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arXiv:cond-mat/0202178 · Physical Review Letters

Lieb Mode in a Quasi-One-Dimensional Bose-Einstein Condensate of Atoms

Andrew David Jackson, G. M. Kavoulakis

Published Jul 30, 2002• 40 citations
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Abstract

We calculate the dispersion relation associated with a solitary wave in a quasi-one-dimensional Bose-Einstein condensate of atoms confined in a harmonic, cylindrical trap in the limit of weak and strong interactions. In both cases, the dispersion relation is linear for long-wavelength excitations and terminates at the point where the group velocity vanishes. We also calculate the dispersion relation of sound waves in both limits of weak and strong coupling.

DOI
10.1103/physrevlett.89.070403

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We calculate the dispersion relation associated with a solitary wave in a quasi-one-dimensional Bose-Einstein condensate of atoms confined in a harmonic, cylindrical trap in the limit of weak and strong interactions. In both cases, the dispersion relation is linear for long-wavelength excitations and terminates at the point where the group velocity vanishes. We also calculate the dispersion relation of sound waves in both limits of weak and strong coupling.
Quasi-one-dimensional solitons that may occur in an elongated Bose-Einstein condensate become unstable at high particle density. We study two basic modes of instability and the corresponding bifurcations to genuinely three-dimensional solitary waves such as axisymmetric vortex rings and nonaxisymmetric solitonic vortices. We calculate the profiles of the above structures and examine their dependence on the velocity of propagation along a cylindrical trap. At sufficiently high velocity, both the vortex ring and the solitonic vortex transform into an axisymmetric soliton. We also calculate the energy-momentum dispersions and show that a Lieb-type mode appears in the excitation spectrum for all particle densities.
We present a calculation of solitary waves propagating in a steady state with constant velocity vvv along a cigar-shaped Bose-Einstein trap approximated as an infinitely elongated cylinder. For sufficiently weak couplings (densities), the main features of the calculated solitons could be captured by effective one-dimensional (1D) models. However, for stronger couplings of practical interest, the relevant solitary waves are found to be hybrids of quasi-1D solitons and 3D vortex rings. An interesting hierarchy of vortex rings occurs as the effective coupling constant is increased through a sequence of critical values. The energy-momentum dispersion of the above structures is shown to exhibit characteristics similar to a mode proposed sometime ago by Lieb within a strictly 1D model, as well as some rotonlike features.
The linearized time-dependent coupled Gross-Pitaevskii equations describing the long-wavelength excitations of Bose-Einstein condensed binary mixtures are solved in harmonic traps for the case where only the binary phase is present. The dispersion law contains two branches whose dependence on the quantum numbers of the modes is the same as for a one-component condensate, but with different prefactors, depending on the ratios of the three sss-wave scattering lengths of the two atomic species. In the general case where the binary phase in the trap coexists with one or both one-component phases, the mode spectrum depends on the geometry of the interphase boundaries due to the boundary conditions there. Measurements of the oscillation frequencies as in recent experiments with modulated traps would yield very detailed information on this system.