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The relationship between natural orbitals, one-body coherences and two-bodycorrelations is explored for bosonic many-body systems of definite parity withtwo occupied single-particle states. We show that the strength of localtwo-body correlations at the parity-symmetry center characterizes the numberstate distribution and controls the structure of non-local two-bodycorrelations. A recipe for the experimental reconstruction of the naturalorbital densities and quantum depletion is derived. These insights into thestructure of the many-body wave-function are applied to the predictedquantum-fluctuations induced decay of dark solitons.

Two-body correlations and natural orbital tomography in ultracold bosonic systems of definite parity