Time dependence of operators in minimal and multipolar nonrelativistic quantum electrodynamics. I. Electromagnetic fields in the neighborhood of an atom

The electromagnetic field operators in the Heisenberg picture are found in both the minimal and multipolar forms of nonrelativistic quantum electrodynamics within the electric dipole approximation. These are made up of free- and source-field terms. The free electric- and magnetic-field operators in the two formalisms differ since the creation/annihilation operators are manifestly distinct. Their explicit forms in the Heisenberg picture are given. However, the source fields are formally the same when expressed in terms of dipole moment operators at positive retarded time. The cancellations that lead to this equivalence are presented. In many applications it is necessary to express the operators at retarded time in terms of creation and annihilation operators at the initial time. Explicit expressions for the fields up to second order in terms of the transition dipole moments are given. These enable the formulas for induced dipole moments to be derived within the context of quantum electrodynamics whereas previous derivations have been based on semiclassical electrodynamics.