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Hydrodynamics of Unitary Fermi Gases

Jasmine Brewer

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

In this thesis we present fully nonlinear dissipative hydrodynamics simulations using the lattice Boltzmann method which we have developed to study hydrodynamics in cold atomic gases. We motivate and derive the specifics of the lattice Boltzmann implementation for this system, and then use our simulations to study collective oscillations in two-dimensional Fermi gases. We show that all dominant frequencies and damping rates of the breathing and quadrupole oscillatory modes agree quantitatively with analytic solutions in the hydrodynamic limit for a gas with ideal equation of state in a harmonic trap. We suggest improvements to our numerical methods which we expect to improve the quantitative agreement of the quadrupole mode frequency with the expected value outside of the hydrodynamic limit. We also suggest techniques for extracting the non-dominant (non-hydrodynamic) damping rate of the quadrupole mode in the hydrodynamic regime with higher precision. Additionally, we propose that the observed non-hydrodynamic damping of the quadrupole mode would be interesting to study experimentally for applications to collective oscillations in the quark-gluon plasma.

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In this thesis we present fully nonlinear dissipative hydrodynamics simulations using the lattice Boltzmann method which we have developed to study hydrodynamics in cold atomic gases. We motivate and derive the specifics of the lattice Boltzmann implementation for this system, and then use our simulations to study collective oscillations in two-dimensional Fermi gases. We show that all dominant frequencies and damping rates of the breathing and quadrupole oscillatory modes agree quantitatively with analytic solutions in the hydrodynamic limit for a gas with ideal equation of state in a harmonic trap. We suggest improvements to our numerical methods which we expect to improve the quantitative agreement of the quadrupole mode frequency with the expected value outside of the hydrodynamic limit. We also suggest techniques for extracting the non-dominant (non-hydrodynamic) damping rate of the quadrupole mode in the hydrodynamic regime with higher precision. Additionally, we propose that the observed non-hydrodynamic damping of the quadrupole mode would be interesting to study experimentally for applications to collective oscillations in the quark-gluon plasma.
We suggest that collective oscillation frequencies of cold trapped gases can be used to test predictions from quantum many-body physics. Our motivation lies in both rigid experimental tests of theoretical calculations and a possible improvement of measurements of particle number, chemical potential or temperature. We calculate the effects of interaction, dimensionality and thermal fluctuations on the collective modes of a dilute Bose gas in the hydrodynamic limit. The underlying equation of state is provided by the non-perturbative Functional Renormalization Group or by Lee–Yang theory. The spectrum of oscillation frequencies could be measured by response techniques. Our findings are generalized to bosonic or fermionic quantum gases with an arbitrary equation of state in the two-fluid hydrodynamic regime. For any given equation of state P (μ, T ) and normal fluid density n n (μ, T ), the collective oscillation frequencies in a d -dimensional isotropic potential are found to be the eigenvalues of an ordinary differential operator. We suggest a method of numerical solution and discuss the zero-temperature limit. Exact results are provided for harmonic traps and certain special forms of the equation of state. We also present a phenomenological treatment of dissipation effects and discuss the possibility of exciting the different eigenmodes individually.
Unitary fermi gases have been widely studied as they provide a tabletop archetype for research on strongly coupled many body systems and perfect fluids. Research into unitary fermi gases can provide insight into may other strongly interacting systems including high temperature superconductor, quark-gluon plasmas, and neutron stars. Within the unitary regime, the equilibrium transport coefficients and thermodynamic properties are universal functions of density and temperature. Thus, unitary fermi gases provide a archetype to study nonperturbative many-body physics, which is of fundamental significance and crosses several fields.\nThis thesis reports on two topics regarding unitary fermi gases. A recent string theory conjecture gives a lower bound for the dimensionless ratio of shear viscosity of entropy, η=s ≥ 4πℏ =kb. Unitary fermi gases are a candidate for prefect fluids, yet η=s is well above the string theory bound. Using a stochastic formulation of hydrodynamics, we calculate a lower bound for this ratio accounting for the momentum dissipation from fluctuations. This lower bound is in good agreement with both theoretical and experimental results.\nThe second question addressed is the simulation of elliptic flow. Elliptic flow, first observed in 2002, is a characteristic of strongly coupled systems and has been studied in both quark-gluon plasmas and unitary fermi gases. As such, simulations of these systems are of interest. We test a variety of lattice Boltzmann models and compare the simulation results to the theoretical and experimental findings.
We study the transition from the hydrodynamic to the collisionless regime in collective modes of three- and two-dimensional Fermi gases by using the semiclassical Boltzmann equation. We use direct numerical simulations as well as the method of phase-space moments to solve the Boltzmann equation and show that the restriction to second-order moments is not accurate enough. By including higher-order moments, we can successfully describe the hydrodynamic to collisionless transition observed in the quadrupole mode in three-dimensional Fermi gases and the frequency shift and damping of the sloshing mode due to the anharmonic shape of the experimental trap potential. In the case of two-dimensional Fermi gases, however, the strong damping of the quadrupole mode observed in a recent experiment remains unexplained.
Very different strongly interacting quantum systems such as Fermi gases, quark-gluon plasmas formed in high-energy ion collisions, and black holes studied theoretically in string theory are known to exhibit quantitatively similar damping of hydrodynamic modes. It is not known if such similarities extend beyond the hydrodynamic limit. Do nonhydrodynamic collective modes in Fermi gases with strong interactions also match those from string theory calculations? In order to answer this question, we use calculations based on string theory to make predictions for modes outside the hydrodynamic regime in trapped Fermi gases. These predictions are amenable to direct testing with current state-of-the-art cold atom experiments.
At finite temperature, the hydrodynamic collective modes of superfluid trapped Fermi gases are coupled to the motion of the normal component, which in the BCS limit behaves like a collisionless normal Fermi gas. The coupling between the superfluid and the normal components is treated in the framework of a semiclassical transport theory for the quasiparticle-distribution function, combined with a hydrodynamic equation for the collective motion of the superfluid component. We develop a numerical test-particle method for solving these equations in the linear-response regime. As a first application we study the temperature dependence of the collective quadrupole mode of a Fermi gas in a spherical trap. The coupling between the superfluid collective motion and the quasiparticles leads to a rather strong damping of the hydrodynamic mode already at very low temperatures. At higher temperatures the spectrum has a two-peak structure, the second peak corresponding to the quadrupole mode in the normal phase.
We present fully nonlinear dissipative fluid dynamics simulations of a strongly interacting trapped two-dimensional Fermi gas using a lattice Boltzmann algorithm. We are able to simulate nonharmonic trapping potentials, temperature-dependent viscosities, as well as a discretized version of the ballistic (noninteracting) behavior. Our approach lends itself to direct comparison with experimental data, opening up the possibility of a precision determination of transport coefficients in the strongly interacting Fermi gas. Furthermore, we predict the presence of a strongly damped (``nonhydrodynamic'') component in the quadrupole mode, which should be observable experimentally.