The BCS–BEC crossover: From ultra-cold Fermi gases to nuclear systems

This report addresses 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 ultra-cold 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 many-body 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 (spin- or 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.