The C 0 J →K (t, s) term (for {|J , K} ∈ {|±⟩, |D⟩}) in Eq. 23 corresponds to the contribution from symmetric bath modes (ν † a,0 and νa,0 ) in the form of diagonal coupling (see the second line of Eq. A2, shifting by different amounts between the polariton states and the dark states (see the second line of Eq. A2). Here, we use C 0 to denote that this contribution originates from the symmetrical phonon ν † a,0 and νa,0 coupling to the polariton and dark states. It is evaluated via the Gaussian integral technique, (for example, Appendix B in Ref. 57), with the details of derivation provided in Sec. IIB of the supplementary material. The final expression of C 0 J →K (t, s) is