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Universal time-fluctuations in near-critical out-of-equilibrium quantum dynamics

Out of equilibrium quantum systems, on top of quantum fluctuations, display complex temporal patterns. Such time fluctuations are generically exponentially small in the system volume and can be therefore safely ignored in most of the cases. However, if one consider small quench experiments, time fluctuations can be greatly enhanced. We show that time fluctuations may become stronger than other forms of equilibrium quantum fluctuations if the quench is performed close to a critical point. For sufficiently relevant operators the full distribution function of dynamically evolving observable expectation values, becomes a universal function uniquely characterized by the critical exponents and the boundary conditions. At regular points of the phase diagram and for non sufficiently relevant operators the distribution becomes Gaussian. Our predictions are confirmed by an explicit calculation on the quantum Ising model.