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arXiv:1706.04348 · Journal of Experimental and Theoretical Physics Letters

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

V. P. Ruban

Published Aug 1, 2017• 8 citations
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Abstract

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.

DOI
10.1134/s0021364017160123

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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.
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.
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.
The dynamics of a quantum vortex toric knot TP,Q and other analogous knots in an atomic Bose condensate at zero temperature in the Thomas–Fermi regime is considered in the hydrodynamic approximation. The condensate has a spatially inhomogeneous equilibrium density profile ρ(z, r) due to the action of an external axisymmetric potential. It is assumed that z*= 0, r*= 1 is the point of maximum of function rρ(z, r), so that δ(rρ) ≈ –(α–)z2/2–(α + )(δr)2/2 for small z and δr. The geometrical configuration of a knot in the cylindrical coordinates is determined by a complex 2πP-periodic function A(ϕ, t) = Z(ϕ, t) + i[R(ϕ, t))–1]. When |A| ≪ 1, the system can be described by relatively simple approximate equations for P rescaled functions Wn(φ)∝A(2πn+φ):iWn,t=−(Wn,φφ+αWn−∈Wn∗)/2−∑j≠n1/(Wn∗−Wj∗){W_n}(\varphi ) \propto A(2\pi n + \varphi ):i{W_{n,t}} = - ({W_{n,\varphi \varphi }} + \alpha {W_n} - \in W_n^*)/2 - \sum\nolimits_{j \ne n} {1/(W_n^* - W_j^*)} Wn​(φ)∝A(2πn+φ):iWn,t​=−(Wn,φφ​+αWn​−∈Wn∗​)/2−∑j=n​1/(Wn∗​−Wj∗​) . For = 0, examples of stable solutions of type W n = θ n (ϕ–γt)exp(–iωt) with a nontrivial topology are found numerically for P = 3. In addition, the dynamics of various unsteady knots with P = 3 is modeled, and the tendency to the formation of a singularity over a finite time interval is observed in some cases. For P = 2 and small ≠ 0, configurations of type W0–W1 ≈ B0exp(iζ) + C(B0, α)exp(–iζ) + D(B0, α)exp(3iζ), where B0 > 0 is an arbitrary constant, ζ = k0ϕ–Ω0t + ζ0, k0 = Q/2, and Ω0 = (–α)/2–2/B02, which rotate about the z axis, are investigated. Wide stability regions for such solutions are detected in the space of parameters (α, B0). In unstable zones, a vortex knot may return to a weakly excited state.
We experimentally and numerically study the collapse dynamics of a quantum vortex in a two-dimensional atomic superfluid following a fast interaction ramp from repulsion to attraction. We find the conditions and timescales for a superfluid vortex to radially converge into a quasistationary density profile, demonstrating the spontaneous formation of a vortex solitonlike structure in an atomic Bose gas. We record an emergent self-similar dynamics caused by an azimuthal modulational instability, which amplifies initial density perturbations and leads to the eventual splitting of a solitonic ring profile or direct fragmentation of a superfluid into disordered, but roughly circular arrays of Townes solitonlike wave packets. These dynamics are qualitatively reproduced by simulations based on the Gross-Pitaevskii equation. However, a discrepancy in the magnitude of amplified density fluctuations predicted by our mean-field analysis suggests the presence of effects beyond the mean-field approximation. Our study sets the stage for exploring out-of-equilibrium dynamics of vortex quantum matter quenched to attractive interactions and their universal characteristics.
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.
We study the nonlinear dynamics of a three-dimensional Bose-Einstein condensate (BEC) excited by a vortex ring phase imprinting. We identify independent, integrated, and stationary modes of the center-of-mass oscillation of the condensate with respect to the vortex ring movement. We show that the oscillation amplitude of the center-of-mass of the condensate depends strongly on the initial radius of the vortex ring, the nonlinear inter-atomic interaction, and the aspect ratio of the trap, while the oscillation frequency is fixed and equals to the frequency of the harmonic trap in the direction of the ring movement. However, when applying Kelvin wave perturbations on the vortex ring, the center-of-mass oscillation of the BEC is changed non-trivially with respect to the perturbation modes, the long-scale perturbation strength as well as the wave number of the perturbations. The parity of the wave number of the Kelvin perturbations plays an important role on the mode of the center-of-mass oscillation of the condensate.