Coherent pulse propagation, a dispersive, irreversible phenomenon

The initial value problem for the propagation of a pulse through a resonant two-level optical medium is solved by the inverse scattering method. In general, an incident pulse decomposes not only into a special class of pulses to which the medium is transparent but also yields radiation which is absorbed by the medium. In this respect ``this problem'' has properties markedly different from other dispersive and reversible wave phenomena some of which are tractable by the inverse scattering method. Indeed, it is remarkable that in the present case the method still applies. In particular, we show that, while there are an infinite number of local conservation laws, the integrated densities, and in particular the energy, are only conserved for a very special class of initial conditions. The theoretical results obtained are in close agreement with all the qualitative features observed in the experiments on coherent pulse propagation. Finally, we also show that causality is preserved. Two new and novel features are introduced and briefly discussed. First, we show that if the homogeneous broadening effect is a function of position in the medium, the pulses may speed up and slow down accordingly, without losing their permanent identities. Second, we have found a new kind of solution mode corresponding to a proper eigenvalue of the scattering problem which is not a bound state.