Josephson tunnelling of dark solitons in a double-well potential

We study the dynamics of matter waves in an effectively one-dimensional Bose–Einstein condensate in a double-well potential. We consider in particular the case when one of the double wells confines excited states. Similar to the known ground state oscillations, the states can tunnel between the wells experiencing the physics known for electrons in a Josephson junction, or be self-trapped. As the existence of dark solitons in a harmonic trap is a continuation of such non-ground state excitations, one can view the Josephson-like oscillations as tunnellings of dark solitons. Numerical existence and stability analysis based on the full equation is performed, where it is shown that such tunnelling can be stable. Through a numerical path-following method, unstable tunnelling is also obtained in different parameter regions. A coupled-mode system is derived and compared to the numerical observations. Regions of (in)stability of Josephson tunnelling are discussed and highlighted. Finally, we outline an experimental scheme designed to explore such dark soliton dynamics in the laboratory.