Polaronic effects in one- and two-band quantum systems

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 intraband and interband transitions. We decouple this Fröhlich-type 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 interband relaxation rate compared to Fermi's golden rule. For a strongly coupled two-band Fröhlich Hamiltonian, the polaron is tightly dressed in each band and can not tunnel between them, leading to an interband self-trapping effect