Einstein-Podolsky-Rosen-entangled Bose-Einstein condensates in state-dependent potentials: A dynamical study

We study the generation of nonlocal correlations by atomic interactions in a pair of bimodal Bose-Einstein condensates in state-dependent potentials including spatial dynamics. The wave functions of the four components are described by combining a Fock state expansion with a time-dependent Hartree-Fock ansatz so that both the spatial dynamics and the local and nonlocal quantum correlations are accounted for. We find that despite the spatial dynamics, our protocol generates enough nonlocal entanglement to perform an Einstein-Podolsky-Rosen steering experiment with two spatially separated condensates of a few thousand atoms.

Figure 1

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

Figure 2

The EPR entanglement witness $E_{\mathrm{EPR}}$ (top row) and the renormalized lengths of the two spins $2\langle {\widehat{S}}_{x\varphi }^a\rangle /N_a$ and $2\langle {\widehat{S}}_{x\varphi }^b\rangle /N_b$ (bottom row) are plotted as a function of the total evolution time $t=2t_{\mathrm{R}}+t_{\mathrm{int}}$ for (a) $N_a=N_b=100$ and (b) $N_a=N_b=500$. The trap frequency is $\omega =2\pi \times 20\text{Hz}$ and the scattering lengths $a_{00}=100.4R_{\mathrm{B}}$, $a_{11}=95.0R_{\mathrm{B}}$, and $a_{01}=98.0R_{\mathrm{B}}$ in units of the Bohr radius. The initial separation between the traps is $\delta z_{\max }=10a_0$ in units of the oscillatory length $a_0=\sqrt{\hslash /m\omega }$ and the ramping time is $\omega t_{\mathrm{R}}=10$. Symbols represent the results of numerical simulations of spatial dynamics in the modulus-phase approximation (see Sec. 2c). In the top row, the blue stars represent $E_{\mathrm{EPR}}$. In the bottom row, the blue stars and yellow squares are for $\langle {\widehat{S}}_{x\varphi }^a\rangle$ and $\langle {\widehat{S}}_{x\varphi }^b\rangle$, respectively. The solid lines are analytic predictions of $E_{\mathrm{EPR}}$ and of the spin lengths derived from a four-mode model valid for a strictly adiabatic evolution (see the text). The two (indistinguishable in the figure) dash-dotted lines in the bottom row represent the normalized density overlap (46) of the wave functions ${\varphi }_{a0}$ and ${\varphi }_{a1}$, on the one hand, and of ${\varphi }_{b0}$ and ${\varphi }_{b1}$, on the other.

Figure 3

The EPR entanglement witness $E_{\mathrm{EPR}}$ (top) and the renormalized lengths of the two spins $2\langle {\widehat{S}}_{x\varphi }^a\rangle /N_a$ and $2\langle {\widehat{S}}_{x\varphi }^b\rangle /N_b$ (bottom) are plotted as a function of the total evolution time $t=2t_{\mathrm{R}}+t_{\mathrm{int}}$ for $N_a=N_b=500$. The other simulation parameters are the same as in Fig. 2 except for a smaller initial separation between the traps $\delta z_{\max }=6a_0$. Symbols represent the results of numerical simulations of spatial dynamics in the modulus-phase approximation (see Sec 2c). In the top panel, the blue stars represent $E_{\mathrm{EPR}}$. In the bottom panel, the blue stars and yellow squares are for $\langle {\widehat{S}}_{x\varphi }^a\rangle$ and $\langle {\widehat{S}}_{x\varphi }^b\rangle$, respectively. The solid lines are from an analytic four-mode model valid for a strictly adiabatic evolution (see the text). The two (indistinguishable in the figure) dash-dotted lines in the bottom row represents the normalized density overlap (46) of the wave functions ${\varphi }_{a0}$ and ${\varphi }_{a1}$, on the one hand, and of ${\varphi }_{b0}$ and ${\varphi }_{b1}$, on the other.