Spontaneous emission in dipole approximation is studied theoretically using both source-field theory and a Schrödinger-picture approach. Using source-field theory we obtain formal equations for the Poynting vector and energy density without making the rotating-wave approximation (RWA) and the Weisskopf-Wigner approximation (WWA). The initial condition at $t=0$ is one in which the atom is in an excited state and the field in the vacuum state. The field operators are fully retarded in source-field theory. To explore the consequences of not making the RWA and WWA, the Poynting vector and energy density are calculated using perturbation theory. We use three methods to calculate the Poynting vector and energy density of the radiated fields, (1) a calculation based on definitions of the fields in terms of creation and annihilation operators, (2) a traditional source-field calculation, and (3) a hybrid source-field model involving a mixture of fully retarded and unretarded fields. We obtain analytic expressions for the Poynting vector and energy density in perturbation theory that are consistent with the results of Power and Thirunamachandran [Phys. Rev. A 45, 54 (1992)] and Compagno, Passante, and Persico [J. Mod. Opt. 37, 1377 (1990)], obtained using a Heisenberg-picture approach. The expressions for the Poynting vector and energy density that we derive in perturbation theory involve integrals over field mode frequencies, which are divergent unless a convergence factor is used to carry out the integrations. The convergence factor contains an effective cutoff frequency ${\omega }_c$, and results are obtained for both finite ${\omega }_c$ and the limit in which ${\omega }_c\to \infty$. It is also proved that the fields calculated using source-field theory always satisfy Poynting's theorem for arbitrary atom-field interactions and initial conditions.

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• Spontaneous emission