  We propose a scheme to control and enhance atomic Bloch oscillations via
photon-mediated interactions in an optical lattice supported by a standing-wave
cavity with incommensurate lattice and cavity wavelengths. Our scheme uses
position-dependent atom-light couplings in an optical cavity to spatially
prepare an array of atoms at targeted lattice sites starting from a thermal
gas. On this initial state we take advantage of dispersive position-dependent
atom-cavity couplings to perform non-destructive measurements of
single-particle Bloch oscillations, and to generate long-range interactions
self-tuned by atomic motion. The latter leads to the generation of dynamical
phase transitions in the deep lattice regime and the amplification of Bloch
oscillations in the shallow lattice regime. Our work introduces new
possibilities accessible in state-of-the-art cavity QED experiments for the
exploration of many-body dynamics in self-tunable potentials.
