  The miscibility-immiscibility phase transition in binary Bose-Einstein
condensates (BECs) can be controlled by a coupling between the two components.
Here we propose a new scheme that uses coupling-induced pattern formation to
test the Kibble-Zurek mechanism (KZM) of topological-defect formation in a
quantum phase transition. For a binary BEC in a ring trap we find that the
number of domains forming the pattern scales with the coupling quench rate with
an exponent as predicted by the KZM. For a binary BEC in an elongated harmonic
trap we find a different scaling law due to the transition being spatially
inhomogeneous. We perform a "simulation" of the harmonically trapped system in
a ring trap to verify the scaling exponent.
