Generation of six-mode cluster states in a coupled cavity array

We consider a chain of six coupled single-mode cavities, each containing an ensemble of four-level atoms, and demonstrate how to prepare stationary six-mode cluster states of the atomic ensembles. We show that a six-mode cluster state can be created in a single step by driving the ensembles with laser pulses of suitably chosen Rabi frequencies and phases. Introducing linear superpositions of the collective bosonic modes of the atomic ensembles, we show that the superposition modes can independently evolve to a desired stationary state with the help of the cavity damping. We illustrate this method by examining three examples of the creation of linear, hexagon, and double square cluster states.