Statistics of vortex trapping in cyclically coupled Bose-Josephson junctions

We investigate the problem of vortex trapping in cyclically coupled Bose-Josephson junctions. Starting with $N$ independent BECs we allow the system to reach a stable circulation by adding a dissipative term in our semi-classical equations of motions. We then ask, ${\it inter \, alia}$ the question: ``Starting with an initial normal distribution of total phases with variance $ \sim \sqrt{N} $ and allowing for phase slips, what is the probability to trap a stable vortex with winding number $2 \pi m$''? We find that the final distribution of winding numbers is narrower than the initial distribution of total phases, indicating an increased probability for no-vortex configurations. The role of dissipation has been studied in determining the final probability distibution. It is also possible to get a non-zero circulation starting with zero total phase around the loop. The final width of the distribution scales as $ \sim d \times N^{\alpha } $, where $ \alpha = 0.47 $ and $ d<1 $ (indicating a shrinking of the final distribution), the actual value of $ d $ depending on the strength of dissipation.