  We model the dynamics of attractively interacting ultracold bosonic atoms in
a quasi-one-dimensional wave-guide with additional harmonic trapping.
Initially, we prepare the system in its ground state and then shift the zero of
the harmonic trap and switch on an additional narrow scattering potential near
the center of the trap. After colliding with the barrier twice, we propose to
measure the number of atoms opposite to the initial condition. Quantum-enhanced
interferometry with quantum bright solitons allows us to predict detection of
an offset of the scattering potential with considerably increased precision as
compared to single-particle experiments. In a future experimental realization
this might lead to measurement of weak forces caused, for example, by small
horizontal gradients in the gravitational potential - with a resolution of
several micrometers given essentially by the size of the solitons. Our
numerical simulations are based on the rigorously proved effective potential
approach developed in [Phys. Rev. Lett. 102, 010403 (2009) and Phys. Rev. Lett.
103, 210402 (2009)]. We choose our parameters such that the prerequisite of the
proof (that the solitons cannot break apart, for energetic reasons) is always
fulfilled, thus exploring a parameter regime inaccessible to the mean-field
description via the Gross-Pitaevskii equation due to Schrodinger-cat states
occurring in the many-particle quantum dynamics.
