Quenches in isolated quantum systems are currently a subject of intense study. Here, we consider quantum few-mode systems that are integrable in their classical mean-field limit and become dynamically unstable after a quench of a system parameter. Specifically, we study a Bose-Einstein condensate (BEC) in a double-well potential and an antiferromagnetic spinor BEC constrained to a single spatial mode. We study the time dynamics after the quench within the truncated Wigner approximation (TWA), focus on the role of motion near separatrices, and find that system relaxes to a steady state due to phase-space mixing. Using the action-angle formalism and a pendulum as an illustration, we derive general analytical expressions for the time evolution of expectation values of observables and their long-time limits. We find that the deviation of the long-time expectation value from its classical value scales as $O(1/lnN)$ , where $N$ is the number of atoms in the condensate. Furthermore, the relaxation of an observable to its steady-state value is a damped oscillation. The damping is Gaussian in time with a time scale of $O[{(lnN)}^{2}]$ . We also give the quantitative dependence of the steady-state value and the damping time on the system parameters. Our results are confirmed with numerical TWA simulations.Quenches in isolated quantum systems are currently a subject of intense study. Here, we consider quantum few-mode systems that are integrable in their classical mean-field limit and become dynamically unstable after a quench of a system parameter. Specifically, we study a Bose-Einstein condensate (BEC) in a double-well potential and an antiferromagnetic spinor BEC constrained to a single spatial mode. We study the time dynamics after the quench within the truncated Wigner approximation (TWA), focus on the role of motion near separatrices, and find that system relaxes to a steady state due to phase-space mixing. Using the action-angle formalism and a pendulum as an illustration, we derive general analytical expressions for the time evolution of expectation values of observables and their long-time limits. We find that the deviation of the long-time expectation value from its classical value scales as $O(1/lnN)$, where $N$ is the number of atoms in the condensate. Furthermore, the relaxation of an observable to its steady-state value is a damped oscillation. The damping is Gaussian in time with a time scale of $O[{(lnN)}^{2}]$. We also give the quantitative dependence of the steady-state value and the damping time on the system parameters. Our results are confirmed with numerical TWA simulations.

arXiv:1705.01702Physical Review AJul 5, 201713 citations

Dynamics of Quantum Systems with Interacting Particles

Elisheva Elbaz

Apr 1, 20160 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

Chaos and ergodicity in a partially integrable 3d Bohmian system: a comparison with 2d systems

Athanasios C. Tzemos, G. Contopoulos

Abstract In the present paper we study the Bohmian dynamics of a partially integrable 3d Bohmian system whose trajectories evolve on spherical surfaces. By use of spherical coordinates (R, ϕ, θ) we study the behaviour of unstable fixed points that generate chaos on the (ϕ, θ) plane and discuss the differences between them and those of planar 2d systems. Finally, we show for the first time that the chaotic trajectories of this system are ergodic although the number of its nodes is very small (two). Thus we can observe ergodicity not only in multinodal 2d systems but also in partially integrable systems with few nodes due to their curved geometry.Abstract In the present paper we study the Bohmian dynamics of a partially integrable 3d Bohmian system whose trajectories evolve on spherical surfaces. By use of spherical coordinates (R, ϕ, θ) we study the behaviour of unstable fixed points that generate chaos on the (ϕ, θ) plane and discuss the differences between them and those of planar 2d systems. Finally, we show for the first time that the chaotic trajectories of this system are ergodic although the number of its nodes is very small (two). Thus we can observe ergodicity not only in multinodal 2d systems but also in partially integrable systems with few nodes due to their curved geometry.

Physica ScriptaMay 29, 20230 citations

O(1/lnN)

, where

N

is the number of atoms in the condensate. Furthermore, the relaxation of an observable to its steady-state value is a damped oscillation. The damping is Gaussian in time with a time scale of

O[{(lnN)}^{2}]

. We also give the quantitative dependence of the steady-state value and the damping time on the system parameters. Our results are confirmed with numerical TWA simulations.

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