Note here that since the polaritonic Hamiltonian, Eq. 15, is real-valued, then the expansion coefficients in the adiabatic-Fock or PFS basis will also be real-valued, i.e., (C k β m ) * = C k β m . In general, any polaritonic observable, even restricting to ground-to-excited elements, are linear combinations of all matrix elements present in the electronic and photonic subs-systems. As such, highly non-trivial mixing of matrix elements can appear when the light-matter coupling becomes large. Here Ô = Âel ⊗ Bph are factorized electronic and photonic operators, respectively, which is not a requirement of Ô but is often the case for simple, non-entangled observables, such as the average photon number Ô = Îel ⊗ â † â. Another example is linear spectroscopy where the polaritonic dipole matrix elements can be expressed as μpol ∼ μel ⊗ Îph + Îel ⊗ qc . In experiment, the relative magnitudes of the excitonic and photonic contributions are highly dependent on the experimental configuration and are usually understood to be dominated by the photonic contributions (i.e., keeping only the qc term in μpol ) and is often called the "visibility spectrum" or "transmission spectrum". In this case, the local information regarding the excitonic spectrum is present only in an indirect sense. However, often one is interested in examining the changes to the excitonic subsystem for purposes of exploring electronic reorganization through the mixing of electronic adiabatic states via hybridization with light. As such, to achieve an observable that will give direct information on the excitonic subsystem, we will ignore the second term in the polaritonic dipole operator to exclude photonic contribution, and in this way, we have traced out the photonic DOFs. An arbitrary matter-projected polaritonic operator can now be written as,