Strong Correlations in

viii Chapter 1 Ultracold Bosonic Atoms in Optical Lattices 1 1.1 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Bosons in Lattices: The Bose Hubbard Model . . . . . . . . . 14 Chapter 2 Quantum Theory of Ultracold Bosons in Lattices 17 2.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Formal Properties of the Wavefunction |ψ(Σ)〉 . . . . . . . . . 23 2.3 Mean Field State and Gaussian Fluctuations . . . . . . . . . . 26 2.4 Application to the Bose Hubbard Hamiltonian . . . . . . . . . 28 2.4.1 Uniform Lattice . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2 Fully Frustrated Lattices . . . . . . . . . . . . . . . . . 32 2.4.3 Uniform Lattice with a Harmonic Trap . . . . . . . . . 36 2.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 xi 2.5.1 Next Neighbor Interactions . . . . . . . . . . . . . . . . 41 2.5.2 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Chapter 3 Spin-Incoherent Luttinger Liquids 47 3.1 Luttinger Liquids . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Spin-Charge Separation . . . . . . . . . . . . . . . . . . . . . . 62 Chapter 4 Spin-Incoherent Luttinger Liquid-Superconductor Hybrid Systems 68 4.1 Single Particle Greens Function . . . . . . . . . . . . . . . . . 73 4.1.1 Tunneling Density of States . . . . . . . . . . . . . . . 76 4.1.2 Pair Correlations . . . . . . . . . . . . . . . . . . . . . 77 4.2 Josephson Current . . . . . . . . . . . . . . . . . . . . . . . . 79