Optimally driving multi-photon transitions in the perturbative single-mode regime

The rate of $m$-photon transitions in matter, induced by an incident light field, depends on the field's $m$th order coherence function. Consequently, the coherence properties of the light field may be shaped to increase the rate of multi-photon transitions. Here, we determine the optimal state of a weak fixed-intensity, narrow-band incident light field, with a restricted maximal photon number, that optimally drives $m$-photon transitions in the case of a short-lived atomic multilevel system. We show that, in this case, no quantum properties of the light field need to be exploited, but that classical mixtures of coherent states are optimal.