Gap solitons of the Wannier and Bloch types in spin-orbit-coupled Bose-Einstein condensates with a moiré lattice

TL;DR

Gap solitons in spin-orbit-coupled Bose-Einstein condensates loaded into moiré lattices are analyzed, focusing on WT GSs bifurcating from flat bands and BT GSs from non-flat bands. The authors derive and solve a three-component Gross-Pitaevskii equation with Rashba SOC strength $\\gamma$ and a moiré potential $V(x,y)$, identify five WT GS families bifurcating from the lowest five bands, and show SOC and lattice parameters control band flatness via $\\Delta_n=\\max(\\mu_n)-\\min(\\mu_n)$ to enable reversible WT$\\leftrightarrow$BT transitions. They perform linear stability analysis via Bogoliubov–de Gennes equations, revealing WT GSs stable at small norm $N$ and BT GSs unstable near band edges; SOC-induced band flattening drives WT-BT transitions, and weakly localized WT GSs emerge even for small lattice depth. They also extend the analysis to quasiperiodic moiré lattices, where GSs persist with localized profiles, indicating a robust mechanism for strongly localized gap solitons in complex SOC-BEC lattices.

Abstract

Gap solitons (GSs) bifurcating from flat bands, which may be represented in terms of Wannier functions, have garnered significant interest due to their strong localization with extremely small norms. Moiré lattices (MLs), with multiple flat bands, offer an appropriate platform for creating such solitons. We explore the formation mechanism and stability of GSs in spin-1 Bose-Einstein condensates under the combined action of the Rashba spin-orbit coupling (SOC) and an ML potential. We identify five Wannier-type GS families bifurcating from the lowest five energy bands in the spectrum induced by the ML with sufficiently large period and depth. These fundamental GSs serve as basic elements for constructing more complex Wannier-type GS states. Reducing the lattice period and depth triggers a transition from the Wannier-type GSs to ones of the Bloch type, the latter exhibiting higher norm thresholds and pronounced spatial broadening near edges of the energy bands. In addition to tuning the lattice-potential parameters, adjusting the SOC strength can also modulate the flatness of energy bands and enhance the localization of gap solitons, enabling reversible transitions between the GSs of the Wannier and Bloch types. Distinctive properties of GSs in the quasiperiodic ML are uncovered too. Thus, we propose the theoretical foundation for the creation of and manipulations with strongly localized GSs.

Figures (8)

• Figure 1: The ML profiles for twisted angles (a) $\theta=\arctan(3/4)\approx 36.9^{\circ}$ and (b) $\theta=\arctan(5/12)\approx 22.6^{\circ}$. The black arrows represent the lattice vectors. The bandgap structures along the high-symmetry lines for the MLs with (c) $\theta=\arctan(3/4)$ and (d) $\theta=\arctan(5/12)$ when $a=0.5\pi$ and $\gamma=0.5$. The box and red arrows in (c) represent the first Brillouin zone and the high-symmetry lines. (e)-(f) The effect of $\gamma$ and $a$ on the ML bandgap structure with $\theta=\arctan(3/4)$. These blue (white) areas represent energy bands (gaps). Here, $V_0=4$.
• Figure 2: WT (Wannier-type) GS families bifurcating from the lowest five flat bands for the ML with $\theta=\arctan(3/4)$, $a=2\pi$, and $V_0=4$. Other parameters are $\gamma=0.5$, $c_0=-1$ and $c_2=1.5$ (the antiferromagnetic system). (a)-(e) The density profiles of three components $\varphi_{\pm1}$ (the first row) and $\varphi_0$ (the second row) for five GS families. The corresponding phases are shown in the corresponding subfigures, where the phase of $\varphi_{+1}$ is in the left subfigure. The drawing grid is $(x,y)\in [-3,3]$. (f) Norm $N$ versus the chemical potential $\mu$ for the five GS families. The solid (dashed) parts of these curves indicate that the corresponding GSs are stable (unstable). The black lines represent five flat bands with $\mu_1=1.6998$, $\mu_2=1.8065$, $\mu_3=3.5533$, $\mu_4=3.5808$, and $\mu_5=3.7703$. Here, the inset shows the norm curve of the GS family bifurcating from $\mu_1$, plotted near the band edge.
• Figure 3: WT (Wannier-type) GS families bifurcating from the lowest two flat bands for the ML with $\theta=\arctan(3/4)$, $a=2\pi$, and $V_0=4$. Other parameters are $\gamma=0.5$, $c_0=1$ and $c_2=-1.5$ (the ferromagnetic system). (a)-(e) The density profiles of three components $\varphi_{+1}$ (the first row), $\varphi_{-1}$ (the second row), and $\varphi_{0}$ (the third row) for five GS families. The corresponding phases are shown in the subfigures. The first and second rows in (a)-(c) share the same color bars. $|\varphi_{-1}|$ in (d) and (e) are equal to $|\varphi_{+1}|$ in (e) and (d), respectively. The drawing grid is $(x,y)\in [-3,3]$. (f) Norm $N$ versus the chemical potential $\mu$ for the five GS families. The norm curves for (d) and (e) are identical. The solid (dashed) parts of these curves indicate that the corresponding GSs are stable (unstable). The black lines represent two flat bands with $\mu_1=1.6998$ and $\mu_2=1.8065$.
• Figure 4: BT (Bloch-type) GS families bifurcating from the lowest two energy bands when $\theta=\arctan(3/4)$, $a=0.5\pi$, $V_0=4$, $\gamma=0.5$, $c_0=1$, and $c_2=-1.5$. (a)-(e) The density profiles of three components $\varphi_{+1}$ (the first row), $\varphi_{-1}$ (the second row), and $\varphi_{0}$ (the third row) for five GS families. The corresponding phases are shown in the subfigures. The first and second rows in (a)-(c) share the same color bars. $|\varphi_{-1}|$ in (d) and (e) are equal to $|\varphi_{+1}|$ in (e) and (d), respectively. Here, the drawing grid is $(x,y)\in [-4,4]$. (f) The norm $N$ versus the chemical potential $\mu$ for the five GS families. The norm curves for (d) and (e) are identical. The solid (dashed) parts of these curves indicate that the corresponding GSs are stable (unstable). The blue area represents the energy band, and the inset shows the magnified view of the region enclosed by the black dashed lines.
• Figure 5: Panels (a)-(c) show the evolution of the maximum densities for the three components as revealed by the numerical simulations of the GSs in Figs. \ref{['4fig:2']}(a)-\ref{['4fig:4']}(a), respectively. The corresponding spatial density profiles of all the three components at t = 100 are displayed in panels (d)-(f).