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 for
the Bose gas in one spatial dimension 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.
Correlations and superfluidity of a one-dimensional Bose gas in a quasiperiodic potential