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 well-known to be governed by a system of coupled semi-linear Schrödinger 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.

%B Journal of Physics A: Mathematical and Theoretical %I IOP Publishing %V 51 %P 405201 %8 sep %G eng %U https://doi.org/10.1088%2F1751-8121%2Faadbc2 %R 10.1088/1751-8121/aadbc2 %0 Book Section %B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %D 2017 %T Effective Non-linear Dynamics of Binary Condensates and Open Problems %A Alessandro Olgiati %E Alessandro Michelangeli %E Gianfausto Dell'Antonio %XWe 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 Schödinger 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.

%B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %I Springer International Publishing %C Cham %P 239–256 %@ 978-3-319-58904-6 %G eng %U https://doi.org/10.1007/978-3-319-58904-6_14 %R 10.1007/978-3-319-58904-6_14 %0 Journal Article %J Nuovo Cimento B 123 (2008) 181-192 %D 2008 %T Equivalent definitions of asymptotic 100% B.E.C. %A Alessandro Michelangeli %X In the mathematical analysis Bose-Einstein condensates, in particular in the study of the quantum dynamics, some kind of factorisation property has been recently proposed as a convenient technical assumption of condensation. After having surveyed both the standard definition of complete Bose-Einstein condensation in the limit of infinitely many particles and some forms of asymptotic factorisation, we prove that these characterisations are equivalent. %B Nuovo Cimento B 123 (2008) 181-192 %G en_US %U http://hdl.handle.net/1963/2546 %1 1573 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-01-10T13:15:49Z\\nNo. of bitstreams: 1\\nsissa_preprint102_07.pdf: 215355 bytes, checksum: a09c310627ed3ed9acf1156596e7dd7e (MD5) %R 10.1393/ncb/i2008-10521-y