  When a d-dimensional quantum system is subjected to a periodic drive, it may
be treated as a (d+1)-dimensional system, where the extra dimension is a
synthetic one. In this work, we take these ideas to the next level by showing
that non-uniform potentials, and particularly edges, in the synthetic dimension
are created whenever the dynamics of system has a memory component. We
demonstrate that topological states appear on the edges of these synthetic
dimensions and can be used as a basis for a wave packet construction. Such
systems may act as an optical isolator which allows transmission of light in a
directional way. We supplement our ideas by an example of a physical system
that shows this type of physics.
