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arXiv:cond-mat/0211469 · Physical Review A

Slow rotation of a superfluid trapped Fermi gas

Mi­chael Urban, Peter Schuck

Published Mar 27, 2003• 18 citations
arXivHTMLPDF

Abstract

The moment of inertia $\ensuremath{\Theta}$ is one of the possible observables for the experimental determination whether a trapped Fermi system has reached the BCS transition or not. In this article we investigate in detail the temperature dependence of $\ensuremath{\Theta}$ below the critical temperature ${T}_{c}.$ Special care is taken to account for the small size of the system, i.e., for the fact that the trapping frequency $\ensuremath{\Elzxh}\ensuremath{\omega}$ is of the same order of magnitude as the gap $\ensuremath{\Delta}.$ It is shown that the usual transport approach, corresponding to the leading order of an expansion in powers of $\ensuremath{\Elzxh},$ is not accurate in this case. It turns out that $\ensuremath{\Theta}$ does not change rapidly if T becomes smaller than ${T}_{c},$ but it rather decreases slowly. Qualitatively this behavior can be explained within the two-fluid model, which again corresponds to the leading order in $\ensuremath{\Elzxh}.$ Quantitatively we find deviations from the two-fluid model due to the small system size.

DOI
10.1103/physreva.67.033611

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The moment of inertia \ensuremathΘ\ensuremath{\Theta}\ensuremathΘ is one of the possible observables for the experimental determination whether a trapped Fermi system has reached the BCS transition or not. In this article we investigate in detail the temperature dependence of \ensuremathΘ\ensuremath{\Theta}\ensuremathΘ below the critical temperature Tc.{T}_{c}.Tc​. Special care is taken to account for the small size of the system, i.e., for the fact that the trapping frequency \ensuremath\Elzxh\ensuremathω\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath\Elzxh\ensuremathω is of the same order of magnitude as the gap \ensuremathΔ.\ensuremath{\Delta}.\ensuremathΔ. It is shown that the usual transport approach, corresponding to the leading order of an expansion in powers of \ensuremath\Elzxh,\ensuremath{\Elzxh},\ensuremath\Elzxh, is not accurate in this case. It turns out that \ensuremathΘ\ensuremath{\Theta}\ensuremathΘ does not change rapidly if T becomes smaller than Tc,{T}_{c},Tc​, but it rather decreases slowly. Qualitatively this behavior can be explained within the two-fluid model, which again corresponds to the leading order in \ensuremath\Elzxh.\ensuremath{\Elzxh}.\ensuremath\Elzxh. Quantitatively we find deviations from the two-fluid model due to the small system size.
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 the gas. This renders the experimental observation of superfluidity effects more difficult and indicates that reliable theoretical predictions concerning other dynamical properties, like the frequencies of collective modes, can only be made by taking into account temperature effects.
We present a detailed study of the BCS pairing transition in a trapped polarized dipolar Fermi gas. In the case of a shallow nearly spherical trap, we find a decrease in the transition temperature as a function of the trap aspect ratio and predict the existence of the optimal trap geometry. The latter corresponds to the highest critical temperature of the BCS transition for a given number of particles. We also derive the phase diagram for an ultracold trapped dipolar Fermi gas in the situation where the trap frequencies can be of the order of the critical temperature of the BCS transition in the homogeneous case, and determine the critical value of the dipole–dipole interaction energy below which the BCS transition ceases to exist. The critical dipole strength is obtained as a function of the trap aspect ratio. Alternatively, for a given dipole strength, there is a critical value of the trap anisotropy for the BCS state to appear. The order parameter calculated at criticality exhibits novel non-monotonic behaviour resulting from the combined effect of the confining potential and the anisotropic character of the interparticle dipole–dipole interaction.
We derive semiclassical transport equations for a trapped atomic Fermi gas in the BCS phase at temperatures between zero and the superfluid transition temperature. These equations interpolate between the two well-known limiting cases of superfluid hydrodynamics at zero temperature and the Vlasov equation at the critical one. The linearized version of these equations, valid for small deviations from equilibrium, is worked out and applied to two simple examples where analytical solutions can be found: a sound wave in a uniform medium and the quadrupole excitation in a spherical harmonic trap. In spite of some simplifying approximations, the main qualitative results of quantum mechanical calculations are reproduced, which are the different frequencies of the quadrupole mode at zero and the critical temperature and strong Landau damping at intermediate temperatures. In addition we suggest a numerical method for solving the semiclassical equations without further approximations.
We discuss collective monopole and quadrupole excitations of a collisionless gas of trapped Fermionic atoms in the superfluid BCS phase, comparing the fully microscopic Bogoliubov--de Gennes and quasiparticle random-phase approximation method with widely used semiclassical methods. In particular, the microscopic treatment allows us to address the questions of temperature dependence and nontrivial dependence on the trap parameters, which cannot be answered within the semiclassical approach. The main result concerning the temperature dependence is a strong Landau damping at intermediate temperature, which disappears in the limits of zero and critical temperature. However, even at zero temperature, considerable deviations from superfluid hydrodynamics are found if the trap frequency is of the same order of magnitude as the pairing gap.
Vortex mass is a key concept in the study of superfluid dynamics, referring to the inertia of vortices in a superfluid, which affects their motion and behavior. Despite being an important quantity, the vortex mass has never been observed experimentally, and remains an unresolved issue in this field. As of now, a large body of research assumes that the vortex mass is a local parameter. In contrast, we present a calculation that suggests a logarithmic dependence on the system size, agreeing with some earlier predictions in the context of Bose gases. We analyze the problem using an effective field theory that describes ultracold atomic Fermi gases over the BEC-BCS crossover at both zero and nonzero temperatures. Our study reveals a strong dependence of the vortex mass on the scattering length; in particular, the vortex mass grows rapidly when moving toward the BCS side. Furthermore, we find that the system-size dependence of the vortex mass results in values an order of magnitude larger than those predicted by other models for realistic system sizes. This implies that the vortex mass could be observable in a wider parameter range than was previously expected. This is particularly relevant considering recent advances in experimental techniques that place the observation of vortex mass in superfluid Fermi gases within reach.