We derive from first principles the experimentally observed effective dynamics of a spinor Bose gas initially prepared as a Bose{\textendash}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{\textendash}spin interaction. The effective dynamics is well-known to be governed by a system of coupled semi-linear Schr{\"o}dinger 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.

}, doi = {10.1088/1751-8121/aadbc2}, url = {https://doi.org/10.1088\%2F1751-8121\%2Faadbc2}, author = {Alessandro Michelangeli and Alessandro Olgiati} } @inbook {Olgiati2017, title = {Effective Non-linear Dynamics of Binary Condensates and Open Problems}, booktitle = {Advances in Quantum Mechanics: Contemporary Trends and Open Problems}, year = {2017}, pages = {239{\textendash}256}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Sch{\"o}dinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.

}, isbn = {978-3-319-58904-6}, doi = {10.1007/978-3-319-58904-6_14}, url = {https://doi.org/10.1007/978-3-319-58904-6_14}, author = {Alessandro Olgiati}, editor = {Alessandro Michelangeli and Gianfausto Dell{\textquoteright}Antonio} }