  The relative phase of two initially independent Bose-Einstein condensates can
be laser cooled to unite the two condensates by putting them into a ring cavity
and coupling them with an internal Josephson junction. First, we show that this
phase cooling process already appears within a semiclassical model. We
calculate the stationary states, find regions of bistable behavior and suggest
a Ramsey-type experiment to measure the build up of phase coherence between the
condensates. We also study quantum effects and imperfections of the system.
