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We provide a comprehensive theoretical study of the radio-frequency (rf) spectra of a two-component Fermi gas with balanced populations in the normal region of the temperature-vs-coupling phase diagram. In particular, rf spectra are analyzed in terms of two characteristic peaks, which can be either distinct or overlapping. On the BEC side of the crossover, these two contributions are associated with a fermionic quasi-particle peak and a bosonic-like contribution due to pairing. On the BCS side of the crossover, the two peaks are instead associated with interactions between particles occurring, respectively, at high or low relative momenta. Through this two-peak analysis, we show how and to what extent the correlation between the widths of the rf spectra and the pair size, previously identified in the superfluid phase at low temperature, can be extended to the normal phase, as well as how the temperature-vs-coupling phase diagram of the BCS-BEC crossover can be partitioned in a number of distinct physical sectors. Several analytic results for the shape and widths of the rf spectra are also derived in appropriate temperature and coupling limits., Comment: 23 pages, 16 figures

In a superfluid system, Off-Diagonal Long-Range Order (ODLRO) is expected to be exhibited in the appropriate reduced density matrices when the relevant particles (either bosons or fermion pairs) are considered to recede sufficiently far apart from each other. This concept is usually exploited to identify the value of the condensate density, without explicit concern, however, on the spatial range over which this asymptotic condition can effectively be achieved. Here, based on a diagrammatic approach that includes beyond-mean-field pairing fluctuations in the broken-symmetry phase at the level of the $t$-matrix also with the inclusion of the Gorkov-Melik-Barkhudarov (GMB) correction, we present a systematic study of the two-particle reduced density matrix for a superfluid fermionic system undergoing the BCS-BEC crossover, when the entities to recede far apart from each other evolve with continuity from largely overlapping Cooper pairs in the BCS limit to dilute composite bosons in the BEC limit. By this approach, we not only provide the coupling and temperature dependence of the condensate density at the level of our diagrammatic approach which includes the GMB correction, but we also obtain the evolution of the spatial dependence of the two-particle reduced density matrix, from a power-law at low temperature to an exponential dependence at high temperature in the superfluid phase, when the inter-particle coupling spans the BCS-BEC crossover. Our results put limitations on the minimum spatial extent of a finite-size system for which superfluid correlations can effectively be established., Comment: 21 pages, 13 figures

We calculate the pair susceptibility of an attractive spin-polarized Fermi gas in the normal phase, as a function of the pair momentum. Close to unitarity, we find a strong enhancement of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing fluctuations over an extended region of the temperature-polarization phase diagram, which manifests itself as a pronounced peak in the pair-momentum distribution at a finite pair momentum. This peak should be amenable to experimental observation at achievable temperatures in a box-like trapping potential, as a fingerprint of FFLO pairing. Our calculations rest on a self-consistent t-matrix approach which, for the unitary balanced Fermi gas, has been validated against experimental data for several thermodynamic quantities., Comment: 11 pages, 7 figures; includes calculations for a realistic box-like trapping potential

A detailed description is given of the phase diagram for a two-component unitary Fermi gas with mass and population imbalance, for both homogeneous and trapped systems. This aims at providing quantitative benchmarks for the normal-to-superfluid phase transition of a mass-imbalanced Fermi gas in the temperature-polarization parameter space. A self-consistent t-matrix approach is adopted, which has already proven to accurately describe the thermodynamic properties of the mass and population balanced unitary Fermi gas. Our results provide a guideline for the ongoing experiments on heteronuclear Fermi mixtures., Comment: 10 pages, 10 figures, final version

In a recent paper [Phys. Rev. A 99, 053617 (2019)], the total number of fermion pairs in a spin-balanced two-component Fermi gas of $^6$Li atoms was experimentally probed in the normal phase above the superfluid critical temperature, in order to investigate the sectors of pseudogap and preformed-pair in the temperature-coupling phase diagram. Here, we present a theoretical account of these experimental results in terms of an ab-initio self-consistent $t$-matrix calculation, which emphasizes the role of the pair-correlation function between opposite-spin fermions at equilibrium. Good agreement is found between the available experimental data and the theoretical results obtained with no adjustable parameter., Comment: 16 pages, 11 figures

The bound states that can occur in a superfluid vortex have recently called for attention owing to the capability of detecting them experimentally. However, a detailed theoretical account for the presence of these vortex bound states is still lacking, for all temperatures in the superfluid phase and couplings along the BCS-BEC crossover. Here, we fill this gap and present a systematic theoretical study based on the Bogoliubov-de~Gennes equations for the bound states that occur over the two characteristic (inner and outer) spatial ranges in which the extension of a superfluid vortex can be partitioned. It is found that the total number of bound states decreases from the BCS (weak-coupling) side of the crossover toward the intermediate-coupling region where they are still present, whereas the bound states disappear upon entering the BEC (strong-coupling) side. A scaling relation is also obtained that connects the number of bound states in the inner spatial range of the vortex to the depth and width of the vortex itself. A criterion is finally provided in terms of the local density of states, to distinguish where a given fermionic superfluid is located in the coupling-temperature phase diagram of the BCS-BEC crossover., Comment: 10 pages, 7 figures

