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Full quantum distribution of contrast in interference experiments between interacting one dimensional Bose liquids

We analyze interference experiments for a pair of independent one dimensional condensates of interacting bosonic atoms at zero temperature. We show that the distribution function of fringe amplitudes contains non-trivial information about non-local correlations within individual condensates and can be calculated explicitly using methods of conformal field theory. We point out interesting relations between these distribution functions, the partition function for a quantum impurity in a one-dimensional Luttinger liquid, and transfer matrices of conformal field theories. We demonstrate the connection between interference experiments in cold atoms and a variety of statistical models ranging from stochastic growth models to two dimensional quantum gravity. Such connection can be used to design a quantum simulator of unusual two-dimensional models described by nonunitary conformal field theories with negative central charges.