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.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.

arXiv:1706.04348Journal of Experimental and Theoretical Physics LettersAug 1, 20178 citations

Parametric Excitation of a Quantum Oscillator

В. С. Попов, A. M. Perelomov

Journal of Experimental and Theoretical PhysicsJan 1, 196924 citations

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

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

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.

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