Energy mechanism of the first-order superradiant phase transition in cavity-BEC system with double asymmetric pump beams

We consider a Bose–Einstein condensate loaded inside an optical cavity and exposed to two crossed coherent pump fields with same imbalance parameter γ. We identify different effects between pure standing wave fields (γ = 1) and the pump beams combining standing wave and running wave (γ ≠ 1). In particular, for γ = 1, the system only hosts a normal phase and a superradiant phase. In contrast, for γ ≠ 1, the system features three distinctive phases: the normal phase (NP), superradiant phase 1 (SR₁), and superradiant phase 2 (SR₂). Importantly, the superradiance is subdivided into different types characterized by the photon phase. Furthermore, we determine perturbatively the phase boundary separating the normal phase and the superradiant phases, and find that there exists a competitive relationship of energy minimum on the overlapping region between SR₁ and SR₂. Interestingly, the transition between the normal phase to SR₁ or SR₂ is identified to be a second-order phase transition, while the transition between SR₁ and SR₂ is a first-order transition. When the first-order phase transition occurs, the phase of the photons changes abruptly from 0 to π/2.