The diagrammatic t-matrix approximation has often been adopted to describe a dilute Fermi gas. This approximation, originally considered by Galitskii for a repulsive inter-particle interaction (Galitskii-1958), has later been widely utilized for an attractive Fermi gas to describe the BCS-BEC crossover from strongly overlapping Cooper pairs in weak coupling to non-overlapping composite bosons in strong coupling. Several variants of the $t$-matrix approximation have been considered in the literature, which are distinguished by the degree of self-consistency allowed in the building blocks of the diagrammatic structure. Here, we perform a systematic and comparative study of all possible variants on the degree of self-consistency for the $t$-matrix approximation in an attractive Fermi gas, which enables us to confront their outcomes for thermodynamic and dynamical quantities on the same footing in an unbiased way. For definiteness, only the normal phase above the superfluid critical temperature is considered. The dispute that can be raised in this context, about the adequateness of introducing progressive degrees of self-consistency over and above the non-self-consistent $t$-matrix approximation for an attractive Fermi gas, parallels the recent interest in the literature on assessing the importance of various degrees of self-consistency in the context of semiconductors and insulators., Comment: 20 pages, 13 figures

The proximity effect, which arises at the interface between two fermionic superfluids with different critical temperatures, is examined with a non-local (integral) equation whose kernel contains information about the size of Cooper pairs that leak across the interface. This integral approach avoids reference to the boundary conditions at the interface that would be required with a differential approach. The temperature dependence of the pair penetration depth on the normal side of the interface is determined over a wide temperature range, also varying the inter-particle coupling along the BCS-BEC crossover independently on both sides of the interface. Conditions are found for which the proximity effect is optimized in terms of the extension of the pair penetration depth.

The equation for the gap parameter represents the main equation of the pairing theory of superconductivity. Although it is formally defined through a single-particle property, physically it reflects the pairing correlations between opposite-spin fermions. Here, we exploit this physical connection and cast the gap equation in an alternative form which explicitly highlights these two-particle correlations, by showing that it is equivalent to a Hugenholtz-Pines condition for fermion pairs. At a formal level, a direct connection is established in this way between the treatment of the condensate fraction in condensate systems of fermions and bosons. At a practical level, the use of this alternative form of the gap equation is expected to make easier the inclusion of pairing fluctuations beyond mean field. As a proof-of-concept of the new method, we apply the modified form of the gap equation to the long-pending problem about the inclusion of the Gorkov-Melik-Barkhudarov correction across the whole BCS-BEC crossover, from the BCS limit of strongly overlapping Cooper pairs to the BEC limit of dilute composite bosons, and for all temperatures in the superfluid phase. Our numerical calculations yield excellent agreement with the recently determined experimental values of the gap parameter for an ultra-cold Fermi gas in the intermediate regime between BCS and BEC, as well as with the available quantum Monte Carlo data in the same regime., Comment: 24 pages, 13 figures

The theoretical description of the critical temperature Tc of a Fermi superfluid dates back to the work by Gor'kov and Melik-Barkhudarov (GMB), who addressed it for a weakly-coupled (dilute) superfluid in the BCS (weak-coupling) limit of the BCS-BEC crossover. The point made by GMB was that particle-particle (pairing) excitations, which are responsible for superfluidity to occur below Tc, and particle-hole excitations, which give rise to screening also in a normal system, get effectively disentangled from each other in the BCS limit, thus yielding a reduction by a factor 2.2 of the value of Tc obtained when neglecting screening effects. Subsequent work on this topic, aimed at extending the original GMB argument away from the BCS limit with diagrammatic methods, has kept this disentangling between pairing and screening throughout the BCS-BEC crossover, without realising that the conditions for it to be valid are soon violated away from the BCS limit. Here, we reconsider this problem from a more general perspective and argue that pairing and screening are intrinsically entangled with each other along the whole BCS-BEC crossover but for the BCS limit considered by GMB. We perform a detailed numerical calculation of the GMB diagrammatic contribution extended to the whole BCS-BEC crossover, where the full wave-vector and frequency dependence occurring in the repeated in-medium two-particle scattering is duly taken into account. Our numerical calculations are tested against analytic results available in both the BCS and BEC limits, and the contribution of the GMB diagrammatic term to the scattering length of composite bosons in the BEC limit is highlighted. We calculate Tc throughout the BCS-BEC crossover and find that it agrees quite well with Quantum Monte Carlo calculations and experimental data available in the unitarity regime., Comment: 21 pages, 11 figures

The properties are considered in detail of a non-local (integral) equation for the superconducting gap parameter, which is obtained by a coarse-graining procedure applied to the Bogoliubov-deGennes (BdG) equations over the whole coupling-vs-temperature phase diagram associated with the superfluid phase. It is found that the limiting size of the coarse-graining procedure, which is dictated by the range of the kernel of this integral equation, corresponds to the size of the Cooper pairs over the whole coupling-vs-temperature phase diagram up to the critical temperature, even when Cooper pairs turn into composite bosons on the BEC side of the BCS-BEC crossover. A practical method is further implemented to solve numerically this integral equation in an efficient way, which is based on a novel algorithm for calculating the Fourier transforms. Application of this method to the case of an isolated vortex, throughout the BCS-BEC crossover and for all temperatures in the superfluid phase, helps clarifying the nature of the length scales associated with a single vortex and the kinds of details that are in practice disposed off by the coarse-graining procedure on the BdG equations.