Limitations of steady state solutions to a two-state model of population oscillations and hole burning

We consider a two-state system driven by an on-resonance, continuous wave pump laser and a much weaker pulsed probe laser that is slightly detuned from the pump laser frequency (usually this detuning is about omega(p)-omega(P)=Delta approximate to 1 kHz). The upper state population is assumed to be slowly decaying, but the off-diagonal element of the density matrix decays rapidly due to homogeneous broadening. This model has been solved by others in rare-earth-element-doped fibers and crystals in a usual steady state approximation for slow optical wave propagation. We show that in general the usual steady state approximation does not apply unless either Delta tau > 1 or (2S+1)gamma(2)tau > 1 where gamma(2) is the decay rate of the excited state population, tau is the pulse length of the probe field, and 2S is the saturation parameter. Both conditions, however, are not satisfied in many population-oscillation- and corresponding group-velocity-reduction-related studies. Our theory and corresponding numerical simulations have indicated that for probe pulses that are much shorter than the lifetime of the upper state, there is no analytical theory for the amplitude, pulse shape, and group velocity of the probe field. In addition, there is no reason to assume that the group velocity remains small when gamma(2)tau < 1 and there is no reason to believe that many pulse length decays can be obtained for such short pulses.