We analyze a superfluid state of two species gas of fermions trapped in a quasi-1D harmonic potential and interacting via attractive s-wave collisions. It is shown that the gap equation posesses a self consistent solution with an antisymmetric gap function. The gap function has a localized soliton or kink in the center of the trap.
Within the framework of the mean-field hydrodynamic model of a degenerate Fermi gas (DFG), we study, by means of numerical methods and variational approximation (VA), the formation of fundamental gap solitons (FGSs) in a DFG (or in a BCS superfluid generated by weak interaction between spin-up and spin-down fermions), which is trapped in a periodic optical-lattice (OL) potential. An effectively one-dimensional (1D) configuration is considered, assuming strong transverse confinement; in parallel, a proper 1D model of the DFG (which amounts to the known quintic equation for the Tonks-Girardeau gas in the OL) is considered too. The FGSs found in the first two bandgaps of the OL-induced spectrum (unless they are very close to edges of the gaps) feature a ( tightly bound ) shape, being essentially confined to a single cell of the OL. In the second bandgap, we also find antisymmetric tightly bound subfundamental solitons (SFSs), with zero at the midpoint. The SFSs are also confined to a single cell of the OL, but, unlike the FGSs, they are unstable. The predicted solitons, consisting of ∼10 4 –10 5 atoms, can be created by available experimental techniques in the DFG of 6 Li atoms.
We will discuss recent results on the theoretical and experimental investigations on solitons in a superfluid Fermi gas along the BCS-BEC crossover. On the theory side, by solving the Bogoliubov de Gennes equations and looking for real and odd solutions for the order parameter, it has been shown that a dark soliton at unitarity posses a large density contrast and fermionic bound states. The superfluid gap is found to be significantly quenched by the presence of the soliton due to the occurrence of Andreev fermionic bound states localized near the nodal plane of the order parameter. By solving the time dependent Bogoliubov de Gennes equations, also the decay and collisions of dark solitons has been investigated. Recently, a cascade of Solitonic Excitations in a Superfluid Fermi gas has been experimentally observed.
We investigate the behavior of dark solitons in a superfluid Fermi gas along the BCS-BEC crossover by solving the Bogoliubov--de Gennes equations and looking for real and odd solutions for the order parameter. We show that in the resonance unitary region, where the scattering length is large, the density profile of the soliton has a deep minimum, differently from what happens in the BCS regime. The superfluid gap is found to be significantly quenched by the presence of the soliton due to the occurrence of Andreev fermionic bound states localized near the nodal plane of the order parameter.
In the context of ultracold fermionic atoms with effective spin S confined to an elongated trap we study the one-dimensional gas interacting via an attractive δ-function potential using the Bethe ansatz solution. There are N = 2 S + 1 fundamental states: The particles can either be unpaired or clustered in bound states of 2, 3, ..., 2 S and 2S + 1 atoms. In a magnetic field, the rich ground state phase diagram consists of these N states and various mixed phases in which combinations of the fundamental states coexist. The phase diagram simplifies considerably in zero-field, where only bound states of N atoms can exist. Due to the harmonic confinement and within the local density approximation, the density profile of bound states decreases along the tube from the center of the trap to its boundaries. In an array of tubes with weak Josephson tunneling superfluid order may arise. In zero-field the response functions determining the superfluid and density wave order are calculated using conformal field theory and the exact Bethe ansatz solution. The response function for superfluidity consists of a power law with distance, while the correlation function for density waves is a power law of distance times a sinusoidal factor oscillating with distance with a period given by two times the Fermi momentum. For S = 1/2 superfluidity is a possibility for all densities and density waves can be excluded. For S ≥ 3/2 superfluidity may occur at low densities but at high densities it gives way to density waves. We discuss the scenario of phase separation where for S ≥ 3/2 the system has superfluid long-range order toward the trap boundaries and density waves at its center.
Superconductivity and superfluidity of fermions require, within the BCS theory, matching of the Fermi energies of the two interacting Fermion species. Difference in the number densities of the two species leads either to a normal state, to phase separation, or---potentially---to exotic forms of superfluidity such as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, Sarma state, or breached pair state. We consider ultracold Fermi gases with polarization, i.e., spin-density imbalance. We show that, due to the gases being trapped and isolated from the environment in terms of particle exchange, exotic forms of superfluidity appear as a shell around the Bardeen-Cooper-Schrieffer (BCS) superfluid core of the gas and, for large density imbalance, in the core as well. We obtain these results by describing the effect of the trapping potential by using the Bogoliubov-de Gennes equations. For comparison to experiments, we calculate also the condensate fraction, and show that, in the center of the trap, a polarized superfluid leads to a small dip in the central density difference. We compare the results to those given by local density approximation and find qualitatively different behavior.
Using coupled equations for the bosonic and fermionic order parameters, we construct families of gap solitons (GSs) in a nearly one-dimensional Bose-Fermi mixture trapped in a periodic optical-lattice (OL) potential, the boson and fermion components being in the states of the Bose-Einstein condensation and Bardeen-Cooper-Schrieffer superfluid, respectively. Fundamental GSs are compact states trapped, essentially, in a single cell of the lattice. Full families of such solutions are constructed in the first two band gaps of the OL-induced spectrum, by means of variational and numerical methods, which are found to be in good agreement. The families include both intragap and intergap solitons, with the chemical potentials of the boson and fermion components falling in the same or different band gaps, respectively. Nonfundamental states, extended over several lattice cells, are constructed too. The GSs are stable against strong perturbations.