Nonlinear polarizations and excitations and their time dependence in discrete multilevel systems

For the interaction of coherent radiation with a nondegenerate, homogeneously broadened multilevel system the nonlinear susceptibilities and excitations and their time dependence are evaluated using a perturbation approach to the density matrix up to fourth order. This is done for the "off-resonance" case, ${(\Delta {\omega }^2+{\Gamma }^2)}^{\frac{1}{2}}\gg \frac{1}{\tau }$, as well as for the "on-resonance" case, ${(\Delta {\omega }^2+{\Gamma }^2)}^{\frac{1}{2}}\ll \frac{1}{\tau }$, with $\tau$ being the pulse duration, by taking into account the temporal change of the electric field amplitude and the phase and energy relaxation times, $T_2$ and $T_1$. It is shown that the transient one-photon excitation known as adiabatic following, is not only important for a pulse duration $\tau <T_2$, but may also be important for $T_2\lesssim \tau <T_1$. In addition, expressions for the transient two-photon excitation in a multilevel system are derived. Finally, the two-level limit is evaluated to all orders of the perturbation including damping effects, which gives results valid also for large incident light intensities. For the "off-resonance" case an expression is obtained which describes not only the transient adiabatic following process, but also the usual one-photon absorption including saturation effects giving rise to power broadening.