  By considering matter wave bright solitons from weakly coupled Bose-Einstein
condensates trapped in a double-well potential, we study the formation of
macroscopic non-classical states, including Schr\"odinger-cat superposition
states and maximally path entangled $N00N$-states. With these macroscopic
states, we examine Mach-Zehnder interferometer in the context of parity
measurements, in order to obtain Heisenberg limit accuracy for linear phase
shift measurement. We reveal that the ratio between two-body scattering length
and intra-well hopping parameter can be measured with the scaling beyond this
limit by using nonlinear phase shift with interacting quantum solitons.
