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SIAM Journal on Mathematical Analysis

On Ground States of Spin-1 Bose–Einstein Condensates with Ioffe–Pritchard Magnetic Field in Two and Three Dimensions

M.H. Li, Xiao Luo, Juncheng Wei, Maoding Zhen

Published Sep 1, 2025• 2 citations

Abstract

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].

DOI: 10.1137/24m1674091

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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].

Since its first observation in 1995, Bose-Einstein condensation (BEC) experiments in dilute gases have been performed with only a few different kinds of magnetic traps. We have performed a BEC experiment with /sup 87/Rb atoms in a Ioffe-Pritchard magnetic potential, which opens up the possibility to study, e.g., Bose-Einstein condensates released in only one dimension. Our trap consists of a clip-like coil, which generates a pure two-dimensional (2D) quadrupole field for radial confinement. The axial confinement is provided by two plane pinch coils inside the clip-coil. These two coils create a bottle field. Two additional Helmholtz coils on the same symmetry axis are used to compensate the homogeneous part of the bottle field. By continuously varying the currents through the coils, the trapping potential can be changed from a 3D spherical harmonic to a 2D linear potential. The change can be done adiabatically or nonadiabatically by switching off the axial confinement slowly, or instantaneously respectively. This opens up the possibility to study 1D nonlinear atom optics. In combination with, e.g., far detuned light sheets, a Fabry-Perot type interferometer for a condensate might be realizable. Our experiments are performed in a vacuum system with two magneto-optical traps, which serves as the source of cold atoms for the magnetic trap. 10/sup 9/ atoms are captured in less than one minute. RF-induced evaporative cooling is used to increase the phase space density of the trapped atomic cloud. The BEC phase transition occurs at a temperature of 550 nK with 10/sup 6/ particles in the trap. Pure condensates with more than 10/sup 5/ atoms were produced.

Summary form only given.Since its first observation in 1995, Bose-Einstein condensation (BEC) experiments in dilute gases have been performed with only a few different kinds of magnetic traps. We have performed a BEC experiment with /sup 87/Rb atoms in a Ioffe-Pritchard magnetic potential, which opens up the possibility to study, e.g., Bose-Einstein condensates released in only one dimension. Our trap consists of a clip-like coil, which generates a pure two-dimensional (2D) quadrupole field for radial confinement. The axial confinement is provided by two plane pinch coils inside the clip-coil. These two coils create a bottle field. Two additional Helmholtz coils on the same symmetry axis are used to compensate the homogeneous part of the bottle field. By continuously varying the currents through the coils, the trapping potential can be changed from a 3D spherical harmonic to a 2D linear potential. The change can be done adiabatically or nonadiabatically by switching off the axial confinement slowly, or instantaneously respectively. This opens up the possibility to study 1D nonlinear atom optics. In combination with, e.g., far detuned light sheets, a Fabry-Perot type interferometer for a condensate might be realizable. Our experiments are performed in a vacuum system with two magneto-optical traps, which serves as the source of cold atoms for the magnetic trap. 10/sup 9/ atoms are captured in less than one minute. RF-induced evaporative cooling is used to increase the phase space density of the trapped atomic cloud. The BEC phase transition occurs at a temperature of 550 nK with 10/sup 6/ particles in the trap. Pure condensates with more than 10/sup 5/ atoms were produced.

Free and Harmonic Trapped Spin-1 Bose–Einstein Condensates in

{{\mathbb{R}}^{3}}

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.

In this paper, we prove the existence of the ground state for the spinor Bose-Einstein condensates with the external Ioffe-Pitchard magnetic field in the one-dimensional case. We also characterize the ground states of spin-1 Bose-Einstein condensates with the external Ioffe-Pitchard magnetic field, i.e., for ferromagnetic systems, we show that under some condition, searching for ground state of ferromagnetic spin-1 BEC with the external Ioffe-Pitchard magnetic field can be reduced to a one-component minimization problem. 10.1017/S0004972712000305