  We study the transport of ultracold impurity atoms immersed in a
Bose-Einstein condensate (BEC) and trapped in a tight optical lattice. Within
the strong-coupling regime, we derive an extended Hubbard model describing the
dynamics of the impurities in terms of polarons, i.e. impurities dressed by a
coherent state of Bogoliubov phonons. Using a generalized master equation based
on this microscopic model we show that inelastic and dissipative phonon
scattering results in (i) a crossover from coherent to incoherent transport of
impurities with increasing BEC temperature and (ii) the emergence of a net
atomic current across a tilted optical lattice. The dependence of the atomic
current on the lattice tilt changes from ohmic conductance to negative
differential conductance within an experimentally accessible parameter regime.
This transition is accurately described by an Esaki-Tsu-type relation with the
effective relaxation time of the impurities as a temperature-dependent
parameter.
