arXiv:1709.05097

Non-stationary vortex ring in a Bose-Einstein condensate with Gaussian density

V. P. Ruban

Published Sep 15, 2017• 0 citations

Abstract

The local induction equation, approximately describing dynamics of a quantized vortex filament in a trapped Bose-Einstein condensate in the Thomas-Fermi regime on a spatially nonuniform density background $ρ({\bf r})$ and taking dimensionless form ${\mathbf R}_t=\varkappa {\mathbf b}+[\nabla\lnρ({\mathbf R})\times {\boldsymbol τ}]$ (where $\varkappa$ is a local curvature of the filament, ${\mathbf b}$ is the unit binormal vector, and ${\boldsymbol τ}$ is the unit tangent vector), is shown to admit a finite-dimensional reduction if the density profile is an isotropic Gaussian, $ρ\propto\exp(-|{\bf r}|^2/2)$. The reduction corresponds to a geometrically perfect vortex ring centered at position ${\bf A}(t)$, with orientation and size both determined by a vector ${\bf B}(t)$. Parameters ${\bf A}$ and ${\bf B}$ exhibit the same dynamics as velocity and position of a Newtonian particle do in 3D: $\dot {\bf A}={\bf B}/|{\bf B}|^2-{\bf B}$, and $\dot {\bf B}={\bf A}$.

DOI: 10.48550/arxiv.1709.05097

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Dynamics of straight vortex filaments in a Bose–Einstein condensate with the Gaussian density profile

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arXiv:1709.05097

Non-stationary vortex ring in a Bose-Einstein condensate with Gaussian density

V. P. Ruban

Published Sep 15, 2017• 0 citations

arXiv HTML PDF

Abstract

ρ({\bf r})

and taking dimensionless form

{\mathbf R}_t=\varkappa {\mathbf b}+[\nabla\lnρ({\mathbf R})\times {\boldsymbol τ}]

(where

\varkappa

is a local curvature of the filament,

{\mathbf b}

is the unit binormal vector, and

{\boldsymbol τ}

is the unit tangent vector), is shown to admit a finite-dimensional reduction if the density profile is an isotropic Gaussian,

ρ\propto\exp(-|{\bf r}|^2/2)

. The reduction corresponds to a geometrically perfect vortex ring centered at position

{\bf A}(t)

, with orientation and size both determined by a vector

{\bf B}(t)

. Parameters

{\bf A}

and

{\bf B}

exhibit the same dynamics as velocity and position of a Newtonian particle do in 3D:

\dot {\bf A}={\bf B}/|{\bf B}|^2-{\bf B}

, and

\dot {\bf B}={\bf A}

DOI: 10.48550/arxiv.1709.05097

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Dynamics of straight vortex filaments in a Bose–Einstein condensate with the Gaussian density profile

V. P. Ruban

The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.

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arXiv:1609.03747Journal of Experimental and Theoretical Physics LettersJun 1, 20166 citations

