Note that [ d a ] is the same as that of Eq. 4 but in the full electron-photon Hilbert space as d ⊗ Îph . As a concrete example, for a polaritonic system composed of two electronic adiabatic states { |g⟩, |e⟩ } and two photonic Fock states { |0⟩, |1⟩ }, the 4 × 4 gradient matrices can be written explicitly, with the a [ Ĥpl ] expressed as,