In this work we study the formation and dynamics of polarons in a system with
a few impurities in a lattice immersed in a Bose-Einstein condensate (BEC).
This system has been experimentally realized using ultracold atoms and optical
lattices. Here we consider a two-band model for the impurity atoms, along with
a Bogoliubov approximation for the BEC, with phonons coupled to impurities via
both intra- and inter-band transitions. We decouple this Fr\"ohlich-like term
by an extended two-band Lang-Firsov polaron transformation using a variational
method. The new effective Hamiltonian with two (polaron) bands differs from the
original Hamiltonian by modified coherent transport, polaron energy shifts and
induced long-range interaction. A Lindblad master equation approach is used to
take into account residual incoherent coupling between polaron and bath. This
polaronic treatment yields a renormalized inter-band relaxation rate compared
to Fermi's Golden Rule. For a strongly coupled two-band Fr\"ohlich Hamiltonian,
the polaron is tightly dressed in each band and can not tunnel between them,
leading to an inter-band self-trapping effect.Comment: 23 pages, 7 figure