(67) It is well known that the above two Hamiltonians do not generate identical polariton eigenspectra under the ultra-strong coupling regime, explicitly breaking down the gauge invariance. This leads to the gauge ambiguity as to which Hamiltonian, Ĥ′ p•A or Ĥd•E , is correct for computing physical quantities when applying P. This is a well-known result in quantum optics that Ĥ′ p•A usually requires a larger subset of the matter states to converge or generate consistent results with Ĥd•E , and apparently, under the complete basis limit, they should be gauge invariant.