Towards the complete description of stationary states of a Bose-Einstein condensate in a one-dimensional quasiperiodic lattice: A coding approach

Abstract

We consider stationary states of an effectively one-dimensional Bose-Einstein condensate in a quasiperiodic lattice. We formulate sufficient conditions for a one-to-one correspondence between the stationary states with a fixed chemical potential and the set of bi-infinite sequences over a finite alphabet. These conditions can be checked numerically. A bi-infinite sequence can be interpreted as a code of the corresponding solution. A numerical example demonstrates the coding approach using an alphabet of three symbols.

Figures (13)

• Figure 1: An island $D_\Delta$ bounded by curves $\alpha^{\pm}$, $\beta^{\pm}$; $h$-curve $\alpha$, $v$-curve $\beta$, and two strips: $h$-strip $H$ and $v$-strip $V$.
• Figure 2: A $\gamma$-donut. For each $\Delta$, the intersection of the $\gamma$-donut and the plane $\mathcal{L}_\Delta$ is a $\gamma$-island.
• Figure 3: On the definition of thickness of a strip
• Figure 4: A sketch illustrating Hypothesis \ref{['hypothesis:strips-mapping']}. A $\gamma$-donut set consists of two donuts, ${T}^{1}$ and ${T}^{2}$. The plane $\mathcal{L}_{\Delta_0}$ contains the islands $D_{\Delta_0}^{1}$ and $D_{\Delta_0}^{2}$ from $T^{1}$ and $T^{2}$, respectively. The planes $\mathcal{L}_{\Delta_{\pm 1}}$ contain the corresponding islands at $\Delta_{1} = \Delta_0\oplus 2\theta \pi$ and $\Delta_{-1} = \Delta_0 \ominus 2\theta \pi$. The images ${\cal P}(D_{\Delta_0}^{1})$ and ${\cal P}(D_{\Delta_0}^{2})$ intersect the $\gamma$-islands $D_{\Delta_{1}}^{1,2}$ and each intersection is an $h$-strip. Similarly, the pre-images ${\cal P}^{-1}(D_{\Delta_0}^{1})$ and ${\cal P}^{-1}(D_{\Delta_0}^{2})$ intersect the $\gamma$-islands $D_{\Delta_{-1}}^{1,2}$, and each intersection is a $v$-strip.
• Figure 5: Numerical verification of Hypothesis \ref{['hypothesis:island-set']} for Example \ref{['exa:example-1']}. Each panel shows the sets ${\mathscr U}^+_{\pi k/4}$ (dark gray), ${\mathscr U}^-_{\pi k/4}$ (light gray) and ${\mathscr U}_{\pi k/4}$ (black). For each $\Delta$, there are exactly three $\gamma$-islands and each $\gamma$-island continuously changes with $\Delta$. Each panel shows the region $(u,u')\in [-2;2]\times[-2;2]$.

Theorems & Definitions (45)

• Definition 1
• Definition 2
• Proposition 1
• Definition 3
• Remark
• Definition 4
• Lemma 1
• Proof
• Remark
• Definition 5