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

Quantum metrology beyond Heisenberg limit with entangled matter wave solitons