Parametric resonance of a Bose-Einstein condensate in a ring trap with periodically driven interactions

We study the instability of a ring Bose-Einstein condensate under a periodic modulation of interatomic interactions. The condensate exhibits temporal and spatial patterns induced by the parametric resonance, which can be characterized by Bogoliubov quasiparticle excitations in the Floquet basis. As the ring geometry significantly limits the number of excitable Bogoliubov modes, we are able to capture the nonlinear dynamics of the condensate using a three-mode model. We further demonstrate the robustness of the temporal and spatial patterns against disorder, which is attributed to the mode-locking mechanism under the ring geometry. Our results can be observed in cold atomic systems and are also relevant to physical systems described by the nonlinear Schrödinger equation.

Figure 1

Illustration of the ring-trap geometry. See text for detailed discussions.

Figure 2

(a) Response of the condensate density ${\left|\psi (0,t)\right|}^2$ in the resonance regime under a driving frequency $\omega =17.59$. (b) Response of the condensate density in the nonresonant regime under a driving frequency $\omega =17.69$. (c) The spatial-temporal patterns in the resonance regime with $\omega =17.59$. The spatial pattern here has a period of $0.04\pi$, which corresponds to a wave vector $k=50$. (d) Fourier spectrum of the condensate density $I\left(\nu \right)=\int {\left|\psi (0,t)\right|}^2{e}^{-i\nu t}dt$ under a driving frequency $\omega =2{\epsilon }_k$. In all panels, $\alpha =0.1$, $\lambda =2\pi /2000$.

Figure 3

Schematics on the scattering processes in a parametric resonance. Solid curves represent Bologliubov spectra in different Floquet bands with indices $m=0$ (blue) and $m=-1,-2$ (red). While circles on the spectra represent available states under the ring geometry, the filled and open circles correspond, respectively, to the occupied and empty states in the scattering process. The black arrows indicate the dominant scattering process in a parametric resonance. The blue dashed arrows indicate higher-order scattering processes, which only occur when $\alpha$ becomes appreciable. The red arrows indicate higher-order scattering processes which are forbidden by the ring geometry.

Figure 4

Comparison of dynamics under the three-mode approximation (top row) and under the full GP evolution (bottom row). (a) and (b) Phase-plane contours of the dynamics under the three-mode approximation, with $\alpha =0.1$ and driving frequencies (a) $\omega =17.59$ and (b) $\omega =17.69$. In both cases, the resonant wave vector is $k=50$. (c) and (d) Phase evolution from the full GP equation (2), with initial occupations ${\rho }_k=1\times {10}^{-4}$ (blue), ${\rho }_k=0.027$ (red), ${\rho }_k=0.028$ (orange), and ${\rho }_k=0.040$ (purple) under driving frequencies (c) $\omega =17.59$ and (d) $\omega =17.69$.

Figure 5

(a) and (b) Comparison of ${\rho }_k$ between results from the GP equation (2) (blue) and under the three-mode approximation (red), with the parameters (a) $\alpha =0.1$ and (b) $\alpha =0.5$. (c) Population of modes other than $p=0,\pm 50$ as a function of $\alpha$, calculated using the GP equation. Here $\rho =1-{\rho }_0-2{\rho }_k$. (d) Fourier spectrum of the condensate density $I\left(\nu \right)$ with $\alpha =0.1$ (blue) and $\alpha =0.5$ (red). Here, $\omega =17.59$.

Figure 6

Fourier spectra of the condensate density $I\left(\nu \right)$: (a) $V_d=0$ with $q=0$ (blue solid) and $q=0.2$ (red dashed); (b) $q=0$ with $V_d=0$ (blue solid) and $V_d=0.05$ (red dashed). The results are similar for $I\left(\nu \right)$ near $\nu =\omega /2$. Insets show zoomed-up figures near $\nu =-\omega /2$.