Control and amplification of Bloch oscillations via photon-mediated interactions

We propose a scheme to control and enhance atomic Bloch oscillations via photon-mediated interactions in a standing-wave cavity. Atoms are trapped in an optical lattice with a wavelength incommensurate with the cavity mode. 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.