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Tạp chí Khoa học Đại học Đông Á

Capillary Waves at Cylindrical Interface of Two Immiscible Bose-Einstein Condensates

Văn Quyết Hoàng

Published Jan 15, 2024• 0 citations
Publisher

Abstract

By means of the hydrodynamic approach within the Gross-Pitaevskii (GP) theory, dispersion relation of Numbu-Goldstone modes at the interface of the system was found out. While component 1 of the system motion parallel to the interface the dispersion relations is of phonon (w ~ k) and furthermore, the system becomes unstable for k < R0 -1 .

DOI
10.59907/daujs.2.4.2023.136

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By means of the hydrodynamic approach within the Gross-Pitaevskii (GP) theory, dispersion relation of Numbu-Goldstone modes at the interface of the system was found out. While component 1 of the system motion parallel to the interface the dispersion relations is of phonon (w ~ k) and furthermore, the system becomes unstable for k < R0 -1 .
The dispersion relation for capillary waves at the boundary of two different Bose condensates is investigated using a trial wave-function approach applied to the Gross-Pitaevskii (GP) equations. The surface tension is expressed by the parameters of the GP equations. In the long wave-length limit the usual dispersion relation is re-derived while for wavelengths comparable to the healing length we predict significant deviations from the ω∝k3/2ω\propto k^{3/2}ω∝k3/2 law which can be experimentally observed. We approximate the wave variables by a frozen order parameter, i.e. the wave function is frozen in the superfluid analogous to the magnetic field in highly conductive space plasmas.
The ripplon modes of two segregated Bose-Einstein condensates (BECs) confined by one and two hard walls are respectively studied by means of the hydrodynamic approach within the Gross-Pitaevskii (GP) theory. For the system at rest we find that due to the spatial restriction the dispersion relations are of the form ω∼k2\omega \sim {k^2}ω∼k2 in low momentum limit for both cases, while for the system in motion parallel to the interface the dispersion relations for both cases are ω∼k\omega \sim kω∼k at low momentum limit and, furthermore, the system becomes unstable.
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.
We study the parametric resonance of capillary waves on the interface between two immiscible Bose-Einstein condensates pushed towards each other by an oscillating force. Guided by analytical models, we solve numerically the coupled Gross-Pitaevskii equations for a two-component Bose-Einstein condensate at zero temperature. We show that, at moderate amplitudes of the driving force, the instability is stabilized due to nonlinear modifications of the oscillation frequency. When the amplitude of the driving force is large enough, we observe a detachment of droplets from the Bose-Einstein condensates, resulting in the generation of quantum vortices (skyrmions). We analytically investigate the vortex dynamics, and conditions of quantized vortex generation.
Dynamical instabilities at the interface between two Bose-Einstein condensates that are moving relative to each other are investigated using mean-field and Bogoliubov analyses. Kelvin-Helmholtz instability is dominant when the interface thickness is much smaller than the wavelength of the unstable interface mode, whereas the counter-superflow instability becomes dominant in the opposite case. These instabilities emerge not only in an immiscible system but also in a miscible system where an interface is produced by external potential. Dynamics caused by these instabilities are numerically demonstrated in rotating trapped condensates.