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Bulletin of the Australian Mathematical Society

ON THE GROUND STATE OF SPIN-1 BOSE–EINSTEIN CONDENSATES WITH AN EXTERNAL IOFFE–PITCHARD MAGNETIC FIELD

Wei Luo, Zhongxue Lü, Zuhan Liu

Published May 28, 2012• 3 citations
Publisher

Abstract

Abstract In this paper, we prove the existence of the ground state for the spinor Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field in the one-dimensional case. We also characterise the ground states of spin-1 Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field; that is, for ferromagnetic systems, we show that, under some condition, searching for the ground state of ferromagnetic spin-1 Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field can be reduced to a ‘one-component’ minimisation problem.

DOI
10.1017/s0004972712000305

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Abstract In this paper, we prove the existence of the ground state for the spinor Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field in the one-dimensional case. We also characterise the ground states of spin-1 Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field; that is, for ferromagnetic systems, we show that, under some condition, searching for the ground state of ferromagnetic spin-1 Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field can be reduced to a ‘one-component’ minimisation problem.
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
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 have calculated the F=1 ground state of a spinor Bose-Einstein condensate trapped harmonic potential with an applied Ioffe-Pitchard magnetic field. The vortex phase diagram is found in the plane spanned by perpendicular and longitudinal magnetic fields. The ferromagnetic condensate has two vortex phases which differ by winding number in the spinor components. The two vortices for the F(z)=-1 antiferromagnetic condensate are separated in space. Moreover, we considered an average local spin |<S-->>| to testify to what extent it is parallel to magnetic field (the nonadiabatic effects). We have shown that the effects are important at vortex cores.
We discuss the occurrence of Bose-Einstein condensation in systems of noninteracting charged particles in three in one dimensions and in presence of an external magnetic field. In the one dimensional, as well as in the magnetic field cases, although not a critical temperature, a characteristic temperature can be found, corresponding to the case in which the ground state density becomes a macroscopic fraction of the total density. The case of relativistic charged scalar and vector particles is studied. The results obtainedgive support to the existence of superconductivity in extremely strong magnetic fields, and leads to the prediction of superconductive-ferromagnetic behavior in the vector field case, which might be of interest in condensed matter as well as in cosmology. Some features of the magnetization in the early universe are conjectured.
We discuss the occurrence of Bose-Einstein condensation in systems of noninteracting charged particles in three and one dimensions and in presence of an external magnetic field. In the one dimensional, as well as in the magnetic field cases, although not a critical temperature, a characteristic temperature can be found, corresponding to the case in which the ground state density becomes a macroscopic fraction of the total density. The case of relativistic charged scalar and vector particles is studied. The results give support to the existence of superconductivity in extremely strong magnetic fields, and lead to the prediction of superconductive-ferromagnetic behavior in the vector field case, which might be of interest in condensed matter as well as in cosmology. Some features of the magnetization in the early universe are conjectured.