where J is the index of the two-level atoms/molecules in the cavity (and there are a total of N of them effectively coupled to the cavity), with corresponding exciton creation operator, σ † J = |e J ⟩⟨g J |, and annihilation operator, σJ = |g J ⟩⟨e J |. Further, due to the model's symmetry, one can introduce the collective excitation operator σ † N = 1 √ N J |e J ⟩⟨g J | and collective de-excitation operator σN = 1 √ N J |g J ⟩⟨e J |. Simi-lar to the JC model, the TC model also has analytical solutions to its eigenstates and eigenenergies in the first excitation subspace. The total ground state is |G, 0⟩ = |g 1 ⟩ ⊗ ...|g J ⟩... ⊗ |g N ⟩ ⊗ |0⟩, the photon dressed ground state is |G, 1⟩, where all the emitters are in the ground state with one photon in the cavity, and the state where the all the molecules are in the ground state except for the J th molecule in the excited state is |E J , 0⟩ = |g 1 ⟩ ⊗ ...|e J ⟩... ⊗ |g N ⟩ ⊗ |0⟩. In the single excitation manifold, the collective "bright state" of the matter is