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VARIATIONAL THEORY FOR TWO-FLUID HYDRODYNAMIC MODES IN TRAPPED FERMI GASES

Edward Taylor

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Strongly interacting two-component Fermi gases have become a powerful tool for studying many-body physics. The interactions between the components can be varied widely, allowing the full crossover from BCS to BEC superfluids to be explored. In the strong interaction region at unitarity, the behaviour of the system becomes independent of microscopic details of the interaction. Hence, these Fermi gases can be used as a testing ground for microscopic theories of strongly-interacting Fermi matter. We propose that the out-of-phase two-fluid modes can be excited and measured in a two-photon Bragg scattering experiment, which measures the density response function. We investigate the structure of the two-fluid density response function at unitarity within our variational approach. The out-of-phase modes are found to have appreciable weight at intermediate temperatures in the superfluid phase and should be observable in Bragg scattering experiments. In this thesis, we study the Landau two-fluid hydrodynamic modes of a trapped Fermi gas superfluid, including the "out-of-phase" oscillations of the superfluid and normal fluid components. These modes are the analogue of second sound in superfluid 4He, and have so far not been observed in trapped Fermi gas superfluids. An important part of this thesis is the development of a new variational method of solving the two-fluid equations for gases confined in a harmonic trap. The solution of the resulting algebraic equations for the two-fluid mode frequencies requires detailed knowledge of the local thermodynamic quantities in a trap. In contrast to "classical" BCS and BEC superfluids such as 3He and 4He, both fermionic and bosonic thermal excitations contribute significantly to the thermodynamics close to unitarity. The fact that these excitations are coupled at finite temperatures means that one cannot write down simple expressions for thermodynamic quantities in terms of the spectrum of these excitations. Our calculations of thermodynamic quantities involve essentially ab initio calculations based on the microscopic theory developed by Nozieres and Schmitt-Rink, applied to a nonuniform trapped gas using a local density approximation.

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In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations reviewed in Chapter 14. In the case of trapped Bose gases, these are coupled differential equations with position-dependent coefficients associated with the local thermodynamic functions. Building on the approach initiated by Zaremba et al. (1999) for trapped atomic Bose gases, in this chapter we develop an alternative variational formulation of two-fluid hydrodynamics. This is based on the work of Zilsel (1950), originally developed to deal with superfluid He. Assuming a simple variational ansatz for the superfluid and normal fluid velocity fields, this approach reduces the problem of finding the hydrodynamic collective mode frequencies to solving coupled algebraic equations for a few variational parameters. These equations contain constants that involve spatial integrals over various equilibrium thermodynamic derivatives. Such a variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid equations numerically. This chapter is mainly based on Taylor and Griffin (2005), Taylor (2008) and Zilsel (1950). In it, we discuss the normal modes of the non-dissipative Landau two-fluid equations for a trapped superfluid. In Section 16.3, we illustrate this formalism by deriving expressions for the frequencies of the dipole and breathing modes of a trapped Bose superfluid. In Chapters 17 and 18, we discuss an extended version of the two-fluid equations that includes hydrodynamic damping. The hydrodynamic damping of the collective modes is calculated in Chapter 19 using a generalized version of the variational approach developed in this chapter. Zilsel's variational formulation Since two-fluid hydrodynamics only describes a system in local equilibrium, all thermodynamic quantities are functions of position and time. Even in static equilibrium, in the presence of a trapping potential, most thermodynamic quantities will be position dependent.

Cambridge University Press eBooks•Feb 19, 2009•0 citations

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Using a variational approach, we solve the equations of two-fluid hydrodynamics for a uniform and trapped Fermi gas at unitarity. In the uniform case, we find that the first and second sound modes are remarkably similar to those in superfluid helium, a consequence of strong interactions. In the presence of harmonic trapping, first and second sound become degenerate at certain temperatures. At these points, second sound hybridizes with first sound and is strongly coupled with density fluctuations, giving a promising way of observing second sound. We also discuss the possibility of exciting second sound by generating local heat perturbations.

