  We consider Bloch oscillations of ultracold atoms stored in a one-dimensional
vertical optical lattice and simultaneously interacting with a unidirectionally
pumped optical ring cavity whose vertical arm is collinear with the optical
lattice. We find that the feedback provided by the cavity field on the atomic
motion synchronizes Bloch oscillations via a mode-locking mechanism, steering
the atoms to the lowest Bloch band. It also stabilizes Bloch oscillations
against noise, and even suppresses dephasing due to atom-atom interactions.
Furthermore, it generates periodic bursts of light emitted into the
counter-propagating cavity mode, providing a non-destructive monitor of the
atomic dynamics. All these features may be crucial for future improvements of
the design of atomic gravimeters based on recording Bloch oscillations.
