  Solitons are long-lived wavepackets that propagate without dispersion and
exist in a wide range of one-dimensional (1D) nonlinear systems. A
Bose-Einstein condensate trapped in a quasi-1D waveguide can support
bright-solitary-matter waves (3D analogues of solitons) when interatomic
interactions are sufficiently attractive that they cancel dispersion.
Solitary-matter waves are excellent candidates for a new generation of highly
sensitive interferometers, as their non-dispersive nature allows them to
acquire phase shifts for longer times than conventional matter-waves
interferometers. However, such an interferometer is yet to be realised
experimentally. In this work, we demonstrate the splitting and recombination of
a bright-solitary-matter wave on a narrow repulsive barrier, which brings
together the fundamental components of an interferometer. We show that both
interference-mediated recombination and classical velocity filtering effects
are important, but for a sufficiently narrow barrier interference-mediated
recombination can dominate. We reveal the extreme sensitivity of
interference-mediated recombination to the experimental parameters,
highlighting the potential of soliton interferometry.
