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Mean field ground state of a spin-1 condensate in a magnetic field

Wenxian Zhang, Su Yi, Li You

Published Jan 1, 2003• 51 citations
OpenAlex

Abstract

Abstract. We revisit the topic of the mean field ground state of a spin-1 atomic condensate inside a uniform magnetic field (B) under the constraints that both the total number of atoms (N) and the magnetization (M) are conserved. In the presence of an internal state (spin component) independent trap, we also investigate the dependence of the so-called single spatial mode approximation (SMA) on the magnitude of the magnetic field and M. Our result indicate that the quadratic Zeeman effect is an important factor in balancing the mean field energy from elastic atom-atom collisions that are known to conserve both N and M.Mean field ground state of a spin-1 condensate in a magnetic field 2 1.

Related papers

Mean field ground state of a spin-1 condensate in a magnetic field

Wenxian Zhang, Su Yi, Li You

We revisit the topic of the mean field ground state of a spin-1 atomic condensate inside a uniform magnetic field ( B ) under the constraints that both the total number of atoms ( N ) and the magnetization ( ) are conserved. In the presence of an internal state (spin component) independent trap, we also investigate the dependence of the so-called single spatial mode approximation on the magnitude of the magnetic field and . Our results indicate that the quadratic Zeeman effect is an important factor in balancing the mean field energy from elastic atom–atom collisions that are known to conserve both N and .

arXiv:cond-mat/0306076•New Journal of Physics•Jun 18, 2003•117 citations

Single-mode approximation in a spinor-1 atomic condensate

S. Yi, Özgür Esat Müstecaplıoğlu, C. P. Sun +1 more

We investigate the validity conditions of the single-mode approximation (SMA) in a spinor-1 atomic condensate when effects due to residual magnetic fields are negligible. For atomic interactions of the ferromagnetic type, the SMA is shown to be exact, with a mode function different from what is commonly used. However, the quantitative deviation is small under current experimental conditions (for ${}^{87}\mathrm{Rb}$ atoms). For antiferromagnetic interactions, we find that the SMA becomes invalid in general. The differences among the mean-field mode functions for the three spin components are shown to depend strongly on the system magnetization. Our results can be important for studies of beyond mean-field quantum correlations, such as fragmentation, spin squeezing, and multipartite entanglement.

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Ground states of trapped spin-1 condensates in magnetic field

Michał Matuszewski

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.

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Fong Yin Lim, Weizhu Bao

We propose efficient and accurate numerical methods for computing the ground-state solution of spin-1 Bose-Einstein condensates subjected to a uniform magnetic field. The key idea in designing the numerical method is based on the normalized gradient flow with the introduction of a third normalization condition, together with two physical constraints on the conservation of total mass and conservation of total magnetization. Different treatments of the Zeeman energy terms are found to yield different numerical accuracies and stabilities. Numerical comparison between different numerical schemes is made, and the best scheme is identified. The numerical scheme is then applied to compute the condensate ground state in a harmonic plus optical lattice potential, and the effect of the periodic potential, in particular to the relative population of each hyperfine component, is investigated through comparison to the condensate ground state in a pure harmonic trap.

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Population and phase dynamics of F=1 spinor condensates in an external magnetic field

Dimas Rodrigues Romano, E. J. V. de Passos

We show that the classical dynamics underlying the mean-field description of homogeneous mixtures of spinor F=1 Bose-Einstein condensates in an external magnetic field is integrable as a consequence of number conservation and axial symmetry in spin space. The population dynamics depends only on the quadratic term of the Zeeman energy and on the strength of the spin-dependent term of the atom-atom interaction. We determine the equilibrium populations as function of the ratio of these two quantities and the miscibility of the hyperfine components in the ground state spinors are thoroughly discussed. Outside the equilibrium, the populations are always a periodic function of time where the periodic motion can be a libration or a rotation. Our studies also indicates the absence of metastability.

arXiv:cond-mat/0408069•Aug 3, 2004•1 citation

Population and phase dynamics of F=1 spinor condensates in an external magnetic field

Dimas Rodrigues Romano, E. J. V. de Passos

We show that the classical dynamics underlying the mean-field description of homogeneous mixtures of spinor $F=1$ Bose-Einstein condensates in an external magnetic field is integrable as a consequence of number conservation and axial symmetry in spin space. The population dynamics depends only on the quadratic term of the Zeeman energy and on the strength of the spin-dependent term of the atom-atom interaction. We determine the equilibrium populations as function of the ratio of these two quantities and the miscibility of the hyperfine components in the ground state spinors are thoroughly discussed. Outside the equilibrium, the populations are always a periodic function of time where the periodic motion can be a libration or a rotation. Our studies also indicate the absence of metastability.

