Abstract:
We analyze the effects of imbalancing the populations of two-component trapped fermions, in the BEC limit of the attractive interaction between different fermions. Starting from the gap equation with two fermionic chemical potentials, we derive a set of coupled equations that describe composite bosons and excess fermions. We include in these equations the processes leading to the correct dimer-dimer and dimer-fermion scattering lengths. The coupled equations are then solved in the Thomas-Fermi approximation to obtain the density profiles for composite bosons and excess fermions, which are relevant to the recent experiments with trapped fermionic atoms

Abstract:
We derive the time-independent Gross-Pitaevskii equation at zero temperature for condensed bosons, which form as bound-fermion pairs when the mutual fermionic attractive interaction is sufficiently strong, from the strong-coupling limit of the Bogoliubov-de Gennes equations that describe superfluid fermions in the presence of an external potential. Three-body corrections to the Gross-Pitaevskii equation are also obtained by our approach. Our results are relevant to the recent advances with ultracold fermionic atoms in a trap.

Abstract:
The linear response to a space- and time-dependent external disturbance of a system of dilute condensed composite bosons at zero temperature, as obtained from the linearized version of the time-dependent Gross-Pitaevskii equation, is shown to result also from the strong-coupling limit of the time-dependent BCS (or broken-symmetry RPA) approximation for the constituent fermions subject to the same external disturbance. In this way, it is possible to connect excited-state properties of the bosonic and fermionic systems by placing the Gross-Pitaevskii equation in perspective with the corresponding fermionic approximations

Abstract:
The present manuscript concerns the calculation of the boson-boson scattering length for the composite bosons that form as bound-fermion pairs in the strong-coupling limit of the BCS-BEC crossover. The material presented in this manuscript was already published as a part of a longer paper on the BCS-BEC crossover problem (P. Pieri and G.C. Strinati, Phys. Rev. B 61, 15370 (2000)). Given the recent experimental advances on the formation of ultracold bosonic molecules from a Fermi gas of atoms by a Feshbach resonance, the calculation of the boson-boson scattering length has now become of particular interest. The present short version of the above paper could thus be helpful to the scientific community working on ultracold atomic gases. Accordingly, the present manuscript is intended for circulation as a preprint only.

Abstract:
A long-standing problem with the many-body approximations for interacting condensed bosons has been the dichotomy between the ``conserving'' and ``gapless'' approximations, which either obey the conservations laws or satisfy the Hugenholtz-Pines condition for a gapless excitation spectrum, in the order. It is here shown that such a dichotomy does not exist for a system of composite bosons, which form as bound-fermion pairs in the strong-coupling limit of the fermionic attraction. By starting from the constituent fermions, for which conserving approximations can be constructed for any value of the mutual attraction according to the Baym-Kadanoff prescriptions, it is shown that these approximations also result in a gapless excitation spectrum for the boson-like propagators in the broken-symmetry phase. This holds provided the corresponding equations for the fermionic single- and two-particle Green's functions are solved self-consistently.

Abstract:
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.

Abstract:
We investigate the density, current, and spin response functions above the critical temperature for a system of three-dimensional fermions interacting via an attractive short-range potential. In the strong-coupling (bosonic) limit of this interaction, we identify the dominant diagrammatic contributions for a ``dilute'' system of composite bosons which form as bound-fermion pairs, and compare them with the usual (Aslamazov-Larkin, Maki-Thompson, and density-of-states) terms occurring in the theory of superconducting fluctuations above the critical temperature for a clean system in the weak-coupling limit. We show that, at the zeroth order in the diluteness parameter for the composite bosons, the Aslamazov-Larkin term still represents formally the dominant contribution to the density and current response functions, while the Maki-Thompson and density-of-states terms are strongly suppressed. Corrections to the Aslamazov-Larkin term are then considered at the next order in the diluteness parameter for the composite bosons. The spin response function is also examined, and it is found to be exponentially suppressed in the bosonic limit only when appropriate sets of diagrams are considered simultaneously.

Abstract:
We analyze the effects of imbalancing the populations of two-component trapped fermions in the BEC (strong-coupling) limit of the attractive interaction between fermions of different components. In particular, we derive a set of coupled equations which describe composite bosons and excess fermions in this limit, starting from the gap equation with two different fermionic chemical potentials. Care is used to include in these equations the processes leading to the correct dimer-dimer and dimer-fermion scattering lengths, which require us to consider beyond-mean-field effects. Numerical results are presented for the density profiles of composite bosons and excess fermions. Results for the formation of vortex patterns in the presence of density imbalance are also presented.

Abstract:
Theoretical treatments of the BCS-BEC crossover need to provide as accurate as possible descriptions of the two regimes where the diluteness condition applies, either in terms of the constituent fermions (BCS limit) or of the composite bosons which form as bound-fermion pairs (BEC limit). This has to occur via a single fermionic theory that bridges across these two limiting representations. In this paper, we set up successive improvements of the fermionic theory, that result into composite bosons described at the level of either the Bogoliubov or the Popov approximations for point-like bosons. This work bears on the recent experimental advances on the BCS-BEC crossover with trapped Fermi atoms, which show the need for accurate theoretical descriptions of BEC side of the crossover.

Abstract:
Upcoming $\gamma$-ray satellites will search for Dark Matter annihilations in Milky Way substructures (or 'clumps'). The prospects for detecting these objects strongly depend on the assumptions made on the distribution of Dark Matter in substructures, and on the distribution of substructures in the Milky Way halo. By adopting simplified, yet rather extreme, prescriptions for these quantities, we compute the number of sources that can be detected with upcoming experiments such as GLAST, and show that, for the most optimistic particle physics setup ($m_\chi=40$ GeV and annihilation cross section $\sigma v = 3 \times 10^{-26}$ cm$^3$ s$^{-1}$), the result ranges from zero to $\sim$ hundred sources, all with mass above $10^{5}M\odot$. However, for a fiducial DM candidate with mass $m_\chi=100$ GeV and $\sigma v = 10^{-26}$ cm$^3$ s$^{-1}$, at most a handful of large mass substructures can be detected at $5 \sigma$, with a 1-year exposure time, by a GLAST-like experiment. Scenarios where micro-clumps (i.e. clumps with mass as small as $10^{-6}M\odot$) can be detected are severely constrained by the diffuse $\gamma$-ray background detected by EGRET.