  Due to its probabilistic nature, a measurement process in quantum mechanics
produces a distribution of possible outcomes. This distribution - or its
Fourier transform known as full counting statistics (FCS) - contains much more
information than say the mean value of the measured observable and accessing it
is sometimes the only way to obtain relevant information about the system. In
fact, the FCS is the limit of an even more general family of observables - the
charged moments - that characterise how quantum entanglement is split in
different symmetry sectors in the presence of a global symmetry. Here we
consider the evolution of the FCS and of the charged moments of a U(1) charge
truncated to a finite region after a global quantum quench. For large scales
these quantities take a simple large-deviation form, showing two different
regimes as functions of time: while for times much larger than the size of the
region they approach a stationary value set by the local equilibrium state, for
times shorter than region size they show a non-trivial dependence on time. We
show that, whenever the initial state is also U(1) symmetric, the leading order
in time of FCS and charged moments in the out-of-equilibrium regime can be
determined by means of a space-time duality. Namely, it coincides with the
stationary value in the system where the roles of time and space are exchanged.
We use this observation to find some general properties of FCS and charged
moments out-of-equilibrium, and to derive an exact expression for these
quantities in interacting integrable models. We test this expression against
exact results in the Rule 54 quantum cellular automaton and exact numerics in
the XXZ spin-1/2 chain.
