Bose-condensate tunneling dynamics: Momentum-shortened pendulum with damping

Abstract

Bose-Einstein condensates in a double-well trap, as well ${}^{3}\mathrm{H}\mathrm{e}\ensuremath{-}\mathrm{B}$ baths connected by micropores, have been shown to exhibit Josephson-like tunneling phenomena. Unlike the superconductor Josephson junction of phase difference $\ensuremath{\varphi}$ that maps onto a rigid pendulum of energy $\mathrm{cos}(\ensuremath{\varphi})$, these systems map onto a momentum-shortened pendulum of energy $\ensuremath{-}\sqrt{1\ensuremath{-}{p}_{\ensuremath{\varphi}}^{2}}\mathrm{cos}(\ensuremath{\varphi})$ and length $\sqrt{1\ensuremath{-}{p}_{\ensuremath{\varphi}}^{2}}$, where ${p}_{\ensuremath{\varphi}}$ is a population imbalance between the wells/baths. We study here the effect of damping on the four distinct modes of the nonrigid pendulum, characterized by distinct temporal mean values, $〈\ensuremath{\varphi}〉$ and $〈{p}_{\ensuremath{\varphi}}〉$. Damping is shown to produce different decay trajectories to the final equilibrium $\ensuremath{\varphi}{=0=p}_{\ensuremath{\varphi}}$ state that are characteristic dynamic signatures of the initial oscillation modes. In particular, damping causes $\ensuremath{\pi}$-state oscillations with $〈\ensuremath{\varphi}〉=\ensuremath{\pi}$ to increase in amplitude and pass through phase-slip states, before equilibrating. Similar behavior has been seen in ${}^{3}\mathrm{H}\mathrm{e}\ensuremath{-}\mathrm{B}$ experiments.