  Quantum integrable models display a rich variety of non-thermal excited
states with unusual properties. The most common way to probe them is by
performing a quantum quench, i.e., by letting a many-body initial state
unitarily evolve with an integrable Hamiltonian. At late times, these systems
are locally described by a generalized Gibbs ensemble with as many effective
temperatures as their local conserved quantities. The experimental measurement
of this macroscopic number of temperatures remains elusive. Here we show that
they can be obtained by probing the dynamical structure factor of the system
after the quench and by employing a generalized fluctuation-dissipation theorem
that we provide. Our procedure allows us to completely reconstruct the
stationary state of a quantum integrable system from state-of-the-art
experimental observations.
