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Physical Review Letters

Analytical Method for Yrast Line States in Interacting Bose-Einstein Condensates

Ying Wu, Xiaoxue Yang, Yi Xiao

Published Mar 12, 2001• 94 citations
Publisher

Abstract

We propose a method to investigate the energy eigenvalue problem and corresponding yrast line for harmonically trapped interacting N-boson systems. This method is particularly simple and effective to obtain the explicit analytical expressions of low-L energy eigenstates with L denoting systems' total angular momentum but with an arbitrarily large N. We have derived the explicit analytical results for L = 0,1,....,9 and discussed the yrast line for these low- L energy eigenstates.

DOI
10.1103/physrevlett.86.2200

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We propose a method to investigate the energy eigenvalue problem and corresponding yrast line for harmonically trapped interacting N-boson systems. This method is particularly simple and effective to obtain the explicit analytical expressions of low-L energy eigenstates with L denoting systems' total angular momentum but with an arbitrarily large N. We have derived the explicit analytical results for L = 0,1,....,9 and discussed the yrast line for these low- L energy eigenstates.
We study the energy eigenvalue problem and the yrast states for a harmonically trapped two-component weak-interacting Bose-Einstein condensate (BEC) using an analytical method. The yrast states and energy eigenvalues for L=0,1,2,3L=0,1,2,3L=0,1,2,3 are given in a special case. We illustrate the different eigenstate behaviors between even L and odd L for the two-component BEC. The odd-even effect of the yrast spectrum in the two-component BEC is also pointed out.
The yrast spectrum for the harmonically trapped two-component Bose-Einstein condensate (BEC), omitting the difference between the two components, has been studied using an analytical method. The energy eigenstates and eigenvalues for L = 0, 1, 2, 3 are given. We illustrate that there are different eigenstate behaviours between the even L and odd L cases for the two-component BEC in two dimensions. Except for symmetric states, there are antisymmetric states for the permutation of the two components, which cannot reduce to those in a single condensate case when the value of L is odd.
For a system of N bosons in a two-dimensional harmonic trap with frequency $\ensuremath{\omega},$ interacting via forces $V\ensuremath{\ll}\ensuremath{\Elzxh}\ensuremath{\omega},$ we develop a method to find the ground states of rotating Bose condensate as a function of two quantum numbers, the total angular momentum and the angular momentum of internal excitations (generalized yrast states). The energies of these condensed vortex states are expressed through the single two-body matrix element of interaction, V.V.V. A broad universality class of predominantly repulsive interactions for which these results hold is described by a simple integral condition on V.V.V. It includes Gaussian, $\ensuremath{\delta}$ function, and log-Coulomb forces.
The yrast spectrum for the harmonically trapped two-component Bose-Einstein condensate (BEC),omitting the difference between the two components,has been studied using an analytical method.The energy eigenstates and eigenvalues for L=0,1,2,3 are given,We illustrate that there are different eigenstate behaviours between the even L and odd L cases for the two-component BEC in two dimensions.Except for symmetric states,there are antisymmetric states for the permutation of the two components,which cannot reduce to those in a single condensate case when the value of L is odd.
We compute numerically the yrast line for harmonically trapped NNN-boson systems with a weak repulsive contact interaction, studying the transition to a vortex state as the angular momentum LLL increases and approaches NNN. There is a smooth crossover between low and high LLL with no indication of a quantum phase transition; at L=NL\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}NL=N there is a kink in the slope of the spectrum. Most strikingly, the energy and wave function appear to be analytical functions of LLL over the entire range $2\ensuremath{\le}L\ensuremath{\le}N$. We confirm the structure of low- LLL states proposed by Mottelson as mainly single-particle excitations with two or three units of angular momentum.