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Physical review. B, Condensed matter

Temperature dependence of quasiparticle sound velocity in a Bose gas

K. K. Singh, Jai Prakash

Published Feb 1, 1978• 12 citations
Publisher

Abstract

The low-temperature dependence of quasiparticle sound velocity in a nonideal Bose gas is investigated. Besides a ${T}^{4}$ ln ${T}^{\ensuremath{-}1}$ increase with temperature - formally identical to that obtained by Khalatnikov and Andreyev for He II by the kinetic-equation approach - a ${T}^{4}$ decrease of the sound velocity is found. Assuming the theoretical expressions to remain formally valid for superfluid helium, comparison is made with experimental results on the variation of sound velocity with temperature.

DOI
10.1103/physrevb.17.1253

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The low-temperature dependence of quasiparticle sound velocity in a nonideal Bose gas is investigated. Besides a T4{T}^{4}T4 ln T\ensuremath−1{T}^{\ensuremath{-}1}T\ensuremath−1 increase with temperature - formally identical to that obtained by Khalatnikov and Andreyev for He II by the kinetic-equation approach - a T4{T}^{4}T4 decrease of the sound velocity is found. Assuming the theoretical expressions to remain formally valid for superfluid helium, comparison is made with experimental results on the variation of sound velocity with temperature.
The superfluidity of low-temperature bosons is well established in the collisional regime. In the collisionless regime, however, the presence of superfluidity is not yet fully clarified, in particular in lower spatial dimensions. Here, we compare the Vlasov-Landau equation, which does not take into account the superfluid nature of the bosonic system, with the Andreev-Khalatnikov equations, which instead explicitly contain a superfluid velocity. We show that recent experimental data of the sound mode in a two-dimensional collisionless Bose gas of 87Rb^{87}\mathrm{Rb}87Rb atoms are in good agreement with both theories, but the sound damping is better reproduced by the Andreev-Khalatnikov equations below the Berezinskii-Kosterlitz-Thouless critical temperature Tc{T}_{c}Tc​, while above Tc{T}_{c}Tc​ the Vlasov-Landau results are closer to the experimental ones. For one-dimensional bosonic fluids, where experimental data are not yet available, we find larger differences between the sound velocities predicted by the two transport theories and, also in this case, the existence of a superfluid velocity reduces the sound damping.
We study the recently proposed effective-field theory for the phonon of an arbitrary nonrelativistic superfluid. After computing the one-loop phonon self-energy, we obtain the low-temperature TTT contributions to the phonon dispersion law at low momentum and see that the real part of those can be parametrized as a thermal correction to the phonon velocity. Because the phonons are the quanta of the sound waves, at low momentum their velocity should agree with the speed of sound. We find that our results match at order T4lnT{T}^{4}\mathrm{ln}TT4lnT with those predicted by Andreev and Khalatnikov for the speed of sound, derived from the superfluid hydrodynamical equations and the phonon kinetic theory. We get also higher-order corrections of order T4{T}^{4}T4, which are not reproduced pushing naively the kinetic theory computation. Finally, as an application, we consider the cold Fermi gas in the unitarity limit and find a universal expression for the low-TTT relative correction to the speed of sound for these systems.
We present an experimental and theoretical study of the phonon mode in a unitary Fermi gas. Using two-photon Bragg spectroscopy, we measure excitation spectra at a momentum of approximately half the Fermi momentum, both above and below the superfluid critical temperature T_{c}. Below T_{c}, the dominant excitation is the Bogoliubov-Anderson (BA) phonon mode, driven by gradients in the phase of the superfluid order parameter. The temperature dependence of the BA phonon is consistent with a theoretical model based on the quasiparticle random phase approximation in which the dominant damping mechanism is via collisions with thermally excited quasiparticles. As the temperature is increased above T_{c}, the phonon evolves into a strongly damped collisional mode, accompanied by an abrupt increase in spectral width. Our study reveals strong similarities between sound propagation in the unitary Fermi gas and bosonic liquid helium.
We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a nonmonotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures TTT, we demonstrate that the coefficient in T2{T}^{2}T2 expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weakly interacting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in detail. At T=0T=0T=0 we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit.
We provide an overview of the recent theoretical and experimental advances in the study of second sound in ultracold atomic gases. Starting from the Landau two fluid hydrodynamic equations we develop the theory of first and second sound in various configurations characterized by different geometries and quantum statistics. These include the weakly interacting 3D Bose gas, the strongly interacting Fermi gas at unitarity in the presence of highly elongated traps and the dilute 2D Bose gas, characterized by the Berezinskii-Kosterlitz-Thouless transition. An explicit comparison with the propagation of second sound in liquid Helium is carried out to elucidate the main analogies and differences. We also make an explicit comparison with the available experimental data and point out the crucial role played by the superfluid density in determining the temperature dependence of the second sound speed.
We calculate the quantum Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ and butterfly velocity vB{v}_{B}vB​ in the dilute Bose gas at temperature TTT deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time-ordered correlators, from which ${\ensuremath{\lambda}}_{L}$ and vB{v}_{B}vB​ are extracted. At very low temperature where elementary excitations are phonon-like, we find ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}{T}^{5}$ and ${v}_{B}\ensuremath{\sim}c$, the sound velocity. At relatively high temperature, we have ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}T$ and ${v}_{B}\ensuremath{\sim}c{(T/{T}_{*})}^{0.23}$. We find that ${\ensuremath{\lambda}}_{L}$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on TTT. The chaos diffusion constant ${D}_{L}={v}_{B}^{2}/{\ensuremath{\lambda}}_{L}$, on the other hand, differs from the energy diffusion constant DE{D}_{E}DE​. We find ${D}_{E}\ensuremath{\ll}{D}_{L}$ at very low temperature and ${D}_{E}\ensuremath{\gg}{D}_{L}$ otherwise.