  Multicomponent quantum Hall systems with internal degrees of freedom provide
a fertile ground for the emergence of exotic quantum liquids. Here we
investigate the possibility of non-Abelian topological order in the half-filled
fractional quantum Hall (FQH) bilayer system driven by the tunneling effect
between two layers. By means of the state-of-the-art density-matrix
renormalization group, we unveil "finger print" evidence of the non-Abelian
Moore-Read Pfaffian state emerging in the intermediate-tunneling regime,
including the ground-state degeneracy on the torus geometry and the topological
entanglement spectroscopy (entanglement spectrum and topological entanglement
entropy) on the spherical geometry, respectively. Remarkably, the phase
transition from the previously identified Abelian $(331)$ Halperin state to the
non-Abelian Moore-Read Pfaffian state is determined to be continuous, which is
signaled by the continuous evolution of the universal part of the entanglement
spectrum, and discontinuities in the excitation gap and the derivative of the
ground-state energy. Our results not only provide a "proof-of-principle"
demonstration of realizing a non-Abelian state through coupling different
degrees of freedom, but also open up a possibility in FQH bilayer systems for
detecting different chiral $p-$wave pairing states.
