Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Communications in Contemporary Mathematics

On the spin-1 Bose–Einstein condensates in the presence of Ioffe–Pritchard magnetic field

Hichem Hajaiej, Rémi Carles

Published Jun 10, 2015• 8 citations
Publisher

Abstract

We study the Cauchy problem of an antiferromagnetic spin-1 Bose–Einstein condensate under Ioffe–Pritchard magnetic field [Formula: see text]. We then address the existence of ground state solutions and characterize the orbit of standing waves.
DOI
10.1142/s0219199715500625

Related papers

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 +1 more

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].

SIAM Journal on Mathematical Analysis•Sep 1, 2025•2 citations

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

Wei Luo, Zhongxue Lü, Zuhan Liu

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.

Bulletin of the Australian Mathematical Society•May 28, 2012•3 citations

On ground state of spin-1 Bose-Einstein condensates with the external Ioffe-Pitchard magnetic

lv zhongxue

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

Bulletin of the Australian Mathematical Society•Oct 22, 2012•0 citations

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

Menghui Li, Xiao Luo, Juncheng Wei +1 more

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.

SIAM Journal on Mathematical Analysis•Jul 1, 2024•6 citations

Ground state of spin-1 Bose-Einstein condensates with spin-orbit coupling in a Zeeman field

Lin Wen, Qing‐Feng Sun, H. Q. Wang +2 more

We systematically investigate the weakly trapped spin-1 Bose-Einstein condensates with spin-orbit coupling in an external Zeeman field. We find that the mean-field ground state favors either a magnetized standing-wave phase or plane-wave phase when the strength of the Zeeman field is below a critical value related to the strength of spin-orbit coupling. The Zeeman field can induce a phase transition between standing-wave and plane-wave phases, and we determine the phase boundary analytically and numerically. The magnetization of these two phases responds to the external magnetic field in a very unique manner: the linear Zeeman effect magnetizes the standing-wave phase along the direction of the magnetic field, but the quadratic Zeeman effect demagnetizes the plane-wave phase. When the strength of Zeeman field surpasses the critical value, the system is completely polarized to a ferromagnetic state or polar state with zero momentum.

arXiv:1208.2514•Physical Review A•Oct 3, 2012•51 citations

Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

Rémi Carles, Hichem Hajaiej

We study the stability of the standing wave solutions of a Gross–Pitaevskii equation describing Bose–Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions.

arXiv:1402.7267•Bulletin of the London Mathematical Society•Apr 26, 2015•30 citations
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].
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
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.
We systematically investigate the weakly trapped spin-1 Bose-Einstein condensates with spin-orbit coupling in an external Zeeman field. We find that the mean-field ground state favors either a magnetized standing-wave phase or plane-wave phase when the strength of the Zeeman field is below a critical value related to the strength of spin-orbit coupling. The Zeeman field can induce a phase transition between standing-wave and plane-wave phases, and we determine the phase boundary analytically and numerically. The magnetization of these two phases responds to the external magnetic field in a very unique manner: the linear Zeeman effect magnetizes the standing-wave phase along the direction of the magnetic field, but the quadratic Zeeman effect demagnetizes the plane-wave phase. When the strength of Zeeman field surpasses the critical value, the system is completely polarized to a ferromagnetic state or polar state with zero momentum.