We show that the stability of three-dimensional Skyrmions in trapped Bose-Einstein condensates depends critically on scattering lengths, atom numbers, trap rotation and trap anisotropy. In particular, for the 87 Rb |F = 1, m f = −1 , |F = 2, m f = 1 hyperfine states stability is sensitive to the scattering lengths at the 2% level, where the differences between them are crucial. In a cigar shaped trap, we find stable Skyrmions with slightly more than 2 × 10 6 atoms, a number which scales with the inverse square root of the trap frequency. These can be stabilized against drift out of the trap by laser pinning.

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.

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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.