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Discrete and Continuous Dynamical Systems

Vortex states of Bose-Einstein condensates with attractive interactions

Tingxi Hu, Lu Lu

Published Jan 1, 2023• 3 citations
Publisher

Abstract

In this paper, we mainly investigate complex-valued vortex states of trapped attractive Bose-Einstein condensates (BECs), which can be described by an $ L^2- $critical constraint minimization problem. We prove that both the existence and nonexistence of vortex states may occur at the threshold $ a^*(m) $ depending on the value of $ V(0) $. When there is no vortex states at the threshold $ a^*(m) $, the limit behavior of vortex states as $ a \nearrow a^*(m) $ is also investigated if the trapping potential $ V(x) $ vanishes on the unit circle $ B(0,1) $ in $ {{\mathbb{R}}}^2 $.

DOI
10.3934/dcds.2023003

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In this paper, we mainly investigate complex-valued vortex states of trapped attractive Bose-Einstein condensates (BECs), which can be described by an L2−L^2-L2−critical constraint minimization problem. We prove that both the existence and nonexistence of vortex states may occur at the threshold a∗(m)a^*(m)a∗(m) depending on the value of V(0)V(0)V(0). When there is no vortex states at the threshold a∗(m)a^*(m)a∗(m), the limit behavior of vortex states as a↗a∗(m)a \nearrow a^*(m)a↗a∗(m) is also investigated if the trapping potential V(x)V(x)V(x) vanishes on the unit circle B(0,1)B(0,1)B(0,1) in R2{{\mathbb{R}}}^2R2.
This article is devoted to studying the model of two-dimensional attractive Bose-Einstein condensates in a trap V(x)V(x)V(x) rotating at the velocity $\Omega $. This model can be described by the complex-valued Gross-Pitaevskii energy functional. It is shown that there exists a critical rotational velocity 000 000 denotes the absolute product for the number of particles times the scattering length, and w>0w>0w>0 is the unique positive solution of Δw−w+w3=0\Delta w-w+w^3=0Δw−w+w3=0 in R2\mathbb{R}^2R2. If V(x)=∣x∣2V(x)=|x|^2V(x)=∣x∣2 and $ 0<\Omega <\Omega^*(=2)$ is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as a↗a∗a \nearrow a^*a↗a∗, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of www.
We study the Bose-Einstein condensates (BEC) in two or three dimensions with attractive interactions, described by L2L^{2}L2 constraint Gross-Pitaevskii energy functional. First, we give the precise description of the chemical potential of the condensate μμμ and the attractive interaction aaa. Next, for a class of degenerated trapping potential with non-isolated critical points, we obtain the existence and the local uniqueness of excited states by precise analysis of the concentrated points and the Lagrange multiplier. To our best knowledge, this is the first result concerning on excited states of BEC in Mathematics. Also, our results show that ka∗ka_*ka∗​ are critical values in two dimension when the concentration occurs for any positive integer kkk with some positive constant a∗a_*a∗​. And we point out that our results on degenerated trapping potential with non-isolated critical points are also new even for the classical Schrödinger equations. Here our main tools are finite-dimensional reduction and various Pohozave identities. The main difficulties come from the estimates on Lagrange multiplier and the different degenerate rate along different directions at the critical points of V(x)V(x)V(x).
The quantized vortex states of a weakly interacting Bose-Einstein condensate of atoms with attractive interatomic interaction in an axially symmetric harmonic oscillator trap are investigated using the numerical solution of the time-dependent Gross-Pitaevskii equation obtained by the semi-implicit Crank-Nicholson method. The collapse of the condensate is studied in the presence of deformed traps with the larger frequency along either the radial or the axial direction. The critical number of atoms for collapse is calculated as a function of the vortex quantum number L. The critical number increases with increasing angular momentum L of the vortex state but tends to saturate for large L.
Abstract. We study ground states of two-dimensional Bose–Einstein condensates with repulsive ([Formula: see text]) or attractive ([Formula: see text]) interactions in a trap [Formula: see text] rotating at velocity [Formula: see text]. It is known that there exist critical parameters [Formula: see text] and [Formula: see text] such that if [Formula: see text], then there is no ground state for any [Formula: see text]; if [Formula: see text], then ground states exist if and only if [Formula: see text]. As a completion of the existing results, in this paper, we focus on the critical case where [Formula: see text] to classify the existence and nonexistence of ground states for any [Formula: see text]. Moreover, for a suitable class of radially symmetric traps [Formula: see text], employing the inductive symmetry method, we prove that up to a constant phase, ground states must be real valued, unique, and free of vortices as [Formula: see text], no matter whether the interactions of the condensates are repulsive or not.
We consider ground states of two-dimensional Bose–Einstein condensates in box-shaped trapping potentials Vext(x) with inhomogeneous attractive interactions am(x), which can be described equivalently by minimizers of Gross–Pitaevskii energy functional in bounded domains. In this paper, we prove that there is a threshold a* > 0 such that minimizers exist for 0 < a < a* and the minimizer does not exist for any a > a*. However, if a = a*, it is shown that whether minimizers exist depends sensitively on the asymptotic behaviors of m(x) near its maximum points. Moreover, based on a detailed analysis on the limit behavior of minimizers as a ↗ a*, we prove local uniqueness of minimizers under some suitable assumptions on m(x).