Current production in ring condensates with a weak link

We consider attractive and repulsive condensates in a ring trap stirred by a weak link, and analyze the spectrum of solitonic trains dragged by the link, by means of analytical expressions for the wave functions, energies and currents. The precise evolution of current production and destruction in terms of defect formation in the ring and in terms of stirring is studied. We find that any excited state can be coupled to the ground state through two proposed methods: either by adiabatically tuning the link's strength and velocity through precise cycles which avoid the critical velocities and thus unstable regions, or by having the link still while setting an auxiliary potential and imprinting a nonlinear phase as the potential is turned off. We also analyze hysteresis cycles through the spectrum of energies and currents.