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Acta Scientiarum Naturalium Universitatis Sunyatseni

Fractional Parentage Coefficients for Spinor Bose-Einstein Condensation

Bao Cheng-guang

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

The total spin_states of a N_body system with spin_one bosons have been investigated. The fractional parentage coefficients for the total spin_states with ｛N｝ and ｛N-1,1｝ symmetries have been derived. These coefficients facilitate greatly the calculation of related matrix elements,they appear as a powerful tool for the establishment of an improved theory for the spinor Bose_Einstein condensation, where the total spin Sand its Z_component S_Zare both conserved.

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The total spin_states of a N_body system with spin_one bosons have been investigated. The fractional parentage coefficients for the total spin_states with ｛N｝ and ｛N-1,1｝ symmetries have been derived. These coefficients facilitate greatly the calculation of related matrix elements,they appear as a powerful tool for the establishment of an improved theory for the spinor Bose_Einstein condensation, where the total spin Sand its Z_component S_Zare both conserved.
The analytical expressions of the fractional parentage coefficients for the total spin states of a spinor NNN-boson system have been derived. Thereby an SSS-conserved theory for the spinor Bose-Einstein condensation has been proposed. A set of equations has been established to describe the first excited band of the condensates. Numerical solution for 23Na^{23}\mathrm{Na}23Na has been given as an example.
Abstract In the independent particle model, the total wave function is expressed as a linear combination of products of single particle wave functions. Fractional parentage coefficients give the part of the expression depending on the symmetry properties of the wave function. An operator is introduced which picks out the antisymmetric part of any n-particle wave function. The operator is given explicit form, and an explicit formula for the fractional parentage coefficients is deduced. The total fractional parentage coefficients can be split into a part depending on the orbital vectors, and a part depending on the spin (and isotopic spin) vectors. The separate parts can have symmetries other than total antisymmetry. Operators are introduced which pick out states of definite symmetry from any state. The operators are given explicit form, and explicit formulae for orbital, or spin, fractional coefficients are deduced. Detailed results are given for the three, four and five particles cases.
We study the uniform solutions to the one-dimensional (1D) spinor Bose–Einstein condensates on a ring. These states explicitly display the associated motion of the super-current and the spin rotation, which give rise to fractional winding numbers according to the various compositions of the hyperfine states. It simultaneously yields a fractional factor to the global phase due to the spin-gauge symmetry. All fractional windings can be denoted as nk/(m+n), with nk<m+n<2F, for arbitrary spin-F Bose–Einstein condensation (BEC). Our method can be applied to explore the fractional vortices by identifying the ring as the boundary of two-dimensional (2D) spinor condensates.