  The phase space is the natural ground to smoothly extrapolate between local
and non-local correlation functions. With this objective, we introduce the
symplectic tomography of many-body quantum gases in tight-waveguides, and
concentrate on the reduced single-particle symplectic tomogram (RSPST) whose
marginals are the density profile and momentum distribution. We present an
operational approach to measure the RSPST from the time evolution of the
density profile after shutting off the interactions in a variety of relevant
situations: free expansion, fall under gravity, and oscillations in a harmonic
trap. From the RSPST, the one-body density matrix of the trapped state can be
reconstructed.
