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

Deep insights into the structure of a many-body state can often be inferred from its natural orbitals (eigenvectors of the reduced one-body density operator) and their populations. These quantities allow e.g.\ to distinguish a Bose-Einstein condensate from a correlated many-body state [1] and were utilized to understand many-body processes such as the decay of dark solitons due to dynamical quantum depletion [2].\\ We explore the relationship between natural orbitals, one-body coherences and two-body correlations for a certain important class of bosonic many-body wave-functions with definite parity [3]. The strength of two-body correlations at the parity-symmetry center is shown (i) to characterize the number state distribution and (ii) to control the structure of non-local two-body correlations. A recipe for the experimental reconstruction of the natural-orbital densities based on two-body correlation measurements is derived. These results are applied to decaying dark solitons.\newline [1] O.\ Penrose, L.\ Onsager. {\it Phys. Rev.} {\bf 104}, 576 (1956). [2] R.\ V.\ Mishmash et al. {\it Phys. Rev. A} {\bf 80}, 053612 (2009); S.\ Kr\"onke, P.\ Schmelcher. {\it Phys. Rev. A} {\bf 91}, 053614 (2015). [3] S.\ Kr\"onke, P.\ Schmelcher. {\it Phys. Rev. A} {\bf 92}, 023631 (2015).