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arXiv:1009.4822 · Physical Review A

Gradient corrections to the local-density approximation for trapped superfluid Fermi gases

András Csordás, O. Almásy, P. Szépfalusy

Published Dec 8, 2010• 16 citations
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Abstract

Two species superfluid Fermi gas is investigated on the BCS side up to the Feshbach resonance. Using the Greens's function technique gradient corrections are calculated to the generalized Thomas-Fermi theory including Cooper pairing. Their relative magnitude is found to be measured by the small parameter $(d/{R}_{\mathrm{TF}}){}^{4}$, where $d$ is the oscillator length of the trap potential and ${R}_{\mathrm{TF}}$ is the radial extension of the density $n$ in the Thomas-Fermi approximation. In particular, at the Feshbach resonance the universal corrections to the local density approximation are calculated and a universal prefactor ${\ensuremath{\kappa}}_{W}=7/27$ is derived for the von Weizs\"acker--type correction ${\ensuremath{\kappa}}_{W}(\ensuremath{\hbar}{}^{2}/2m)({\ensuremath{\nabla}}^{2}{n}^{1/2}/{n}^{1/2})$.

DOI
10.1103/physreva.82.063609

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Two species superfluid Fermi gas is investigated on the BCS side up to the Feshbach resonance. Using the Greens's function technique gradient corrections are calculated to the generalized Thomas-Fermi theory including Cooper pairing. Their relative magnitude is found to be measured by the small parameter (d/RTF)4(d/{R}_{\mathrm{TF}}){}^{4}(d/RTF​)4, where ddd is the oscillator length of the trap potential and RTF{R}_{\mathrm{TF}}RTF​ is the radial extension of the density nnn in the Thomas-Fermi approximation. In particular, at the Feshbach resonance the universal corrections to the local density approximation are calculated and a universal prefactor \ensuremathκW=7/27{\ensuremath{\kappa}}_{W}=7/27\ensuremathκW​=7/27 is derived for the von Weizs"acker--type correction \ensuremathκW(\ensuremathℏ2/2m)(\ensuremath∇2n1/2/n1/2){\ensuremath{\kappa}}_{W}(\ensuremath{\hbar}{}^{2}/2m)({\ensuremath{\nabla}}^{2}{n}^{1/2}/{n}^{1/2})\ensuremathκW​(\ensuremathℏ2/2m)(\ensuremath∇2n1/2/n1/2).
We present a generalization of fermionic fluiddynamics to the case of two trapped fermion species with a contact interaction. Within a mean field approximation, we derive coupled equations of motion for the particle densities, particle currents, and anomalous pair density. For an inhomogeneous system, the equilibrium situation with vanishing currents is described by a generalized Thomas-Fermi relation that includes the superfluid gap, together with a new nonlocal gap equation that replaces the usual BCS one. These equations are numericaly solved resorting to a local density approximation (LDA). Density and gap profiles are analyzed in terms of the scattering length, revealing that the current frame can exhibit microscopic details of quantum origin that are frequently absent in more macroscopic scenarios.
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.
We investigate the single-particle properties at T=0T=0T=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--Bose-Einstein condensate (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 67, 063612 (2003)] to include the effect of a harmonic trap. We do not use the local density approximation (LDA), but directly solve the Bogoliubov--de Gennes (BdG) coupled equations. While our explicit numerical solutions are for a weak (narrow) Feshbach resonance, we argue that the single-particle BdG excitation spectrum will exhibit the same features for a strong (broad) Feshbach resonance. Using these equations, we find self-consistent values for the spatially dependent local density n(r)n(\mathbf{r})n(r) as well as the composite BCS order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(\mathbf{r})$, the latter describing both the Cooper-pair and molecular condensate contributions. Using these results, we calculate the single-particle density of states in the crossover region, and from this determine the true single-particle energy gap (Eg)({E}_{g})(Eg​) of the trapped Fermi superfluid. This is associated with the in-gap (or Andreev) states in the low-density region at the edge of the trap. We calculate the laser-induced tunneling current $I(\ensuremath{\omega})$ into another hyperfine state, as measured in recent rf spectroscopy experiments. This rf spectrum gives a direct probe of the quasiparticle spectrum. We show how the high-energy part of $I(\ensuremath{\omega})$ gives information about $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r=0)$ at the center of the trap (which is comparable to the Fermi energy ${\ensuremath{\epsilon}}_{F}$ in the crossover region). We show that $I(\ensuremath{\omega})$ is very dependent on the spatial profile of the pair potential $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r)$ that is used. We also emphasize that the narrow ``unpaired atom" peak in the rf data gives information about Eg{E}_{g}Eg​ and the low-energy $(⪡{\ensuremath{\epsilon}}_{F})$ in-gap states of a Fermi superfluid. While our calculations are limited to T=0T=0T=0, we use them to discuss the recent data of Chin et al. and the LDA calculations of T\"orm\"a and co-workers. The LDA, while useful, can lead to an incorrect physical picture of the low-density surface region of the Fermi superfluid.
We study the extended Thomas-Fermi (ETF) density functional of the superfluid unitary Fermi gas. This functional includes a gradient term which is essential to describe accurately the surface effects of the system, in particular with a small number of atoms, where the Thomas-Fermi (local density) approximation fails. We find that our ETF functional gives density profiles which are in good agreement with recent Monte Carlo results and also with a more sophisticated superfluid density functional based on Bogoliubov-de Gennes equations. In addition, by using extended hydrodynamics equations of superfluids, we calculate the frequencies of collective surface oscillations of the unitary Fermi gas, showing that quadrupole and octupole modes strongly depend on the number of trapped atoms.
We apply the formulation developed in a recent paper [Y. Ohashi and A. Griffin, Phys. Rev. A 72, 013601 (2005)] for single-particle excitations in the BCS-BEC crossover to the case of a broad Feshbach resonance. At T=0T=0T=0, we solve the Bogoliubov--de Gennes coupled equations taking into account a Bose condensate of bound states (molecules). In the case of a broad resonance, the density profile n(r)n(r)n(r), as well as the profile of the superfluid order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r)$, are spatially spread out to the Thomas-Fermi radius, even in the crossover region. This order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r)$ suppresses the effects of low-energy Andreev bound states on the rf tunneling current. As a result, the peak energy in the rf spectrum is found to occur at an energy equal to the superfluid order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r=0)$ at the center of the trap, in contrast to the case of a narrow resonance, and in agreement with recent measurements. The local density approximation is found to give a good approximation for the rf-tunneling spectrum.