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arXiv:2106.14369 · SIAM Journal on Mathematical Analysis

Existence and Asymptotic Behavior of Ground States for Rotating Bose–Einstein Condensates

Yujin Guo, Yong Luo, Shuangjie Peng

Published Mar 21, 2023• 7 citations
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Abstract

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.

DOI
10.1137/22m1474680

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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.
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 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 ground, symmetric and central vortex states, as well as their energy and chemical potential diagrams, in rotating Bose-Einstein condensates (BEC) analytically and numerically.We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometrical optics.We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and find that symmetric and central vortex states are independent of the angular rotational momentum.We perform numerical experiments computing these states using a continuous normalized gradient flow (CNGF) method with a backward Euler finite difference (BEFD) discretization.Ground, symmetric and central vortex states, as well as their energy configurations, are reported in 2D and 3D for a rotating BEC.Through our numerical study, we find various configurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of different states in a strong replusive interaction regime.Finally we report the critical angular velocity at which the ground state loses symmetry, numerical verification of dimension reduction from 3D to 2D, errors for the Thomas-Fermi approximation, and spourous numerical ground states when the rotation speed is larger than the minimal trapping frequency in the xy plane.
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).
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 $.
I review free and trapped Bose‐Einstein condensates (BECs) [1, 2, 3, 4, 5]. In an isotropic harmonic potential V(r) = 12Mω2r2, the single‐particle ground state introduces a new intrinsic scale of length [the ground‐state size d = ℏ/(Mω)] and energy [the ground‐state energy E0 = 32ℏω]. When the trap rotates [6, 7, 8], there is a critical angular velocity for the appearance of one or a few quantized vortices. For more rapid rotation [9, 10, 11, 12, 13] the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus the condensate becomes effectively two‐dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest‐Landau‐level” regime, and a simple description again becomes possible [14, 15]. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field [16, 17, 18, 19, 20].