Fig. presents the behavior of the accumulated spectral function 𝒜(𝜔), which is calculated by evaluating Eq. 29 numerically using trapezoidal integration within the region of 𝜔 c ≤ 𝜔 k ≤ 5𝜔 c using 4 × 10 6 grid points, where numerical convergence is carefully checked. One can see that 𝒜(𝜔) peaks at 𝜔 > 𝜔 c when 𝒟/𝑐 is large, and gradually moves to 𝜔 = 𝜔 c when 𝒟/𝑐 decreases and Γ ′ 10 dominates the cavity loss for 𝑘 ‖ > 0.