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Derivation of the dipolar Gross-Pitaevskii energy

Arnaud Triay
(CEREMADE and Universite de Paris Dauphine)
Analysis Seminar
Thursday, 19 April, 2018, at 16:15-17:00, in Aud. D3 (1531-215)
Abstract:
The Gross-Pitaevskii theory effectively describes the ground state and the evolution of a dilute and ultracold gas of bosons. A very vast literature exists on the derivation of this theory from the principles of quantum mechanics, nevertheless it remains a challenging task is to address the case of non-positive interactions such as dipole-dipole potentials. We will show, using the so-called quantum de Finetti theorem, the convergence of the ground state and of the ground state energy of the (linear) $N$ body Hamiltonian towards those of the dipolar GP functional. The latter, in addition to the usual cubic interaction, has a long range dipolar term $K\star |u|^2 |u|^2$. Our results hold under the assumption that the two-particle interaction is scaled in the form $N^{3\beta-1}w(N^\beta x)$ for some $0\leq\beta< \beta_{max}$ with $\beta_{max} = 1/3 + s/(45 + 42s)$ where $s$ is related to the growth of the trapping potential.
Contact person: Søren Fournais
89566 / i31