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arXiv:cond-mat/0511578 · Journal of Physics B Atomic Molecular and Optical Physics

Sound wave propagation in strongly elongated fermion clouds at finite collisionality

P. Capuzzi, Patrizia Vignolo, F. Federici, Monica Tosi

Published May 2, 2006• 7 citations
arXivHTMLPDF

Abstract

We evaluate the transition from zero-sound to first-sound behaviour with increasing collisionality in the propagation of density waves through an ultracold gaseous mixture of fermionic atoms in the normal state confined inside a cigar-shaped harmonic trap. We study for this purpose the evolution of the one-body distribution functions associated with a density perturbation generated in the central region of the cloud, as obtained by numerical solution of the Vlasov–Landau equations. We examine a variety of trap anisotropies and of repulsive or attractive interaction strengths between the components of the mixture. The speed of propagation of the density disturbance is found to decrease in both cases as the magnitude of the coupling strength is increased. The results are compared with the values of the speed of zero sound and of first sound, as obtained analytically from the limit of vanishing collisionality and from linearized hydrodynamics. The main effects of the quasi-one-dimensional confinement are the stabilization of zero-sound excitations in the attractive regime before collapse and the lowering of the hydrodynamic sound velocity by a factor relative to three-dimensional behaviour.

DOI
10.1088/0953-4075/39/10/s03

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We evaluate the transition from zero-sound to first-sound behaviour with increasing collisionality in the propagation of density waves through an ultracold gaseous mixture of fermionic atoms in the normal state confined inside a cigar-shaped harmonic trap. We study for this purpose the evolution of the one-body distribution functions associated with a density perturbation generated in the central region of the cloud, as obtained by numerical solution of the Vlasov–Landau equations. We examine a variety of trap anisotropies and of repulsive or attractive interaction strengths between the components of the mixture. The speed of propagation of the density disturbance is found to decrease in both cases as the magnitude of the coupling strength is increased. The results are compared with the values of the speed of zero sound and of first sound, as obtained analytically from the limit of vanishing collisionality and from linearized hydrodynamics. The main effects of the quasi-one-dimensional confinement are the stabilization of zero-sound excitations in the attractive regime before collapse and the lowering of the hydrodynamic sound velocity by a factor relative to three-dimensional behaviour.
We use hydrodynamic equations to study sound propagation in a superfluid Fermi gas at zero temperature inside a strongly elongated cigar-shaped trap, with main attention to the transition from the BCS to the unitary regime. First, we treat the role of the radial density profile in the limit of a cylindrical geometry and then evaluate numerically the effect of the axial confinement in a configuration in which a hole is present in the gas density at the center of the trap. We find that in a strongly elongated trap the speed of sound in both the BCS and the unitary regime differs by a factor 3∕5\sqrt{3∕5}3∕5​ from that in a homogeneous three-dimensional superfluid. The predictions of the theory could be tested by measurements of sound-wave propagation in a setup such as that exploited by Andrews et al. [Phys. Rev. Lett. 79, 553 (1997)] for an atomic Bose-Einstein condensate.
Abstract We consider a weakly-interacting fermionic gas of alkali-metal atoms characterized by two hyperfine states which are Rabi coupled. By using a Born approximation for the repulsive interaction we determine the zero-temperature equation of state of this Fermi gas in D spatial dimensions ( D = 1, 2, 3). Then, adopting the Landau–Vlasov equation and hydrodynamic equtions, we investigate the speed of first sound and zero sound. We show that the two sounds, which occur respectively in collisional and collisionless regimes, crucially depend on the interplay between interaction strength and Rabi coupling. Finally, we discuss for some experimentally relevant cases the effect of a trapping harmonic potential on the density profles of the fermionic system.
