Quantum phases in a Bose–Einstein condensate loaded in two crossed high-finesse optical cavities with orthogonal polarizations

We consider a Bose–Einstein condensate (BEC) coupled to two crossed optical cavities with orthogonal polarization modes via the scalar and vectorial polarizabilities of the atoms. We determine the ground-state phase diagram and investigate the behaviors of superradiant order parameters and the photon numbers in both cavities under various conditions. With the set parameters, we find that the system accommodates three distinct phases: a normal phase with no photons in both cavities ($\mathrm{NP}$), a superradiant phase with photons only in cavity 1 (${\mathrm{SR}}_1$) and a superradiant phase with photons in both cavities (${\mathrm{SR}}_{12}$). We explore the effects of the coupling angle on the phase boundaries under different magnitudes of cavity detuning in the two cavities. We find that the coupling angle strongly modifies the boundary between ${\mathrm{SR}}_1$ and ${\mathrm{SR}}_{12}$. In particular, when the coupling angle is equal to $90°$, the phase of ${\mathrm{SR}}_{12}$ is completely suppressed.