Emergence of continuous rotational symmetries in ultracold atoms coupled to optical cavities

We investigate the physics of a gas of ultracold atoms coupled to three single-mode optical cavities and transversely pumped with a laser. Recent work has demonstrated that, for two cavities, the $\mathbb{Z}_{2}$ symmetries of each cavity can be combined into a global $U(1)$ symmetry. Here, we show that when adding an extra cavity mode, the low-energy description of this system can additionally exhibit an $SO(3)$ rotational symmetry which can be spontaneously broken. This leads to a superradiant phase transition in all the cavities simultaneously, and the appearance of Goldstone and amplitude modes in the excitation spectrum. We determine the phase diagram of the system, which shows the emergence and breaking of the continuous symmetries and displays first- and second-order phase transitions. We also obtain the excitation spectrum for each phase and discuss the atomic self-organized structures that emerge in the different superradiant phases. We argue that coupling the atoms equally to $n$ different modes will in general generate a global $SO(n)$ symmetry if the mode frequencies can be tuned to the same value.