Mobility edges and reentrant localization induced by superradiance

Abstract

We study a Bose-Einstein condensate trapped by a ladder lattice in a high-fitness cavity. The ladder lattice is loaded in the <inline-formula><tex-math id="M1">\begin{document}$x\text-y$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M1.png"/></alternatives></inline-formula> plane and the cavity is along the <i>x</i> direction. A pump laser shines on atoms from the <i>z</i> direction. Under the mean-field approximation, we consider the emergence of the quasi-periodic potentials induced by superradiance in the ladder lattice, which is described by <inline-formula><tex-math id="M2">\begin{document}$\hat{H}_{\text{MF}}=\hat{H}_{\text{Lad}}+\hat{V}_{\text{eff}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M2.png"/></alternatives></inline-formula> with the effective potential <inline-formula><tex-math id="M3">\begin{document}$\hat{V}_{\text{eff}}(\alpha)={\displaystyle \sum\nolimits_{i = 1}^{N}}\displaystyle \sum\nolimits_{\sigma = 1,2}\left[\lambda_{\rm{D}}\cos({2\pi\beta i})+U_{\rm{D}}\cos^{2}({2\pi\beta i})\right]\hat{c}^{†}_{i,\sigma}\hat{c}_{i,\sigma}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M3.png"/></alternatives></inline-formula>. We find that the quasi-periodic potential can induce the reentrant localization transition and the regime with mobility edges. In the smaller <inline-formula><tex-math id="M4">\begin{document}$U_{\rm{D}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M4.png"/></alternatives></inline-formula> case, the system exhibits a localization transition. The transition is associated with an intermediate regime with mobility edges. When <inline-formula><tex-math id="M5">\begin{document}$U_{\rm{D}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M5.png"/></alternatives></inline-formula> goes beyond a critical value <inline-formula><tex-math id="M6">\begin{document}$U_{\rm{D}}^{(\rm c)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M6.png"/></alternatives></inline-formula>, with the increase of <inline-formula><tex-math id="M7">\begin{document}$\lambda_{\rm{D}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M7.png"/></alternatives></inline-formula>, the system undergoes a reentrant localization transition. This indicates that after the first transition, some of the localized eigenstates change back to the extended ones for a range of <inline-formula><tex-math id="M8">\begin{document}$\lambda_{\rm{D}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M8.png"/></alternatives></inline-formula>. For a larger <inline-formula><tex-math id="M9">\begin{document}$\lambda_{\rm{D}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M9.png"/></alternatives></inline-formula>, the system experiences the second localization transition, then all states become localized again. Finally, the local phase diagram of the system is also discussed. This work builds a bridge between the reentrant localization and the superradiance, and it provides a new perspective for the reentrant localization.