  We present an analysis of Bose-Fermi mixtures in optical lattices for the
case where the lattice potential of the fermions is tilted and the bosons (in
the superfluid phase) are described by Bogoliubov phonons. It is shown that the
Bogoliubov phonons enable hopping transitions between fermionic Wannier-Stark
states; these transitions are accompanied by energy dissipation into the
superfluid and result in a net atomic current along the lattice. We derive a
general expression for the drift velocity of the fermions and find that the
dependence of the atomic current on the lattice tilt exhibits negative
differential conductance and phonon resonances. Numerical simulations of the
full dynamics of the system based on the time-evolving block decimation
algorithm reveal that the phonon resonances should be observable under the
conditions of a realistic measuring procedure.
