Vortex patterns and sheets in segregated two component Bose–Einstein condensates

Abstract

We study minimizers of a Gross-Pitaevskii energy describing a two-component Bose-Einstein condensate set into rotation. We consider the case of segregation of the components in the Thomas-Fermi regime, where a small parameter ε conveys a singular perturbation. We estimate the energy as a term due to a perimeter minimization and a term due to rotation. In particular, we prove a new estimate concerning the error of a Modica Mortola type energy away from the interface. For large rotations, we show that the interface between the components gets long, which is a first indication towards vortex sheets.

Differential Equations of Quantum Mechanics

“…the energies of the ground states and resonances), the phenomenon known as the infrared problem. 7 If we think of H κ as a perturbation of the Hamiltonian H κ=0 of the decoupled system in which the particles and photons do not interact, then we run into another, related, manifestation of the infrared problem: the bound state energies of the Hamiltonian H κ=0 are not isolated. 8 The standard perturbation theory fails in treating such eigenvalues and one needs a new theory.…”

“…The mathematical formulation of this property leads to the problem of asymptotic completeness which we have already met in the previous section. 7 The photon clouds lead to renormalizations of various physical quantities (such as the mass of electron and energies of the ground state and resonances), which would diverge if the ultra-violet cut-off (UV) is removed.…”

“…There are many similarities between the Ginzburg-Landau and Gross-Pitaevskii equations and more specifically between the phenomena of superconductivity and the Bose-Einstein condensations (BEC) described by these equations, respectively. The key results for the BEC vortices in the Gross-Petaevski equation are due to A. Aftalion, R. Jerrard, M. Correggi, N. Rougerie, J. Yngvasson et al, see the book [3] and the more recent papers [4,5,6,7,9,83,77].…”

Domain walls in the coupled Gross-Pitaevskii equations with the harmonic potential

“…Mixtures of Bose-Einstein condensates exhibit a wealth of phenomena studied extensively both in physics [6,7,19,32,34,35] and mathematics [1,2,3,16,17]. Topological defects such as domain walls and vortices appear in the segregation problems common in soft condensed matter [12,25,28,30,36,38].…”

Description of the ground state for a model of two-component rotating Bose–Einstein condensates.

“…However it is mathematically not so simple to rigourously validate this intuition and it requires, in particular, to prove in a precise quantitative way that almost minimizers of Modica-Mortola type energy functionals have their energy concentrated near the interface. We report here on joint work with Amandine Aftalion [3] where this analysis is carried out.…”