Bright solitary waves in a Bose-Einstein condensate contain thousands of identical atoms held together despite their only weakly attractive contact interactions. They nonetheless behave like a compound object, staying whole in collisions, with their collision properties strongly affected by intersoliton quantum coherence. We show that separate solitary waves decohere due to phase diffusion, dependent on their effective ambient temperature, after which their initial mean-field relative phases are no longer well defined or relevant for collisions. In this situation, collisions occur predominantly repulsively and can no longer be described within mean-field theory. When considering the timescales involved in recent solitary wave experiments where nonequilibrium phenomena play an important role, these features could explain the predominantly repulsive collision dynamics observed in most condensate soliton train experiments.

We predict hyper-entanglement generation during binary scattering of mesoscopic bound states, solitary waves in Bose-Einstein condensates containing thousands of identical Bosons. This requires collisions in elongated, cigar shaped traps, in which the residual presence of transverse dimensions gives rise to effective interactions that are cubic and quintic in the particle density, in a dimensionally reduced description. Under these integrability breaking conditions, we show that the post-collision state of an initially fragmented soliton pair can be hyper-entangled in spatial degrees of freedom and atom number within solitons, for realistic parameters. For this, we model collisions of quantum solitons in the quintic model beyond the mean-field, using the truncated Wigner approximation.

It is an open fundamental question how the classical appearance of our environment arises from the underlying quantum many-body theory. We propose that phenomena involved in the quantum-to-classical transition can be probed in collisions of bright solitary waves in Bose- Einstein condensates, where thousands of atoms form a large compound object at ultra cold temperatures. For the experimentally most relevant quasi-1D regime, where integrability is bro- ken through effective three-body interactions, we find that ensembles of solitary waves exhibit complex interplay between phase coherence and entanglement generation in beyond mean-field simulations using the truncated Wigner method: An initial state of two solitons with a well de- fined relative phase looses that phase coherence in the ensemble, with its single particle two-mode density matrix exhibiting similar dynamics as a decohering two mode superposition. This apparent decoherence is a prerequisite for the formation of entangled superpositions of different atom num- bers in a subsequent soliton collision. The necessity for the solitons to first decohere is explained based on the underlying phase-space of the quintic mean field equation. We show elsewhere that superpositions of different atom numbers later further evolve into spatially entangled solitons. Loss of ensemble phase coherence followed by system internal entanglement generation appear in an unusual order in this closed system, compared to a typical open quantum system.

We predict hyper-entanglement generation by binary scattering of mesoscopic bound states, considering solitary waves in Bose-Einstein condensates containing thousands of identical Bosons. 
For it to occur, the underlying many-body Hamiltonian must not be integrable, and the pre-collision quantum state of the solitons needs to be fragmented.
Under these conditions, we show that the post-collision state will be hyper-entangled in spatial degrees of freedom and atom number within solitons, for realistic parameters. The effect links aspects of non-linear systems and quantum-coherence and the entangled post-collision state challenges present entanglement criteria for identical particles. Our results are based on simulations of colliding quantum solitons in a quintic interaction model beyond the mean-field, using the truncated Wigner approximation.