An Infinite Ladder Coupled to a Quantum Mode; An Exactly Solvable Quantum Model

Abstract

In spite of intensive study, the interface between quantum mechanics and classical mechanics still offers questions to be answered. The transition is best investigated with models where both the classical and quantum mechanical versions are solvable. In quantum optics one in this case talks about semiclassical and quantum behaviour which differ in that in the latter case the electromagnetic field is quantized. A well known case is the Jaynes-Cummings model¹ in which a two-level system is coupled with a boson mode. One obtains that the model reproduces the corresponding semiclassical behaviour only if the initial conditions are suitable chosen. Otherwise interesting non-classical phenomena like revivals can occur². In this paper we study a related quantum mechanical model for which the corresponding semiclassical behaviour clearly differs from the two-level case. The two-level system is replaced with an infinite ladder of states and the single quantum field mode is resonantly coupled to each step in the ladder. We solve the problem exactly the case with equal couplings. The corresponding semiclassical model has been used for example to describe optical transitions in molecules. Because of the equal coupling constants the present model dramatically differs from the model of coupled harmonic oscillators.