Studies of bosons in optical lattices in a harmonic potential

We present a theoretical study of bose condensation and specific heat of non-interacting bosons in finite lattices in harmonic potentials in one, two, and three dimensions. We numerically diagonalize the Hamiltonian to obtain the energy levels of the systems. Using the energy levels thus obtained, we investigate the temperature dependence, dimensionality effects, lattice size dependence, and evolution to the bulk limit of the condensate fraction and the specific heat. Some preliminary results on the specific heat of fermions in optical lattices are also presented. The results obtained are contextualized within the current experimental and theoretical scenario.Comment: Revised version, References updated, a new section on Fermions added, 14 pages, 16 figure

Detecting multi-atomic composite states in optical lattices

We propose and discuss methods for detecting quasi-molecular complexes which are expected to form in strongly interacting optical lattice systems. Particular emphasis is placed on the detection of composite fermions forming in Bose-Fermi mixtures. We argue that, as an indirect indication of the composite fermions and a generic consequence of strong interactions, periodic correlations must appear in the atom shot noise of bosonic absorption images, similar to the bosonic Mott insulator [S. F\"olling, et al., Nature {\bf 434}, 481 (2005)]. The composites can also be detected directly and their quasi-momentum distribution measured. This method -- an extension of the technique of noise correlation interferometry [E. Altman et al., Phys. Rev. A {\bf 79}, 013603 (2004)] -- relies on measuring higher order correlations between the bosonic and fermionic shot noise in the absorption images. However, it fails for complexes consisting of more than three atoms.Comment: 9 revtex page

Representations of involutory subalgebras of affine Kac-Moody algebras

We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of $\mathbb{N}$-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity