From an extensive calculation of static properties of a trapped Fermi superfluid at zero temperature using a density-functional formulation, we demonstrate a universal behavior of its observables, such as energy, chemical potential, radius etc., over the crossover from the BCS limit to unitarity leading to scaling over many orders of magnitude in fermion number. This scaling allows to predict the static properties of the system, with a large number ($\sim 10^5$) of fermions, over the crossover with an error of 1-2%, from the knowledge of those for a small number ($\sim 10$) of fermions.
The Josephson relation for the superfluid density is derived for a Fermi superfluid in the BCS-BEC crossover. This identity extends the original Josephson relation for Bose superfluids. It gives a simple exact relation between the superfluid density $ρ_s$ and the broken-symmetry Cooper pair order parameter $Δ_0$ in terms of the infrared limit of the pair fluctuation propagator. The same expression holds through the entire BCS-BEC crossover, describing the superfluid density of a weak-coupling BCS superfluid as well as the superfluid density of a Bose condensate of dimer molecules.
We consider the evolution of superfluid properties of a three dimensional p-wave Fermi gas from weak (BCS) to strong (BEC) coupling as a function of scattering volume. We analyse the order parameter, quasi-particle excitation spectrum, chemical potential, average Cooper pair size and the momentum distribution in the ground state ($T = 0$). We also discuss the critical temperature $T_{\rm c}$, chemical potential and number of unbound, scattering and bound fermions in the normal state ($T = T_{\rm c}$). Lastly, we derive the time-dependent Ginzburg-Landau equation for $T \approx T_{\rm c}$ and e…
Critical velocities have been observed in an ultracold superfluid Fermi gas throughout the BEC-BCS crossover. A pronounced peak of the critical velocity at unitarity demonstrates that superfluidity is most robust for resonant atomic interactions. Critical velocities were determined from the abrupt onset of dissipation when the velocity of a moving one dimensional optical lattice was varied. The dependence of the critical velocity on lattice depth and on the inhomogeneous density profile was studied.
We consider cigar shaped Fermi superfluid in the BEC-BCS crossover. Using polytropic form of equation of state, we derive low energy multibranch bosonic excitations and the corresponding density fluctuations in three different regimes along the crossover, namely weak-coupling BCS, unitarity and molecular BEC regimes. Bragg spectroscopy can be used to probe the multibranch nature of the low energy bosonic excitations by measuring dynamic structure factor. Therefore, we calculate dynamic structure factor in those three different regimes. In the Bragg spectroscopy, an actual observable is momentu…
We study the stationary Josephson effect for neutral fermions across the BCS-BEC crossover, by solving numerically the Bogoliubov-de Gennes equations at zero temperature. The Josephson current is found to be considerably enhanced for all barriers at about unitarity. For vanishing barrier, the Josephson critical current approaches the Landau limiting value which, depending on the coupling, is determined by either pair-breaking or sound-mode excitations. In the coupling range from the BCS limit to unitarity, a procedure is proposed to extract the pairing gap from the Landau limiting current.
We introduce a quasi-analytic nonlinear Schrödinger equation with beyond mean-field corrections to describe the dynamics of a zero-temperature dilute superfluid Fermi gas in the crossover from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) regime, where $k_F|a| \ll 1$ with $a$ the s-wave scattering length and $k_F$ the Fermi momentum, through the unitarity limit, $k_Fa \to \pm \infty$, to the Bose regime where $k_Fa >0$. The energy of our model is parametrized using the known asymptotic behavior in the BCS, Bose, and the unitarity limits and is in excellent agreement with accurate Green fun…
We derive an expression for the superfluid density of a uniform two-component Fermi gas through the BCS-BEC crossover in terms of the thermodynamic potential in the presence of an imposed superfluid flow. Treating the pairing fluctuations in a Gaussian approximation following the approach of Nozières and Schmitt-Rink, we use this definition of $ρ_s$ to obtain an explicit result which is valid at finite temperatures and over the full BCS-BEC crossover. It is crucial that the BCS gap $Δ$, the chemical potential $μ$, and $ρ_s$ all include the effect of fluctuations at the same level in a self-con…
We show that both DC and AC Josephson effects with superfluid Fermi atoms in the BCS-BEC crossover can be described at zero temperature by a nonlinear Schrodinger equation (NLSE). By comparing our NLSE with mean-field extended BCS calculations, we find that the NLSE is reliable in the BEC side of the crossover up to the unitarity limit. The NLSE can be used for weakly-linked atomic superfluids also in the BCS side of the crossover by taking the tunneling energy as a phenomenological parameter.
In the case of tight transverse confinement (cigar-shaped trap) the three-dimensional (3D) nonlinear Schrödinger equation, describing superfluid Fermi atoms at unitarity (infinite scattering length $|a|\to \infty$), is reduced to an effective one-dimensional form by averaging over the transverse coordinates. The resultant effective equation is a 1D nonpolynomial Schrodinger equation, which produces results in good agreement with the original 3D one. In the limit of small and large fermion number $N$ the nonlinearity is of simple power-law type. A similar reduction of the 3D theory to a two-dim…
We develop a variational approach to calculate the density response function at finite temperatures of the lowest-lying two-fluid modes in a trapped two-component Fermi superfluid close to a Feshbach resonance. The out-of-phase oscillations, which are the analogue in trapped gases of second sound in uniform superfluids, have so far not been observed in cold-atom experiments. At unitarity, we show that these modes are observable at finite temperatures via two-photon Bragg scattering, whose spectrum is related to the imaginary part of density response function. This provides direct evidence for …
The equation of state (EOS) of a Fermi superfluid is investigated in the BCS-BEC crossover at zero temperature. We discuss the EOS based on Monte-Carlo (MC) data and asymptotic expansions and the EOS derived from the extended BCS (EBCS) mean-field theory. Then we introduce a time-dependent density functional, based on the bulk EOS and Landau's superfluid hydrodynamics with a von Weizsäcker-type correction, to study the free expansion of the Fermi superfluid. We calculate the aspect ratio and the released energy of the expanding Fermi cloud showing that MC EOS and EBCS EOS are both compatible w…
We develop a gauge-invariant linear response theory for relativistic Bardeen-Cooper-Schrieffer (BCS) superfluids based on a consistent-fluctuation-of-the order-parameter (CFOP) approach. The response functions from the CFOP approach satisfy important generalized Ward identities. The gauge invariance of the CFOP theory is a consequence of treating the gauge transformation and the fluctuations of the order parameter on equal footing so collective-mode effects are properly included. We demonstrate that the pole of the response functions is associated with the massless Goldstone boson. Important p…
We investigate the zero-temperature properties of a superfluid Bose-Fermi mixture by introducing a set of coupled Galilei-invariant nonlinear Schrödinger equations valid from weak-coupling to unitarity. The Bose dynamics is described by a Gross-Pitaevskii-type equation including beyond-mean-field corrections possessing the correct weak-coupling and unitarity limits. The dynamics of the two-component Fermi superfluid is described by a density-functional equation including beyond-mean-field terms with correct weak-coupling and unitarity limits. The present set of equations is equivalent to the e…
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 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 …
We determine the effects of quantum fluctuations about the T=0 mean field solution of the BCS-BEC crossover in a dilute Fermi gas using the functional integral method. These fluctuations are described in terms of the zero point motion of collective modes and the virtual scattering of gapped quasiparticles. We calculate their effects on various measurable properties, including chemical potential, ground state energy, the gap, the speed of sound and the Landau critical velocity. At unitarity, we find excellent agreement with quantum Monte Carlo and experimental results. In the BCS limit, we show…
We calculate the frequencies of the Tkachenko oscillations of a vortex lattice in a harmonically trapped superfluid Fermi gas. We use the elasto-hydrodynamic theory by properly accounting for the elastic constants, the Thomas-Fermi density profile of the atomic cloud, and the boundary conditions. Thanks to the Fermi pressure, which is responsible for larger cloud radii with respect to the case of dilute Bose-Einstein condensed gases, large vortex lattices are achievable in the unitary limit of infinite scattering length, even at relatively small angular velocities. This opens the possibility o…
