Atoms coupled to optical cavities provide a novel platform for understanding high-orbital exotic phenomena in strongly correlated materials. In this study, we investigate strongly correlated ultracold bosonic gases that are coupled to two orthogonally arranged optical cavities and driven by a blue-detuned running-wave laser field. In contrast, with the single-cavity configuration, which couples atoms to $d$ and $p$ orbitals, our results demonstrate that atoms initially in the $s$-orbital state can be scattered into both $p_x$- and $p_y$-orbital states. This process can occur in either a symmetric or asymmetric manner, depending on the frequencies of the two cavities. For the symmetric configuration, we observe that atoms are scattered into the $p_x$ and $p_y$ orbitals equally. In the asymmetric case, photons emitted into one cavity mode suppress the scattering into the orthogonal mode. Notably, the coupling of atoms with multiple cavity modes leads to the emergence of $p$-orbital self-organized phases, accompanied by the stripe and checkerboard orbital-density waves that break different symmetries.

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• Cold gases in optical lattices
• Orbital order
• Phase diagrams
• Bose-Einstein condensates
• Ultracold gases
• Atomic Properties
• Dynamical mean field theory