  We propose to utilize density-density correlations in the image of an
expanding gas cloud to probe complex many body states of trapped ultra-cold
atoms. In particular we show how this technique can be used to detect
superfluidity of fermionic gases and reveal broken spin symmetries in
Mott-states of atoms in optical lattices. The feasibility of the method is
investigated by analysis of the relevant signal to noise ratio including
experimental imperfections.