arXiv:0905.0257•Physical Review A•Nov 2, 2009•60 citations
Strongly interacting two-component Fermi gases have become a powerful tool for studying many-body physics. The interactions between the components can be varied widely, allowing the full crossover from BCS to BEC superfluids to be explored. In the strong interaction region at unitarity, the behaviour of the system becomes independent of microscopic details of the interaction. Hence, these Fermi gases can be used as a testing ground for microscopic theories of strongly-interacting Fermi matter. We propose that the out-of-phase two-fluid modes can be excited and measured in a two-photon Bragg scattering experiment, which measures the density response function. We investigate the structure of the two-fluid density response function at unitarity within our variational approach. The out-of-phase modes are found to have appreciable weight at intermediate temperatures in the superfluid phase and should be observable in Bragg scattering experiments. In this thesis, we study the Landau two-fluid hydrodynamic modes of a trapped Fermi gas superfluid, including the "out-of-phase" oscillations of the superfluid and normal fluid components. These modes are the analogue of second sound in superfluid 4He, and have so far not been observed in trapped Fermi gas superfluids. An important part of this thesis is the development of a new variational method of solving the two-fluid equations for gases confined in a harmonic trap. The solution of the resulting algebraic equations for the two-fluid mode frequencies requires detailed knowledge of the local thermodynamic quantities in a trap. In contrast to "classical" BCS and BEC superfluids such as 3He and 4He, both fermionic and bosonic thermal excitations contribute significantly to the thermodynamics close to unitarity. The fact that these excitations are coupled at finite temperatures means that one cannot write down simple expressions for thermodynamic quantities in terms of the spectrum of these excitations. Our calculations of thermodynamic quantities involve essentially ab initio calculations based on the microscopic theory developed by Nozieres and Schmitt-Rink, applied to a nonuniform trapped gas using a local density approximation.
We develop a variational approach to calculate the density response function at finite temperatures of the lowest-lying two-fluid modes in a trapped two-component Fermi superfluid close to a Feshbach resonance. The out-of-phase oscillations, which are the analogue in trapped gases of second sound in uniform superfluids, have so far not been observed in cold-atom experiments. At unitarity, we show that these modes are observable at finite temperatures via two-photon Bragg scattering, whose spectrum is related to the imaginary part of density response function. This provides direct evidence for superfluidity and a promising way to test microscopic results for thermodynamics at unitarity.
We present the results of a variational calculation of the frequencies of the low-lying Landau two-fluid hydrodynamic modes in a trapped Fermi superfluid gas at unitarity. Landau's two-fluid hydrodynamics is expected to be the correct theory of Fermi superfluids at finite temperatures close to unitarity, where strong interactions give rise to collisional hydrodynamics. Two-fluid hydrodynamics predicts the existence of in-phase modes in which the superfluid and normal fluid components oscillate together, as well as out-of-phase modes where the two components move against each other. We prove that, at unitarity, the dipole and breathing in-phase modes are locally isentropic. Their frequencies are independent of temperature and are the same above and below the superfluid transition, a feature due as much to the harmonic trapping potential as to the thermodynamic properties at unitarity. The out-of-phase modes, in contrast, are strongly dependent on temperature and hence can be used to test the thermodynamic properties and superfluid density of a Fermi gas at unitarity. We give numerical results for the frequencies of these modes as function of temperature in an isotropic trap at unitarity.
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.
In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations. This region can now be probed over the entire BCS--Bose-Einstein-condensate crossover in trapped Fermi superfluids with a Feshbach resonance, including the unitarity region. Building on the approach initiated by Zaremba, Nikuni, and Griffin in 1999 for trapped atomic Bose gases, we present a variational formulation of two-fluid hydrodynamic collective modes based on the work of Zilsel in 1950 developed for superfluid helium. Assuming a simple variational Ansatz for the superfluid and normal fluid velocities, the frequencies of the hydrodynamic modes are given by solutions of coupled algebraic equations, with constants only involving spatial integrals over various equilibrium thermodynamic derivatives. This variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid differential equations. Our two-fluid results are shown to reduce to those of Pitaevskii and Stringari for a pure superfluid at T=0T=0T=0.
In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations reviewed in Chapter 14. In the case of trapped Bose gases, these are coupled differential equations with position-dependent coefficients associated with the local thermodynamic functions. Building on the approach initiated by Zaremba et al. (1999) for trapped atomic Bose gases, in this chapter we develop an alternative variational formulation of two-fluid hydrodynamics. This is based on the work of Zilsel (1950), originally developed to deal with superfluid He. Assuming a simple variational ansatz for the superfluid and normal fluid velocity fields, this approach reduces the problem of finding the hydrodynamic collective mode frequencies to solving coupled algebraic equations for a few variational parameters. These equations contain constants that involve spatial integrals over various equilibrium thermodynamic derivatives. Such a variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid equations numerically. This chapter is mainly based on Taylor and Griffin (2005), Taylor (2008) and Zilsel (1950). In it, we discuss the normal modes of the non-dissipative Landau two-fluid equations for a trapped superfluid. In Section 16.3, we illustrate this formalism by deriving expressions for the frequencies of the dipole and breathing modes of a trapped Bose superfluid. In Chapters 17 and 18, we discuss an extended version of the two-fluid equations that includes hydrodynamic damping. The hydrodynamic damping of the collective modes is calculated in Chapter 19 using a generalized version of the variational approach developed in this chapter. Zilsel's variational formulation Since two-fluid hydrodynamics only describes a system in local equilibrium, all thermodynamic quantities are functions of position and time. Even in static equilibrium, in the presence of a trapping potential, most thermodynamic quantities will be position dependent.
Using a variational approach, we solve the equations of two-fluid hydrodynamics for a uniform and trapped Fermi gas at unitarity. In the uniform case, we find that the first and second sound modes are remarkably similar to those in superfluid helium, a consequence of strong interactions. In the presence of harmonic trapping, first and second sound become degenerate at certain temperatures. At these points, second sound hybridizes with first sound and is strongly coupled with density fluctuations, giving a promising way of observing second sound. We also discuss the possibility of exciting second sound by generating local heat perturbations.