Instability-induced localization of matter waves in moving optical lattices

By using a one-dimensional nonpolynomial nonlinear mean-field model, we numerically analyze the process of the loading of the Bose-Einstein condensate into a moving one-dimensional optical lattice. We demonstrate that the recently observed dynamical instability of the Bloch states of a repulsive condensate in a moving optical lattice can lead to formation of trains of spatially localized wave packets that can be associated with matter-wave gap solitons. We study the characteristic features of this matter-wave localization under realistic conditions by modeling the dynamics of the condensate beyond the onset of dynamical instability.

Figure 1

(Color online) Schematics of the location (in momentum space) of the condensate wave packet relative to the lattice band-gap structure. Points (a) and (b) correspond to the modulationally stable and unstable Bloch states, respectively. Below: schematic time diagram of the adiabatic loading and acceleration of the condensate.

Figure 2

(Color online) (a) Dynamical formation of localized structures during the evolution of the homogeneous Bloch state shown in (b). (b) Modulationally unstable homogeneous nonlinear Bloch state (solid filled lines) for $\mu =2.49$ and $V_0=5$ found from Eq. (1) and modulated by the function $\sim \cos (z∕5)$, plotted together with the lattice potential (dash-dotted filled lines) reduced by the factor of 10. (c) The condensate density profile at $t=1200$ (the maximum density in the localized state is reached at $t\approx 1100$). All parameters are in dimensionless units.

Figure 3

(Color online) Dynamical formation of localized atomic wave packets during the evolution of a modulationally unstable inhomogeneous Bloch state ($\mu =1.1$,$V_0=2$). Shown are snapshots of the condensate density profile (a) at the start $V_0=0$,$v=0$ (dash-dotted line) and at the end $V_0=2$,$v=0$ (black solid line) of the ramping-up time $t=t_R=200$ and (b) at the end of acceleration time $t=t_R+t_A=500$ ($V_0=2$,$v=1$). The spatial oscillations in the condensate density cannot be resolved on the scale of the $z$ axis in (a) and (b). (c) Close-up of the central area of the condensate density at $t=600$, i.e., after $t_E=100$ evolution at the band edge ($V_0=2$,$v=1$). All parameters are in dimensionless units.

Figure 4

(Color online) (a) Density profile of a gap soliton of the NPSE for $\mu =1.97$ (solid filled line) and the corresponding lattice potential with $V_0=2$ (dash-dotted filled). The lattice amplitude on the plot is reduced by the factor of $5$. (b) Detail of the density of the train shown in Fig. 3c in the vicinity of a localized structure centered at $z_0=43$. All parameters are in dimensionless units.