  One of the most striking features of quantum mechanics is the appearance of
phases of matter with topological origins. These phases result in remarkably
robust macroscopic phenomena such as the edge modes in integer quantum Hall
systems, the gapless surface states of topological insulators, and elementary
excitations with non-abelian statistics in fractional quantum Hall systems and
topological superconductors. Many of these states hold promise in the
applications to quantum memories and quantum computation. Artificial quantum
systems, with their precise controllability, provide a versatile platform for
creating and probing a wide variety of topological phases. Here we investigate
topological phenomena in one dimension, using photonic quantum walks. The
photon evolution simulates the dynamics of topological phases which have been
predicted to arise in, for example, polyacetylene. We experimentally confirm
the long-standing prediction of topologically protected localized states
associated with these phases by directly imaging their wavefunctions. Moreover,
we reveal an entirely new topological phenomenon: the existence of a
topologically protected pair of bound states which is unique to periodically
driven systems. Our experiment demonstrates a powerful new approach for
controlling topological properties of quantum systems through periodic driving.
