Ultracold atomic Fermi gases present an opportunity to study strongly interacting fermionic systems in a controlled and uncomplicated setting. The ability to tune attractive interactions has led to the discovery of superfluidity in these systems with an extremely high transition temperature with respect to the Fermi temperature near T/TF=0.2. This superfluidity is the electrically neutral analogue of superconductivity; however, superfluidity in atomic Fermi gases occurs in the limit ofstrong interactions and defies a conventional Bardeen–Cooper– Schrieffer (BCS) description. For these strong interactions, it is predicted that the onset of pairing and superfluidity can occur at different temperatures. Thus, for a range of temperatures, a pseudogap region may exist, in which the system retains some of the characteristics of the superfluid phase—such as a BCS-like dispersion and a partially gapped density of states—but does not exhibit superfluidity. By making two independent measurements—the direct observation of pair condensation in momentum space and a measurement of the single-particle spectral function using an analogue to photoemission spectroscopy6—we directly probe the pseudogap phase. Our measurements reveal a BCS-like dispersion with back-bending near the Fermiwavevector kF, which persists well above the transition temperature for pair condensation

The crossover from weak to strong coupling for a three dimensional continuum model of fermions interacting via an attractive contact potential is studied above the superconducting critical temperature Tc. The pair-fluctuation propagator, the one-loop self-energy, and the spectral function are investigated in a systematic way from the superconducting fluctuation regime (weak coupling) to the bosonic regime (strong coupling). Analytic and numerical results are reported. In the strongcoupling regime, where the pair fluctuation propagator has bosonic character, two quite different peaks appear in the spectral function at a given wave vector, a broad one at negative frequencies and a narrow one at positive frequencies. The broad peak of the spectral function at negative frequencies is asymmetric about its maximum, with its spectral weight decreasing by increasing coupling and temperature. In this regime, two crossover temperatures T * 1 (at which the two peaks in the spectral function merge in one peak) and T * 0 (at which the maximum of the lower peak crosses zero frequency) can be identified, with Tc ≪ T * 0 < T * 1 . By decreasing coupling, the two-peak structure evolves smoothly. In the weak-coupling regime, where the fluctuation propagator has diffusive Ginzburg-Landau character, the overall line-shape of the spectral function is more symmetric and the two crossover temperatures approach Tc. The systematic analysis of the spectral function identifies specific features which allow one to distinguish by ARPES whether a system is in the weak-or strong-coupling regime. Connection of the results of our analysis with the phenomenology of cuprate superconductors is also attempted and rests on the recently introduced two-gap model , according to which a crossover from weak to strong coupling is realized when moving in the Brillouin zone away from the nodal points toward the M points where the d-wave gap acquires its maximum value.PACS numbers: 74.25.Jb

This report adresses topics and questions of common interest in the fields of ultra-cold gases and nuclear physics in the context of the BCS-BEC crossover. By this crossover, the phenomena of Bardeen-Cooper-Schrieffer (BCS) superfluidity and Bose-Einstein condensation (BEC), which share the same kind of spontaneous symmetry breaking, are smoothly connected through the progressive reduction of the size of the fermion pairs involved as the fundamental entities in both phenomena. This size ranges, from large values when Cooper pairs are strongly overlapping in the BCS limit of a weak inter-particle attraction, to small values when composite bosons are non-overlapping in the BEC limit of a strong inter-particle attraction, across the intermediate unitarity limit where the size of the pairs is comparable with the average inter-particle distance.The BCS-BEC crossover has recently been realized experimentally, and essentially in all of its aspects, with ultracold Fermi gases. This realization, in turn, has raised the interest of the nuclear physics community in the crossover problem, since it represents an unprecedented tool to test fundamental and unanswered questions of nuclear manybody theory. Here, we focus on the several aspects of the BCS-BEC crossover, which are of broad joint interest to both ultra-cold Fermi gases and nuclear matter, and which will likely help to solve in the future some open problems in nuclear physics (concerning, for instance, neutron stars). Similarities and differences occurring in ultra-cold Fermi gases and nuclear matter will then be emphasized, not only about the relative phenomenologies but also about the theoretical approaches to be used in the two contexts. Common to both contexts is the fact that at zero temperature the BCS-BEC crossover can be described at the mean-field level with reasonable accuracy. At finite temperature, on the other hand, inclusion of pairing fluctuations beyond mean field represents an essential ingredient of the theory, especially in the normal phase where they account for precursor pairing effects.After an introduction to present the key concepts of the BCS-BEC crossover, this report discusses the mean-field treatment of the superfluid phase, both for homogeneous and inhomogeneous systems, as well as for symmetric (spinor isospin-balanced) and asymmetric (spin-or isospin-imbalanced) matter. Pairing fluctuations in the normal phase are then considered, with their manifestations in thermodynamic and dynamic quantities. The last two Sections provide a more specialized discussion of the BCS-BEC crossover in ultra-cold Fermi gases and nuclear matter, respectively. The separate discussion in the two contexts aims at cross communicating to both communities topics and aspects which, albeit arising in one of the two fields, share a strong common interest.

We consider the problem of the crossover from BCS superconductivity to Bose-Einstein condensation in three dimensions for a system of fermions with an attractive interaction, for which we adopt the simplifying assumption of a suitably regularized point-contact interaction. We examine in a critical way the fermionic (self-consistent) T-matrix approximation which has been widely utilized in the literature to describe this crossover above the superconducting critical temperature, and show that it fails to yield the correct behaviour of the system in the strong-coupling limit, where composite bosons form as tightly bound fermion pairs. We then set up the correct approximation for a "dilute" system of composite bosons and show that an entire new class of diagrams has to be considered in the place of the fermionic T-matrix approximation for the self-energy. This new class of diagrams correctly describes both the weak-and strong-coupling limits, and consequently results into an improved interpolation scheme for the intermediate (crossover) region. In this context, we provide also a systematic mapping between the corresponding diagrammatic theories for the composite bosons and the constituent fermions. As a preliminary result to demonstrate the numerical effect of our new class of diagrams on physical quantities, we calculate the value of the scattering length for composite bosons in the strong-coupling limit and show that it is considerably modified with respect to the result obtained within the self-consistent fermionic T-matrix approximation.