Solitons are long-lived wavepackets that propagate without dispersion and exist in a wide range of onedimensional (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.Bright-solitary waves, referred to as solitons in this work, are wavepackets that propagate in a quasi-1D geometry without dispersion, owing to a self-focussing nonlinearity. They are of fundamental interest in a broad range of settings due to their ubiquity in nonlinear systems, which occur prolifically in nature 1, 2 . In Bose-Einstein 1 arXiv:1906.06083v1 [cond-mat.quant-gas] 14 Jun 2019 condensates (BECs) the nonlinearity is provided by interatomic interactions governed by the s-wave scattering length, which can be tuned using a magnetic Feshbach resonance 3 . Bright solitons in BECs of 7 Li, 85 Rb, 39 K and 133 Cs have so far been experimentally demonstrated 4-10 . Understanding and probing the coherent phase carried by matter-wave solitons is an area of particular relevance for BEC physics, both because it is important in determining the stability of soliton-soliton collisions 10-14 and because there is a great interest in using solitons for atom interferometry 15-22 .Matter-wave interferometers have emerged as a means of achieving unprecedented sensitivity in interferometric measurements 23-26 . However, they have typically been limited by either interatomic collisions or dispersion of the atomic wavepackets, which cause dephasing and a reduced signal to noise, respectively 27 . Previous works have successfully reduced the impact of interatomic collisions through the control of interatomic interactions 28, 29 , or by generating squeezed states 30, 31 . However, dispersion remains a limitation. A soliton-based interferometer has the potential to overcome dispersion, allowing for much longer phase-accumulation times, albeit for an increased quantum noise 32 . To date, only one experiment has demonstrated interferometry with a soliton 8 , in which Bragg pulses were used for splitting and recombination. However, interferom...

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