement and $E_{\text{HZ}}^{\text{spin}\left(1\right)}<0.5$ indicates EPR steering. Main graph: black dashed curve, $N_1=5\text{and}{N}_2=100$; blue dotted curve, $N_1=20$ and $N_2=100$. The curves are for values of local coupling that optimize $E_{\text{HZ}}^{\left(1\right)}$ for each mode pair, in which case for $N_2\gg N_1$ the $E_{\text{HZ}}^{\text{spin}\left(1\right)}$ reaches the value of $E_{\text{HZ}}^{\left(1\right)}$ displayed in Fig. 7. The inset reveals the individual degree of HZ entanglement $E_{\text{HZ}}^{\left(1\right)}$ for the mode pairs $a_1$,$b_1$ and $a_2$,$b_2$, as explained in the text.

Figure 16

Enhancing entanglement for the repulsive regime: entanglement of the ground state for a two-well potential, at $T=0\text{K}$, for the four-mode model of Fig. 2. Here $\kappa ={\kappa }_1={\kappa }_2$. The parameters are the same as for Fig. 14, but the entanglement parameter is calculated for the rotated modes $a^{\prime }\text{and}{b}^{\prime }$ of Eq. (32). For large $N_2$ the strength of the entanglement measure is enough to confirm EPR steering via the criterion (25).

Figure 17

Effect of critical temperatures for the parameters of Fig. 14 when $Ng/\kappa \approx -2.23$.

Figure 18

Effect of an uncorrelated coherent atomic oscillator field for mode 2 in a coherent state with amplitude $\alpha$ in the optimal case of Fig. 7 when $N_{1}g/{\kappa }_1\approx -2.03$.