To obtain a matrix representation of ĤQ and μ, we construct a diabatic basis by dressing the exciton states with photonic Fock states within a double excitation subspace as described in Section. III.A of Paper-I (Ref. 1). For a system of N two-level excitons coupled to a single cavity mode, the total number of quantum states within the double excitation subspace is K = 2N + 3 + N (N -1)/2, which scales as O(N 2 ).