  Recent success in manipulating ultra-cold atomic systems allows to probe
different strongly correlated regimes in one-dimension. Regimes such as the
(spin-coherent) Luttinger liquid and the spin-incoherent Luttinger liquid can
be realized by tuning the inter-atomic interaction strength and trap
parameters. We identify the noise correlations of density fluctuations as a
robust observable (uniquely suitable in the context of trapped atomic gases) to
discriminate between these two regimes. Finally, we address the prospects to
realize and probe these phenomena experimentally using optical lattices.
