Highly non-Gaussian states created via cross-Kerr nonlinearity

We propose a feasible scheme for generation of strongly non-Gaussian states using the cross-Kerr nonlinearity. The resultant states are highly non-classical states of an electromagnetic field and exhibit negativity of their Wigner function, sub-Poissonian photon statistics and amplitude squeezing. Furthermore, the Wigner function has a distinctly pronounced 'banana' or 'crescent' shape specific for the Kerr-type interactions, which so far has not been demonstrated experimentally. We show that creating and detecting such states should be possible with the present technology using electromagnetically induced transparency in a four-level atomic system in N-configuration.