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Acta Physica Sinica

Energy level-splitting of ground state in spinor Bose-Einstein condensate with large number of atoms

Yan Xu, Jia Duo-Je, Zhaoxin Li, Hou Feng-Chao, Lei Tan, Zhang Lu-Yin, (1)兰州大学理论物理研究所,兰州 730000; (2)山东科技大学理学院，青岛 266510; (3)西北师范大学理论物理研究所，兰州 730030

Published Jan 1, 2009• 3 citations
Publisher

Abstract

We study the mode deviation effect of a spin-1 Bose-Einstein condensate with large number of atoms in optical trap beyond the single-mode approximation. Based on the effective Hamiltonian with nondegenerate level of the collective spin states, we get the mode deviation factor and explicitly calculate the level splitting energy as function of the atom number N.

DOI
10.7498/aps.58.55

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We study the mode deviation effect of a spin-1 Bose-Einstein condensate with large number of atoms in optical trap beyond the single-mode approximation. Based on the effective Hamiltonian with nondegenerate level of the collective spin states, we get the mode deviation factor and explicitly calculate the level splitting energy as function of the atom number N.
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.
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 spin evolution of spin-3 52^{52}52Cr Bose-Einstein condensate
The spin evolution of a Bose-Einstein condensate starting from a mixture of two or three groups of 52^{52}52Cr (spin-3) atoms in an optical trap has been studied theoretically. The initial state is so chosen that the system does not distinguish up and down. In this choice, the deviation caused by the single-mode approximation is reduced. Moreover, since the particle number is given very small (N=20), the deviation caused by the neglect of the long-range dipole force is also reduced. Making use of these two simplifications, a theoretical calculation beyond the mean field theory is performed. The numerical results are help to evaluate the unknown strength g0g_0g0​.
This paper studies theoretically the spin evolution of a Bose—Einstein condensate starting from a mixture of two or three groups of 52 Cr (spin-3) atoms in an optical trap. The initial state is so chosen that the condensate has total magnetization zero so that the system does not distinguish up and down. It is assumed that the system is very dilute (particle number is very small), the temperature is very low, and the frequency of the harmonic trap is large enough. In these situations, the deviation caused by the neglect of the dipole—dipole interaction and by using the single-mode approximation is reduced. A theoretical calculation beyond the mean field theory is performed and the numerical results are helpful for the evaluation of the unknown strength g 0 .
Compared to single-component Bose-Einstein condensates, spinor Bose-Einstein condensates display much richer dynamics. In addition to density oscillations, spinor Bose-Einstein condensates exhibit intriguing spin dynamics that is associated with population transfer between different hyperfine components. This paper analyzes the validity of the widely employed single-mode approximation when describing the spin dynamics in response to a quench of the system Hamiltonian. The single-mode approximation assumes that the different hyperfine states all share the same time-independent spatial mode. This implies that the resulting spin Hamiltonian only depends on the spin interaction strength and not on the density interaction strength. Taking the spinor sodium Bose-Einstein condensate in the f=1f=1f=1 hyperfine manifold as an example and working within the mean-field theory framework, it is found numerically that the single-mode approximation misses, in some parameter regimes, intricate details of the spin and spatial dynamics. We develop a physical picture that explains the observed phenomenon. Moreover, using that the population oscillations described by the single-mode approximation enter into the effective potential felt by the mean-field spinor, we derive a semiquantitative condition for when dynamical mean-field induced corrections to the single-mode approximation are relevant. Our mean-field results have implications for a variety of published and planned experimental studies.
We theoretically study atom number fluctuations between the halves of a double-well trap containing a Bose-Einstein condensate. The basic tool is the two-mode approximation, which assumes that only two one-particle states are involved. An analytical harmonic-oscillator-like model is developed and verified numerically for both stationary fluctuations in the ground state of the system, and for the fluctuations resulting from splitting of a single trap by dynamically erecting a barrier in the middle. With increasing strength of the atom-atom interactions and/or increasing height of the potential barrier, the fluctuations tend to evolve from Poissonian to sub-Poissonian. Limits of validity of the two-mode model and its relations to the phase-atom-number approach of Leggett and Sols [A. J. Leggett and F. Sols, Found. Phys. 21, 353 (1991)] are discussed in detail.