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International Journal of Modern Physics B

NONDEGENERATED SPIN STATES OF A TRAPPED SPINOR BOSE GAS BEYOND MEAN FIELD THEORY

Duoje Jia, Yan Xu, Zhaoxin Li, FENGCHAO HOU, Zhaoyang Chen

Published Jan 30, 2008• 0 citations
Publisher

Abstract

An effective Hamiltionian with level splitting of the collective spin states is derived for a spinor Bose condensate of trapped atoms with hyperfine spin f = 1 beyond the single-mode approximation. The splitting of these spin states arises from quantum polarization between the mf = +1 and mf = -1 spin channels, and is characterized by small magnetization exhibited by the trapped gas. For big ferromagnetic 87 Rb and antiferromagnetic 23 Na Bose gases, the splitting fraction in total energy increases as -N-1 and decreases as N-2 when the atom number N varies, with typical order of 10-6 and 10-4, respectively. In the case of small condensates, the splitting fraction increases as -N-2, with typical order of 10-5~10-6, both for 87 Rb and 23 Na atoms. In both cases of the big and small condensates, the splitting energy is analogous to the anisotropic energy of a magnet, being proportional to the square of the pseudo-magnetization.

DOI
10.1142/s021797920803851x

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An effective Hamiltionian with level splitting of the collective spin states is derived for a spinor Bose condensate of trapped atoms with hyperfine spin f = 1 beyond the single-mode approximation. The splitting of these spin states arises from quantum polarization between the mf = +1 and mf = -1 spin channels, and is characterized by small magnetization exhibited by the trapped gas. For big ferromagnetic 87 Rb and antiferromagnetic 23 Na Bose gases, the splitting fraction in total energy increases as -N-1 and decreases as N-2 when the atom number N varies, with typical order of 10-6 and 10-4, respectively. In the case of small condensates, the splitting fraction increases as -N-2, with typical order of 10-5~10-6, both for 87 Rb and 23 Na atoms. In both cases of the big and small condensates, the splitting energy is analogous to the anisotropic energy of a magnet, being proportional to the square of the pseudo-magnetization.
The deviation effect of spinor mode from the single-mode for a spin-1 Bose gas of trapped atoms is studied beyond the mean field theory. Based on the effective Hamiltonian with nondegenerated level of the collective spin states, the splitting level of the system energy due to the deviation effect has been calculated. For the large condensates of 87 Rb and 23 Na with atom number N>105, the splitting fraction of the energy, arising from the magnetization exhibited by the trapped Bose gas, is found to have a typical order of (10-4~10-8), decreasing as N-2 for 87 Rb and increasing as -N-2 for 23 Na , respectively.
The achievement in cooling alkali atomic gases, such as 87^{87}87Rb, 23^{23}23Na and 7^{7}7Li, to quantum degeneracy opens up a way to study magnetism in spinor bosons, because these constituent atoms usually have a hyperfine spin degree of freedom. This article reviews several basic problems related to the ferromagnetic phase transition in spinor atomic Bose gases from a theoretical perspective. After a brief discussion on various possible origins of the ferromagnetic interaction, the phase diagram of the ferromagnetically coupled spinor bosons is investigated. It is found that the ferromagnetic transition occurs always above Bose-Einstein condensation and the Bose condensate is fully polarized. The low-lying collective excitations of the spinor condensate, including spin and density modes, are discussed. The spectrum of the density mode is of the Bogliubov form and the spin wave spectrum has a k2k^2k2-formed dispersion relation at long wavelengths. The spin-wave stiffness coefficient contains contribut
Three hyperfine states of Bose-condensed sodium atoms, recently optically trapped, can be described as a spin-1 Bose gas. We study the behavior of this system in a magnetic field, and construct the phase diagram, where the temperature of the Bose condensation TBEC{T}_{\mathrm{BEC}}TBEC​ increases with magnetic field. In particular the system is ferromagnetic below TBEC{T}_{\mathrm{BEC}}TBEC​ and the magnetization is proportional to the condensate fraction in a vanishing magnetic field. Second derivatives of the magnetization with regard to temperature or magnetic field are discontinuous along the phase boundary.
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.