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SIAM Journal on Mathematical Analysis

Free and Harmonic Trapped Spin-1 Bose–Einstein Condensates in \({{\mathbb{R}}^{3}}\)

Menghui Li, Xiao Luo, Juncheng Wei, Maoding Zhen

Published Jul 1, 2024• 6 citations
Publisher

Abstract

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.

DOI
10.1137/23m1572222

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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. New developments of optical trapping techniques make it possible to confine atoms independently of their spin orientation in experiments and thus result in spinor condensates. As a continuation of our previous work [M. Li, X. Luo, J. Wei, and M. Zhen, SIAM J. Math. Anal., 56 (2024), pp. 4375–4414], we investigate physical states as well as qualitative properties describing the aggregation and extinction of atoms, of spin-1 Bose–Einstein condensate in an Ioffe–Pritchard magnetic field, through two conserved quantities, the number of atoms and the total magnetization. Unlike the related free case which is well studied in [Y. Z. Kong, Q. X. Wang, and D. Zhao, Calc. Var. Partial Differential Equations, 60 (2021), 152] and our previous work, the presence of the Ioffe–Pritchard magnetic field, which competes dramatically with the harmonic trapping, requires new ideas to capture the physical states and analyze their qualitative properties. Based on the ferromagnetic or antiferromagnetic characterization of spin-1 Bose–Einstein condensate, our results support some experimental observations in [T. Ho and S. Yip, Phys. Rev. Lett., 84 (2000), pp. 4031–4034; C. Law, H. Pu, and N. Bigelow, Phys. Rev. Lett., 81 (1998), pp. 5257–5261] and some numerical analysis on ground states reported in [W. Bao, I. Chern, and Y. Zhang, J. Comput. Phys., 253 (2013), pp. 189–208; W. Bao and F. Lim, SIAM J. Sci. Comput., 30 (2008), pp. 1925–1948; W. Bao and H. Wang, SIAM J. Numer. Anal., 45 (2007), pp. 2177–2200].
We consider a spin-1 Bose-Einstein condensate trapped in a harmonic potential under the influence of a homogeneous magnetic field. We investigate spatial and spin structure of the mean-field ground states under constraints on the number of atoms and the total magnetization. We show that the trapping potential can make the antiferromagnetic condensate separate into three distinct phases and ferromagnetic condensate into two distinct phases. In the ferromagnetic case, the magnetization is located in the center of the harmonic trap, while in the antiferromagnetic case magnetized phases appear in the outer regions. We describe how the transition from the Thomas-Fermi regime to the single-mode approximation regime with decreasing number of atoms results in the disappearance of the domains. We suggest that the ground states can be created in experiment by adiabatically changing the magnetic-field strength.
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.
Ultra-cold spinor Bose gases present rich physics due to the special combination of superfluidity and magnetism in a quantum system. This system was first realized in 1998 by confining a Bose-Einstein condensate in an all-optical trap.\nIn this thesis, we consider the fluctuations of observable quantities in a spin-1 Bose gas. Understanding how fluctuations arise due to the spin excitations of the condensate is important because it offers an insight into how measurement noise reveals the many-body physics of the system.\nTo begin, we present the mean-field and Bogoliubov theory of a uniform spin-1 Bose gas subject to a constant magnetic field, describing the condensate and its low-energy collective excitations.\nWe then develop a formalism to describe the fluctuations in a general density-like observable. We start from the two-point correlation function and cast it in the form of a generalised static structure factor determined by the three Bogoliubov quasiparticle excitation branches. We derive analytic results for the fluctuation amplitudes and the temperature-dependent static structure factors for observables of total density and the three spin densities.\nFor all four magnetic phases, we analyse the spinor order parameter and quasiparticle spectra while numerically mapping out the fluctuation amplitudes and static structure factors for the total and spin density operators.\nWe describe the fluctuations in experimental measurements made within finite cells, which is an important step to making meaningful predictions for experiments. We consider cylindrical cells and gaussian cells as two limiting cases. We apply this analysis to an experimentally realisable system of a quasi-2D spinor gas in a harmonic trap, comparing extensive numeric results with analytic limits.
We study the ground state of spin-2 dipolar rotating Bose–Einstein condensates in a harmonic potential. As the strength of dipolar interaction enhances, our results show that the vortex number becomes smaller and smaller at a fixed component. In addition, the vortex stripe occurs and its number increases. The components m[Formula: see text]1 or ±2 display the same density and the total density shows layered distribution. Increasing the rotational frequency to a higher value, the vortex number and vortex row number increase. We also plot the dependence of angular momentum Lz as a function of dipole strength c[Formula: see text], showing that Lz decreases with increasing c[Formula: see text].