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arXiv:cond-mat/0505204 · Physical Review A

First excited band of a spinor Bose-Einstein condensate

Bao Cheng-guang, Zhibing Li

Published Oct 26, 2005• 22 citations
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Abstract

The analytical expressions of the fractional parentage coefficients for the total spin states of a spinor $N$-boson system have been derived. Thereby an $S$-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 $^{23}\mathrm{Na}$ has been given as an example.

DOI
10.1103/physreva.72.043614

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Shinjang dashösi ilmiy jurnili•Jan 1, 2010•0 citations
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.
For spinor Bose-Einstein condensates both the total spin SSS and its ZZZ component SZ{S}_{Z}SZ​ should be conserved. However, in existing theories, only the conservation of SZ{S}_{Z}SZ​ has been taken into account. To remedy this, here we attempt to take the conservation of both SSS and SZ{S}_{Z}SZ​ into account. For this purpose, a total spin state with the good quantum numbers SSS and SZ{S}_{Z}SZ​ is introduced in the trial wave function; thereby a generalized Gross-Pitaevskii equation has been derived. With this equation, the ground bands of the 23Na^{23}\mathrm{Na}23Na and 87Rb^{87}\mathrm{Rb}87Rb condensates have been studied, where the levels that are distinct in SSS split. It was found that the level density is extremely dense in the bottom of the ground band of 23Na^{23}\mathrm{Na}23Na, i.e., in the vicinity of the ground state. On the contrary, for 87Rb^{87}\mathrm{Rb}87Rb, the levels are extremely dense in the top of the ground band.
The possible excitation of angular degrees of freedom of pseudo-spin-1/2 $N\ensuremath{-}boson$ systems has been considered. For this purpose, the analytical forms of the fractional parentage coefficients have been introduced to deal with the symmetry problems involved. A set of equations for the Yrast band has been derived, and related numerical calculation has been performed. The yrast band is found to contain N∕2N∕2N∕2 subbands specified by the total spin SSS ranging from N∕2N∕2N∕2 to 0 or 1/2, the one with a larger SSS is lower. The states in a subband are further specified by the total orbital angular momentum LLL. Each state is a fragmented state (except the lowest one), where a part of particles have been excited and may form vortex. Specifically, the structure of the yrast band appears to obey the Hund's first and second rules.
The magnetic radiation of the fully condensed states of spinor condensates has been studied. A narrow characteristic spectral line together with a strong background at its violet side have been found. For the condensate of N=1012N={10}^{12}N=1012 atoms of 23Na^{23}\mathrm{Na}23Na confined in an isotropic and harmonic trap with frequency ${\ensuremath{\omega}}_{o}=300\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$, for instance, the photon energy of the characteristic line is $127.9\ensuremath{\hbar}{\ensuremath{\omega}}_{o}$ with width $<1.\ensuremath{\hbar}{\ensuremath{\omega}}_{o}$ and the probability of transition is $1.44\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}34}\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$. The photon energy and probability have been found to be proportional to N2∕5{N}^{2∕5}N2∕5 and N17∕5{N}^{17∕5}N17∕5, respectively. It implies that magnetic radiations with higher photon energies and much larger probabilities can be obtained if numerous atoms are trapped. A technique developed by the authors, namely, the analytical forms of the fractional parentage coefficients, was used to calculate analytically the matrix elements between the spin states that conserve the total spin and its ZZZ component.
The interaction between two species of atoms in the condensate has been deduced and the trial wave functions that conserve the total spin have been constructed for the states of the ground band. A couple of generalized Gross-Pitaevskii equations have been derived for the spatial wave functions and the energies. The mixture of 23Na^{23}\mathrm{Na}23Na and 87Rb^{87}\mathrm{Rb}87Rb has been calculated as an example. The density of states is found to exhibit a ladder shape.
We present a theoretical study of Bose-Einstein condensation(BEC) of trapped ideal gas of spin-1 ultracold atoms in tight-binding bands corresponding to simple cubic(SC) optical lattices.Under the constraints that both the total number of atoms (N) and the total magnetization (M) are conserved,we first investigate the phase transitions process of noninteracting spin-1 atomic system,by investigating the critical condensation temperature and condensate fraction,we find BEC with spin degrees of freedom shows richer phase transitions than BEC of scalar atoms.We then extend our studies to the case with a uniform external magnetic field included,the phase transitions for this case make interesting changes again.