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Journal of Physics Conference Series

Superfluid Thomas—Fermi approximation for trapped fermi gases

E. Susana Hernández, P. Capuzzi, L. Szybisz

Published Feb 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/150/3/032028

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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.
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 ${\ensuremath{\kappa}}_{W}=7/27$ is derived for the von Weizs\"acker--type correction ${\ensuremath{\kappa}}_{W}(\ensuremath{\hbar}{}^{2}/2m)({\ensuremath{\nabla}}^{2}{n}^{1/2}/{n}^{1/2})$.
A novel Thomas-Fermi (TF) approach to inhomogeneous superfluid Fermi-systems is presented and shown that it works well also in cases where the Local Density Approximation (LDA) breaks down. The novelty lies in the fact that the semiclassical approximation is applied to the pairing matrix elements not implying a local version of the chemical potential as with LDA. Applications will be given to the generic fact that if a fermionic superfluid in the BCS regime overflows from a narrow container into a much wider one, pairing is substantially reduced at the overflow point. Two examples pertinent to the physics of the outer crust of neutron stars and superfluid fermionic atoms in traps will be presented. The TF results will be compared to quantal and LDA ones.
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.
Two issues will be discussed in this contribution: (i) the generic fact that if a fermionic superfluid in the BCS regime overflows from a narrow container into a much wider one, pairing is much suppressed at the overflow point. Two examples pertinent to the physics of the outer crust of neutron stars and superfluid fermionic atoms in traps will be presented, (ii) A novel Thomas-Fermi approach to inhomogeneous superfluid Fermi-systems is presented and shown that it works well in cases where the local density approximation breaks down.
Strongly interacting two-component Fermi gases have become a powerful tool for studying many-body physics. The interactions between the components can be varied widely, allowing the full crossover from BCS to BEC superfluids to be explored. In the strong interaction region at unitarity, the behaviour of the system becomes independent of microscopic details of the interaction. Hence, these Fermi gases can be used as a testing ground for microscopic theories of strongly-interacting Fermi matter. We propose that the out-of-phase two-fluid modes can be excited and measured in a two-photon Bragg scattering experiment, which measures the density response function. We investigate the structure of the two-fluid density response function at unitarity within our variational approach. The out-of-phase modes are found to have appreciable weight at intermediate temperatures in the superfluid phase and should be observable in Bragg scattering experiments. In this thesis, we study the Landau two-fluid hydrodynamic modes of a trapped Fermi gas superfluid, including the "out-of-phase" oscillations of the superfluid and normal fluid components. These modes are the analogue of second sound in superfluid 4He, and have so far not been observed in trapped Fermi gas superfluids. An important part of this thesis is the development of a new variational method of solving the two-fluid equations for gases confined in a harmonic trap. The solution of the resulting algebraic equations for the two-fluid mode frequencies requires detailed knowledge of the local thermodynamic quantities in a trap. In contrast to "classical" BCS and BEC superfluids such as 3He and 4He, both fermionic and bosonic thermal excitations contribute significantly to the thermodynamics close to unitarity. The fact that these excitations are coupled at finite temperatures means that one cannot write down simple expressions for thermodynamic quantities in terms of the spectrum of these excitations. Our calculations of thermodynamic quantities involve essentially ab initio calculations based on the microscopic theory developed by Nozieres and Schmitt-Rink, applied to a nonuniform trapped gas using a local density approximation.