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arXiv:2211.14808

Nonexistence of Vortices for Rotating Two-Component Focusing Bose Gases

Yongshuai Gao, Yujin Guo, Yan Li, Luo Yong

Published Nov 27, 2022• 0 citations
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Abstract

This paper is concerned with ground states of two-component Bose gases confined in a harmonic trap $V(x)=x_1^2+Λ^2 x_2^2$ rotating at the velocity $Ω>0$, where $Λ\ge 1$ and $(x_1, x_2)\in R^2$. We focus on the case where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies interaction $-β$ are both attractive, i.e, $a_1, a_2$ and $β$ are all positive. It is shown that for any $0<Ω<Ω^*:=2$, ground states exist if and only if $00$ is the unique positive solution of $-Δw+ w-w^3=0$ in $R^2$. By developing the argument of refined expansions, we further prove the nonexistence of vortices for ground states as $β\nearrowβ^*$, where $0<Ω<Ω^*$ and $0

DOI
10.48550/arxiv.2211.14808

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Vortices and Persistent Currents in Rotating Bose Gases - a Diagonalization Approach

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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
This paper considers ground states of two-component Bose gases confined in an anharmonic trap rotating at the velocity 0<Ω<∞0<Ω<\infty0<Ω<∞, 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. We prove the axially symmetry and the refined spike profiles of ground states as β↗β∗:=a∗+(a∗−a1)(a∗−a2)β\nearrowβ^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}β↗β∗:=a∗+(a∗−a1​)(a∗−a2​)​, where 000000 is the unique positive solution of −Δu+u−u3=0-Δu+u-u^3=0−Δu+u−u3=0 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.
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 study the existence of ground states for two-component attractive Bose--Einstein condensates with displaced harmonic trapping potentials V1(x)=∣x−x1∣2 and V2(x)=∣x−x2∣2, V_1(x)=|x-x_1|^2 \ \text{and}\ V_2(x)=|x-x_2|^2, V1​(x)=∣x−x1​∣2 and V2​(x)=∣x−x2​∣2, where x1≠x2∈R2x_1\neq x_2\in\mathbb R^2x1​=x2​∈R2. For intraspecies interactions a1,a2∈(0,a∗)a_1,a_2\in (0,a^*)a1​,a2​∈(0,a∗), we focus on the critical interspecies coupling β=β∗:=a∗+(a∗−a1)(a∗−a2) β=β^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)} β=β∗:=a∗+(a∗−a1​)(a∗−a2​)​ and prove that there exists a separation threshold dc∈(0,∞)d_c\in(0,\infty)dc​∈(0,∞) such that the corresponding constrained minimization problem admits no minimizer when $0<|x_1-x_2|
As a continuation of our prior article, we study new pattern formations of ground states (u1,u2)(u_1,u_2)(u1​,u2​) for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in R2\mathbb {R}^2R2, where the intraspecies interaction (−a,−b)(-a,-b)(−a,−b) and the interspecies interaction −β-\beta−β are both attractive, i.e., aaa, bbb, and β\betaβ are all positive. If 0<b<a∗:=∥w∥220<b<a^*:=\|w\|^2_20<b<a∗:=∥w∥22​ and 0<β<a∗0<\beta <a^*0<β<a∗ are fixed, where www is the unique positive solution of Δw−w+w3=0\Delta w-w+w^3=0Δw−w+w3=0 in R2\mathbb {R}^2R2, the semi-trivial behavior of (u1,u2)(u_1,u_2)(u1​,u2​) as a↗a∗a\nearrow a^*a↗a∗ is proved in the sense that u1u_1u1​ concentrates at a unique point and while u2≡0u_2\equiv 0u2​≡0 in R2\mathbb {R}^2R2. However, if 0<b<a∗0<b<a^*0<b<a∗ and a∗≤β<β∗=a∗+(a∗−a)(a∗−b)a^*\le \beta <\beta ^*=a^*+\sqrt {(a^*-a)(a^*-b)}a∗≤β<β∗=a∗+(a∗−a)(a∗−b)​, the refined spike profile and the uniqueness of (u1,u2)(u_1,u_2)(u1​,u2​) as a↗a∗a\nearrow a^*a↗a∗ are analyzed, where (u1,u2)(u_1,u_2)(u1​,u2​) must be unique, u1u_1u1​ concentrates at a unique point, and meanwhile u2u_2u2​ can either blow up or vanish, depending on how β\betaβ approaches a∗a^*a∗.
In this thesis, I explore the behavior of rotating ultra-cold Bose gases, by diagonalizing the Hamiltonian. This method has the advantage of being exact in the limit of weak interactions, and thus is a useful complement to the more commonly used Gross-Pitaevskii (GP) equation, which relies on a mean-field approximation. It is generally known that an ultra-cold Bose gas shows some remarkable properties under rotation, such as quantized vortices and persistent currents, and these are one main reason why we choose to study rotation. Abstract A repulsively interacting Bose-Einstein condensate (BEC) in a harmonic trapping potential will form singly quantized vortices when rotated. Papers I-V are examples of different kinds of generalizations to this basic system, each showing its particular non-trivial new features. Abstract In Paper I we make the trapping potential weakly anharmonic, and confirm the existence of multiply quantized vortices in both repulsively and attractively interacting condensates, which were previously predicted in Gross-Pitaevskii calculations. Abstract In Papers II and III, we generalize to a two-component system in a harmonic trap. The vortices that form under rotation will in general be coreless, i.e. one component will form a vortex and rotate around the other component. In Paper II we give some analytical expressions for the dispersion relation and occupation numbers at low angular momenta, and in Paper III we show how multiply quantized vortices can be stable in these systems. Abstract Finally we investigate bosons in potentials which can support persistent currents, i.e. rotating states that do not decay in a finite amount of time. These are possible if there is an energy barrier separating the rotating state from the non-rotating ground state. In Paper IV, we have a two-component condensate in a one-dimensional potential and we calculate among other things the minimum interaction strength where persistent currents can occur. In Paper V, we study an experimentally relevant two-dimensional annulus. Again we find that persistent currents are possible above a certain interaction strength, and we find this strength as a function of the width of the annulus, as well as how it depends on the relative population of the two species.