Generation of four-photon coherent states in dispersive cavity QED

I discuss the generation of certain kinds of four-photon coherent states in the context of dispersive cavity QED. These states, eigenstates of fourth power of the photon annihilation operator, may be expressed as various superpositions of four coherent states symmetrically located on a circle in phase space. I also point out that for low amplitude initial coherent states, number states containing one, two, or three photons can be produced in the cavity. The decay of these states can be monitored by doing parity measurements on the field from which the lifetime of a number state can be obtained.