Bifurcation trees of Stark-Wannier ladders for accelerated Bose-Einstein condensates in an optical lattice

In this paper we show that in the semiclassical regime of periodic potential large enough, the Stark-Wannier ladders become a dense energy spectrum because of a cascade of bifurcations while increasing the ratio between the effective nonlinearity strength and the tilt of the external field; this fact is associated to a transition from regular to quantum chaotic dynamics. The sequence of bifurcation points is explicitly given.

Figure 1

Here we plot the values of the energy $\mu /f$ associated to stationary solution sets $S$ such that $minS=0$; we can see a cascade of bifurcations when $\nu /f$ increases. This picture occurs for each rung of the ladder.

Figure 2

Here we plot the absolute value of the stationary solutions associated to the solution sets (a) $S_1$, (b) $S_2$, and (c) $S_3$; $x_j$ denotes the center of the $j\mathrm{th}$ well of the periodic potential.