  The Su-Schrieffer-Heeger (SSH) model, which captures the most striking
transport properties of the conductive organic polymer $trans$-polyacetylene,
provides perhaps the most basic model system supporting topological
excitations. The alternating bond pattern of polyacetylene chains is captured
by the bipartite sublattice structure of the SSH model, emblematic of
one-dimensional chiral symmetric topological insulators. This structure
supports two distinct nontrivial topological phases, which, when interfaced
with one another or with a topologically trivial phase, give rise to
topologically-protected, dispersionless boundary states. Using $^{87}$Rb atoms
in a momentum-space lattice, we realize fully-tunable condensed matter
Hamiltonians, allowing us to probe the dynamics and equilibrium properties of
the SSH model. We report on the experimental quantum simulation of this model
and observation of the localized topological soliton state through quench
dynamics, phase-sensitive injection, and adiabatic preparation.
