  One of the most fundamental difference between classical and quantum
mechanics is observed in the particle tunneling through a localized potential:
the former predicts a discontinuous transmission coefficient ($T$) as a
function in incident velocity between one (complete penetration) and zero
(complete reflection), while the later always changes smoothly as a wave
nature. Here we report a systematic study of the quantum tunneling property for
a bright soliton in ultracold atoms, which behaves as a classical particle
(matter wave) in the limit of small (large) incident velocity. In the
intermediate regime, the classical and quantum properties are combined via a
finite (but not full) discontinuity in the tunneling transmission coefficient.
We demonstrate that the formation of a localized bound state is essential to
describe such inelastic collisions, showing a nontrivial nonlinear effect on
the quantum transportation of a bright soliton.
