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Bifurcation between 2-component and 3-component ground states of spin-1 Bose-Einstein condensates in uniform magnetic fields ∗

Li Li, I‐Liang Chern

Published Feb 1, 2013• 4 citations
OpenAlex

Abstract

We study antiferromagnetic spin-1 Bose-Einstein condensates under external uniform magnetic fields. The bifurcation between two-component and three-component regimes of ground states with respect to the magnetization M and the quadratic Zeeman effect q is justified. The proof is based on the technique of “mass redi stribution” introduced in the authors previous work, which gives interesting inequalities and equalities satisfied by ground states. Some open problems arising naturally from our investigation are also discussed in the end.

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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
We study antiferromagnetic spin-1 Bose-Einstein condensates under external uniform magnetic fields. The bifurcation between two-component and three-component regimes of ground states with respect to the magnetization M and the quadratic Zeeman effect q is justified. The proof is based on the technique of “mass redi stribution” introduced in the authors previous work, which gives interesting inequalities and equalities satisfied by ground states. Some open problems arising naturally from our investigation are also discussed in the end.
For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied uniform magnetic field, its ground state (ψ1,ψ0,ψ−1)(ψ_1,ψ_0,ψ_{-1})(ψ1​,ψ0​,ψ−1​) undergoes a phase transition from a two-component state (ψ0≡0ψ_0 \equiv 0ψ0​≡0) to a three-component state (ψj≠0ψ_j\ne 0ψj​=0 for all jjj) at a critical value of the magnetic field. This phenomenon has been observed in numerical simulations as well as in experiments. In this paper, we provide a mathematical proof based on a simple principle found by the authors: a redistribution of the mass densities between different components will decrease the kinetic energy.
For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied uniform magnetic field, its ground state (ψ1,ψ0,ψ−1)(\psi_1,\psi_0,\psi_{-1})(ψ1​,ψ0​,ψ−1​) undergoes a phase transition from a two-component state (ψ0≡0\psi_0 \equiv 0ψ0​≡0) to a three-component state (ψj≠0\psi_j\ne 0ψj​=0 for all jjj) at a critical value of the magnetic field. This phenomenon has been observed in numerical simulations as well as in experiments. In this paper, we provide a mathematical proof based on a simple principle found by the authors: a redistribution of the mass densities between different components will decrease the kinetic energy.
We justify some characterizations of the ground states of spin-1 Bose-Einstein condensates exhibited from numerical simulations. For ferromagnetic systems, we show the validity of the single-mode approximation (SMA). For an antiferromagnetic system with nonzero magnetization, we prove the vanishing of the mF=0m_F=0mF​=0 component. In the end of the paper some remaining degenerate situations are also discussed. The proofs of the main results are all based on a simple observation, that a redistribution of masses among different components will reduce the kinetic energy.
In this thesis we investigate the static, dynamic, and thermodynamic properties\nof atomic spin-1 Bose gases in external magnetic fields. At low magnetic fields\nthe properties of single-component, or scalar condensates, are essentially\nunaffected but can become significantly altered for spinor Bose condensates as\nshown by our studies.\n\nWe first study the Bose-Einstein condensation of trapped spin-1 Bose gases by\nemploying the Hartree-Fock approximation and the two-fluid model within a mean\nfield approximation. Our detailed investigation reveals that the\nferromagnetically interacting spin-1 condensates exhibit triple condensations\nwhile the antiferromagnetically interacting ones show double condensations.\n\nThe ground state structure of homogeneous and trapped spin-1 Bose condensates\nwith ferromagnetic and antiferromagnetic interactions at zero temperature in\nmagnetic fields are then investigated systematically. We further illuminate the\nimportant effect of quadratic Zeeman shift which causes a preferred occupation\nof the $|m_F=0\nangle$ state through spin exchange collisions, $2|m_F=0\nangle\nleftrightarrow |m_F=1\nangle + |m_F=-1\nangle$.\n\nWe also present detailed studies of the off-equilibrium coherent dynamics of\nspin-1 Bose condensates in magnetic fields within the single spatial mode\napproximation. Dynamical instabilities of the off-equilibrium oscillations are\nshown to be responsible for the formation of multiple domains as recently\nobserved in several 87^{87}87Rb experiments.\n\nFinally, we discuss briefly excited condensate states, or soliton-like states,\nin cigar-shaped spin-1 Bose condensates with an effective quasi-1D description,\nusing the developed nonpolynomial Schr"odinger equation.
The possible ground states of spin-2 Bose-Einstein condensates in an external magnetic field are obtained analytically and classified systematically according to the population of the condensed atoms at the hyperfine sublevels. It is shown that the atoms can populate simultaneously at four hyperfine sublevels in a weak magnetic field with only the linear Zeeman energy, in contrast to that in a stronger magnetic field with the quadratic Zeeman energy, where condensed atoms can at most populate at three hyperfine sublevels in the ground states. Any spin configuration we obtained will give a closed subspace in the order parameter space of the condensates.
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