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arXiv:1509.05687 · Physical Review A

Asymptotic expressions for the hyperfine populations in the ground state of spin-1 condensates against a magnetic field

Y. M. Liu, Y. Z. He, Bao Cheng-guang

Published Oct 20, 2015• 2 citations
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Abstract

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 $B$ have been derived. The ranges of $B$ 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 $^{87}\mathrm{Rb}$, the two analytical expressions, one for a weak field and the other one for a strong field, together cover the whole range of $B$ from zero to infinity. For Na, the analytical expressions are valid only if $B$ is very weak or sufficiently strong.

DOI
10.1103/physreva.92.043617

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Based on perturbation theory up to second order, analytical asymptotic expressions for the variation of the population of hyperfine component \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=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.
Lower bound for the population of hyperfine component μ=0μ=0μ=0 particles in the ground state of spin-1 condensates
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.
The variation of the hyperfine populations $N{\ensuremath{\rho}}_{\ensuremath{\mu}}$ of the ground state (g.s.) of spin-2 condensates against a magnetic field BBB is studied. A NNN-body theory which is rigorous in dealing with the spin degrees of freedom has been adopted. The conservation of the total magnetization MMM is assumed. A combined hyperfine population $N{\ensuremath{\rho}}_{\text{comb}}=N({\ensuremath{\rho}}_{0}\ensuremath{-}3{\ensuremath{\rho}}_{2}\ensuremath{-}3{\ensuremath{\rho}}_{\ensuremath{-}2})$, which is an observable, is defined. The lower bound for this population has been derived analytically. It turns out that the lower bound is identical to the actual population when B=0B=0B=0 and $\ensuremath{\infty}$, and the population is monotonically uprising in between. Thus, via the analytical form of the lower bound, the variation of the population against BBB can be roughly known in advance. Numerical examples are given to demonstrate the applicability of the bound. When M=0M=0M=0, the g.s. in the polar phase was found to be highly unstable against BBB because the neighboring level density is extremely high. When M=NM=NM=N, the g.s. in the cyclic phase was also found to be highly unstable. This is because the appearance of BBB, even if it is very weak, will break the high degeneracy of the g.s. and result in having numerous levels emerging in the neighborhood of the g.s.
We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin F=1.F=1.F=1. Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.
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.
The spin mixing dynamics of spin-1 Bose–Einstein condensates with zero magnetization and under an external magnetic field is investigated in a systematic way. Beyond the mean field theory, the time-dependent solutions of the Hamiltonian under the single mode approximation are obtained via a procedure of diagonalization. Thereby the time-dependent populations of spin-0 component starting from various initial states are obtained and analyzed. The features of evolution are compared in detail with those with the field removed so as to emphasize the effect of the field. Since the solutions are exact for the Hamiltonian, we can study the long-term behavior of the evolution, where periodicity and inherent symmetry emerge. In particular, some special cases are found in which the evolution is highly sensitive to the field, very strong oscillation in population can be thereby induced. A new mode of oscillation of the population characterized by having two frequencies, one is high and one is low, is found when the field is sufficiently strong.
We propose efficient and accurate numerical methods for computing the ground-state solution of spin-1 Bose-Einstein condensates subjected to a uniform magnetic field. The key idea in designing the numerical method is based on the normalized gradient flow with the introduction of a third normalization condition, together with two physical constraints on the conservation of total mass and conservation of total magnetization. Different treatments of the Zeeman energy terms are found to yield different numerical accuracies and stabilities. Numerical comparison between different numerical schemes is made, and the best scheme is identified. The numerical scheme is then applied to compute the condensate ground state in a harmonic plus optical lattice potential, and the effect of the periodic potential, in particular to the relative population of each hyperfine component, is investigated through comparison to the condensate ground state in a pure harmonic trap.