We have theoretically studied vortex waves of Bose-Einstein condensates in elongated harmonic traps. Our focus is on the axisymmetric varicose waves and helical Kelvin waves of singly quantized vortex lines. Growth and decay dynamics of both types of vortex waves are discussed. We propose a method to experimentally create these vortex waves on demand.

arXiv:0809.0993Physical Review ANov 4, 200823 citations

Quantized Vortex Lines in BECs with a Generalized Equation of State

Tommaso Macrì

Springer proceedings in physicsJan 1, 20200 citations

APS

Quantized Kelvin modes of a vortex in a nonuniform elongated BEC

Alexander L. Fetter

Published Mar 1, 2004• 0 citations

OpenAlex

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Kelvin mode of a vortex in a nonuniform Bose-Einstein condensate

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In a uniform fluid, a quantized vortex line with circulation $h∕M$ can support long-wavelength helical traveling waves $\ensuremath{\propto}{e}^{i(kz\ensuremath{-}{\ensuremath{\omega}}_{k}t)}$ with the well-known Kelvin dispersion relation ${\ensuremath{\omega}}_{k}\ensuremath{\approx}(\ensuremath{\hbar}{k}^{2}∕2M)\mathrm{ln}(1∕\ensuremath{\mid}k\ensuremath{\mid}\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the vortex-core radius. This result is extended to include the effect of a nonuniform harmonic trap potential, using a quantum generalization of the Biot-Savart law that determines the local velocity $\mathbf{V}$ of each element of the vortex line. The normal-mode eigenfunctions form an orthogonal Sturm-Liouville set. Although the line's curvature dominates the dynamics, the transverse and axial trapping potential also affect the normal modes of a straight vortex on the symmetry axis of an axisymmetric Thomas-Fermi condensate. The leading effect of the nonuniform condensate density is to increase the amplitude along the axis away from the trap center. Near the ends, however, a boundary layer forms to satisfy the natural Sturm-Liouville boundary conditions. For a given applied frequency, the next-order correction renormalizes the local wave number $k(z)$ upward near the trap center, and $k(z)$ then increases still more toward the ends.In a uniform fluid, a quantized vortex line with circulation $h∕M$ can support long-wavelength helical traveling waves $\ensuremath{\propto}{e}^{i(kz\ensuremath{-}{\ensuremath{\omega}}_{k}t)}$ with the well-known Kelvin dispersion relation ${\ensuremath{\omega}}_{k}\ensuremath{\approx}(\ensuremath{\hbar}{k}^{2}∕2M)\mathrm{ln}(1∕\ensuremath{\mid}k\ensuremath{\mid}\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the vortex-core radius. This result is extended to include the effect of a nonuniform harmonic trap potential, using a quantum generalization of the Biot-Savart law that determines the local velocity $\mathbf{V}$ of each element of the vortex line. The normal-mode eigenfunctions form an orthogonal Sturm-Liouville set. Although the line's curvature dominates the dynamics, the transverse and axial trapping potential also affect the normal modes of a straight vortex on the symmetry axis of an axisymmetric Thomas-Fermi condensate. The leading effect of the nonuniform condensate density is to increase the amplitude along the axis away from the trap center. Near the ends, however, a boundary layer forms to satisfy the natural Sturm-Liouville boundary conditions. For a given applied frequency, the next-order correction renormalizes the local wave number $k(z)$ upward near the trap center, and $k(z)$ then increases still more toward the ends.

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Quantized Vortex Lines in BECs with a Generalized Equation of State

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Springer proceedings in physicsJan 1, 20200 citations

In a uniform fluid, a quantized vortex line with circulation

h∕M

can support long-wavelength helical traveling waves $\ensuremath{\propto}{e}^{i(kz\ensuremath{-}{\ensuremath{\omega}}_{k}t)}$ with the well-known Kelvin dispersion relation ${\ensuremath{\omega}}_{k}\ensuremath{\approx}(\ensuremath{\hbar}{k}^{2}∕2M)\mathrm{ln}(1∕\ensuremath{\mid}k\ensuremath{\mid}\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the vortex-core radius. This result is extended to include the effect of a nonuniform harmonic trap potential, using a quantum generalization of the Biot-Savart law that determines the local velocity

\mathbf{V}

of each element of the vortex line. The normal-mode eigenfunctions form an orthogonal Sturm-Liouville set. Although the line's curvature dominates the dynamics, the transverse and axial trapping potential also affect the normal modes of a straight vortex on the symmetry axis of an axisymmetric Thomas-Fermi condensate. The leading effect of the nonuniform condensate density is to increase the amplitude along the axis away from the trap center. Near the ends, however, a boundary layer forms to satisfy the natural Sturm-Liouville boundary conditions. For a given applied frequency, the next-order correction renormalizes the local wave number

k(z)

upward near the trap center, and

k(z)

then increases still more toward the ends.