where Γ 01 is the rate for the |0 k ⟩ → |1 k ⟩ photonic Fock states transition due to thermal excitation, and Γ 10 = 1/𝜏 c is the cavity loss rate along the 𝑘 ⊥ direction (associated with |1 k ⟩ → |0 k ⟩), which was assumed to be identical for all k modes. To effectively account for the additional photon loss associated with the 𝑘 ‖ direction, we modify the detailed balance relation by replacing the original 𝑒 -𝛽ℏ𝜔 k with 𝒫 eff (𝜔 k ), defined as follows.