Studies of superfluidity in ultracold trapped Fermi gases are attracting physicists from a wide range of sub-disciplines including nuclear, condensed matter and particle physics. The excitement in the field is due, in large part, to the remarkable tuneability of these Fermi gases. One can tune the attractive interaction strength continuously from weak to strong (thereby effecting a transition from a BCS to Bose Einstein condensed (BEC) superfluid). One can introduce polarization into the gases at will, which may lead to long-sought-after, but not yet confirmed, exotic superfluid phases. In this talk we discuss the relevance of the cold Fermi gases to other physics subdisciplines. We then summarize how BCS-BEC crossover in the ultracold gases connects with a particularly important topic in condensed matter: high temperature superconductivity. We emphasize some striking similarities relating to the very unusual normal or 'pseudogap' phase of each of the two systems. In the process we give a summary of some of the latest exciting experimental developments in the two fields.

This thesis reports on the use of scanning tunnelling microscopy to elucidate the atomic-scale electronic structure of a charge density wave, revealing that it has a d-symmetry form factor, hitherto unobserved in nature. It then details the development of an entirely new class of scanned probe: the scanning Josephson tunnelling microscope. This scans the Josephson junction formed between a cuprate superconducting microscope tip and the surface of a cuprate sample, thereby imaging the superfluid density of the sample with nanometer resolution. This novel method is used to establish the existence of a spatially modulated superconducting condensate, something postulated theoretically over half a century ago but never previously observed.

The field of cold atomic gases faced a revolution in 1995 when Bose-Einstein condensation was achieved. Since then, there has been an impressive progress, both experimental and theoretical. The quest for ultra-cold Fermi gases started shortly after the 1995 discovery, and quantum degeneracy in a gas of fermionic atoms was obtained in 1999. The Pauli exclusion principle plays a crucial role in many aspects of ultra-cold Fermi gases, including inhibited interactions with applications to precision measurements, and strong correlations. The path towards strong interactions and pairing of fermions opened up with the discovery in 2003 that molecules formed by fermions near a Feshbach resonance were surprisingly stable against inelastic decay, but featured strong elastic interactions. This remarkable combination was explained by the Pauli exclusion principle and the fact that only inelastic collisions require three fermions to come close to each other. The unexpected stability of strongly interacting fermions and fermion pairs triggered most of the research which was presented at this summer school. It is remarkable foresight (or good luck) that the first steps to organize this summer school were already taken before this discovery. It speaks for the dynamics of the field how dramatically it can change course when new insight is obtained. The contributions in this volume provide a detailed coverage of the experimental techniques for the creation and study of Fermi quantum gases, as well as the theoretical foundation for understanding the properties of these novel systems.

Recent experimental and theoretical progress has elucidated the tunable crossover, in ultracold Fermi gases, from BCS-type superconductors to BEC-type superfluids. The BCS-BEC Crossover and the Unitary Fermi Gas is a collaborative effort by leading international experts to provide an up-to-date introduction and a comprehensive overview of current research in this fast-moving field. It is now understood that the unitary regime that lies right in the middle of the crossover has remarkable universal properties, arising from scale invariance, and has connections with fields as diverse as nuclear physics and string theory. This volume will serve as a first point of reference for active researchers in the field, and will benefit the many non-specialists and graduate students who require a self-contained, approachable exposition of the subject matter.

This primer is aimed at elevating graduate students of condensed matter theory to a level where they can engage in independent research. Topics covered include second quantisation, path and functional field integration, mean-field theory and collective phenomena.

Superfluidity – and closely related to it, superconductivity – are very general phenomena that can occur on vastly different energy scales. Their underlying theoretical mechanism of spontaneous symmetry breaking is even more general and applies to a multitude of physical systems. In these lecture notes, a pedagogical introduction to the field-theory approach to superfluidity is presented. The connection to more traditional approaches, often formulated in a different language, is carefully explained in order to provide a consistent picture that is useful for students and researchers in all fields of physics. After introducing the basic concepts, such as the two-fluid model and the Goldstone mode, selected topics of current research are addressed, such as the BCS-BEC crossover and Cooper pairing with mismatched Fermi momenta.