Exact dynamics of a model for a three-level quantum system interacting with a continuous spectrum

Abstract

Previous work of the authors on a three-level quantum system is extended to allow the radiation field in interaction with the system to have a continuous spectrum of possible frequencies. The limiting procedure involved in the passage from discrete to continuous spectra is complicated by the need to express sums over different discrete spectra as integrals with well-behaved limits. Exact expressions are found, for a spontaneous emission problem, for the time evolution of the probabilities that the system be in (each) one of its three states and representative calculations of these probabilities are presented. The emergence of irreversible behavior upon constructing the thermodynamic limit of the problem is plainly seen and this demonstration permits for the first time a discussion of the relative effectiveness of competing decay channels in three-level quantum systems without imposing any ad hoc assumptions (such as exponential decay). Rather, the actual form of the decay emerges as a consequence of the structure and parameters of the Hamiltonian defining the model, and hence one can examine the variety of circumstances in which the evolution can reasonably be described as exponential. The results obtained should be of great use in clarifying certain outstanding conceptual problems in radiation physics, particularly those which deal with the universality of exponential decay in three- (and two-) level quantum systems in interaction with a radiation field and the conditions under which nonexponential or even nonergodic behavior can emerge in such dissipative quantum systems.