  We describe the ground state of a large, dilute, neutral atom Bose- Einstein
condensate (BEC) doped with N strongly coupled mutually indistinguishable,
bosonic neutral atoms (referred to as "impurity") in the polaron regime where
the BEC density response to the impurity atoms remains significantly smaller
than the average density of the surrounding BEC. We find that N impurity atoms
(N is not one) can self-localize at a lower value of the impurity-boson
interaction strength than a single impurity atom. When the 'bare' short-range
impurity-impurity repulsion does not play a significant role, the
self-localization of multiple bosonic impurity atoms into the same single
particle orbital (which we call co-self-localization) is the nucleation process
of the phase separation transition. When the short-range impurity-impurity
repulsion successfully competes with co-self-localization, the system may form
a stable liquid of self-localized single impurity polarons.
