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Physical Review A

Fluid-dynamical scheme for equilibrium properties of two trapped fermion species with pairing interactions

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

Published Oct 31, 2008• 7 citations
Publisher

Abstract

We present a generalization of the fluid-dynamical scheme developed for nuclear physics to the case of two trapped fermion species with pairing interactions. To establish a macroscopic description of the mass and momentum conservation laws, we adopt a generalization of the usual Thomas-Fermi approach that includes the pairing energy. We analyze the equilibrium density and gap profiles for an equal population mixture of harmonically trapped $^{6}\mathrm{Li}$ atoms for different choices of the local equation of state. We examine slight departures from equilibrium within our formulation, finding that density oscillations can propagate as first sound coupled to pairing vibrations, that in a homogeneous fermion system exhibit a Bogoliubov-like quasiparticle spectrum. In this case, the dispersion relation for the coupled modes displays a rich scenario of stable, unstable, and damped regimes.

DOI
10.1103/physreva.78.043619

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We present a generalization of the fluid-dynamical scheme developed for nuclear physics to the case of two trapped fermion species with pairing interactions. To establish a macroscopic description of the mass and momentum conservation laws, we adopt a generalization of the usual Thomas-Fermi approach that includes the pairing energy. We analyze the equilibrium density and gap profiles for an equal population mixture of harmonically trapped 6Li^{6}\mathrm{Li}6Li atoms for different choices of the local equation of state. We examine slight departures from equilibrium within our formulation, finding that density oscillations can propagate as first sound coupled to pairing vibrations, that in a homogeneous fermion system exhibit a Bogoliubov-like quasiparticle spectrum. In this case, the dispersion relation for the coupled modes displays a rich scenario of stable, unstable, and damped regimes.
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.
We discuss the ground state and the small-amplitude excitations of a degenerate vapour of fermionic atoms placed in two hyperfine states inside a spherical harmonic trap. An equations-of-motion approach is set up to discuss the hydrodynamic dissipation processes from the interactions between the two components of the fluid beyond mean-field theory and to emphasize analogies with spin dynamics and spin diffusion in a homogeneous Fermi liquid. The conditions for the establishment of a collisional regime via scattering against cold-atom impurities are analyzed. The equilibrium density profiles are then calculated for a two-component vapour of 40K atoms: they are little modified by the interactions for presently relevant values of the system parameters, but spatial separation of the two components will spontaneously arise as the number of atoms in the trap is increased. The eigenmodes of collective oscillation in both the total particle number density and the concentration density are evaluated analytically in the special case of a symmetric two-component vapour in the collisional regime. The dispersion relation of the surface modes for the total particle density reduces in this case to that of a one-component Fermi vapour, whereas the frequencies of all other modes are shifted by the interactions.
We explore a few-fermion mixture consisting of two components that are repulsively interacting and confined in a one-dimensional harmonic trap. Different scenarios of population imbalance ranging from the completely imbalanced case where the physics of a single impurity in the Fermi sea is discussed to the partially imbalanced and equal population configurations are investigated. For the numerical calculations the multiconfigurational time-dependent Hartree method is employed, extending its application to few-fermion systems. Apart from numerical calculations we generalize our ansatz for a correlated pair wave function proposed recently [I. Brouzos and P. Schmelcher, Phys. Rev. Lett. 108, 045301 (2012)] for bosons to mixtures of fermions. From weak to strong coupling between the components the energies, the densities and the correlation properties of one-dimensional systems change vastly with an upper limit set by fermionization where for infinite repulsion all fermions can be mapped to identical ones. The numerical and analytical treatments are in good agreement with respect to the description of this crossover. We show that for equal populations each pair of different component atoms splits into two single peaks in the density while for partial imbalance additional peaks and plateaus arise for very strong interaction strengths. The case of a single-impurity atom shows rich behavior of the energy and density as we approach fermionization and is directly connected to recent experiments [G. Z\"urn et al., Phys. Rev. Lett. 108, 075303 (2012)].
A unitary Fermi gas has a surprisingly rich spectrum of large amplitude modes of the pairing field alone, which defies a description within a formalism involving only a reduced set of degrees of freedom, such as quantum hydrodynamics or a Landau-Ginzburg-like description. These modes are very slow, with oscillation frequencies well below the pairing gap, which makes their damping through quasiparticle excitations quite ineffective. In atomic traps these modes couple naturally with the density oscillations, and the corresponding oscillations of the atomic cloud are an example of a new type of collective mode in superfluid Fermi systems. They have lower frequencies than the compressional collective hydrodynamic oscillations, have a nonspherical momentum distribution, and could be excited by a quick time variation of the scattering length.
We study via mean-field theory the pairing between fermions of different masses, especially at the unitary limit. At equal populations, the thermodynamic properties are identical with the equal mass case provided an appropriate rescaling is made. At unequal populations, for sufficiently light majority species, the system does not phase separate. For sufficiently heavy majority species, the phase separated normal phase have a density larger than that of the superfluid. For atoms in harmonic traps, the density profiles for unequal mass fermions can be drastically different from their equal-mass counterparts.
In summary, in this Chapter we have given a brief overview on the dynamical properties of a trapped Fermi gas, both in the collisionless and in the collisional regime. While in a one-component system interactions are negligible (and nevertheless the system presents a rich spectrum), in the two-component case they can influence in a relevant way the equilibrium state and the dispersion relation. We have described mainly the case of a normal fluid, but we have pointed out that the long-wavelength behavior of a superfluid can be described in terms of the properties of the normal fluid in the collisional regime.