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Refined Mass Concentration of Rotating Bose-Einstein Condensates with Attractive Interactions

Yujin Guo, Yong Luo, Wen Yang

Published Jan 28, 2019• 3 citations
OpenAlex

Abstract

This article is devoted to studying the model of two-dimensional attractive Bose-Einstein condensates in a trap $V(x)$ 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 $0 0$ denotes the absolute product for the number of particles times the scattering length, and $w>0$ is the unique positive solution of $\Delta w-w+w^3=0$ in $\mathbb{R}^2$. If $V(x)=|x|^2$ 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 $a \nearrow a^*$, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of $w$.

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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 Ω\Omega Ω. 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<Ω<Ω∗(=2) 0<\Omega <\Omega^*(=2)0<Ω<Ω∗(=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 a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevskii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in Correggi et al. [J. Math. Phys. 53, 095203 (2012)] that such a transition occurs when the angular velocity is of order ε−4, with ε−2 denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and ε ≪ 1 (Thomas-Fermi regime). In this paper, we identify a finite value Ωc such that if Ω = Ω0/ε4 with Ω0 > Ωc, the condensate is in the giant vortex phase. Under the same condition, we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.
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).
This paper is concerned with ground states of two-component Bose gases confined in a harmonic trap V(x)=x12+Λ2x22V(x)=x_1^2+Λ^2 x_2^2V(x)=x12​+Λ2x22​ rotating at the velocity Ω>0Ω>0Ω>0, where Λ≥1Λ\ge 1Λ≥1 and (x1,x2)∈R2(x_1, x_2)\in R^2(x1​,x2​)∈R2. We focus on the case where the intraspecies interaction (−a1,−a2)(-a_1,-a_2)(−a1​,−a2​) and the interspecies interaction −β-β−β are both attractive, i.e, a1,a2a_1, a_2a1​,a2​ and βββ are all positive. It is shown that for any 0<Ω<Ω∗:=20<Ω<Ω^*:=20<Ω<Ω∗:=2, ground states exist if and only if 000000 is the unique positive solution of −Δw+w−w3=0-Δw+ w-w^3=0−Δw+w−w3=0 in R2R^2R2. By developing the argument of refined expansions, we further prove the nonexistence of vortices for ground states as β↗β∗β\nearrowβ^*β↗β∗, where 0<Ω<Ω∗0<Ω<Ω^*0<Ω<Ω∗ and $0
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.
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 $.
We study a rotating Bose-Einstein condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of two-dimensional Gross-Pitaevskii theory. We write the coupling constant for the interactions between the gas atoms as 1∕ε2 and we are interested in the limit ε→0 (Thomas-Fermi limit) with the angular velocity Ω depending on ε. We derive rigorously the leading asymptotics of the ground state energy and the density profile when Ω tends to infinity as a power of 1∕ε. If Ω(ε)=Ω0∕ε a “hole” (i.e., a region where the density becomes exponentially small as 1∕ε→∞) develops for Ω0 above a certain critical value. If Ω(ε)⪢1∕ε the hole essentially exhausts the container and a “giant vortex” develops with the density concentrated in a thin layer at the boundary. While we do not analyze the detailed vortex structure we prove that rotational symmetry is broken in the ground state for const∣logε∣<Ω(ε)≲const∕ε.