We investigate scattering bright solitons off a potential using both analytical and numerical methods. Our paper focuses on low kinetic energies for which differences between the mean-field description via the Gross-Pitaevskii equation (GPE) and the quantum behavior are particularly large. On the
N-particle quantum level, adding an additional harmonic confinement leads to a simple signature to distinguish quantum superpositions from statistical mixtures. While the nonlinear character of the GPE does not allow quantum superpositions, the splitting of GPE solitons takes place only partially. When the potential strength is increased, the fraction of the soliton which is transmitted or reflected jumps noncontinuously. We explain these jumps via energy conservation and interpret them as indications for quantum superpositions on the
N-particle level. On the GPE level, we also investigate the transition from this stepwise behavior to the continuous case.
We observe nonlinear scattering of 39 K atomic bright solitons launched in a one-dimensional (1D) speckle disorder. We directly compare it with the scattering of non-interacting particles in the same disorder. The atoms in the soliton tend to be collectively either reflected or transmitted, in contrast with the behavior of independent particles in the singlescattering regime, thus demonstrating a clear nonlinear effect in scattering. The observed strong fluctuations in the reflected fraction, between zero and 100%, are interpreted as a consequence of the strong sensitivity of the system to the experimental conditions and in particular to the soliton velocity. This behavior is reproduced in a mean-field framework by Gross-Pitaevskii simulations, and mesoscopic quantum superpositions of the soliton being fully reflected and fully transmitted are not expected for our parameters. We discuss the conditions for observing such superpositions, which would find applications in atom interferometry beyond the standard quantum limit.