Physical Review A•Oct 12, 2004•31 citations
Abstract. We revisit the topic of the mean field ground state of a spin-1 atomic condensate inside a uniform magnetic field (B) under the constraints that both the total number of atoms (N) and the magnetization (M) are conserved. In the presence of an internal state (spin component) independent trap, we also investigate the dependence of the so-called single spatial mode approximation (SMA) on the magnitude of the magnetic field and M. Our result indicate that the quadratic Zeeman effect is an important factor in balancing the mean field energy from elastic atom-atom collisions that are known to conserve both N and M.Mean field ground state of a spin-1 condensate in a magnetic field 2 1.
We revisit the topic of the mean field ground state of a spin-1 atomic condensate inside a uniform magnetic field ( B ) under the constraints that both the total number of atoms ( N ) and the magnetization ( ) are conserved. In the presence of an internal state (spin component) independent trap, we also investigate the dependence of the so-called single spatial mode approximation on the magnitude of the magnetic field and . Our results indicate that the quadratic Zeeman effect is an important factor in balancing the mean field energy from elastic atom–atom collisions that are known to conserve both N and .
We investigate the validity conditions of the single-mode approximation (SMA) in a spinor-1 atomic condensate when effects due to residual magnetic fields are negligible. For atomic interactions of the ferromagnetic type, the SMA is shown to be exact, with a mode function different from what is commonly used. However, the quantitative deviation is small under current experimental conditions (for 87Rb{}^{87}\mathrm{Rb}87Rb atoms). For antiferromagnetic interactions, we find that the SMA becomes invalid in general. The differences among the mean-field mode functions for the three spin components are shown to depend strongly on the system magnetization. Our results can be important for studies of beyond mean-field quantum correlations, such as fragmentation, spin squeezing, and multipartite entanglement.
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.
We propose efficient and accurate numerical methods for computing the ground-state solution of spin-1 Bose-Einstein condensates subjected to a uniform magnetic field. The key idea in designing the numerical method is based on the normalized gradient flow with the introduction of a third normalization condition, together with two physical constraints on the conservation of total mass and conservation of total magnetization. Different treatments of the Zeeman energy terms are found to yield different numerical accuracies and stabilities. Numerical comparison between different numerical schemes is made, and the best scheme is identified. The numerical scheme is then applied to compute the condensate ground state in a harmonic plus optical lattice potential, and the effect of the periodic potential, in particular to the relative population of each hyperfine component, is investigated through comparison to the condensate ground state in a pure harmonic trap.
We show that the classical dynamics underlying the mean-field description of homogeneous mixtures of spinor F=1 Bose-Einstein condensates in an external magnetic field is integrable as a consequence of number conservation and axial symmetry in spin space. The population dynamics depends only on the quadratic term of the Zeeman energy and on the strength of the spin-dependent term of the atom-atom interaction. We determine the equilibrium populations as function of the ratio of these two quantities and the miscibility of the hyperfine components in the ground state spinors are thoroughly discussed. Outside the equilibrium, the populations are always a periodic function of time where the periodic motion can be a libration or a rotation. Our studies also indicates the absence of metastability.
We show that the classical dynamics underlying the mean-field description of homogeneous mixtures of spinor F=1F=1F=1 Bose-Einstein condensates in an external magnetic field is integrable as a consequence of number conservation and axial symmetry in spin space. The population dynamics depends only on the quadratic term of the Zeeman energy and on the strength of the spin-dependent term of the atom-atom interaction. We determine the equilibrium populations as function of the ratio of these two quantities and the miscibility of the hyperfine components in the ground state spinors are thoroughly discussed. Outside the equilibrium, the populations are always a periodic function of time where the periodic motion can be a libration or a rotation. Our studies also indicate the absence of metastability.