We present mean-field calculations of the equilibrium state in a gaseous mixture of bosonic and spin-polarized fermionic atoms with repulsive or attractive interspecies interactions, confined inside a cigar-shaped trap under conditions such that the radial thickness of the two atomic clouds is approaching the magnitude of the s-wave scattering lengths. In this regime the kinetic pressure of the fermionic component is dominant. Full demixing under repulsive boson-fermion interactions can occur only when the number of fermions in the trap is below a threshold, and collapse under attractive interactions is suppressed within the range of validity of the mean-field model. Specific numerical illustrations are given for values of system parameters obtaining in 7Li-6Li clouds.
This thesis is devoted to the study of the hydrodynamic behavior of the unitary Fermi gas trapped by a highly elongated harmonic potential. Propagation of sound is one of the most exciting features exhibited by interacting many-body systems. It provides crucial information on the dynamic behavior of the system as well as on key thermodynamic quantities. The propagation of sound is particularly interesting in superfluids where two-fluid hydrodynamic theory predicts the occurrence of two different sounds: first sound, where the normal and superfluid component oscillate in phase, and second sound, where the two components oscillate with opposite phase. In the thesis, we investigate the propagation of sound waves of the unitary Fermi gas in a cylindrical geometry by solving the equations of two-fluid hydrodynamics in the `1D' scenario at finite temperature. The relevant thermodynamic functions entering the hydrodynamic equations are discussed in the superfluid and normal regimes in terms of universal scaling functions. Both the first sound and second sound solutions are calculated as a function of temperature and the role of the superfluid density is explicitly pointed out. The density fluctuations in the second sound wave are found to be large enough to be measured as a consequence of the finite thermal expansion coefficient of the gas, which is the strategy used in a recent experiment carried out at Innsbruck where second sound was detected in the unitary Fermi gas. We also provide an investigation of the temperature dependence of the collective oscillations of first sound nature exhibited by a highly elongated harmonically trapped Fermi gas at unitarity, including the region below the critical temperature for superfluidity. Differently from the lowest axial breathing mode, the hydrodynamic frequencies of the higher-nodal excitations show a temperature dependence, which is calculated starting from Landau two-fluid theory and using the available experimental knowledge of the equation of state.
We evaluate in a mean-field model the equilibrium stability conditions of a gaseous mixture of bosonic and spin-polarized fermionic atoms inside a pancake-shaped or a cigar-shaped harmonic trap, under conditions such that the trap thickness approaches the magnitude of the s -wave scattering lengths but the atoms still experience collisions in three dimensions. With decreasing dimensionality, the Fermi pressure plays an increasingly dominant role. Full demixing in the case of repulsive boson-fermion interactions can be induced by squeezing the thickness of the clouds in a pancake-shaped trap or by lowering the number of trapped fermions in a cigar-shaped trap. Collapse under attractive interspecies interaction in quasi-one-dimensional confinement is inhibited within the range of validity of a mean-field model.
We study the zero sound and the first sound in a dilute and ultracold disk-shaped normal Fermi gas with a strong harmonic confinement along the axial direction and uniform in the two planar directions. Working at zero temperature we calculate the chemical potential $\ensuremath{\mu}$ of the fermionic fluid as a function of the uniform planar density $\ensuremath{\rho}$ and find that $\ensuremath{\mu}$ changes its slope in correspondence to the filling of harmonic axial modes (shell effects). Within the linear response theory, and under the random phase approximation, we calculate the velocity cs0{c}_{s}^{0}cs0​ of the zero sound. We find that also cs0{c}_{s}^{0}cs0​ changes its slope in correspondence of the filling of the harmonic axial modes and that this effect depends on the Fermi-Fermi scattering length aF{a}_{F}aF​. In the collisional regime, we calculate the velocity cs{c}_{s}cs​ of first sound showing that cs{c}_{s}cs​ displays jumps at critical densities fixed by the scattering length aF{a}_{F}aF​. Finally, we discuss the experimental achievability of these zero sound and first sound waves with ultracold alkali-metal atoms.