  We theoretically investigate two-particle and many-particle Anderson
localizations of a spin-orbit coupled ultracold atomic Fermi gas trapped in a
quasi-periodic potential and subjected to an out-of-plane Zeeman field. We
solve exactly the two-particle problem in a finite length system by exact
diagonalization and solve approximately the many-particle problem within the
mean-field Bogoliubov-de Gennes approach. At a small Zeeman field, the
localization properties of the system are similar to that of a Fermi gas with
conventional $s$-wave interactions. As the disorder strength increases, the
two-particle binding energy increases and the fermionic superfluidity of many
particles disappears above a threshold. At a large Zeeman field, where the
interatomic interaction behaves effectively like a $p$-wave interaction, the
binding energy decreases with increasing disorder strength and the resulting
topological superfluidity shows a much more robust stability against disorder
than the conventional $s$-wave superfluidity. We also analyze the localization
properties of the emergent Majorana fermions in the topological phase. Our
results could be experimentally examined in future cold-atom experiments, where
the spin-orbit coupling can be induced artificially by using two Raman lasers,
and the quasi-periodic potential can be created by using bichromatic optical
lattices.
