Creating multimode squeezed states and Greenberger-Horne-Zeilinger entangled states using atomic coherent effects

We propose a scalable scheme for a unitary multimode operator using an optical cavity with an atomic ensemble. We exemplify three-mode and four-mode cases and engineer the squeeze operators that are decoupled from the atomic degrees of freedom. The squeeze parameters can be large since they are proportional to the number of atoms. Using the input-output theory we show that ideal squeezed states and perfect squeezing could be approached at the output. At the same time, we show that it is possible to obtain tripartite and quadripartite Greenberger-Horne-Zeilinger entangled states for continuous variables. The responsible mechanism for both the multimode squeezing and the genuine multipartite entanglement is based on the atomic coherence controlled parametric interactions. The scalability of the scheme is simply obtained by including more transitions in the atomic system.