Controlling chaos of a Bose-Einstein condensate loaded into a moving optical Fourier-synthesized lattice

We study the chaotic properties of steady-state traveling-wave solutions of the particle number density of a Bose-Einstein condensate with an attractive interatomic interaction loaded into a traveling optical lattice of variable shape. We demonstrate theoretically and numerically that chaotic traveling steady states can be reliably suppressed by small changes of the traveling optical lattice shape while keeping the remaining parameters constant. We find that the regularization route as the optical lattice shape is continuously varied is fairly rich, including crisis phenomena and period-doubling bifurcations. The conditions for a possible experimental realization of the control method are discussed.

Figure 1

Potential function $p(x;m)$ [cf. Eq. (5)] for $m=0$ (thick line), 0.995 (medium line), and $1-{10}^{-14}$ (thin line). The quantities plotted are dimensionless.

Figure 2

(Color online) Top panel: Relative error $[p(x;m)-p^{\left(2\right)}(x;m)]∕p(x;m)$ with $p$ as in Eq. (5) and $p^{\left(2\right)}$ corresponding to its two-term Fourier expansion as in Eq. (8). Shape parameter in the range $m∊[0,0.9]$. Bottom panel: Functions $p(x;m=0.9)$ (thick line) and $p^{\left(2\right)}(x;m=0.9)$ (thin line), showing their proximity. The quantities plotted are dimensionless.

Figure 3

(Color online) Top panel: Relative error $[p(x;m)-p^{\left(3\right)}(x;m)]∕p(x;m)$, with $p$ as in Eq. (5) and $p^{\left(3\right)}$ corresponding to its three-term Fourier expansion as in Eq. (8). Shape parameter in the range $m∊[0,0.99]$. Bottom panel: Functions $p(\xi ;m=0.99)$ (thick line) and $p^{\left(3\right)}(\xi ;m=0.99)$, (thin line) showing their proximity. The quantities plotted are dimensionless.

Figure 4

(Color online) Top panel: Relative error $[U(m,v)-\tilde{U}(m,v)]∕U(m,v)$ [cf. Eqs. (12, 13)] for the parameters in the ranges $m∊[0,0.99]$ and $v∊[1,2.5]$. Bottom panel: Functions $U(m,v=2)$ (thick line) and $\tilde{U}(m,v=2)$ (thin line) in the range $m∊[0.9,0.99]$. The quantities plotted are dimensionless.

Figure 5

(Color online) Plots of the chaotic threshold function $\tilde{U}(m,v)$ [see Eqs. (13, 14)]. Top panel: $\tilde{U}$ vs $m$ and $v$. Middle panel: $\tilde{U}$ vs $m$ for $v=1.5$ (thick line), 1.9 (medium line), and 2.03 (thin line). Bottom panel: $\tilde{U}$ vs $v$ for $m=0$ (thick line), 0.8 (medium line), and 0.9 (thin line). The quantities plotted are dimensionless.

Figure 6

Bifurcation diagrams of particle number density $R^2$ as a function of the shape parameter $m$ for parameter values $\gamma =0.05$, ${\tilde{V}}_0=2$, $v=2.03$, and $g=-0.75$. Bottom panel shows the detail corresponding to the range where crises appear. The quantities plotted are dimensionless.