  We show that one-dimensional quasi-periodic optical lattice systems can
exhibit edge states and topological phases which are generally believed to
appear in two-dimensional systems. When the Fermi energy lies in gaps, the
Fermi system on the optical superlattice is a topological insulator
characterized by a nonzero topological invariant. The topological nature can be
revealed by observing the density profile of a trapped fermion system, which
displays plateaus with their positions uniquely determined by the ration of
wavelengths of the bichromatic optical lattice. The butterfly-like spectrum of
the superlattice system can be also determined from the finite-temperature
density profiles of the trapped fermion system. This finding opens an
alternative avenue to study the topological phases and Hofstadter-like spectrum
in one-dimensional optical lattices.
