Mott transitions of exciton polaritons and indirect excitons in a periodic potential

We derive an effective Bose-Hubbard model that predicts a phase transition from Bose-Einstein condensate to Mott insulator in two different systems subject to applied periodic potentials: microcavity exciton polaritons and indirect excitons. Starting from a microscopic Hamiltonian of electrons and holes, we derive an effective Bose-Hubbard model for both systems and evaluate the on-site Coulomb interaction U and hopping transition amplitudes t. Experimental parameters required for observing a phase transition between a Bose-Einstein condensate and a Mott insulator are discussed. Our results suggest that strong periodic potentials and polaritons with a very large excitonic component are required for observing the phase transition. The form of the indirect exciton interaction is derived including direct and exchange components of the Coulomb interaction. For indirect excitons, the system crosses over from a Bose-Hubbard model into a double layer Fermi-Hubbard model as a function of increasing bilayer separation. The Fermi-Hubbard model parameters are calculated, and the criteria for the location of this crossover are derived. We conjecture that a crossover between a Bose Mott insulator to a Fermi Mott insulator should occur with increasing bilayer separation.