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arXiv:1809.01099 · Physical review. B./Physical review. B

Propagation and attenuation of sound in one-dimensional quantum liquids

К. А. Матвеев, Anton V. Andreev

Published Oct 29, 2018• 8 citations
arXivHTMLPDF

Abstract

At low temperatures, elementary excitations of a one-dimensional quantum liquid form a gas that can move as a whole with respect to the center of mass of the system. This internal motion attenuates at exponentially long time scales. As a result, in a broad range of frequencies, the liquid is described by two-fluid hydrodynamics, and the system supports two sound modes. The physical nature of the two sounds depends on whether the particles forming the quantum liquid have a spin degree of freedom. For particles with spin, the modes are analogous to the first and second sound modes in superfluid $^{4}\mathrm{He}$, which are the waves of density and entropy, respectively. When dissipative processes are taken into account, we find that at low frequencies, the second sound is transformed into heat diffusion, while the first sound mode remains weakly damped and becomes the ordinary sound. In a spinless liquid, the entropy and density oscillations are strongly coupled and the resulting sound modes are hybrids of the first and second sound. As the frequency is lowered and dissipation processes become important, the crossover to a single-fluid regime occurs in two steps. First the hybrid modes transform into predominantly density and entropy waves, similar to the first and second sound, and then the density wave transforms to the ordinary sound and the entropy wave becomes a heat-diffusion mode. Finally, we account for the dissipation due to viscosity and intrinsic thermal conductivity of the gas of excitations, which controls attenuation of the sound modes at high frequencies.