We investigate unconventional superfluidity in a gas of Fermi atoms with an anisotropic p-wave Feshbach resonance. Including the p-wave Feshbach resonance as well as the associated three kinds of quasi-molecules with finite orbital angular momenta $L_z=\pm1,0$, we calculate the transition temperature of the superfluid phase. As one passes through the p-wave Feshbach resonance, we find the usual BCS-BEC crossover phenomenon. The p-wave BCS state continuously changes into the BEC of bound molecules with L=1. Our calculation includes the effect of fluctuations associated with Cooper-pairs and mol…
The work by T. K. Ghosh and K. Machida [cond-mat/0510160 and Phys. Rev. A 73, 013613 (2006)] on the sound velocity in a cylindrically confined Fermi superfluid obeying a power-law equation of state is shown to make use of an improper projection of the sound wave equation. This inaccuracy fully accounts for the difference between their results and those previously reported by Capuzzi et al. [cond-mat/0509323 and Phys. Rev. A 73, 021603(R) (2006)]. In this Comment we show that both approaches lead exactly to the same result when the correct weight function is used in the projection. Plots of the…
We explore the zero temperature phase behavior of a two-dimensional two-component atomic Fermi gas with population and mass imbalance in the regime of the BEC-BCS crossover. Working in the mean-field approximation, we show that the normal and homogeneous balanced superfluid phases are separated by an inhomogeneous superfluid phase of Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) type. We obtain an analytical expression for the line of continuous transitions separating the normal and inhomogeneous FFLO phases. We further show that the transition from the FFLO phase to the homogeneous balanced superflu…
We present the results of a variational calculation of the frequencies of the low-lying Landau two-fluid hydrodynamic modes in a trapped Fermi superfluid gas at unitarity. Landau's two-fluid hydrodynamics is expected to be the correct theory of Fermi superfluids at finite temperatures close to unitarity, where strong interactions give rise to collisional hydrodynamics. Two-fluid hydrodynamics predicts the existence of in-phase modes in which the superfluid and normal fluid components oscillate together, as well as out-of-phase modes where the two components move against each other. We prove th…
We introduce a nonlinear Schroedinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where $a n^{1/3}\ll 1$ with $a$ the inter-atomic s-wave scattering length and $n$ the bosonic density, to the unitarity limit, where $a\to +\infty$. We call this equation the {unitarity Schroedinger equation} (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core…
We compute the shear viscosity of a superfluid atomic Fermi gas in the unitarity limit. The unitarity limit is characterized by a divergent scattering length between the atoms, and it has been argued that this will result in a very small viscosity. We show that in the low temperature T limit the shear viscosity scales as xi^5/T^5, where the universal parameter 'xi' relates the chemical potential and the Fermi energy, mu=xi E_F. Combined with the high temperature expansions of the viscosity our results suggest that the viscosity has a minimum near the critical temperature T_c. A naive extrapola…
We investigate the collective oscillations in the BCS-BEC crossover region of a trapped Fermi superfluid at T=0. We explicitly include the effect of a Feshbach resonance, which leads to a tunable pairing interaction adjusted by the threshold energy of the resonance, as well as the associated mol ecules. In a previous paper,we obtained solutions of the Bogoliubov-de Gennes coupled equations, describing a trapped Fermi superfluid ga s. Using these Bogoliubov quasi-particle excitations, we calculate the density correlation function in the Hartree-Fock-Bogoliubov generalized random phase approxima…
We investigate fermion pairing in the unitary regime for a mass ratio corresponding to a 6Li-40K mixture using Quantum Monte Carlo methods. The ground-state energy and the average light and heavy particle excitation spectrum for the unpolarized superfluid state are nearly independent of the mass ratio. In the majority light system, the polarized superfluid is close to the energy of a phase separated mixture of nearly fully polarized normal and unpolarized superfluid. For a majority of heavy particles, we find an energy minimum for a normal state with a ratio of ~ 3:1 heavy to light particles. …
