Preparation of a four-atom Greenberger-Horne-Zeilinger state

I propose a method to prepare four atoms in an entangled state of the type discussed by Greenberger, Horne, and Zeilinger [in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer Academic, Dordrecht, 1989), p. 107]. The method involves two micromaser cavities, each supporting two modes initially prepared in a nonlocalized two-photon state by an atom undergoing a two-photon transition. Subsequently, two atoms pass through each cavity, becoming entangled in a Greenberger-Horne-Zeilinger state leaving the cavities in the vacuum.