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International Journal of Modern Physics B

THE MODE-DEVIATION EFFECT OF TRAPPED SPINOR BOSE GAS BEYOND MEAN FIELD THEORY

Yan Xu, Duoje Jia, Zhaoyang Chen, Yuan Gao, Fashen Li

Published Apr 10, 2004• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217979204024719

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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.
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.
The recent realization of Bose-Einstein condensation in atomic gases opens new possibilities for observation of macroscopic quantum phenomena. There are two important features of the system - weak interaction and significant spatial inhomogeneity. Because of this inhomogeneity a non-trivial "zeroth-order" theory exists, compared to the "first-order" Bogolubov theory. This theory is based on the mean-field Gross-Pitaevskii equation for the condensate ψ-function. The equation is classical in its essence but contains the ℏ constant explicitly. Phenomena such as collective modes, interference, tunneling, Josephson-like current and quantized vortex lines can be described using this equation. The study of deviations from the zeroth-order theory arising from zero-point and thermal fluctuations is also of great interest. Thermal fluctuations are described by elementary excitations which define the thermodynamic behaviour of the system and result in Landau-type damping of collective modes. Fluctuations of the phase of the condensate wave function restrict the monochromaticity of the Josephson current. Fluctuations of the numbers of quanta result in the quantum collapse-revival of the collective oscillations. This phenomenon is considered in some details. Collapse time for the JILA experimental conditions turns out to be of the order of seconds.
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.