Spin squeezing and Einstein-Podolsky-Rosen entanglement of two bimodal condensates in state-dependent potentials

We propose and analyze a scheme to entangle the collective spin states of two spatially separated bimodal Bose-Einstein condensates. Using a four-mode approximation for the atomic field, we show that elastic collisions in a state-dependent potential simultaneously create spin-squeezing in each condensate and entangle the collective spins of the two condensates. We investigate mostly analytically the nonlocal quantum correlations that arise in this system at short times and show that Einstein-Podolsky-Rosen (EPR) entanglement is generated between the condensates. At long times we point out macroscopic entangled states and explain their structure. The scheme can be implemented with condensates in state-dependent microwave potentials on an atom chip.

Figure 1

Sequence allowing us to entangle condensate $a$ (right well) and $b$ (left well) via controlled collisional interaction in state-dependent trapping potentials. The interaction phase, where the squeezing and the correlations are created, is depicted in the central panel.

Figure 2

Real part of the reduced density matrix in the Fock basis ${\left({\rho }_a\right)}_{n_a,n_a^{\prime }}$ at different times (in units of $1/{\chi }_{ab}$) for ${\chi }_{ab}=\chi$ and $N=100$ atoms in each of the two ensembles. We start with the density matrix of a phase state (11). The entanglement dynamics damps out the off-diagonal coefficients after which striped patterns appear. The interpretation of these patterns is given in the text. The color scale ranges from red (largest positive), green, light blue (zero), to dark blue (negative).

Figure 3

Entropy of entanglement between ensembles $a$ and $b$ as a function of time for different values of the ratio $\chi /{\chi }_{ab}$. The number of atoms in $a$ and $b$ is even: $N=50$. The insets show the total density matrix in phase state basis ${\rho }_{{\varphi }_a,{\varphi }_b}$ [see Eq. (25)] at times ${\chi }_{ab}t=\pi /3$, $2\pi /3$, $\pi$, $4\pi /3$, and $(2\pi -\frac{1}{2\sqrt{N}})$. The phases ${\varphi }_a$, ${\varphi }_b$ range in $[\pi /2,5\pi /2]$. We note the appearance of simple superpositions of phase states.

Figure 4

Top row: Evolution of $E_{\mathrm{EPR}}$ (34) for three different ratios $\chi /{\chi }_{ab}$. The number of atoms in each ensemble is $N=50$ (black dotted line), $N=500$ (blue dashed line), or $N=5000$ (red solid line). The quadrature angle $\alpha$ has been optimized numerically, while $\beta$ is fixed by the conditions (35), (38), (39). Bottom row: Dependence of $E_{\mathrm{EPR}}$ on the angle $\alpha$ while $\beta$ is still given by the conditions (35), (38), (39). In these curves the time is fixed to the best time (minimum of $E_{\mathrm{EPR}}$) extracted from the top-row curves, although the time dependence is smooth in the interesting region (37).

Figure 5

Top: Evolution of $E_{\mathrm{EPR}}$ for $\chi =0.5{\chi }_{ab}$ and quadratures chosen according to (41). The atom numbers are the same as in Fig. 4. Bottom: At the best time, dependence of $E_{\mathrm{EPR}}$ on the angle $\alpha$, while $\beta$ is still given by the conditions (41).

Figure 6

Top row: Evolution of the entanglement witness $E_{\mathrm{ent}}$ (42) for three different ratios $\chi /{\chi }_{ab}$. The number of atoms in each ensemble is $N=50$ (black dotted line), $N=500$ (blue dashed line), and $N=5000$ (red solid line). The quadrature angle $\alpha$ has been optimized numerically, while $\beta$ is fixed by the conditions (35), (38), (39). Bottom row: Dependence of $E_{\mathrm{ent}}$ on the angle $\alpha$ while $\beta$ is still given by the conditions (35), (38), (39). In these curves the time is fixed to the best time (minimum of $E_{\mathrm{ent}}$) extracted from the top-row curves.

Figure 7

Angular dependence of the squeezing parameter ${\xi }_{\alpha }$ (43) (solid red line) together with $E_{\mathrm{EPR}}$ (dashed blue line, already shown in Fig. 4 bottom row) for two values of the ratio $\eta ={\chi }_{ab}/2\chi$. For both ${\xi }_{\alpha }$ and $E_{\mathrm{EPR}}$ the time is fixed to the time-minimizing $E_{\mathrm{EPR}}$, as in Fig. 4. The atom number is $N=5000$ in each condensate. The horizontal red dotted line shows the analytical value of the squeezing limit (47).