  Quantum simulators allow to explore static and dynamical properties of
otherwise intractable quantum many-body systems. In many instances, however, it
is the read-out that limits such quantum simulations. In this work, we
introduce a new paradigm of experimental read-out exploiting coherent
non-interacting dynamics in order to extract otherwise inaccessible
observables. Specifically, we present a novel tomographic recovery method
allowing to indirectly measure second moments of relative density fluctuations
in one-dimensional superfluids which until now eluded direct measurements. We
achieve this by relating second moments of relative phase fluctuations which
are measured at different evolution times through known dynamical equations
arising from unitary non-interacting multi-mode dynamics. Applying methods from
signal processing we reconstruct the full matrix of second moments, including
the relative density fluctuations. We employ the method to investigate
equilibrium states, the dynamics of phonon occupation numbers and even to
predict recurrences. The method opens a new window for quantum simulations with
one-dimensional superfluids, enabling a deeper analysis of their equilibration
and thermalization dynamics.
