In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations. This region can now be probed over the entire BCS--Bose-Einstein-condensate crossover in trapped Fermi superfluids with a Feshbach resonance, including the unitarity region. Building on the approach initiated by Zaremba, Nikuni, and Griffin in 1999 for trapped atomic Bose gases, we present a variational formulation of two-fluid hydrodynamic collective modes based on the work of Zilsel in 1950 developed for superfluid helium. Assuming a simple variational Ansatz for the superfluid and normal fluid velocities, the frequencies of the hydrodynamic modes are given by solutions of coupled algebraic equations, with constants only involving spatial integrals over various equilibrium thermodynamic derivatives. This variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid differential equations. Our two-fluid results are shown to reduce to those of Pitaevskii and Stringari for a pure superfluid at
T=0.
We elucidate the role of the effective range in the Bose-Einstein-condensate (BEC) to Bardeen-Cooper-Schrieffer (BCS) crossover regime of two-component fermions in three dimensions. In contrast to ultracold Fermi gases near the broad Feshbach resonance, where the interaction can be characterized by the contact-type interaction, the interaction range in general becomes important in the density-induced BEC-BCS crossover discussed in the context of condensed-matter and nuclear systems. By characterizing the nonlocal interaction in terms of the low-energy constants such as scattering length
a and effective range
r, we show how the crossover phenomena are affected by nonzero effective ranges. In particular, we show that the superfluid order parameter is strongly suppressed in the high-density regime and the sound velocity exhibits a nonmonotonic behavior reflecting the mechanical stability of the system. Moreover, we point out that the high-momentum tail associated with the contact parameter can be visible when the magnitude of momentum
k is much less than
1/r. We also discuss the relation between the high-momentum tail and the nuclear contact formalism.
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 Nozi\`eres and Schmitt-Rink, we use this definition of ${\ensuremath{\rho}}_{s}$ 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 $\ensuremath{\Delta}$, the chemical potential $\ensuremath{\mu}$, and ${\ensuremath{\rho}}_{s}$ all include the effect of fluctuations at the same level in a self-consistent manner. We show that the normal fluid density ${\ensuremath{\rho}}_{n}\ensuremath{\equiv}n\ensuremath{-}{\ensuremath{\rho}}_{s}$ naturally separates into a sum of contributions from Fermi BCS quasiparticles $({\ensuremath{\rho}}_{n}^{F})$ and Bose collective modes $({\ensuremath{\rho}}_{n}^{B})$. The expression for ${\ensuremath{\rho}}_{n}^{F}$ is just Landau's formula for a BCS Fermi superfluid but now calculated over the BCS-BEC crossover. The expression for the Bose contribution ${\ensuremath{\rho}}_{n}^{B}$ is more complicated and only reduces to Landau's formula for a Bose superfluid in the extreme BEC limit, where all the fermions have formed stable Bose pairs and the Bogoliubov excitations of the associated molecular Bose condensate are undamped. In a companion paper, we present numerical calculations of ${\ensuremath{\rho}}_{s}$ using an expression equivalent to the one derived in this paper, over the BCS-BEC crossover, including unitarity, and at finite temperatures.
We study a gas of fermions undergoing a wide resonance
s-wave BCS--Bose-Einstein-condensate (BEC) crossover, in the BEC regime at zero temperature. We calculate the chemical potential and the speed of sound of this Bose-Einstein-condensed gas, as well as the condensate depletion, in the low-density approximation. We discuss how higher-order terms in the low-density expansion can be constructed. We demonstrate that the standard BCS-BEC gap equation is invalid in the BEC regime and is inconsistent with the results obtained here. The low-density approximation we employ breaks down in the intermediate BCS-BEC crossover region. Hence our theory is unable to predict how the chemical potential and the speed of sound evolve once the interactions are tuned towards the BCS regime. As a part of our theory, we derive the well-known result for the bosonic scattering length diagrammatically and check that there are no bound states of two bosons.
The collective mode frequencies in isotropic and deformed traps are calculated for general polytropic equation of states, P proportional n(gamma+1), and expressed in terms of gamma and the trap geometry. For molecular and standard Bose-Einstein condensates and Fermi gases near Feshbach resonances, the effective power gamma approximately 0.5-1.3 is calculated from Jastrow type wave-function ansatz, and from the crossover model of Leggett. The resulting mode frequencies are calculated for these phases around the BEC-BCS crossover.
A trapped Fermi gas near a collisional resonance provides a unique laboratory for testing many-body theories in a variety of fields. The ultracold Fermi gas produced in our lab is comprised of the lowest two spin states of
6Li. At 834 G there is a collisional or Feshbach resonance between the two spin states. The scattering length between trapped atoms of opposing spins far exceeds the interparticle spacing of the gas. On resonance, a strongly interacting, unitary, Fermi gas is created which exhibits universal behavior. The unitary Fermi gas is a prototype for other exotic systems in nature from nuclear matter to neutron stars and high temperature superconductors. \n For magnetic fields less than 834 G the scattering length is positive, and pairs Fermi atoms can form molecular dimers. These dimers, comprised of two fermions, are bosons. At ultracold temperatures the molecular bosons populate the lowest energy level and form a Bose Einstein Condensate (BEC). For magnetic fields greater than 834G the scattering length between fermions in opposing spin states is negative, like Cooper pairs formed between electrons in a superconductor. The Bardeen, Cooper, and Shriefer (BCS) theory was developed to describe the pairing effect in the context of superconductors. In our experiment we produce an ultracold unitary gas. By tuning the magnetic field to either side of the Feshbach resonance we can transform the gas into a weakly interacting BEC or BCS superfluid. Therefore, the region near a Feshbach resonance is called the BEC-BCS crossover. \n This dissertation presents a precision measurement of the hydrodynamic sound velocity in an ultracold Fermi gas near a Feshbach resonance. The sound velocity is measured at various magnetic fields both above and below resonance. Moreover, we are able compare our measurements to theoretical descriptions of hydrodynamic sound propagation. Further, our measurement of sound velocity exactly reproduces the non-perturbative case, eliminating the need to consider nonlinear effects. At resonance the sound velocity exhibits universal scaling with the Fermi velocity to within 1.8\\% over a factor of 30 in density. In a near zero temperature unitary gas the average sound velocity at the axial center was measured,
c(0)/vF = 0.364(0.005), as well as the universal constant,
beta = -0.565(0.015). The measurement of sound velocity in an ultracold gas throughout the BEC-BCS crossover provides further evidence of the continuous connection between the physics of the BEC, unitary, and BCS systems.