which can be intuitively understood as the matter dipole polarizing the cavity field, and then the polarized cavity field acting back on the matter dipole, causing additional energy. Note that the DSE is different than the quadratic terms z 2 j Â2 /2m j in Ĥp•A (Eq. 45), which is commonly referred to as the Â2 term or diamagnetic term. Mathematically, the PZW gauge transformation operator shifts away (along the matter momentum direction) the Â2 terms in the p • A Hamiltonian, and causes a new shift (along the photonic momentum direction) that results in the DSE term in Ĥd•E . Thus, the DSE is an essential component to make sure that Ĥd•E (Eq. 50) and Ĥp•A (Eq. 45) are gauge invariant. Ignoring it under the weak coupling limit (g c /ω c ≪ 1) will not cause a significant numerical error but will break the gauge invariance. In the strong and ultra-strong coupling regimes, ignoring the DSE can cause an unstable ground state, especially under the long-wavelength approximation. As discussed at length in Ref. 39, the loss of the DSE term causes the ground state to be unbounded from below. Additionally, without the DSE term, the Maxwell equations in matter are no longer satisfied. In this manner, it is essential to include the DSE term in the strong and ultra-strong coupling regimes to accurately capture the physics of the system.