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arXiv:0709.2757 · Physical Review A

Tkachenko modes in a superfluid Fermi gas at unitarity

Gentaro Watanabe A, Marco Cozzini, S. Stringari

Published Feb 7, 2008• 3 citations
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Abstract

We calculate the frequencies of the Tkachenko oscillations of a vortex lattice in a harmonically trapped superfluid Fermi gas. We use the elastohydrodynamic theory and properly account for the elastic constants, the Thomas-Fermi density profile of the atomic cloud, and the boundary conditions. Thanks to the Fermi pressure, which is responsible for larger cloud radii with respect to the case of dilute Bose-Einstein condensed gases, large vortex lattices are achievable in the unitary limit of infinite scattering length, even at relatively small angular velocities. This opens the possibility of experimentally observing vortex oscillations in the regime where the dispersion relation approaches the Tkachenko law for incompressible fluids and the mode frequency is almost comparable to the trapping frequencies.

DOI
10.1103/physreva.77.021602

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We calculate the frequencies of the Tkachenko oscillations of a vortex lattice in a harmonically trapped superfluid Fermi gas. We use the elastohydrodynamic theory and properly account for the elastic constants, the Thomas-Fermi density profile of the atomic cloud, and the boundary conditions. Thanks to the Fermi pressure, which is responsible for larger cloud radii with respect to the case of dilute Bose-Einstein condensed gases, large vortex lattices are achievable in the unitary limit of infinite scattering length, even at relatively small angular velocities. This opens the possibility of experimentally observing vortex oscillations in the regime where the dispersion relation approaches the Tkachenko law for incompressible fluids and the mode frequency is almost comparable to the trapping frequencies.
The elastic oscillations of the vortex lattice of a cold Bose gas (Tkachenko modes) are shown to play a crucial role in the saturation of the compressibility sum rule, as a consequence of the hybridization with the longitudinal degrees of freedom. The presence of the vortex lattice is responsible for a q2 behavior of the static structure factor at small wave vectors q, which implies the absence of long range order in 2D configurations at zero temperature. Sum rules are used to calculate the Tkachenko frequency in the presence of harmonic trapping. Results are derived in the Thomas-Fermi regime and compared with experiments as well as with previous theoretical estimates.
We derive the frequency spectrum of the lowest compressional oscillations of a three-dimensional harmonically trapped Fermi superfluid in the presence of a vortex lattice, treated in the diffused vorticity approximation within a hydrodynamic approach. We consider the general case of a superfluid at T=0T=0T=0 characterized by a polytropic equation of state $(\ensuremath{\sim}{n}^{\ensuremath{\gamma}})$, which includes both the Bose-Einstein condensed regime of dimers $(\ensuremath{\gamma}=1)$ and the unitary limit of infinite scattering length $(\ensuremath{\gamma}=2∕3)$. Important limiting cases are considered, including the centrifugal limit, the isotropic trapping, and the cigar geometry. The conditions required to enter the lowest Landau level and quantum Hall regimes at unitarity are also discussed.
We directly image Tkachenko waves in a vortex lattice in a dilute-gas Bose-Einstein condensate. The low (sub-Hz) resonant frequencies are a consequence of the small but nonvanishing elastic shear modulus of the vortex-filled superfluid. The frequencies are measured for rotation rates as high as 98% of the centrifugal limit for the harmonically confined gas. Agreement with a hydrodynamic theory worsens with increasing rotation rate, perhaps due to the increasing fraction of the volume displaced by the vortex cores. We also observe two low-lying m=0 longitudinal modes at about 20 times higher frequency.
The frequencies of the collective oscillations of a harmonically trapped Fermi gas interacting with large scattering lengths are calculated at zero temperature using hydrodynamic theory. Different regimes are considered, including the molecular Bose-Einstein condensate and the unitarity limit for collisions. We show that the frequency of the radial compressional mode in an elongated trap exhibits a pronounced non-monotonous dependence on the scattering length, reflecting the role of the interactions in the equation of state.
We consider a periodic vortex lattice in a rotating Bose-Einstein-condensed gas, where the centrifugal potential is exactly compensated by the external harmonic trap. By introducing a gauge transformation which makes the Hamiltonian periodic, we numerically solve the two-dimensional (2D) Gross-Pitaevskii equation finding the exact mean field ground state. In particular, we explore the crossover between the Thomas-Fermi regime, holding for large values of the coupling constant, and the lowest Landau level limit, corresponding to the weakly interacting case. Explicit results are given for the equation of state, the vortex core size, as well as the elastic shear modulus, which is crucial for the calculation of the Tkachenko frequencies.
We derive the finite-temperature oscillation modes of a harmonically confined Bose-Einstein condensed gas undergoing rigid body rotation supported by a vortex lattice in the condensate. The hydrodynamic modes separate into two classes corresponding to center of mass and relative oscillations of the thermal cloud and the condensate. These classes are independent of each other in the case where the thermal cloud is inviscid for all modes studied, except the radial pulsations which couple them because the pressure perturbations of the condensate and the thermal cloud are governed by different adiabatic indices. If the thermal cloud is viscous, the two classes of oscillations are coupled, i.e., each type of motion involves simultaneously mass and entropy currents. The relative oscillations are damped by the mutual friction between the condensate and the thermal cloud mediated by the vortex lattice. The damping is large for the values of the drag-to-lift ratio of the order of unity and becomes increasingly ineffective in either limit of small or large friction. An experimental measurement of a subset of these oscillation modes and their damping can provide information on the values of the phenomenological mutual friction coefficients and the quasiparticle-vortex scattering processes in dilute atomic Bose gases.