Quantum Theory of Near-Field Electrodynamics pp 515-551 | Cite as

# Photon Emission From Atoms: Elements of the Nonrelativistic QED Description

## Abstract

We begin this chapter with a brief introduction to the theory of spontaneous emission, and we add some remarks on the Lamb shift. For mesoscopic (and microscopic) objects, the field retardation across the transverse current density domain of the source usually will be small. The Lamb shift can be determined without the inclusion of retardation effects, but a calculation of the spontaneous emission requires that the field retardation is taken into account. That the spontaneous emission process depends on retardation is already clear from Einstein’s phenomenological approach, where it turns out that the time rate of spontaneous emission (given by the well known Einstein A-coefficient [324, 301]) is proportional to *c*_{0}^{ − 3}. For *c*_{0} → *∞*, *A* → 0. For electric dipole-allowed transitions the spontaneous decay rate of the electron can be obtained including retardation to first order, only. It is indicated that there exists a close link between the field driving the spontaneous emission and the electric radiation reaction field studied in a classical framework in Chap. 17. After having studied the propagator connection between the photon-field operators and the polarization and magnetization field operators of the source, we turn our attention towards the field radiation from single-particle sources. Via a second-quantization of the source current density the important flip operators are introduced. In a somewhat sketchy manner we determine the classical electric-dipole Hamiltonian starting from the Poincaré Hamiltonian. The associated Hamilton operator we obtain by means of a rigorous so-called long-wavelength unitary transformation of the Coulomb Hamiltonian. In order to illustrate some of the basic aspects of the field-atom interaction we analyze in some detail the electrodynamics of so-called two-level atoms. We set up Heisenberg equations of motions for the atomic flip operator and the annihilation operator of the various field modes, and with these coupled dynamical equations in hand, and on the basis of certain approximations, we calculate the rate of spontaneous emission. We also briefly discuss the Lamb shift parameter, and the somewhat complicated physics we need to address in order to obtain an accurate value for it. In the last part of the chapter, we study the radiated transverse field, and we demonstrate that the spontaneous decay rate can be expressed in terms of the imaginary part of the transverse field propagator evaluated for coincident source and observation points.