We perform a detailed study of the collective mode across the whole BEC-BCS crossover in fermionic gases at zero temperature, covering the whole range of energy beyond the linear regime. This is done on the basis of the dynamical BCS model. We recover first the results of the linear regime in a simple form. Then specific attention is payed to the non linear part of the dispersion relation and its interplay with the continuum of single fermionic excitations. In particular we consider in detail the merging of collective mode into the continuum of single fermionic excitations. This occurs not onl…
Both the trapping geometry and the interatomic interaction strength of a dilute ultracold fermionic gas can be well controlled experimentally. When the interactions are tuned to strong attraction, Cooper pairing of neutral atoms takes place and a BCS superfluid is created. Alternatively, the presence of Feshbach resonances in the interatomic scattering allows populating a molecular (bound) state. These molecules are more tightly bound than the Cooper pairs and can form a Bose-Einstein condensate (BEC). In this contribution, we describe both the BCS and BEC regimes, and the crossover, from a fu…
In this paper we study the superfluid density of the two component Fermi gas in optical lattices with population imbalance. Three different type of phases, the BCS-state (Bardeen, Cooper, and Schrieffer), the FFLO-state (Fulde, Ferrel, Larkin, and Ovchinnikov), and the Sarma state, are considered. We show that the FFLO superfluid density differs from the BCS/Sarma superfluid density in an important way. Although there are dynamical instabilities in the FFLO phase, when the interaction is strong or densities are high, on the weak coupling limit the FFLO phase is found to be stable.
We investigate theoretically the formation of a vortex lattice in a superfluid two-spin component Fermi gas in a rotating harmonic trap, in a BCS-type regime of condensed non-bosonic pairs. Our analytical solution of the superfluid hydrodynamic equations, both for the 2D BCS equation of state and for the 3D unitary quantum gas, predicts that the vortex free gas is subject to a dynamic instability for fast enough rotation. With a numerical solution of the full time dependent BCS equations in a 2D model, we confirm the existence of this dynamic instability and we show that it leads to the format…
We investigate the single-particle properties at T=0 of a trapped superfluid gas of Fermi atoms with a Feshbach resonance. A tunable pairing interaction associated with the Feshbach resonance leads to the BCS-BEC crossover, where the character of superfluidity continuously changes from the BCS-type to a BEC of composite bosons. In this paper, we extend our previous work for a uniform superfluid Fermi gas [Y. Ohashi and A. Griffin, Phys. Rev. A {\bf 67}, 063612 (2003)] to include the effect of a harmonic trap. We directly solve the Bogoliubov-de Gennes coupled equations, and find self-consisten…
We present a measurement of the potential energy of an ultracold trapped gas of $^{40}$K atoms in the BCS-BEC crossover and investigate the temperature dependence of this energy at a wide Feshbach resonance, where the gas is in the unitarity limit. In particular, we study the ratio of the potential energy in the region of the unitarity limit to that of a non-interacting gas, and in the T=0 limit we extract the universal many-body parameter $β$. We find $β= -0.54^{+0.05}_{-0.12}$; this value is consistent with previous measurements using $^{6}$Li atoms and also with recent theory and Monte Carl…
We investigate a two-species Fermi gas with a large mass ratio interacting by an interspecies short-range interaction. Using the Born-Oppenheimer approximation, we determine the interaction energy of two heavy fermions immersed in the Fermi sea of light fermions as a function of the s-wave scattering length. In the BCS limit, we recover the perturbative calculation of the effective interaction between heavy fermions. The p-wave projection of the effective interaction is attractive in the BCS limit while it turns out to be repulsive near the unitarity limit. We find that the p-wave attraction r…
We investigate the dynamical properties of a superfluid gas of trapped fermionic atoms in the BCS phase. As a simple example we consider the reaction of the gas to a slow rotation of the trap. It is shown that the currents generated by the rotation can be understood within a two-fluid model similar to the one used in the theory of superconductors, but with a position dependent ratio of normal and superfluid densities. The rather general result of this paper is that already at very low temperatures, far below the critical one, an important normal-fluid component appears in the outer regions of …