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V. P. Ruban

The dynamics of deformations of a quantum vortex ring in a Bose-Einstein condensate with periodic equilibrium density $\rho(z)= 1-\epsilon\cos z$ has been considered within the local induction approximation. Parametric instabilities of the normal modes with azimuthal numbers $\pm m$ have been revealed at the energy integral $E$ near values $E_m^{(p)}=2m\sqrt{m^2-1}/p$ , where $p$ is the resonance order. Numerical simulations have shown that already at $\epsilon\sim 0.03$ a rapid growth of unstable modes with $m=2$ , $p=1$ to magnitudes of order of unity is typical, which is then followed, after a few large oscillations, by fast return to a weakly excited state. Such behavior corresponds to an integrable Hamiltonian of the form $H\propto \sigma(E_2^{(1)}-E)(|b_+|^2 + |b_-|^2) -\epsilon(b_+ b_- + b_+^* b_-^*) +u(|b_+|^4 +|b_-|^4) + w |b_+|^2|b_-|^2$ for two complex envelopes $b_\pm(t)$ . The results have been compared to parametric instabilities of vortex ring in condensate with density $\rho(z,r)=1-r^2-\alpha z^2$ , which take place at $\alpha\approx 8/5$ and at $\alpha\approx 16/7$ .The dynamics of deformations of a quantum vortex ring in a Bose-Einstein condensate with periodic equilibrium density $\rho(z)= 1-\epsilon\cos z$ has been considered within the local induction approximation. Parametric instabilities of the normal modes with azimuthal numbers $\pm m$ have been revealed at the energy integral $E$ near values $E_m^{(p)}=2m\sqrt{m^2-1}/p$, where $p$ is the resonance order. Numerical simulations have shown that already at $\epsilon\sim 0.03$ a rapid growth of unstable modes with $m=2$, $p=1$ to magnitudes of order of unity is typical, which is then followed, after a few large oscillations, by fast return to a weakly excited state. Such behavior corresponds to an integrable Hamiltonian of the form $H\propto \sigma(E_2^{(1)}-E)(|b_+|^2 + |b_-|^2) -\epsilon(b_+ b_- + b_+^* b_-^*) +u(|b_+|^4 +|b_-|^4) + w |b_+|^2|b_-|^2$ for two complex envelopes $b_\pm(t)$. The results have been compared to parametric instabilities of vortex ring in condensate with density $\rho(z,r)=1-r^2-\alpha z^2$, which take place at $\alpha\approx 8/5$ and at $\alpha\approx 16/7$.

Jun 14, 20170 citations

ρ({\bf r})

and taking dimensionless form

{\mathbf R}_t=\varkappa {\mathbf b}+[\nabla\lnρ({\mathbf R})\times {\boldsymbol τ}]

(where

\varkappa

is a local curvature of the filament,

{\mathbf b}

is the unit binormal vector, and

{\boldsymbol τ}

is the unit tangent vector), is shown to admit a finite-dimensional reduction if the density profile is an isotropic Gaussian,

ρ\propto\exp(-|{\bf r}|^2/2)

. The reduction corresponds to a geometrically perfect vortex ring centered at position

{\bf A}(t)

, with orientation and size both determined by a vector

{\bf B}(t)

. Parameters

{\bf A}

and

{\bf B}

exhibit the same dynamics as velocity and position of a Newtonian particle do in 3D:

\dot {\bf A}={\bf B}/|{\bf B}|^2-{\bf B}

, and

\dot {\bf B}={\bf A}

Parametric instability of a vortex ring oscillations in a

z

-periodic Bose-Einstein condensate and the recurrence to starting state

The dynamics of deformations of a quantum vortex ring in a Bose-Einstein condensate with periodic equilibrium density

\rho(z)= 1-\epsilon\cos z

has been considered within the local induction approximation. Parametric instabilities of the normal modes with azimuthal numbers

\pm m

have been revealed at the energy integral

E

near values

E_m^{(p)}=2m\sqrt{m^2-1}/p

, where

p

is the resonance order. Numerical simulations have shown that already at

\epsilon\sim 0.03

a rapid growth of unstable modes with

m=2

p=1

to magnitudes of order of unity is typical, which is then followed, after a few large oscillations, by fast return to a weakly excited state. Such behavior corresponds to an integrable Hamiltonian of the form

H\propto \sigma(E_2^{(1)}-E)(|b_+|^2 + |b_-|^2) -\epsilon(b_+ b_- + b_+^* b_-^*) +u(|b_+|^4 +|b_-|^4) + w |b_+|^2|b_-|^2

for two complex envelopes

b_\pm(t)

. The results have been compared to parametric instabilities of vortex ring in condensate with density

\rho(z,r)=1-r^2-\alpha z^2

, which take place at

\alpha\approx 8/5

and at

\alpha\approx 16/7