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arXiv:1812.08002 · Physical review. B./Physical review. B

Exact surface-wave spectrum of a dilute quantum liquid

Peter V. Pikhitsa, Uwe R. Fischer

Published May 10, 2019• 3 citations
arXivHTMLPDF

Abstract

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 $k$ 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 $k$ 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 $m$ is mass of bosons and $c$ 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).

DOI
10.1103/physrevb.99.184504

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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→\ensuremath∞k\ensuremath{\rightarrow}\ensuremath{\infty}k\ensuremath→\ensuremath∞. The small wavelength excitations are shown to be bound to the surface with a maximal binding energy \ensuremathΔ=18(17\ensuremath−3)2mc2\ensuremath≃0.158mc2\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}\ensuremathΔ=81​(17​\ensuremath−3)2mc2\ensuremath≃0.158mc2, 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 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.
The localized low-energy interfacial excitations, or interfacial Nambu-Goldstone modes, of phase-segregated binary mixtures of Bose-Einstein condensates are investigated analytically. To this end a double-parabola approximation (DPA) is performed on the Lagrangian density in Gross-Pitaevskii theory for a system in a uniform potential. This DPA entails a model in which analytic expressions are obtained for the excitations underlying capillary waves or ripplons for arbitrary strength KKK (>1>1>1) of the phase segregation. The dispersion relation $\ensuremath{\omega}(k)\ensuremath{\propto}{k}^{3/2}$ is derived directly from the Bogoliubov--de Gennes equations in the limit that the wavelength $2\ensuremath{\pi}/k$ is much larger than the interface width. The proportionality constant in the dispersion relation provides the static interfacial tension. A correction term in $\ensuremath{\omega}(k)$ of order k5/2{k}^{5/2}k5/2 is calculated analytically within the DPA model. The combined result is tested against numerical diagonalization of the exact Bogoliubov--de Gennes equations. Satisfactory agreement is obtained in the range of physically relevant wavelengths. The ripplon dispersion relation is relevant to state-of-the-art experiments using (quasi)uniform optical-box traps. Furthermore, within the DPA model explicit expressions are obtained for the structural deformation of the interface due to the passing of the capillary wave. It is found that the amplitude of the wave is enhanced by an amount that is quadratic in the ratio of the phase velocity $\ensuremath{\omega}/k$ to the sound velocity ccc. For generic mixtures consisting of condensates with unequal healing lengths, an additional modulation is predicted of the common value of the condensate densities at the interface.
The Bogoliubov treatment of a weakly interacting Bose gas is extended by allowing macroscopic occupation of many single-particle quantum states. This generalization of having more than a single condensed mode gives rise to a nonuniform condensate and is being considered as a possible mechanism for the description of a dense strongly interacting Bose gas at low temperatures. The model with a repulsive $\ensuremath{\delta}$-function interparticle potential gives an energy spectrum of elementary excitations or quasiparticles with both phonon and roton features. The values obtained for the roton minimum energy $\ensuremath{\Delta}$ and the speed of (first) sound agree reasonably well with inelastic-neutron-scattering data.
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.