Decoherence and correlations in systems of trapped ultra-cold quantum gases

Since the achievement of Bose-Einstein condensation (BEC), the progress in matterwave physics has been immense. Among the many recent achievements there is the miniaturization of atom traps, demonstration of the superfluid-Mott insulator quantum-phase transition in optical lattices and the experimental demonstration of the BEC-BCS crossover in ultra-cold gases. Miniaturization of atom traps using micro-structured wires on a chip is one important step towards an on-chip cold-atom device. These so-called \atom chips" provide high control and versatility for trapping and guiding the ultra-cold atomic clouds. Particularly interesting is the use of these microchips to build mesoscopic devices for cold atomic clouds as, for instance, in the case of an atom-cloud interferometer. However, these mesoscopic devices require coherent transport of the atom cloud. A general method to treat decoherence due to current fluctuations in multi-wire atomchip traps is presented in the rst part of this thesis. The decoherence rate Γ shows a strong dependence on the distance between the wire and the atom cloud, r0, scaling as Γ ~ r-4 0 for a single atom waveguide. Considering an interferometer device, a strong dependence of the decoherence rate on the trap geometry is found. Studying many-body eects in ultra-cold quantum gases is another important research eld. Experiments using ultra-cold quantum gases in optical lattices have demonstrated the superuid-Mott insulator quantum phase transition and manybody entanglement. Optical lattices are based on a periodic modulation of the light intensity, generated by retro-reected laser beams. Correlations of the atomic cloud between dierent lattice sites of the optical lattice play a central role in these manybody experiments. The dierent phases of the superuid-Mott insulator system can be characterized by the dierent behavior of the inter-lattice site correlations. There are several numerical methods such as Quantum Monte Carlo (QMC) simulations, Density Matrix Renormalization Group (DMRG) simulations, exact-diagonalization, or the Gutzwiller ansatz, to investigate the dynamics of an ultra-cold gas in an optical lattice theoretically. The Gutzwiller method, corresponding to the mean-eld solution, allows for the treatment of large lattice sizes. Mean-eld approaches have proven to be very useful to describe many-body physics. However, diculties arise in the correct description of the behavior of the decay of inter-lattice site correlations. Based on the Gutzwiller approach, we have developed a method which allows the successive inclusion of inter-lattice site correlations. Comparing the results for the particle-number uctuations and the correlation function obtained from pure Gutzwiller calculations, to calculations which perturbatively include short-range correlations and calculations using \quasi-exact methods", showed a considerable improvement relative to the pure Gutzwiller results due to the inclusion of short-range correlations. Many-body eects do not only arise in periodic potentials, but become increasingly important at ultra-low temperatures. The formation of Bose-Einstein condensates requires an overlap of the atom wavefunctions and, hence, the formation of a single condensate wavefunction. Another example of a many-body state is the superuid-BCS state, commonly used as a description of superconductivity. Here, fermions in dierent hyperne states form Cooper pairs. Experiments with ultracold quantum gases enable a variation of the interparticle interaction, e.g. , by using a Feshbach resonance. Using Feshbach resonances to tune the interaction strength has enabled the experimental observation of the crossover from a superuid-BCS state to a Bose-Einstein condensate of molecules. A useful way to characterize the dierent states of ultra-cold quantum gases is to investigate the particle-number uctuations. In this thesis we suggest to divide the atomic cloud into bins and consider the atom-number uctuations in these bins. We calculate the full counting statistics for dierent physical systems of ultra-cold gases (e.g. bosonic gases, fermionic gases, and spin mixtures). In particular, we consider the BCS-state as a rst trial example to show that there is a strong variation in the particle-number statistics at the crossover from a superuid-BCS state to a Bose-Einstein condensate of molecules.