Sign of coupling in barrier-separated Bose-Einstein condensates and stability of double-ring systems

We revisit recent claims about the instability of nonrotating tunnel coupled annular Bose-Einstein condensates leading to the emergence of angular momentum Josephson oscillation [Phys. Rev. Lett. 98, 050401 (2007)]. It was predicted that all stationary states with uniform density become unstable in certain parameter regimes. By careful analysis, we arrive at a different conclusion. We show that there is a stable nonrotating and uniform ground state for any value of the tunnel coupling and repulsive interactions. The instability of an excited state with $\pi$ phase difference between the condensates can be interpreted in terms of the familiar snake instability. We further discuss the sign of the tunnel coupling through a separating barrier, which carries significance for the nature of the stationary states. It is found to always be negative for physical reasons.

Figure 1

Energy difference ${\Delta }_{\mathrm{AS}}=E_{\mathrm{A}}-E_{\mathrm{S}}$ between the lowest antisymmetric and symmetric eigenstates of a quadratic-plus-quartic double-well potential, plotted as a function of the dimensionless effective interaction strength $\mathrm{g̃}$. The fact that ${\Delta }_{\mathrm{AS}}⩾0$ indicates that the symmetric (nodeless) state remains the ground state even in the limit where the atoms interact strongly. Double-well parameters [see Eq. (7)] are ${\xi }_0=5$ and $h=0.002$ (solid curve), 0.02 (dashed curve), and 0.05 (dot-dashed curve).

Figure 2

Lowest symmetric (solid curve) and antisymmetric (dashed curve) condensate wave functions obtained for a double-well potential [see Eq. (7)] with ${\xi }_0=5$, $h=0.05$, and $\mathrm{g̃}=30$ (left panel) or 300 (right panel). Notice the greater delocalization of atoms between the two wells when the interaction strength is high. This arises because repulsive interactions result in an effective lowering of the tunnel barrier.