Ultracold atoms trapped by optical lattice potentials or confined to low dimensions can realize interesting strongly correlated quantum states. When the atoms are released from the trap they expand freely, collectively carrying with them a memory of the intricate many-body correlations that characterized their state in the trap.

The key to reading this memory is a mapping, that occurs via many-body interference effects during time of flight, between correlations in the trap and observable fluctuations in the expanding cloud. I will show that such "quantum noise interferometry" can be used to probe pairing, as well as spin and density correlations in ultracold Bose and Fermi systems. I will also discuss a related approach, which utilizes the interference between two expanding clouds. The statistical fluctuations of the emerging fringe pattern can be used to detect highly non local correlation functions. I will point out interesting relations between interference experiments with cold atoms and a variety of models ranging from a quantum impurity in a one-dimensional Luttinger liquid to two-dimensional quantum gravity.