There are many different notations for the methods developed by changing the highest level of excitation for each term in the cluster operator. In this review, we will use the notation whereby CCSD-n-jm, which implies that the electronic DOFs are treated up to double excitations in the cluster operator, the photonic excitation is limited to n levels, and the mixed excitation is set to j electronic and m photonic. As per usual CC theory, the cutoff of excitation level leads to effects that include yet higher excitations through the exponential treatment of the cluster operator T , thus effectively outperforming similar methods like CI with the same excitation level cutoff. However, due to the incredibly high cost of the CC method in general, including more than two Fock states has been a challenge even for small molecular systems, 157,164 and limited study has been performed including up to 10 Fock states for a halffilled four-site Hubbard model with direct comparison to the full configuration interaction result. 163 This will have many negative implications on highly negativelydetuned cavities or for purposes of multi-photon upconversion, where higher numbers of photons are required to extract the physics.