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Bulletin of the Korean Mathematical Society

NOTE ON THE GROUND STATES OF TWO-COMPONENT BOSE-EINSTEIN CONDENSATES WITH AN INTERNAL ATOMIC JOSEPHSON JUNCTION

Zhongxue Lü, Zuhan Liu

Published Sep 30, 2013• 1 citation
Publisher

Abstract

In this paper, we consider two-component Bose-Einstein condensates with an internal atomic Josephson junction in the general case, i.e., 0 < p < $\frac{2}{(d-2)^+}$ . We prove existence and uniqueness results for the ground states, and obtain some properties of the ground states with large parameters.

DOI
10.4134/bkms.2013.50.5.1441

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In this paper, we consider two-component Bose-Einstein condensates with an internal atomic Josephson junction in the general case, i.e., 0 < p < 2(d−2)+\frac{2}{(d-2)^+}(d−2)+2​ . We prove existence and uniqueness results for the ground states, and obtain some properties of the ground states with large parameters.
In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.
Abstract. The ground states of two-component Bose–Einstein condensates with a Josephson junction can be described by minimizers of the problem [Formula: see text] where [Formula: see text] is the Gross–Pitaevskii energy functional [Formula: see text] [Formula: see text] [Formula: see text] By introducing some technical blow-up arguments, we classify how the Rabi frequency [Formula: see text] and potential [Formula: see text] affect the existence and nonexistence of minimizers at one of the endpoints [Formula: see text] or [Formula: see text], where [Formula: see text] and [Formula: see text] can be given explicitly. This improves the results of [W. Bao and Y. Cai, East Asian J. Appl. Math., 1 (2011), pp. 49–81] and [Y. Guo et al., J. Funct. Anal., 276 (2019), pp. 183–230], where [Formula: see text], or both [Formula: see text] and [Formula: see text] are nonendpoints. We also investigate the uniqueness of minimizers for certain local homogeneous potentials as [Formula: see text]. Different from the case of [Formula: see text] in [Y. Guo et al., J. Funct. Anal., 276 (2019), pp. 183–230], [Formula: see text] causes an uncontrolled error term in estimating the difference of Euler–Lagrange multipliers. To overcome this difficulty, we first establish the relationship between the [Formula: see text]-norm of minimizers and the partial derivatives of [Formula: see text] for almost all [Formula: see text] and then show that the Euler–Lagrange multipliers are independent of the choice of minimizers.
We investigate the existence of ground states for two-component Bose--Einstein condensates with intraspecies interactions a1,a2∈(0,a∗)a_1, a_2 \in (0, a^*)a1​,a2​∈(0,a∗) and interspecies interaction β>0β>0β>0. By carefully investigating an associated auxiliary minimization problem, we prove the existence of a unique, continuous, critical surface γ=γ(a1,a2)γ= γ(a_1, a_2)γ=γ(a1​,a2​) between β∗:=(a∗−a1)(a∗−a2)β_*:= \sqrt{(a^* - a_1) (a^* - a_2)}β∗​:=(a∗−a1​)(a∗−a2​)​ and β∗:=a∗−a1+a22β^*:= a^* - \frac{a_1 + a_2}{2}β∗:=a∗−2a1​+a2​​. This surface provides a complete classification for the existence of minimizers: ground states exist for β∈(0,γ)β\in (0,γ)β∈(0,γ) and do not exist for β>γβ> γβ>γ. Additionally, when the trapping potentials are continuous at a common zero point, there is no ground state for β=γβ= γβ=γ. Our results close the gap case β∈[β∗,β∗]β\in [β_*, β^*]β∈[β∗​,β∗] when a1≠a2a_1\ne a_2a1​=a2​ in literature, significantly extending the results in \cite{Bao-Cai-1,guoBlowupSolutionsTwo2017a}.
In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F>=3) BEC are discussed.
The ground state solutions of two-component Bose-Einstein condensates (BECs) in Bessel optical lattices (BLs) are studied by means of the balance condition between BLs strength and inter-atomic interactions. We consider a quasi-two-dimensional (2D) BECs system，strongly confined in longitudinal direction and weakly trapped in the radial direction in the transverse plane，which obeys 2D Gross-Pitaevskii equation (GPE) derived from its 3D counterpart. Analytically we obtained the atom number distribution，atom number and energy of ground state and give the parameter ranges. Compared to single-component BEC，two-component BECs exhibit a rich variety of ground state structures. These structures depend upon various parameters，in particular the inter-atomic interactions and the BLs strength. Neglecting the inter-component atomic interaction，the corresponding results of single-component situation are given.