DOI
10.1103/physrevb.98.155441

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At low temperatures, elementary excitations of a one-dimensional quantum liquid form a gas that can move as a whole with respect to the center of mass of the system. This internal motion attenuates at exponentially long time scales. As a result, in a broad range of frequencies, the liquid is described by two-fluid hydrodynamics, and the system supports two sound modes. The physical nature of the two sounds depends on whether the particles forming the quantum liquid have a spin degree of freedom. For particles with spin, the modes are analogous to the first and second sound modes in superfluid 4He^{4}\mathrm{He}4He, which are the waves of density and entropy, respectively. When dissipative processes are taken into account, we find that at low frequencies, the second sound is transformed into heat diffusion, while the first sound mode remains weakly damped and becomes the ordinary sound. In a spinless liquid, the entropy and density oscillations are strongly coupled and the resulting sound modes are hybrids of the first and second sound. As the frequency is lowered and dissipation processes become important, the crossover to a single-fluid regime occurs in two steps. First the hybrid modes transform into predominantly density and entropy waves, similar to the first and second sound, and then the density wave transforms to the ordinary sound and the entropy wave becomes a heat-diffusion mode. Finally, we account for the dissipation due to viscosity and intrinsic thermal conductivity of the gas of excitations, which controls attenuation of the sound modes at high frequencies.
Water flowing down a pipe experiences viscous drag that causes it to lose energy and slow down. A superfluid, such as liquid helium (3He and 4He) or electrons in a superconductor, can flow without friction when in contact with a wall. This fascinating phenomenon, or superfluidity, was discovered by Kapitza [1], Allen and Misener [2] in 1937. The formation of a superfluid is related to the formation of a Bose--Einstein condensate (BEC). 4He is a boson as its total spin is zero, an integer. Condensation in fermionic 3He liquid occurs through composite bosons of paired atoms in a process similar to electron pairing in a superconductor. To explain the peculiar properties of superfluid helium, Landau considered a superfluid to be comprised of two parts: a ‘pure’ superfluid component with zero entropy and viscosity, and a normal component that behaves like ordinary fluid. An interesting manifestation of this two-fluid model is the existence of two types of oscillations, the ‘first and second sounds’ as named by Landau. The ‘first sound’ is a wave that propagates at constant entropy per particle, just like sounds in ordinary fluids. It propagates with pressure and density oscillations. ‘Second sound’, unlike what its name suggests, is not audible as it propagates at nearly constant pressure (isobaric oscillation). Here, entropy density takes the role of pressure/density in normal sound waves. Historically, the observation of the second sound in superfluid helium, and the measurement of its speed, played a crucial role in understanding the nature of superfluid helium. The speed of the second sound can be used to determine the superfluid fraction, i.e. the ratio of the superfluid density to the total fluid density [3,4]. 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Recently, the group of Professor Rudolf Grimm from the University of Innsbruck (Austria), in collaboration with a theoretical team from the Trento University (Italy) led by Professors Sandro Stringari and Lev Pitaevskii, successfully measured the speed of second sound in a resonantly interacting Fermi gas using a simple and elegant method. Their results were published in [5]. To observe the second sound, the authors created a local heat pulse at the center of an elongated trapped gas by creating a short burst of modulation to the power of a light beam intercepting this gas (see Fig. 1b). They then measured the speed of the second sound by observing the movement of a thermal-induced density dip along the elongated gas (Fig. 1c). Taking advantage of the inhomogeneous density distributions of the superfluid atoms in the trap, they ingeniously determined the temperature dependence of the superfluid fraction without needing to vary the temperature of the samples. The observation of the second sound in the strongly interacting Fermi gases opens up a new horizon for studying the superfluid properties with tunable interactions. Knowing the speed of second sound can provide a lot of valuable information, allowing physicists working on many-body theories to falsify their models and theories. It demonstrates the importance of using experimentally measured quantities, such as the equation of state employed here, to calibrate and to supplement theoretical investigations. Significant new advances will follow in the understanding of superfluidity of many strongly interacting particles, where faithful theoretical modeling and calculations are impossible at the microscopic level.
We study sound in Galilean invariant systems of one-dimensional fermions. At low temperatures, we find a broad range of frequencies in which in addition to the waves of density there is a second sound corresponding to the ballistic propagation of heat in the system. The damping of the second sound mode is weak, provided the frequency is large compared to a relaxation rate that is exponentially small at low temperatures. At lower frequencies, the second sound mode is damped, and the propagation of heat is diffusive.
In superfluid systems several sound modes can be excited, such as, for example, first and second sound in liquid helium. Here, we excite running and standing waves in a uniform two-dimensional Bose gas and we characterize the propagation of sound in both the superfluid and normal regimes. In the superfluid phase, the measured speed of sound is in good agreement with the prediction of a two-fluid hydrodynamic model, and the weak damping is well explained by the scattering with thermal excitations. In the normal phase we observe a stronger damping, which we attribute to a departure from hydrodynamic behavior.
Sound propagation is a macroscopic manifestation of the interplay between the equilibrium thermodynamics and the dynamical transport properties of fluids. Here, for a two-dimensional system of ultracold fermions, we calculate the first and second sound velocities across the whole BCS-BEC crossover, and we analyze the system response to an external perturbation. In the low-temperature regime we reproduce the recent measurements [Phys. Rev. Lett. 124, 240403 (2020)] of the first sound velocity, which, due to the decoupling of density and entropy fluctuations, is the sole mode excited by a density probe. Conversely, a heat perturbation excites only the second sound, which, being sensitive to the superfluid depletion, vanishes in the deep BCS regime and jumps discontinuously to zero at the Berezinskii-Kosterlitz-Thouless superfluid transition. A mixing between the modes occurs only in the finite-temperature BEC regime, where our theory converges to the purely bosonic results.
We consider a system of one-dimensional fermions moving in one direction, such as electrons at the edge of a quantum Hall system. At sufficiently long timescales the system is brought to equilibrium by weak interactions between the particles, which conserve their total number, energy, and momentum. The time evolution of the system near equilibrium is described by hydrodynamics based on the three conservation laws. We find that the system supports three sound modes. In the low-temperature limit one mode is a pure oscillation of particle density, analogous to the ordinary sound. The other two modes involve oscillations of both particle and entropy densities. In the presence of disorder, the first sound mode is strongly damped at frequencies below the momentum relaxation rate, whereas the other two modes remain weakly damped.