  Since the experimental realization of synthetic gauge fields for neutral
atoms, the simulation of topologically non-trivial phases of matter with
ultracold atoms has become a major focus of cold atom experiments. However,
several obvious differences exist between cold atom and solid state systems,
for instance the finite size of the atomic cloud and the smooth confining
potential. In this article we show that sharp boundaries are not required to
realize quantum Hall or quantum spin Hall physics in optical lattices and, on
the contrary, that edge states which belong to a smooth confinement exhibit
additional interesting properties, such as spatially resolved splitting and
merging of bulk bands and the emergence of robust auxiliary states in bulk gaps
to preserve the topological quantum numbers. In addition, we numerically
validate that these states are robust against disorder. Finally, we analyze
possible detection methods, with a focus on Bragg spectroscopy, to demonstrate
that the edge states can be detected and that Bragg spectroscopy can reveal how
topological edge states are connected to the different bulk bands.
