In the population transfer correlation function C ±→∓ (t, s) from one polariton state to the other, the symmetric bath mode k = 0 contributes to both diagonal and off-diagonal coupling. As a result, it is no longer possible to separate the two contributions cleanly like the previously discussed cases, leading to a slightly more complex evaluation via the Gaussian integral approach. The details are provided in supplementary material Sec. IIC. The final result of C ±→∓ (t, s) is expressed as