Open Access

Nanoengineering and laser optics allow for the fabrication of a wide range of systems that subject fermionic particles to geometric restrictions. In addition to strong correlations, the fermions may couple to internal or external bosonic fields, such as quantized lattice vibrations or light fields. This thesis considers the theoretical description of two such systems. One is a molecular junction, i.e., a small organic molecule contacted by metallic electrodes or leads. Itinerant electrons induce molecular vibrations and deformations, corresponding to phonon modes of considerable energy. The thesis investigates the effects of this local electron-phonon interaction on the electric and thermoelectric transport through the junction. Starting with an Anderson-Holstein quantum dot model, our ansatz is based on the application of a variational Lang-Firsov transformation that accounts for the polaronic character of the dot state. We solve the steady-state Kadanoff-Baym equations and derive a self-consistent approximation to the polaronic self-energy that accounts for finite densities and multi-phonon scattering processes. The optimal variational parameter is determined numerically by minimizing the thermodynamical potential. This allows a detailed study of the electronic dot spectral function for all interaction strengths and adiabaticity regimes. For instance, we discuss how a voltage dependent polaronic renormalization of the dot-lead coupling and the dot level causes negative differential conductance and novel conductance features. The investigation of the second system is motivated by recent experiments on the Bose-Einstein condensation of excitons in small semiconducting cuprous oxide crystals. At ultra cold temperatures three species of para- and orthoexcitons are caught in stress induced potential traps. Their decay luminescence is the primary method of detection. This thesis considers the thermodynamics of this system in terms of a multicomponent gas of weakly interacting bosons in external potentials. The coupled equations of motion are solved within a Hartree-Fock-Bogoliubov-Popov approximation. For typical experimental parameters the density distributions of the interacting species are calculated numerically. Based on the luminescence formula by Shi and Verechaka we discuss, e.g., how the spectrum of the direct decay of thermal paraexcitons may reveal the formation of a nonluminescent paraexciton condensate as well as the spatial separation of strongly repulsive orthocondensates. First results for an extended luminescence theory are presented, which takes into account the polariton effect.