Under the classical limit, the d • E Hamiltonian can be obtained by applying the classical version of Û (Eq. 48), which is the Göppert-Mayer gauge transformation, on the classical p • A Hamiltonian in Eq 35. The details can be found in Ref. 47 (page 73) or Ref. 48 (page 53). Interestingly, the classical version of the d • E Hamiltonian does not contain the dipole self-energy term. This is because the DSE arises as a consequence of the quantum commutation relation among field operators. In the semiclassical picture, the electric and magnetic fields are time-dependent potentials that commute with the semiclassical PZW operator