  The quantum Hall phase diagram of the half-filled bilayer system in the
second Landau level is studied as a function of tunneling and layer separation
using exact diagonalization. We make the striking prediction that bilayer
structures would manifest two distinct branches of incompressible fractional
quantum Hall effect (FQHE) corresponding to the Abelian 331 state (at moderate
to low tunneling and large layer separation) and the non-Abelian Pfaffian state
(at large tunneling and small layer separation). The observation of these two
FQHE branches and the quantum phase transition between them will be compelling
evidence supporting the existence of the non-Abelian Pfaffian state in the
second Landau level.
