Effective non-linear spinor dynamics in a spin-1 Bose-Einstein condensate

We derive from first principles the experimentally observed effective dynamics of a spinor Bose gas initially prepared as a Bose-Einstein condensate and then left free to expand ballistically. In spinor condensates, which represent one of the recent frontiers in the manipulation of ultra-cold atoms, particles interact with a two-body spatial interaction and a spin-spin interaction. The effective dynamics is governed by a system of coupled semi-linear Schr\"odinger equations: we recover this system, in the sense of marginals in the limit of infinitely many particles, with a mean-field re-scaling of the many-body Hamiltonian. When the resulting control of the dynamical persistence of condensation is quantified with the parameters of modern observations, we obtain a bound that remains quite accurate for the whole typical duration of the experiment.