where J(ω) is defined in Eq. 8, and n(ω) is defined in Eq. 11. The detailed derivations are provided in Sec. II of the Supplementary Material. On the other hand, due to the energy differences, the scattering rate constants for LP → UP (Γ -→+ ) and LP → DS (Γ -→{D k } ) are much smaller in magnitude than Γ +→-and Γ +→{D k } , respectively. As a result, the LP linewidth is barely influenced by inter-branch scatterings.