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arXiv:cond-mat/0107160 · Physical Review A

Waves on an interface between two phase-separated Bose-Einstein condensates

I. E. Mazets

Published Feb 27, 2002• 42 citations
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Abstract

We examine waves localized near a boundary between two weakly segregated Bose-Einstein condensates. In the case of a wavelength of the order of or larger than the thickness of the overlap region the variational method is used. The dispersion laws for the two oscillation branches are found in analytic form. The opposite case of a wavelength much shorter than the healing length in the bulk condensate is also discussed.

DOI
10.1103/physreva.65.033618

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We examine waves localized near a boundary between two weakly segregated Bose-Einstein condensates. In the case of a wavelength of the order of or larger than the thickness of the overlap region the variational method is used. The dispersion laws for the two oscillation branches are found in analytic form. The opposite case of a wavelength much shorter than the healing length in the bulk condensate is also discussed.
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 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 perform a mean-field study of the binary Bose-Einstein condensate mixtures as a function of the mutual repulsive interaction strength. In the phase segregated regime, we find that there are two distinct phases: the weakly segregated phase characterized by a ``penetration depth'' and the strongly segregated phase characterized by a healing length. In the weakly segregated phase the symmetry of the shape of each condensate will not take that of the trap because of the finite surface tension, but its total density profile still does. In the strongly segregated phase even the total density profile takes a different symmetry from that of the trap because of the mutual exclusion of the condensates. The lower critical condensate-atom number to observe the complete phase segregation is discussed. A comparison to recent experimental data suggests that the weakly segregated phase has been observed.
Bogoliubov-de Gennes equations and the excitation spectrum of a two-component Bose-Einstein condensate (BEC) are derived with an arbitrary interaction between bosons, including long-range and short-range forces. Spectra of optical and acoustic plasmons (APs) are obtained. For the Coulomb interaction the AP has a quadratic long-wavelength dispersion in contrast with the linear dispersion in the normal-state two-component plasma. The nonconverting BEC mixture segregates into two phases for some two-body interactions. Gross-Pitaevskii equations are solved for the phase-segregated BEC. A possibility of boundary-surface and other localized excitations is studied.
This paper deals with the study of the behavior of the wave functions of a two-component Bose–Einstein condensate near the interface, in the case of strong segregation. This yields a system of two coupled ordinary differential equations for which we want to have estimates on the asymptotic behavior, as the strength of the coupling tends to infinity. As in phase separation models, the leading order profile is a hyperbolic tangent. We construct an approximate solution and use the properties of the associated linearized operator to perturb it into a genuine solution for which we have an asymptotic expansion. We prove that the constructed heteroclinic solutions are linearly nondegenerate, in the natural sense, and that there is a spectral gap, independent of the large interaction parameter, between the zero eigenvalue (due to translations) at the bottom of the spectrum and the rest of the spectrum. Moreover, we prove a uniqueness result which implies that, in fact, the constructed heteroclinic is the unique minimizer (modulo translations) of the associated energy, for which we provide an expansion.