Laboratoire de Physique Théorique
Toulouse - UMR 5152

One dimensional (1D) strongly correlated quantum systems are nowadays the subject of intense theoretical and experimental activity due to the incredible experimental control offered by ultra-cold atoms setups. In such systems, the momentum distribution is a powerful probe of both gas statistics and of the intertwined effect of interactions between particles and the effective dimensionality they move in.

A remarkable momentum distribution feature, in all dimensions, is the presence of universal power-law tails n(k) k^-4 for a gas where interactions can be schematized as contact ones (as it is the case for most standard cold gases experiments). The weight of such tails, denoted as Tan’s contact, can be put into relation with several many-body quantities, ranging from the interaction energy to the depletion rate by inelastic collisions.

In this work, we provide a complete characterization of the Tan’s contact for one-dimensional bosons under harmonic confinement for arbitrary interaction strength, number of particles, temperature and trap frequency. Using a combination of thermal Bethe-ansatz solutions with local-density approximation and exact numerical calculations, we demonstrate that the contact is a universal function of only two scaling parameters and determine the scaling function. As a main result, we find that for any finite value of the interaction strength the contact displays a maximum versus temperature. Moreover we have studied the small deviations at the scaling properties of the contact that occur for few bosons at zero temperature. In this case, we propose a simple function which takes into account for particle finite-number effects and yields an analytical expression for the Tan’s contact and for the total energy of the gas.