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arXiv:1009.5438 · Physical Review A

Exact yrast spectra of cold atoms on a ring

Eriko Kaminishi, Rina Kanamoto, Jun Sato, Tetsuo Deguchi

Published Mar 1, 2011• 14 citations
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Abstract

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 $\text{U}(1)$ conformal field theories if the total angular momentum is given by an integral multiple of particle number; i.e., if the system is supercurrent.

DOI
10.1103/physreva.83.031601

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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.
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.
We present an operator-algebraic approach to derive the low-lying quasidegenerate spectrum of weakly interacting trapped N bosons with total angular momentum $\ensuremath{\Elzxh}L$ for the case of $L/N\ensuremath{\ll}1.$ We demonstrate that the lowest-lying excitation spectrum is given by ${27gn}_{3}{(n}_{3}\ensuremath{-}1)/34,$ where g is the strength of the repulsive contact interaction and n3{n}_{3}n3​ is the number of excited octupole quanta. Our method provides constraints for these quasidegenerate many-body states and gives higher excitation energies that depend linearly on N.
We demonstrate that the quantum dynamics of yrast states in the one-dimensional (1D) Bose gas gives an illustrative example to equilibration of an isolated quantum many-body system. We first formulate the energy spectrum of yrast states in terms of the dressed energy by applying the method of finite-size corrections. We then review the exact time evolution of quantum states constructed from yrast states shown by the Bethe ansatz. In time evolution the density profile of an initially localized quantum state constructed from yrast states collapses into a flat profile in the case of a large particle number such as N = 1000, while recurrence of the localized state occurs in the case of a small particle number such as N = 20. We suggest that the dynamical relaxation behavior for the large N case is consistent with the viewpoint of typicality for generic quantum states: the expectation values of local operators evaluated in most of quantum states are very close to those of the micro-canonical ensemble.
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.
Through an extensive numerical study, we find that the low-lying, quasidegenerate eigenenergies of weakly interacting trapped N bosons with total angular momentum L are given in the case of small L/NL/NL/N and sufficiently small L by $E=L\ensuremath{\Elzxh}\ensuremath{\omega}+g[N(N\ensuremath{-}L/2\ensuremath{-}1)+1.59n(n\ensuremath{-}1)/2],$ where $\ensuremath{\omega}$ is the frequency of the trapping potential and g is the strength of the repulsive contact interaction. The last term arises from the pairwise repulsive interaction among n octupole excitations and describes the lowest-lying excitation spectra from the yrast line. In this case, the quadrupole modes do not interact with themselves and, together with the octupole modes, exhaust the low-lying spectra that are separated from others by energy gaps linear in N.