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Calculus of Variations and Partial Differential Equations

Ground states of spin-1 BEC with attractive mean-field interaction trapped in harmonic potential in $${\mathbb {R}}^2$$

Yuzhen Kong, Qingxuan Wang, Dun Zhao

Published Jul 2, 2021• 6 citations
Publisher
DOI
10.1007/s00526-021-02015-4

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Ground states of spin-1 BEC with attractive mean-field interaction trapped in harmonic potential in R2{\mathbb {R}}^2R2
Free and Harmonic Trapped Spin-1 Bose–Einstein Condensates in R3{{\mathbb{R}}^{3}}R3
Abstract. We investigate physical states of spin-1 Bose–Einstein condensate in [Formula: see text] with mean-field interaction constant [Formula: see text] and spin-exchange interaction constant [Formula: see text], two conserved quantities, the number of atoms [Formula: see text], and the total magnetization [Formula: see text] are involved in. First, in the free case, existence and asymptotic behavior of ground states are analyzed according to the relations among [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Furthermore, we show that the corresponding standing wave is strongly unstable. When the atoms are trapped in a harmonic potential, we prove the existence of ground states and excited states along with some precisely asymptotics. Besides, we get that the set of ground states is stable under the associated Cauchy flow while the excited state corresponds to a strongly unstable standing wave. Our results not only show some characteristics of three-dimensional spin-1 BEC under the effect between the spin-dependent interaction and the external magnetic field, but also support some experimental observations as well as numerical results on spin-1 BEC.
Abstract Numerically we investigate the ground state of effective one-dimensional spin–orbit (SO) and Rabi coupled two pseudo-spinor Bose–Einstein condensates (BECs) under the effect of harmonic traps. For both signs of the interaction, density profiles of SO and Rabi coupled BECs in harmonic potentials, which simulate a real experimental situation are obtained. The harmonic trap causes a strong reduction of the multi-peak nature of the condensate and it increases its density. For repulsive interactions, the increase of SO coupling results in an uncompressed less dense condensate and with increased multi-peak nature of the density. The increase of Rabi coupling leads to a density increase with an almost constant number of multi-peaks. For both signs of the interaction and negative values of Rabi coupling, the condensate develops a notch in the central point and it seems to a dark-in-bright soliton. In the case of the attractive nonlinearity, an interesting result is the increase of the collapse threshold under the action of the SO and Rabi couplings.
This paper will focus on theoretical study of BEC in one dimension with weakly attractive interactions in particular in a symmetric harmonic oscillator potential.In this case, the ground state may not be symmetric state, which is in contrast to a BEC with repulsive interaction.The behaviors of the wave function under the influences of different external trapping potentials usually used in experiments that lead to produced Bose-Einstein condensation in ultra cold gases is analysis.Several values of harmonic trapping potentials are used.These analyses give us the overall view of the region of confinement that the external trapping potentials have employed and the influence of some terms in Gross-Pitaevskii equation.