  We analyze the optomechanics of an atomic Bose-Einstein condensate
interacting with the optical lattice inside a laser-pumped optical cavity and
subject to a uniform bias force such as gravity. An atomic wave packet in a
tilted lattice undergoes Bloch oscillations; in a cavity the backaction of the
atoms on the light leads to a time-dependent modulation of the intracavity
lattice at the Bloch frequency. When the Bloch frequency is on the order of the
cavity damping rate we find transport of the atoms either up or down the
lattice. The transport dynamics can be interpreted as a manifestation of
dynamical backaction-induced sideband damping/amplification of the
optomechanical Bloch oscillator. Depending on the sign of the pump-cavity
detuning, atoms are transported either with or against the bias force
accompanied by an up- or down-conversion of the frequency of the pump laser
light. We also evaluate the prospects for using the optomechanical Bloch
oscillator to make continuous measurements of forces by reading out the Bloch
frequency. In this context we establish the significant result that the optical
spring effect is absent and the Bloch frequency is not modified by the
backaction.
