  A growing expertise to engineer, manipulate and probe different cold-atom
analogs of electronic condensed matter systems allows to probe properties of
exotic pairing. We study paired states of spin-imbalanced ultracold atomic
system of fermions with attractive short-range interactions in one-dimensional
traps. Calculations are done using the Bethe Ansatz technique and the trap is
incorporated into the solution via a local density approximation. The
thermodynamic-Bethe-Ansatz equations are solved numerically and different local
density profiles are calculated for zero and finite temperatures. A procedure
to identify the homogeneous-system phase diagram using local density profiles
in the trap is also proposed. Such scheme would be immediately useful for the
experiments.
