Speaker
Description
In this talk, we consider the minimization problem for the Gross–Pitaevskii energy
$E(u) = ∫_D (|∇u|² + V(x)|u|² − Ω·ū·L_z u + γ|u|⁴) dx,~~~~~~~~~~~ (E)$
under the mass constraint $‖u‖_{L²(D)} = 1$. Here, D is a domain of ℝ², V is a confining potential, $L_z = −i(x ∂_y − y ∂_x)$ is the angular momentum operator, $Ω$ is a rotational velocity parameter and $γ ≥ 0$ describes the defocusing nonlinear interactions of the condensate.
After introducing the underlying physical model and some mathematical properties in the continuous case, we will bring our attention to a finite volume discretization of $(E)$. We will begin by recalling the definition of the usual finite volume Laplacian before focusing on the angular momentum operator.
We will then state our main result, showing that minimizers of the discrete energy converge to minimizers of the continuous energy as the mesh diameter tends to zero. We then present the main ideas of the proof before showing numerical results, which notably show the presence of quantum vortices in the minimizers of $(E)$.
In the last part, we will then focus on a normalized gradient flow scheme for the computation of minimizers. In particular, we will state an exponential convergence result towards local minimizers. We will also discuss how the global phase invariance, as well as the presence or absence of rotational symmetry, affects this local convergence property and give numerical illustrations of these phenomena.
The research presented in this talk is the subject of a paper currently being written by the speaker in collaboration with Quentin Chauleur and Guillaume Dujardin.