Matter waves in anharmonic periodic potentials

We study anharmonic (optical or magnetic) periodic potentials and demonstrate that they may exhibit unusual features in the band-gap spectra and the nature of the matter-wave Bloch waves and solitons. We reveal that the band gaps may be strongly modified and even close, and show that changing the potential shape may change the symmetry of the gap-edge Bloch waves and affect the symmetry of higher-order gap solitons. We introduce a mixed-gap matter-wave interface soliton which belongs simultaneously to two different spectral gaps and describe interface localized states in chirped periodic potentials. Finally, we generalize our results to two-dimensional periodic potentials and examine the band-gap structure, matter-wave gap solitons, and lattice vortex states.

Figure 1

Examples of the 1D potential (5) for (a) $s=-0.8$, (b) $s=0$, and (c) $s=+0.8$ for $V_0=1$.

Figure 2

Examples of the 2D potential (6) for (a) $s=-0.8$, (b) $s=0$, and (c) $s=+0.8$ for $V_0=1$.

Figure 3

Appearance of gaps and bands in $\mu$ as $s$ is varied for $V_0=1$. Dashed lines correspond to $s$ values of potentials shown in Fig. 1. The horizontal dotted line is at constant $\mu$, highlighting that for different values of $s$ this value of $\mu$ sits in different gaps (see text).

Figure 4

(Color online) Variation of overlap integral (7) with $s$ for different symmetry Bloch waves at $V_0=1$. Numbers correspond to the particular Bloch waves shown in Fig. 5.

Figure 5

(Color online) (a) and (b) Band diagram at $s=0$ and $s=0.8$, respectively, for $V_0=1$. (c) and (d) First- and second-gap-edge Bloch waves. The dotted line corresponds to the potential. Numbers indicate the band edge in (a) and (b) to which the particular Bloch wave corresponds. The symmetry of the bottom and top second-gap-edge Bloch waves is reversed between the panels.

Figure 6

(Color online) Variation of gap widths with $s$ for first gap (solid line), second gap (dashed line), and third gap (dotted line) for $V_0=1$ and $V_0=4$.

Figure 7

(a) Change in width of first gap as $s$ and $V_0$ are varied. Contour intervals are in steps of 0.4, and the solid line indicates the maximum width for given $s$ and $V_0$. (b) Similar analysis for the second gap; however, now two gaps exist for given $V_0$. The dashed line shows the minimum. Contour intervals are in steps of 0.2. The dotted line shows the calculation of gap closure from the overlap, Eq. (7).

Figure 8

Low-density second-gap-edge Bloch-wave evolution in the full nonlinear model (3). (a) and (b) $s=0$ ($\mu =2.33$ and 1.95, respectively) and (c) and (d) $s=0.8$ ($\mu =2.96$ and 2.72, respectively). Top panels: upper gap edge. Bottom panels: lower gap edge.

Figure 9

Gap solitons at (a) and (b) $s=-0.8$ ($\mu =0.5$ and 1.25, respectively); (c) and (d) $s=0$ ($\mu =1.55$ and 2.2, respectively); and (e) and (f) $s=0.8$ ($\mu =2.15$ and 2.86, respectively) for $V_0=1$. Left panels: gap solitons in first gap. Right panels: gap solitons in second gap.

Figure 10

Evolution of gap solitons in second gap. (a) Second-gap soliton shown in Fig. 9f ($s=0.8$, $V_0=1$, $\mu =2.86$). (b), (c), and (d) Second-gap solitons in deeper potential $\left(V_0=4\right)$ for (b) $s=0.8$, $\mu =8.55$, (c) $s=0$, $\mu =4.4$, and (d) $s=-0.8$, $\mu =1.5$.

Figure 11

(a) and (c) Examples of “mixed-gap” solitons existing at interface between $s=-0.7$ and $s=0.7$ potentials (corresponding dotted lines) for $V_0=4$ for different $\mu$. (b) Unstable evolution of stationary state shown in (a). (d) Evolution of peak density of the stationary solutions shown in (a) and (c).

Figure 12

Dependence of the first and second gaps on $s$ for the 2D potential (6) with $V_0=4$.

Figure 13

Band diagram at (a) $s=-0.4$ and (b) $s=0.5$. Shaded regions highlight the first gaps; no higher gaps exist for these $s$ values. The inset shows the 2D potential at $s=-0.4$ and the corresponding Brillouin zone.

Figure 14

(a) Gap width as a function of $V_0$ and $s$. The thick contour shows the boundary of the existence of the first gap; other contour lines are in gap-width increments of 0.8. (b) Variation of gap width versus $s$ for $V_0=4$ [corresponding to the dashed line in (a)] for the first and second gaps.

Figure 15

Gap solitons at (a) $s=-0.4$, $\mu =1.9$ and (b) $s=0.5$, $\mu =6$. In both cases, $V_0=4$.

Figure 16

(Color online) Examples of vortex solitons in the first band gap for (a) $s=-0.4$, $\mu =1.9$; (b) $s=0$, $\mu =3.5$; and (c) $s=0.5$, $\mu =6$. In each case the top panels show the density of the wave function and the bottom panels the corresponding phase. Values are given by the color bars on the right. The thin white line in the lower panels outlines a cut at ${\left|\psi \right|}^2=0.1$. In each case, $V_0=4$.