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arXiv:1012.1816 · Europhysics Letters (EPL)

Solitary waves and yrast states in Bose-Einstein condensed gases of atoms

A. D. Jackson, J. Smyrnakis, M. Magiropoulos, G. M. Kavoulakis

Published Jul 12, 2011• 15 citations
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Abstract

Considering a Bose-Einstein condensed gas confined in one dimension with periodic boundary conditions, we demonstrate that, very generally, solitary-wave and rotational excitations coincide. This exact equivalence allows us to establish connections within a number of effects that are present in these two problems, many of which have been studied in the mean-field approximation and in the Tonks-Girardeau limit of hard-core bosons.

DOI
10.1209/0295-5075/95/30002

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Considering a Bose-Einstein condensed gas confined in one dimension with periodic boundary conditions, we demonstrate that, very generally, solitary-wave and rotational excitations coincide. This exact equivalence allows us to establish connections within a number of effects that are present in these two problems, many of which have been studied in the mean-field approximation and in the Tonks-Girardeau limit of hard-core bosons.
For a weakly interacting Bose gas rotating in a harmonic trap we relate the yrast states of small systems (that can be treated exactly) to the thermodynamic limit (derived within the mean-field approximation). For a few dozens of atoms, the yrast line shows distinct quasiperiodic oscillations with increasing angular momentum that originate from the internal structure of the exact many-body states. These finite-size effects disappear in the thermodynamic limit, where the Gross-Pitaevskii approximation provides the exact energy to leading order in the number of particles NNN. However, the exact yrast states reveal significant structure not captured by the mean-field approximation: Even in the limit of large NNN, the corresponding mean-field solution accounts for only a fraction of the total weight of the exact quantum state.
Motivated by recent progress in trapping Bose-Einstein-condensed atoms in toroidal potentials, we examine solitary-wave solutions of the nonlinear Schr\"odinger equation subject to periodic boundary conditions. When the circumference of the ring is much larger than the size of the wave, the density profile is well approximated by that of an infinite ring; however, the density and the velocity of propagation cannot vanish simultaneously. When the size of the ring becomes comparable to the size of the wave, the density variation becomes sinusoidal and the velocity of propagation saturates to a constant value.
We show that quantum solitons in the Lieb-Liniger Hamiltonian are precisely the yrast states. We identify such solutions with Lieb's type II excitations from weak to strong interactions, clarifying a long-standing question of the physical meaning of this excitation branch. We demonstrate that the metastable quantum phase transition previously found in mean-field analysis of the weakly interacting Lieb-Liniger Hamiltonian [Phys. Rev. A 79, 063616 (2009)] extends into the medium- to strongly interacting regime of a periodic one-dimensional Bose gas. Our methods are exact diagonalization, finite-size Bethe ansatz, and the boson-fermion mapping in the Tonks-Girardeau limit.