Note that Ĥ′ p•A = P Ĥp•A P = P Û † ĤM Û P + Ĥph . It is well known that the above two Hamiltonians do not generate the same polariton eigenspectrum 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 viable to compute physical quantities when applying P. This is attributed to the fact 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 are gauge invariant (see Figs. ). Further, this fundamentally different behavior of Ĥ′ p•A and Ĥd•E upon state truncation is also attributed to the fundamental asymmetry of the operators p and μ = j z j xj . Fig. (a)-(d) shows these convergence properties of both the dipole and Coulomb gauge Hamiltonians, demonstrating that for this Shin-Metiu model system, Ĥ′ p•A requires 9x more matter states to converge (see panels (c) and (d)).