State reduction and $|n〉$-state preparation in a high-$Q$ micromaser

The long lifetime of photons in a high-$Q$ micromaser cavity is used to generate a $|n〉$-state maser field by injecting atoms, one at a time. Every atom is probed for its excitation after leaving the cavity so that the precise number of photons in the field is known. We calculate the probability of obtaining $n$ photons after $m$ atoms have passed the cavity.