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Parametric instability of a vortex ring oscillations in a $z$-periodic Bose-Einstein condensate and the recurrence to starting state

V. P. Ruban

Published Jun 14, 2017• 0 citations
OpenAlex

Abstract

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 $\alpha\approx 16/7$.

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Parametric instability of oscillations of a vortex ring in a z-periodic Bose condensate and return to the initial state

V. P. Ruban

The dynamics of deformations of a quantum vortex ring in a Bose condensate with the periodic equilibrium density ρ(z) = 1 − ϵ cos z has been considered in the local induction approximation. Parametric instabilities of normal modes with the azimuthal numbers ±m at the energy integral E near the values $$E_m^{\left( p \right)} = 2m\sqrt {{m^2} - 1} /p$$ , where p is the order of resonance, have been revealed. Numerical experiments have shown that the amplitude of unstable modes with m = 2 and p = 1 can sharply increase already at ϵ ~ 0.03 to values about unity. Then, after several fast oscillations, fast return to a weakly perturbed state occurs. Such a behavior corresponds to the integrable Hamiltonian H ∝ σ(E 2 (1) − E)(|b +|2 + |b -|2)-ϵ(b + b - + b +*b -*)+u(|b +|4 + |b -|4)+w|b +|2|b -|2 for two complex envelopes b ±(t). The results have been compared to parametric instabilities of the vortex ring in the condensate with the density ρ(z, r) = 1 − r 2 − αz 2, which occur at α ≈ 8/5 and 16/7.

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Instabilities of vortex-ring-bright coherent structures in harmonically trapped two-component three-dimensional Bose-Einstein condensates are studied numerically within the coupled Gross-Pitaevskii equations and interpreted analytically. Interestingly, the filled vortex core with a sufficiently large amount of the bright component is observed to reduce the parametric interval of stability of the vortex ring. We have identified the mechanisms of several linear instabilities and one nonlinear parametric instability in this connection. Two of the linear instabilities are qualitatively different from ones reported earlier and are associated with azimuthal modes of $m=0$ and $m=1$, i.e., deviations of the vortex from the stationary ring shape. Our nonlinear parametric resonance instability occurs between the $m=0$ and $m=2$ modes and signals the exchange of energy between them.

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A three-dimensional study of the ring vortex solitons is conducted for both attractive and repulsive Bose-Einstein condensates subject to harmonic potential confinement. A family of stationary ring vortex solitons, which is defined by the radial excitation number and the winding number of the intrinsic vorticity, are obtained numerically for a given atomic interaction strength. We find that stabilities of the ground and radially excited states of the ring vortex soliton are dependent on the winding number differently. The ground state of the ring vortex soliton with the large winding number is unstable dynamically against random perturbation. The radially excited state of the ring vortex soliton with large winding number corresponds to the increased collapse threshold and therefore can be made stable for sufficiently small atomic interaction strengths. The ground and radially excited states also demonstrate different dynamic evolutions under large atomic interaction strengths. The former exhibits simultaneous symmetric splitting in the transverse plane, while the latter displays periodic expand-merge cycles in the longitudinal direction.

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Dynamics and pattern formation of ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable interactions

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We investigate the dynamics and pattern formation of two ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable intercomponent interaction via numerical simulation of the time-dependent Gross-Pitaevskii equation. Both the black and gray ring dark solitons are considered for cases where the modulation frequency of the intercomponent interaction is resonant or nonresonant with the one of the trapping potential. Our results show that in the presence of periodic modulation of the intercomponent interaction not only are the lifetimes of the ring dark solitons largely extended but also their decaying dynamics are dramatically affected. Before snaking instability sets in, new ring dark solitons are formed, and both the numbers and depths of the ring dark solitons exhibit collective oscillations. With the development of instability, the system exhibits different decaying processes, and a variety of decay profiles, such as vortex necklace, distorted octagon, vortex-antivortex ring, and cross, are formed, showing a strong dependence on the modulation frequency of the intercomponent interaction and the initial depth of the ring dark soliton.

