  We consider laser-pumped one-dimensional two-component bosons in a parabolic
trap embedded in a high-finesse optical cavity. Above a threshold pump power,
the photons that populate the cavity modify the effective atom trap and mediate
a coupling between the two components of the Bose-Einstein condensate. We
calculate the ground state of the laser-pumped system and find different stages
of self-organization depending on the power of the laser. The modified
potential and the laser-mediated coupling between the atomic components give
rise to rich many-body physics: an increase of the pump power triggers a
self-organization of the atoms while an even larger pump power causes
correlations between the self-organized atoms -- the BEC becomes fragmented and
the reduced density matrix acquires multiple macroscopic eigenvalues. In this
fragmented superradiant state, the atoms can no longer be described as
two-level systems and the mapping of the system to the Dicke model breaks down.
