Hidden continuous symmetry and Nambu-Goldstone mode in a two-mode Dicke model
In this paper we investigate the ground- and excited-state properties of a two-mode Dicke model, which is realized in an ensemble of many four-level atoms interacting simultaneously with two quantized cavity fields and two pairs of Raman lasers. We reveal rich phase diagrams, including a first-order quantum phase transition from an electric superradiant phase to a magnetic superradiant phase. In particular, when the two collective atom-photon coupling strengths are equal, we find a hidden continuous $\text{U}(1)$ symmetry for the two-mode Dicke model. If these identical coupling strengths are sufficiently strong, a quantum phase called the electromagnetic superradiant phase is predicted. In such a phase, the predicted continuous $\text{U}(1)$ symmetry breaks down and the gapless excitation spectrum (the Nambu-Goldstone mode), which could be measured experimentally by combining Bragg spectroscopy and cavity-enhanced Bragg scattering, is thus achieved. Our results pave the way to explore the physics for a model with continuous symmetry in quantum optics without using the rotating-wave approximation.