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Ring dark solitons in three-dimensional Bose-Einstein condensates

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In this work we present a systematic study of the three-dimensional extension of the ring dark soliton, examining its existence, stability, and dynamics in isotropic harmonically trapped Bose-Einstein condensates. Detuning the chemical potential from the linear limit, the ring dark soliton becomes unstable immediately but can be fully stabilized by an external cylindrical potential. The ring has a large number of unstable modes which are analyzed through spectral stability analysis. Furthermore, a few typical destabilization dynamical scenarios are revealed with a number of interesting vortical structures emerging, such as the two or four coaxial parallel vortex rings. In the process of considering the stability of the structure, we also develop a modified version of the degenerate perturbation theory method for characterizing the spectra of the coherent structure. This semianalytical method can be reliably applied to any soliton with a linear limit to explore its spectral properties near this limit. The good agreement of the resulting spectrum is illustrated via a comparison with the full numerical Bogolyubov--de Gennes spectrum. The application of the method to the two-component ring dark-bright soliton is also discussed.

arXiv:1910.02272•Physical Review A•Nov 25, 2019•29 citations
Parametric instability of a vortex ring oscillations in a zzz-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 ρ(z)=1−ϵcos⁡z\rho(z)= 1-\epsilon\cos zρ(z)=1−ϵcosz has been considered within the local induction approximation. Parametric instabilities of the normal modes with azimuthal numbers ±m\pm m±m have been revealed at the energy integral EEE near values Em(p)=2mm2−1/pE_m^{(p)}=2m\sqrt{m^2-1}/pEm(p)​=2mm2−1​/p, where ppp is the resonance order. Numerical simulations have shown that already at ϵ∼0.03\epsilon\sim 0.03ϵ∼0.03 a rapid growth of unstable modes with m=2m=2m=2, p=1p=1p=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∝σ(E2(1)−E)(∣b+∣2+∣b−∣2)−ϵ(b+b−+b+∗b−∗)+u(∣b+∣4+∣b−∣4)+w∣b+∣2∣b−∣2H\propto \sigma(E_2^{(1)}-E)(|b_+|^2 + |b_-|^2) -\epsilon(b_+ b_- + b_+^* b_-^*) +u(|b_+|^4 +|b_-|^4) + w |b_+|^2|b_-|^2H∝σ(E2(1)​−E)(∣b+​∣2+∣b−​∣2)−ϵ(b+​b−​+b+∗​b−∗​)+u(∣b+​∣4+∣b−​∣4)+w∣b+​∣2∣b−​∣2 for two complex envelopes b±(t)b_\pm(t)b±​(t). The results have been compared to parametric instabilities of vortex ring in condensate with density ρ(z,r)=1−r2−αz2\rho(z,r)=1-r^2-\alpha z^2ρ(z,r)=1−r2−αz2, which take place at α≈8/5\alpha\approx 8/5α≈8/5 and at α≈16/7\alpha\approx 16/7α≈16/7.
The dynamics of deformations of a quantum vortex ring in a Bose condensate with the periodic equilibrium density ρ(z) = 1 − ϵ cos z has been considered in the local induction approximation. Parametric instabilities of normal modes with the azimuthal numbers ±m at the energy integral E near the values Em(p)=2mm2−1/pE_m^{\left( p \right)} = 2m\sqrt {{m^2} - 1} /pEm(p)​=2mm2−1​/p , where p is the order of resonance, have been revealed. Numerical experiments have shown that the amplitude of unstable modes with m = 2 and p = 1 can sharply increase already at ϵ ~ 0.03 to values about unity. Then, after several fast oscillations, fast return to a weakly perturbed state occurs. Such a behavior corresponds to the integrable Hamiltonian H ∝ σ(E 2 (1) − E)(|b +|2 + |b -|2)-ϵ(b + b - + b +*b -*)+u(|b +|4 + |b -|4)+w|b +|2|b -|2 for two complex envelopes b ±(t). The results have been compared to parametric instabilities of the vortex ring in the condensate with the density ρ(z, r) = 1 − r 2 − αz 2, which occur at α ≈ 8/5 and 16/7.
Instabilities of vortex-ring-bright coherent structures in harmonically trapped two-component three-dimensional Bose-Einstein condensates are studied numerically within the coupled Gross-Pitaevskii equations and interpreted analytically. Interestingly, the filled vortex core with a sufficiently large amount of the bright component is observed to reduce the parametric interval of stability of the vortex ring. We have identified the mechanisms of several linear instabilities and one nonlinear parametric instability in this connection. Two of the linear instabilities are qualitatively different from ones reported earlier and are associated with azimuthal modes of m=0m=0m=0 and m=1m=1m=1, i.e., deviations of the vortex from the stationary ring shape. Our nonlinear parametric resonance instability occurs between the m=0m=0m=0 and m=2m=2m=2 modes and signals the exchange of energy between them.
