We study a Bose-Einstein condensate trapped by a ladder lattice in a high-fitness cavity. The ladder lattice is loaded in the
$x\text-y$
plane and the cavity is along the
xdirection. A pump laser shines on atoms from the
zdirection. 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
$\hat{H}_{\text{MF}}=\hat{H}_{\text{Lad}}+\hat{V}_{\text{eff}}$
with the effective potential
$\hat{V}_{\text{eff}}(\alpha)={\displaystyle \sum\nolimits_{i = 1}^{N}}\displaystyle \sum\nolimits_{\sigma = 1,2}\left[\lambda_{\rm{D}}\cos({2\pi\beta i})+U_{\rm{D}}\cos^{2}({2\pi\beta i})\right]\hat{c}^{†}_{i,\sigma}\hat{c}_{i,\sigma}$
. We find that the quasi-periodic potential can induce the reentrant localization transition and the regime with mobility edges. In the smaller
$U_{\rm{D}}$
case, the system exhibits a localization transition. The transition is associated with an intermediate regime with mobility edges. When
$U_{\rm{D}}$
goes beyond a critical value
$U_{\rm{D}}^{(\rm c)}$
, with the increase of
$\lambda_{\rm{D}}$
, 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
$\lambda_{\rm{D}}$
. For a larger
$\lambda_{\rm{D}}$
, 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.
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