Exact Solution for a Trapped Fermi Gas with Population Imbalance and BCS Pairing

The problem of a two-component Fermi gas in a harmonic trap, with an imbalanced population and a pairing interaction of zero total momentum, is mapped onto the exactly solvable reduced BCS model. For a one-dimensional trap, the complete ground state diagram is determined with various topological features in ground state energy spectra. In addition to the conventional two-shell density profile of a paired core and polarized outer wings, a three-shell structure as well as a double-peak superfluid distribution are unveiled.