A three-dimensional study of the ring vortex solitons is conducted for both attractive and repulsive Bose-Einstein condensates subject to harmonic potential confinement. A family of stationary ring vortex solitons, which is defined by the radial excitation number and the winding number of the intrinsic vorticity, are obtained numerically for a given atomic interaction strength. We find that stabilities of the ground and radially excited states of the ring vortex soliton are dependent on the winding number differently. The ground state of the ring vortex soliton with the large winding number is unstable dynamically against random perturbation. The radially excited state of the ring vortex soliton with large winding number corresponds to the increased collapse threshold and therefore can be made stable for sufficiently small atomic interaction strengths. The ground and radially excited states also demonstrate different dynamic evolutions under large atomic interaction strengths. The former exhibits simultaneous symmetric splitting in the transverse plane, while the latter displays periodic expand-merge cycles in the longitudinal direction.
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(r)ρ({\bf r})ρ(r) and taking dimensionless form Rt=ϰb+[∇ln⁡ρ(R)×τ]{\mathbf R}_t=\varkappa {\mathbf b}+[\nabla\lnρ({\mathbf R})\times {\boldsymbol τ}]Rt​=ϰb+[∇lnρ(R)×τ] (where ϰ\varkappaϰ is a local curvature of the filament, b{\mathbf b}b is the unit binormal vector, and τ{\boldsymbol τ}τ is the unit tangent vector), is shown to admit a finite-dimensional reduction if the density profile is an isotropic Gaussian, ρ∝exp⁡(−∣r∣2/2)ρ\propto\exp(-|{\bf r}|^2/2)ρ∝exp(−∣r∣2/2). The reduction corresponds to a geometrically perfect vortex ring centered at position A(t){\bf A}(t)A(t), with orientation and size both determined by a vector B(t){\bf B}(t)B(t). Parameters A{\bf A}A and B{\bf B}B exhibit the same dynamics as velocity and position of a Newtonian particle do in 3D: A˙=B/∣B∣2−B\dot {\bf A}={\bf B}/|{\bf B}|^2-{\bf B}A˙=B/∣B∣2−B, and B˙=A\dot {\bf B}={\bf A}B˙=A.
We investigate the dynamics and pattern formation of two ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable intercomponent interaction via numerical simulation of the time-dependent Gross-Pitaevskii equation. Both the black and gray ring dark solitons are considered for cases where the modulation frequency of the intercomponent interaction is resonant or nonresonant with the one of the trapping potential. Our results show that in the presence of periodic modulation of the intercomponent interaction not only are the lifetimes of the ring dark solitons largely extended but also their decaying dynamics are dramatically affected. Before snaking instability sets in, new ring dark solitons are formed, and both the numbers and depths of the ring dark solitons exhibit collective oscillations. With the development of instability, the system exhibits different decaying processes, and a variety of decay profiles, such as vortex necklace, distorted octagon, vortex-antivortex ring, and cross, are formed, showing a strong dependence on the modulation frequency of the intercomponent interaction and the initial depth of the ring dark soliton.
In this work we present a systematic study of the three-dimensional extension of the ring dark soliton, examining its existence, stability, and dynamics in isotropic harmonically trapped Bose-Einstein condensates. Detuning the chemical potential from the linear limit, the ring dark soliton becomes unstable immediately but can be fully stabilized by an external cylindrical potential. The ring has a large number of unstable modes which are analyzed through spectral stability analysis. Furthermore, a few typical destabilization dynamical scenarios are revealed with a number of interesting vortical structures emerging, such as the two or four coaxial parallel vortex rings. In the process of considering the stability of the structure, we also develop a modified version of the degenerate perturbation theory method for characterizing the spectra of the coherent structure. This semianalytical method can be reliably applied to any soliton with a linear limit to explore its spectral properties near this limit. The good agreement of the resulting spectrum is illustrated via a comparison with the full numerical Bogolyubov--de Gennes spectrum. The application of the method to the two-component ring dark-bright soliton is also discussed.