  We explore the phase diagram of a two-component ultracold atomic Fermi gas
interacting with zero-range forces in the limit of weak-coupling. We focus on
the dependence of the pairing gap and the free energy on the variations in the
number densities of the two species while the total density of the system is
held fixed. As the density asymmetry is increased, the system exhibits a
transition from a homogenous Bardeen-Cooper-Schrieffer (BCS) phase to phases
with spontaneously broken global space symmetries. One such realization is the
deformed Fermi surface superfluidity (DFS) which exploits the possibility of
deforming the Fermi-surfaces of the species into ellipsoidal form at zero total
momentum of Cooper pairs. The critical asymmetries at which the transition from
the DFS to the unpaired state occurs are larger than those for the BCS phase.
In this precritical region the DFS phase lowers the condensation energy of the
asymmetric BCS state. We compare quantitatively the DFS phase to another
realization of superconducting phases with broken translational symmetry - the
single plane wave Larkin-Ovchinnikov-Fulde-Ferrell (LOFF)phase, which is
characterized by a non-vanishing center-of-mass momentum of the Cooper pairs.
The possibility of the detection of the DFS phase in the time-of-flight
experiments is discussed and quantified for the case of $^6$Li atoms trapped in
two different hyperfine states.
