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

We revisit recent claims about the instability of non-rotating tunnel coupled annular Bose-Einstein condensates leading to the emergence of angular-momentum Josephson oscillation [Phys. Rev. Lett. 98, 050401 (2007), arXiv:quant-ph/0609133v2]. 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 non-rotating 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.