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Numerical Evaluation of the Two-Fluid Hydrodynamic Coefficients and Temperature-Dependent Second Sound in a Finite-Temperature Bose Gas

Lohith Lohith

Published Aug 21, 2026• 0 citations
Publisher

Abstract

We present a numerically stable evaluation of the two-fluid hydrodynamic coefficients for a dilute Bose gas of rubidium-87 at finite temperature. Our contribution is threefold: (i) we compute the condensate gradient stiffness $B$, the entropy density $S$, and the specific heat $c_v$ directly from the one-loop Bogoliubov action; (ii) we outline the key steps of the density-response matching that yields the condensate--normal fluid coupling $\lambda$, making the calculation transparent; and (iii) we quantify the systematic error incurred by the common approximation $\epsilon_k/E_k \to 1$ in $\lambda(T)$. We provide a numerical temperature scan of the hydrodynamic coefficients for rubidium-87 across a broad range, using stable dimensionless variables that ensure ultraviolet convergence. For $^{87}\mathrm{Rb}$ at $T=100\,\mathrm{nK}$ and $n_0=10^{20}\,\mathrm{m^{-3}}$, we find $\lambda = (2.59 \pm 0.52) \times 10^{-22} \, \mathrm{kg\,s^{-1}}$ and $c_2 = 1.09\,\mathrm{mm/s}$ (the phonon-limit value $c_s/\sqrt{3}$). The temperature dependence of $c_2$ exhibits a non-monotonic shape with a maximum followed by a drop to zero near $T_c$, qualitatively consistent with the universal two-fluid behavior observed in the homogeneous $^{39}$K Bose gas experiment of Hilker et al. \cite{Hilker2022}. The quantitative results are reliable for $T\lesssim 0.5T_c$; higher-temperature entries indicate the expected qualitative trend. These results provide a quantitative benchmark for comparison with future experiments on rubidium gases.

DOI
10.5281/zenodo.22047265

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The two-fluid model is fundamental for the description of superfluidity. In the nearly incompressible liquid regime, it successfully describes first and second sound, corresponding, respectively, to density and entropy waves, in both liquid helium and unitary Fermi gases. Here, we study the two sounds in the opposite regime of a highly compressible fluid, using an ultracold ^{39}K Bose gas in a three-dimensional box trap. We excite the longest-wavelength mode of our homogeneous gas, and observe two distinct resonant oscillations below the critical temperature, of which only one persists above it. In a microscopic mode-structure analysis, we find agreement with the hydrodynamic theory, where first and second sound involve density oscillations dominated by, respectively, thermal and condensed atoms. Varying the interaction strength, we explore the crossover from hydrodynamic to collisionless behavior in a normal gas.
We derive coupled equations of motion for the condensate (superfluid) and noncondensate (normal fluid) degrees of freedom in a trapped Bose gas at finite temperatures. Our results are based on the Hartree-Fock-Popov approximation for the time-dependent condensate wave function, and thermodynamic local equilibrium for the noncondensate atoms. In the special case of a uniform weakly interacting gas, our hydrodynamic equations are shown to be consistent with the two-fluid equations of Landau. The collective modes in a parabolically trapped Bose gas include the analog of the out-of-phase second-sound mode in superfluid 4He{}^{4}\mathrm{He}4He.
We extend our recent work on the two-fluid hydrodynamics of the condensate and non-condensate in a trapped Bose gas by including the dissipation associated with viscosity and thermal conduction. For purposes of illustration, we consider the hydrodynamic modes in the case of a uniform Bose gas. A finite thermal conductivity and shear viscosity give rise to a damping of the first and second sound modes in addition to that found previously due to the lack of diffusive equilibrium between the condensate and non-condensate. The relaxational mode associated with this equilibration process is strongly coupled to thermal fluctuations and reduces to the usual thermal diffusion mode above the Bose-Einstein transition. In contrast to the standard Landau two-fluid hydrodynamics, we predict a damped mode centered at zero frequency, in addition to the usual second sound doublet.
We develop the moment method for Bose-Einstein condensates at finite temperatures that enable us to study collective sound modes from the hydrodynamic to the collisionless regime. In particular, we investigate collective excitations in a weakly interacting dilute Bose gas by applying the moment method to the Zaremba-Nikuni-Griffin equation, which is the coupled equation of the Boltzmann equation with the generalized Gross-Pitaevskii equation. Utilizing the moment method, collective excitations in the crossover regime between the hydrodynamic and collisionless regimes are investigated in detail. In the crossover regime, the second sound mode loses the weight of the density response function because of the significant coupling with incoherent modes, whereas the first sound shows a distinct but broad peak structure. We compare the result obtained by the moment method with that of the Landau two-fluid equations and show that the collective mode predicted by the Landau two-fluid equations well coincides with the result from the moment method even far from the hydrodynamic regime, whereas clear distinction also emerges in the relatively higher momentum regime.
We theoretically investigate sound modes in a weakly-interacting collisional Bose gas in DDD dimensions. Using the Landau's two-fluid hydrodynamics and working within the Bogoliubov theory, we observe the hybridization of the first and second sound modes for D≥2D\ge 2D≥2. To model the recent measurements of the sound velocities in 2D, obtained around the Berezinskii-Kosterlitz-Thouless transition temperature and outside the very weakly-interacting regime, we derive a refined calculation of the superfluid density, finding a fair agreement with the experiment. In the 1D case, for which experimental results are currently unavailable, we find no hybridization, triggering the necessity of future experimental investigations. Our analysis provides a systematic understanding of sound propagation in a collisional weakly-interacting Bose gas in DDD dimensions.
Using Landau's theory of two-fluid hydrodynamics we investigate the sound modes propagating in a uniform weakly interacting superfluid Bose gas for values of temperature up to the critical point. In order to evaluate the relevant thermodynamic functions needed to solve the hydrodynamic equations, including the temperature dependence of the superfluid density, we use the Bogoliubov theory at low temperatures and the results of a perturbative approach based on the Beliaev diagrammatic technique at higher temperatures. Special focus is given on the hybridization phenomenon between first and second sound which occurs at low temperatures of the order of the interaction energy and we discuss explicitly the behavior of the two sound velocities near the hybridization point.