Erlangen 2022 – wissenschaftliches Programm
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Q: Fachverband Quantenoptik und Photonik
Q 57: Quantum Gases II
Q 57.1: Poster
Donnerstag, 17. März 2022, 16:30–18:30, P
An algebraic geometric study of the solution space of the 1D Gross-Pitaevskii equation — •David Reinhardt1, Matthias Meister1, Dean Lee2, and Wolfgang P. Schleich1 — 1Institute of Quantum Technologies, German Aerospace Center (DLR), Ulm, Germany — 2Michigan State University, Facility for Rare Isotope Beams and Department of Physics and Astronomy, East Lansing, Michigan, USA
The stationary solutions of the Schrödinger equation considering box or periodic boundaries show a clear correspondence to solutions found for the non-linear Gross-Pitaevskii equation commonly used to model Bose-Einstein condensates. However, in the non-linear case there exists an additional class of solutions for periodic boundaries first identified by L.D. Carr et al. [1]. These nodeless complex symmetry breaking solutions have no corresponding counterpart in the linear case. To examine how these solutions behave in the limit of vanishing non-linearity we consider an algebraic geometric picture. Therefore, we treat both equations in the hydrodynamic framework, resulting in a first-order differential equation for the density determined by a quadratic polynomial in the linear case and by a cubic polynomial in the non-linear case, respectively. Our approach allows for a clear geometric interpretation of the solution space in terms of the nature and location of the roots of these polynomials.
[1] L.D. Carr, C.W. Clark, W.P. Reinhardt, Phys. Rev. A 62, 063610 & 063611 (2000)