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arXiv:2502.01104

Surface tension of Bose-Einstein condensate at finite temperature

Nguyễn Văn Thụ

Published Feb 3, 2025• 0 citations
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Abstract

We examine the influence of nonzero temperature on a bounded surface of a dilute Bose gas confined by a hard wall, utilizing the Gross-Pitaevskii theory and the Bogoliubov-de Gennes equation to describe surface excitations. The theoretical calculations are compared with experimental data for liquid helium II, demonstrating excellent agreement. An empirical relation is found for the temperature-dependence of the surface tension. Furthermore, our findings indicate that the contribution of surface excitations remains below 0.7\% in the most of recent experiments on Bose-Einstein condensate.

DOI
10.48550/arxiv.2502.01104

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We examine the influence of nonzero temperature on a bounded surface of a dilute Bose gas confined by a hard wall, utilizing the Gross-Pitaevskii theory and the Bogoliubov-de Gennes equation to describe surface excitations. The theoretical calculations are compared with experimental data for liquid helium II, demonstrating excellent agreement. An empirical relation is found for the temperature-dependence of the surface tension. Furthermore, our findings indicate that the contribution of surface excitations remains below 0.7% in the most of recent experiments on Bose-Einstein condensate.
We investigate the effects of quantum fluctuations (QFs) and finite-range interatomic interactions on the ground state and wall tension of a Bose-Einstein condensate (BEC) at zero temperature by means of modified Gross-Pitaevskii equations. The QFs are shown to stiffen the condensate near the hard wall, narrowing the density profile and raising the wall tension above its mean-field value by an amount proportional to square-root of the gas parameter. Finite-range corrections always reduces the wall tension relative to the QFs correction value regardless of the effective range. We find the ordering γ0<γII<γIγ_0 < γ_{\rm II} < γ_{\rm I}γ0​<γII​<γI​ in the dilute regime and identify a crossover condition under which the two beyond-mean-field contributions mutually cancel. These results provide an experimentally accessible way for probing beyond-LHY physics through precision measurements of the wall tension in ultracold Bose gases.
We consider a dilute gas of bosons with repulsive contact interactions, described on the mean-field level by the Gross-Pitaevski\ifmmode \check{i}\else \v{i}\fi{} equation, and bounded by an impenetrable ``hard'' wall (either rigid or flexible). We solve the Bogoliubov-de Gennes equations for excitations on top of the Bose-Einstein condensate analytically, by using matrix-valued hypergeometric functions. This leads to the exact spectrum of gapless Bogoliubov excitations localized near the boundary. The dispersion relation for the surface excitations represents for small wave numbers kkk a ripplon mode with fractional power law dispersion for a flexible wall, and a phonon mode (linear dispersion) for a rigid wall. For both types of excitation we provide, for the first time, the exact dispersion relations of the dilute quantum liquid for all kkk along the surface, extending to $k\ensuremath{\rightarrow}\ensuremath{\infty}$. The small wavelength excitations are shown to be bound to the surface with a maximal binding energy $\mathrm{\ensuremath{\Delta}}=\frac{1}{8}{(\sqrt{17}\ensuremath{-}3)}^{2}m{c}^{2}\ensuremath{\simeq}0.158\phantom{\rule{0.16em}{0ex}}m{c}^{2}$, which both types of excitation asymptotically approach, where mmm is mass of bosons and ccc bulk speed of sound. We demonstrate that this binding energy is close to the experimental value obtained for surface excitations of helium II confined in nanopores, reported in Phys. Rev. B 88, 014521 (2013).
A Bose-Einstein condensate of bosons with repulsion, described by the Gross-Pitaevskii equation and restricted by an impenetrable “hard wall” (either rigid or flexible) which is intended to suppress the “snake instability” inherent for dark solitons, is considered. The Bogoliubov-de Gennes equations to find the spectra of gapless Bogoliubov excitations localized near the “domain wall” and therefore split from the bulk excitation spectrum of the Bose-Einstein condensate are solved. The “domain wall” may model either the surface of liquid helium or of a strongly trapped Bose-Einstein condensate. The dispersion relations for the surface excitations are found for all wavenumbers along the surface up to the ”free-particle” behavior , the latter was shown to be bound to the “hard wall” with some “universal” energy .
We derive and discuss the temperature dependence of the condensate and noncondensate density profile of a Bose-Einstein condensate gas with Feshbach resonance in a parabolic trap. These quantities are calculated self-consistently using the generalized Hartree-Fock-Bogoliubov (HFB) equations within the Bogoliubov approximation. At zero temperature, the HFB equation can be solved by means of a variation method that gives the low-excitation spectrum. Moreover, within the two-body collision theory, we estimate the relationship between the number of atoms in the condensate and the external magnetic field B . The result is in good agreement with data from recent experiments.
Motivated by recent observations of phase-segregated binary Bose-Einstein condensates, we propose a method to calculate the excess energy due to the interface tension of a trapped configuration. By this method one should be able to numerically reproduce the experimental data by means of a simple Thomas-Fermi approximation, combined with interface excess terms and the Laplace equation. Using the Gross-Pitaevskii theory, we find expressions for the interface excesses which are accurate in a very broad range of the interspecies and intraspecies interaction parameters. We also present finite-temperature corrections to the interface tension which, aside from the regime of weak segregation, turn out to be small.
We have applied the Popov version of the Hartree-Fock-Bogoliubov (HFB) approximation to calculate the finite-temperature excitation spectrum of a Bose-Einstein condensate (BEC) of 87Rb{}^{87}\mathrm{Rb}87Rb atoms. For lower values of the temperature, we find excellent agreement with recently published experimental data for the JILA time-averaged orbiting potential trap. In contrast to recent comparison of the results of HFB-Popov theory with experimental condensate fractions and specific heats, there is disagreement of the theoretical and recent experimental results near the BEC phase transition temperature.