Strongly Interacting Fermions in Optical Lattices

Abstract

This thesis explores certain extraordinary phenomena that occur when a gas of neutral atoms is cooled to the coldest temperatures in the universe --- much colder, in fact, than the electromagnetic radiation that permeates the vacuum of interstellar space. At those extreme temperatures, quantum effects dominate and the collective behaviour of the atoms can have unexpected consequences. For example, Bose-Einstein condensation may occur where the atoms lose their individual identities to coalesce into a macroscopic quantum particle. Although such ultracold atomic gases are interesting in their own right, much of the excitement generated in this field is due to the possibility that studying these gases could shed light on intractable problems in other areas of physics. This is predominantly due to the uniquely high degree of control over various physical parameters that ultracold atomic gases afford to experimentalists. Recent technological advances exploit this advantage to study quantum phenomena in a detail that would not be possible in other systems. For instance, atoms can be made to attract or repel each other, the strength of this interaction can be set to almost any value, and external potentials of various geometries and periodicities can be introduced. In this way, atoms can be used to model phenomena as diverse as the quark-gluon plasmas arising in high-energy particle physics, the colour superfluids conjectured to exist in the core of neutron stars, and the high-temperature superconductivity exhibited by electrons on the ion lattice of certain compounds. Indeed, ultracold atomic gases also have a demonstrated applicability to quantum information and computation. Due to a subtle interplay between electronic and nuclear spins known as the hyperfine interaction, atoms can have either an integer or half-integer total spin quantum number, making them either bosonic or fermionic at low temperatures, respectively. With the exception of chapter 7, the work presented here concerns fermionic atoms in periodic potential formed by interfering laser beams. Indeed, the standing light wave created by the interfering beams gives rise to a lattice potential because of the Stark effect which couples the electronic energy levels of the atoms to the spatially undulating electric field. Furthermore, fermionic atoms can be prepared in two different hyperfine states corresponding to the the \spin-up" and \spin-down" quantum states, and as such mimic electrons moving in the lattice structure of solids. This system is well described by the famous Hubbard model which we introduce in chapter 2 and, under certain conditions, undergoes a phase transition into the N eel state which believed to be a precursor to superconductivity in certain high-temperature superconductors. In chapter 3, we calculate precisely how the N eel state may be achieved in an ultracold fermionic atom gas. When the number of spin-up and spin-down atoms is unequal the system becomes spin-canted and exhibits both ferro- and antiferromagnetic characteristics, as we show in chapter 4. We also nd there are topological excitations present in the quantum spin texture known as merons which have never unambiguously been observed before. In order to form a Bose-Einstein condensate, fermionic atoms must rst form pairs, and can do so in two contrasting ways. The relationship between these two qualitatively di erent forms of pairing is described in chapter 5, and we examine how these two types of pairs transform into one another in an optical lattice in chapter 6. Finally, chapter 7 is a detailed eld-theoretic study of pairing as it occurs in an ultracold Bose gas. There, we nd there is an intriguing bosonic analogy of the two forms of fermion pairing and explore the properties of these pairs