Pump operator for lasers with multi-level excitation

In a multi-level laser the pump process involves several intermediate levels. The details of the atomic excitation are often irrelevant so that an effective heat bath description and a two-level-laser model are applicable. The heat bath approximation may be too crude, however; it does not predict phenomena such as intensity-noise squeezing and sub-Poissonian field statistics. We show how, in the general situation, the multi-level description can still be reduced to a two-level description in terms of a non-Markovian master equation where a rate matrix, rather than a single rate, accounts for the excitation process. The resulting dependence of the excitation rates on the state of the laser field constitutes a correction to the standard laser model. We hold it responsible for the dynamic noise reduction in multi-level lasers.