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arXiv:1407.1351

Lower bound for the population of hyperfine component $μ=0$ particles in the ground state of spin-1 condensates

C. G. Bao

Published Jul 5, 2014• 0 citations
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Abstract

An analytical expression for the lower bound of the average number of hyperfine component $μ=0$ particles in the ground state of spin-1 condensates (denoted as $\overset{\_\_}{ρ_{0}}$) under a magnetic field has been derived. In the derivation the total magnetization $M$ is kept rigorously conserved. Numerical examples are given to show the applicability of the analytical expression. It was found that, in a broad domain of parameters specified in the paper, the lower bound is very close to the actual $\overset{\_\_}{ρ_{0}}$. Thereby, in this domain, $\overset{\_\_}{% ρ_{0}}$ can be directly evaluated simply by using the analytical expression.

DOI
10.48550/arxiv.1407.1351

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An analytical expression for the lower bound of the average number of hyperfine component μ=0μ=0μ=0 particles in the ground state of spin-1 condensates (denoted as ρ0__\overset{\_\_}{ρ_{0}}ρ0​__​) under a magnetic field has been derived. In the derivation the total magnetization MMM is kept rigorously conserved. Numerical examples are given to show the applicability of the analytical expression. It was found that, in a broad domain of parameters specified in the paper, the lower bound is very close to the actual ρ0__\overset{\_\_}{ρ_{0}}ρ0​__​. Thereby, in this domain, \overset{\_\_}{% ρ_{0}} can be directly evaluated simply by using the analytical expression.
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Based on perturbation theory up to second order, analytical asymptotic expressions for the variation of the population of hyperfine component $\ensuremath{\mu}=0$ particles in the ground state of spin-1 condensates against a magnetic field BBB have been derived. The ranges of BBB in which the asymptotic expressions are applicable have been clarified via a comparison of the numerical results from the analytical expressions and from a diagonalization of the Hamiltonian in a complete spin space. It was found that, for 87Rb^{87}\mathrm{Rb}87Rb, the two analytical expressions, one for a weak field and the other one for a strong field, together cover the whole range of BBB from zero to infinity. For Na, the analytical expressions are valid only if BBB is very weak or sufficiently strong.
The possible ground states of spin-2 Bose-Einstein condensates in an external magnetic field are obtained analytically and classified systematically according to the population of the condensed atoms at the hyperfine sublevels. It is shown that the atoms can populate simultaneously at four hyperfine sublevels in a weak magnetic field with only the linear Zeeman energy, in contrast to that in a stronger magnetic field with the quadratic Zeeman energy, where condensed atoms can at most populate at three hyperfine sublevels in the ground states. Any spin configuration we obtained will give a closed subspace in the order parameter space of the condensates.
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