A quantum for a quantum : Quantum simulators in exotic lattices

Abstract: This thesis centers on how to use exotic lattices in different ways to design potential quantum simulators. These exotic lattices are either real physical objects in the form of ultra cold atoms in  bipartite optical lattice systems or lattice models revealed in state space. For optical lattices, the potential wells forms a lattice, such that the atoms of the system will be highly localized at the potential minima, and where the bipartite nature of the lattice ensures that the lattice sites will alternate between two different types.  When studying the atoms in the optical lattices they can be described by a Bose-Hubbard model, where it turns out that bipartite lattice systems provides a route to realizing frustration through competing nearest and next nearest neighbor couplings in both the superfluid phase and the Mott insulating phase.For state space lattices are no longer objects in real space, but instead they live in state space. Though such state space lattice models can be represented as single particle systems, they still hold a potential for realizing quantum simulators. For this type of exotic lattices, the focus is on how to study quantum optical models in terms of their Fock space lattices (FSLs). Such models only have a few degrees-of-freedom which together with symmetries of these system, make it simple to identify the emerging FSLs with know lattice models from the condensed matter. Thus shedding new light on the quantum optical systems at hand. The three-mode Jaynes-Cummings model in the large detuned limit, is used as an example to display the strength of this method. It is the growth of a systems state space, that determines whether the system is a potential quantum simulator or not. For a system to be a quantum simulator, the growth of the phase space has to be such, that it becomes computational hard to find the systems energy. This means, that if we can design a state space lattice, which grows exponential, then we have a potential candidate for a quantum simulator. The Bethe lattice is one such example that grows exponentially.