where b † ν and bν are the creation and annihilation operators for the ν th phonon mode, respectively, with frequency ω ν , phonon coupling strength g J ν and molecular excitation operator σ † J = |e J ⟩⟨g J |. Both the GTC Hamiltonian and the HTC Hamiltonian have been extensively used in recent theoretical simulations in molecular polariton systems. The details will be discussed in Sec. 6.1.