Collectively induced many-vortices topology via rotatory Dicke quantum phase transition

We examine the superradiance of a Bose-Einstein condensate pumped with a Laguerre-Gaussian laser of high winding number, e.g., $\ell = 7$. The laser beam transfers its orbital angular momentum (OAM) to the condensate at once due to the collectivity of the superradiance. An $\ell$-fold rotational symmetric structure emerges with the take place of rotatory superradiance. $\ell$ number of single-charge vortices appear at the arms of this structure. Even though the pump and the condensate profiles initially have cylindrical symmetry, we observe that it is broken to $\ell$-fold rotational symmetry during the superradiance. Breaking of the cylindrical symmetry into the $\ell$-fold symmetry and OAM transfer to the BEC become possible after the same critical pump strength. Reorganization of the condensate resembles the ordering in the experiment by Esslinger and colleagues [Nature, {\bf 264}, 1301 (2010)]. We show that the critical point for the onset of the reorganization follows the form of the Dicke quantum phase transition.