On the relaxation to quantum-statistical equilibrium of the Wigner–Weisskopf atom in a one-dimensional radiation field. VII. Emission in a finite system in the presence of an extra photon

In this paper, we present the exact solution to a problem previously unsolved in radiation theory: the emission of a two-level atom in a (one-dimensional) radiation field in the presence of an extra photon. The solution is obtained directly from the Schrödinger equation of the problem using techniques suggested by the work of Muskhelishvili on singular integral equations. The solution corresponding to a finite system as well as the one corresponding to a system infinite in extent are given, although our primary concern in this paper is the finite-system problem. For a particular choice of initial condition, the probability at time t that the two-level atom is in the excited state is found, and the effects of system size and choice of coupling function are studied numerically for a given coupling constant. Our results are compared with those obtained in an earlier paper of the series, wherein the spontaneous emission of a Wigner-Weisskopf atom in a (one-dimensional) field of radiation for a system finite in extent was studied. The physical effects calculated and the conclusions drawn from our comparative studies are all in accord with simple intuition regarding the problem. The paper concludes with some brief remarks on problems in radiation theory which are accessible to study given the methods laid down in this paper.