Long-Living Bloch Oscillations of Matter Waves in Periodic Potentials

The dynamics of matter waves in linear and nonlinear optical lattices subject to a spatially uniform linear force is studied both analytically and numerically. It is shown that by properly designing the spatial dependence of the scattering length it is possible to induce long-living Bloch oscillations of gap-soliton matter waves in optical lattices. This occurs when the effective nonlinearity and the effective mass of the soliton have opposite signs for all values of the crystal momentum in the Brillouin zone. The results apply to all systems modeled by the periodic nonlinear Schrödinger equation, including propagation of light in photonic and photorefractive crystals with tilted band structures.

Figure 1

(a) Curves $G_{0,1}(V)$ separating domains where ${\chi }_{0,1}(V)\gtrless 0$, for $g=-0.777$. Points $A$, $B$, $C$, correspond to the parameters used in Fig. 2b (point $A$), Figs. 2a, 3a, (point $B$), and Fig. 3b (point $C$). Long-living BO exist for parameters inside the shadowed domain. (b) The inverse effective mass $M^{-1}$ (solid line) and the effective nonlinearity $\chi$ (dashed-dotted line) vs $q$, for the point $B$ of panel (a). (c) $M\chi$ vs $q$ for parameters corresponding to point $B$ in panel (a). Points $q_M\approx 0.547$ and $q_{\chi }\approx 0.514$ in panels (b),(c), denote the values of $q$ where $\chi$ and $M^{-1}$ are, respectively, equal to zero.

Figure 2

Long-living (a) and decaying (b) BO for $G=1.0$ [point $B$ in Fig. 1a] and $G=0.95$ [point $A$ in Fig. 1a], respectively. In both cases the initial condition is a stationary gap-soliton located at $X_0=-65\pi$ with $E=-0.938$ near the lower edge. Other parameters are $\gamma =-0.001$ and $V=3.0$. The period and the amplitude of the BO are $T_s=2·{10}^3$ and $X_s\approx 32.5\pi$ (for the ${}^7\mathrm{Li}$ condensate, mentioned in the text, these correspond to 45.2 ms and $32.5\mu \mathrm{m}$). (c) The profile of the gap soliton in panel (a) at the fifth right turning point (thin lines) compared with the stationary state (thick lines) with the same $N=2.420$ ($\approx 3800$ atoms in physical units) and energy $E=-0.732$ close to the upper edge of the band. (d) Energy of the gap soliton vs time for long-living ($G=1.0$, solid line) and decaying ($G=0.95$, dashed line) BO. The lowest allowed band is $E\in [-0.937,-0.733]$.

Figure 3

Same as in Fig. 2 but with the BO started from a gap soliton near the top of the band at $E=-0.732$, with initial position $X_0=0$. Panels (a) and (b) refer, respectively, to points $B$ and $C$, of Fig. 1, while panels (c) and (d) show corresponding quantities as in Fig. 2. In (c) we compare solitons at the fifth left turning point with the stationary state at $E=-0.938$ near the lower edge, while in panel (d) the dashed line refers to $G=1.05$.