  An intrinsic feature of disordered and out-of-equilibrium materials, such as
glasses, is the dependence of their properties on their history. An important
example is rheological memory, in which disordered solids obtain properties
based on their mechanical history. Here, we employ x-ray photon correlation
spectroscopy (XPCS) with \textit{in situ} rheometry to characterize memory
formation in a nanocolloidal soft glass due to cyclic shear. During a cycle,
particles undergo irreversible displacements composed of a combination of
shear-induced diffusion and strain fields. The magnitudes of these
displacements decrease with each cycle before reaching a steady state where the
microstructure has become trained to achieve enhanced reversibility. The
displacements resemble a random walk in which the directions in each cycle are
independent of those in preceding cycles. Associated with these displacements
is a gradual evolution in the amplidute of the residual stress after each cycle
towards a steady state value. Memory of this training is revealed by
measurements in which the amplitude of the shear is changed after steady state
is reached. The magnitude of the particle displacements as well as the change
in residual stress vary non-monotonically with the new strain amplitude, having
minima near the training amplitude, thereby revealing both microscopic and
macroscopic signatures of memory.
