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Physical Review A

Vortex in the yrast band of condensates of pseudo-spin-1/2 bosonic atoms

Zhibing Li, Bao Cheng-guang

Published Jul 13, 2006• 5 citations
Publisher

Abstract

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 $N∕2$ subbands specified by the total spin $S$ ranging from $N∕2$ to 0 or 1/2, the one with a larger $S$ is lower. The states in a subband are further specified by the total orbital angular momentum $L$. 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.

DOI
10.1103/physreva.74.013606

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The possible excitation of angular degrees of freedom of pseudo-spin-1/2 N\ensuremath−bosonN\ensuremath{-}bosonN\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 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.
We propose a methodology to construct excited states with a fixed angular momentum, namely, ``yrast excited states'' of finite-size one-dimensional bosonic systems with periodic boundary conditions. The excitation energies such as the first yrast excited energy are calculated through the system-size asymptotic expansion and expressed analytically by dressed energy. Interestingly, they are grouped into sets of almost degenerate energy levels. The low-lying excitation spectrum near the yrast state is consistent with the U(1)\text{U}(1)U(1) conformal field theories if the total angular momentum is given by an integral multiple of particle number; i.e., if the system is supercurrent.
We present an exact diagonalization study of the breathing mode collective excitations for a rotating Bose-Einstein condensate of N=10N = 10N=10 spinless bosons interacting via repulsive finiterange Gaussian potential and harmonically confined in quasi-two-dimension. The yrast state and the low-lying excited states are variationally obtained in given subspaces of the quantized total angular momentum LLL employing the beyond lowest Landau level approximation in slowly rotating regime with 0≤L<2N0\le L< 2N0≤L<2N. For a given LLL, the low-energy eigenspectra (bands) are obtained in weakly to moderately interacting regime. Further, for a given interaction, the split in low-lying eigenenergies with increasing LLL is the precursor to spontaneous symmetry breaking of the axisymmetry associated with the entry of the first vortex. With increase in repulsive interaction, the value of the first breathing mode increases for stable total angular momentum states L=0L = 0L=0 and NNN, but decreases for intermediate 0<L<N0 < L < N0<L<N metastable states. The position of the observed first breathing modes in the eigenspectrum remains unchanged as the interaction is varied over several orders of magnitude.
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.
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.
The low energy quantal spectrum is considered as a function of the total angular momentum for a system of weakly interacting bosonic atoms held together by an external isotropic harmonic potential. It is found that besides the usual condensation into the lowest state of the oscillator the system exhibits two additional kinds of condensate and associated thermodynamic phase transitions. These new phenomena are derived from the degrees of freedom of ``partition space'' which describe the multitude of different ways in which the angular momentum can be distributed among the atoms while remaining all the time in the lowest state of the oscillator.