Recently an experiment on superradiant transition of a Bose condensate in two crossed beam cavities has been reported by Léonard et al. in Nature 543, 87 (2017). The surprise is they find that across the superradiant transition, the cavity light can be emitted in any superposition of these two cavity modes. This indicates an additional $U(1)$ symmetry that does not exist in the full Hamiltonian. In this paper we show that this symmetry is an emergent symmetry in the vicinity of the phase transition. We identify all the necessary conditions that are required for this emergent $U(1)$ symmetry and show that this experiment is a special case that satisfies these conditions. We further show that the superradiant transition in this system can also be driven to a first order one when the system is tuned away from the point having the emergent symmetry.

最近，Léonard 等人报道了两个交叉束腔中玻色凝聚体超辐射跃迁的实验。在 Nature 543, 87 (2017) 中。令人惊讶的是，他们发现在超辐射跃迁中，腔光可以以这两种腔模式的任何叠加形式发射。这表明额外的$ü(\mathrm{1个})$完全哈密顿量中不存在的对称性。在本文中，我们表明这种对称性是相变附近出现的对称性。我们确定了这一紧急情况所需的所有必要条件$ü(\mathrm{1个})$对称性并表明该实验是满足这些条件的特例。我们进一步表明，当系统调离具有紧急对称性的点时，该系统中的超辐射跃迁也可以被驱动到一阶跃迁。