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arXiv:cond-mat/0202182 · Physical Review Letters

Vortex Rings and Lieb Modes in a Cylindrical Bose-Einstein Condensate

Stavros Komineas, N. Papanicolaou

Published Jul 30, 2002• 49 citations
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Abstract

We present a calculation of a solitary wave propagating along a cylindrical Bose-Einstein trap. For sufficiently strong couplings of experimental interest, it is found to be a hybrid of a one-dimensional (1D) soliton and a three-dimensional vortex ring. The calculated energy-momentum dispersion exhibits characteristics similar to those of a mode proposed sometime ago by Lieb within a 1D model, as well as some rotonlike features.

DOI
10.1103/physrevlett.89.070402

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We present a calculation of a solitary wave propagating along a cylindrical Bose-Einstein trap. For sufficiently strong couplings of experimental interest, it is found to be a hybrid of a one-dimensional (1D) soliton and a three-dimensional vortex ring. The calculated energy-momentum dispersion exhibits characteristics similar to those of a mode proposed sometime ago by Lieb within a 1D model, as well as some rotonlike features.
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.
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.
Interactions of solitary waves in a cylindrically confined Bose-Einstein condensate are investigated by simulating their head-on collisions. Slow vortex rings and fast solitons are found to collide elastically contrary to the situation in the three-dimensional homogeneous Bose gas. Strongly inelastic collisions are absent for low density condensates but occur at higher densities for intermediate velocities. The scattering behavior is rationalized by use of dispersion diagrams. During inelastic collisions, spherical shell-like structures of low density are formed and they eventually decay into depletion droplets with solitary-wave features. The relation to similar shells observed in a recent experiment by Ginsberg et al. [Phys. Rev. Lett. 94, 040403 (2005)] is discussed.
We analyse the dynamical properties of three-dimensional solitary waves in cylindrically trapped Bose–Einstein condensates. Families of solitary waves bifurcate from the planar dark soliton and include the solitonic vortex, the vortex ring and more complex structures of intersecting vortex lines known collectively as Chladni solitons. The particle-like dynamics of these guided solitary waves provides potentially profitable features for their implementation in atomtronic circuits, and play a key role in the generation of metastable loop currents. Based on the time-dependent Gross–Pitaevskii equation we calculate the dispersion relations of moving solitary waves and their modes of dynamical instability. The dispersion relations reveal a complex crossing and bifurcation scenario. For stationary structures we find that for the solitonic vortex is the only stable solitary wave. More complex Chladni solitons still have weaker instabilities than planar dark solitons and may be seen as transient structures in experiments. Fully time-dependent simulations illustrate typical decay scenarios, which may result in the generation of multiple separated solitonic vortices.