Transition to instability in a periodically kicked Bose-Einstein condensate on a ring

A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor, where the nonlinearity stems from the mean-field interactions between the condensed atoms. For weak interactions, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. In the stable regime, the system remains predominantly in the two lowest energy states and may be mapped onto a spin model, from which we obtain an analytic expression for the beat frequency and discuss the route to instability. We numerically explore a parameter regime for the occurrence of instability and reveal the characteristic density profile for both condensed and noncondensed atoms. The Arnold diffusion to higher energy levels is found to be responsible for the transition to instability. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.

Figure 1

Plots of average energy $E\left(t\right)$ versus the number of kicks $t$ for three values of $g$. The kick strength $K=0.8$.

Figure 2

Fourier transformation of the energy evolutions in Fig. 1. The unit for the frequency is $1∕T$.

Figure 3

Plots of beat and oscillation frequencies versus the interaction strength (a) and kick strength (b), where the scatters are the results from numerical simulation using the GP equation and lines from analytic expressions Eqs. (15, 16).

Figure 4

Periodic stroboscopic plots of the projection of spin on the $S_z=0$ plane. The thick line and dots correspond to the orbits with initial spin $\vec{S}=(0,0,1)$. $K=0.8$ (a) $g=0.0$; (b) $g=0.1$.

Figure 5

(a) Plots of relative phase versus the number of kicks $t$, where $g=0.1$, $K=0.8$. (b) Schematic plot of the phase shift. $n{T}^{-(+)}$ represents the moment just before (after) the $n\text{th}$ kick.

Figure 6

(a) Plot of population difference versus the number of kicks $t$, where $g=0.1$, $K=0.8$. (b) Details of (a) at the middle of a beat period. The population difference decreases in two consecutive kicks.

Figure 7

Plots of average energy $E\left(t\right)$ versus the number of kicks $t$ for $K=0.1$, $g=0.1$. $T_{AR}$ is determined through Eq. (17).

Figure 8

Plots of population difference with respect to the number of kicks in the presence of decoherence. $K=0.8$, $g=0.2$. Thin lines correspond to $\Gamma =0$. Thick lines correspond to (a) $2\pi \Gamma ∕g=0.001$, (b) $2\pi \Gamma ∕g=1$.

Figure 9

(a) Semilog plot of the mean number of noncondensed atoms versus the number of kicks $t$. The thicker lines are fitting functions. $K=0.8$, $g=0.1$ (dashed line, fitting function $0.0003t^{1.3}$), $g=1.5$ (dotted line, fitting function $0.0011t^2$), $g=2.0$ [dash-dotted line, fitting function $0.32\exp \left(0.1t\right)$]. The inset shows the interaction dependence of the growth rate. The scatters are from numerical simulation and the solid line is the fitting function $0.33{(g-1.96)}^{1∕2}$. (b) Phase diagram of the transition to instability.

Figure 10

Plots of condensate and noncondensate densities, where $K=0.8$. (a), (b) $g=0.1$; (c), (d) $g=2.0$.

Figure 11

Periodic stroboscopic plots of population difference with respect to the relative phase between the first two modes. $K=0.8$. (a)–(d) the two-mode model, where (a) $g=0.1$, circle dots corresponds to $g=0$; (b) $g=1.5$; (c) $g=1.9$; (d) $g=2.0$; The thicker line and larger dots on the phase portraits represent the trajectories with initial conditions $S_z=1$, $\alpha =0$. (e)–(h) the four-mode approximation. The portrait is the projection on the first two modes of the trajectory with initial condition $S_z=1$, $\alpha =0$. (e) $g=0.1$; (f) $g=1.6$; (g) $g=2.2$; (h) $g=2.5$.

Figure 12

Nonlinear effects on dynamically localized states. $K=5$, $T=1$. (a) Plots of average energy $E\left(t\right)$ versus the number of kicks $t$, where the dash-dotted line corresponds to the classical diffusion. $g=0$ (dash), $g=1$ (dot), $g=5$ (solid). (b) Semilog plot of the mean number of noncondensed atoms versus the number of kicks $t$. $g=1$ (dot), $g=5$ (solid).