Ultra-cold atoms in optical lattices: simulating quantum spin systems

Abstract

Optical lattices provide ideal experimental tools for simulating a wide variety of physical systems. They exhibit excellent coherence properties, their microscopic dynamics are well-understood, and they can be dynamically controlled with great precision. Recent advances in cooling and trapping gases of dipolar molecules and Rydberg atoms have further increased the versatility of optical lattices as quantum simulators, particularly for solid state systems possessing long-range interactions. In this thesis we further examine the microscopic properties of cold atoms trapped in an optical lattice, going beyond the case where the lattice axes are orthogonal. We present a transformation of the single-particle energy eigenstates to a complete, orthonormal basis of so-called 'generalised Wannier functions' that are localised to each lattice site. Generalised Wannier functions, as opposed to ordinary Wannier functions, are formed by mixing states between bands and are required when the bands are degenerate. We modify an algorithm devised by Marzari and Vanderbilt (Phys. Rev. B 56(20) 12847) for calculating maximally localised generalised Wannier functions, specifically to determine these functions for optical lattice systems of arbitrary geometry, including those with degenerate lower bands. Moreover, we overcome an issue with the initialisation of the original algorithm to find the maximally localised set without fail. We then present results for a variety of optical lattice systems in one- and two-dimensions, including hexagonal and Kagomé geometries. In each case we use the generalised Wannier functions to calculate the hopping and interaction parameters, and thus determine a reduced form of the system's Hamiltonian that contains only local interactions. We also study two spin-1/2 systems with long-range interactions as paradigm examples of the models that can be simulated using optical lattices loaded with dipolar molecules. We first examine the critical properties of a 1D spin-1/2 Ising chain possessing exponentially decaying interactions as the range of the interactions is increased. Using the infinite density matrix renormalisation group method, we calculate the location of the critical point and observe a smooth approach to the classical mean-field result in the limit of infinite-ranged interactions. We go on to consider the quantum dynamics of a single spin-1/2 coupled to a bath of interacting spins as a model for decoherence in solid state quantum memories. The large degree of symmetry of the bath, described by the infinite-ranged Lipkin-Meshkov-Glick model, allows us to find parameter regimes where the initial qubit state is revived at well-defined times after the qubit preparation. We observe that these times may become independent of the bath size for large baths and thus enable faithful qubit storage even in the presence of strong coupling to the bath.