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
K (
>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 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
c. 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.
Abstract We develop the Quadruple-Parabola Approximation (QPA), an analytical, piecewise-quadratic trial-function model for the static, planar, zero-temperature interface of a phase-separated binary Bose--Einstein condensate (BEC) mixture with unequal healing lengths, $\xii_1\neq\xii_2$. The construction extends the Double- and Triple-Parabola Approximations (DPA, TPA) by assigning independent curvatures and matching positions to the two components whilepreserving exact linear solvability. The DPA, TPA, and QPA trial families are nested by construction, so that evaluating every trial profile in the exact Gross--Pitaevskii (GP) functional establishes the variational ordering $\si_{\rm DPA}\ge\si_{\rm TPA}\ge\si_{\rm QPA}\ge\si_{\rm GP}$ for arbitrary $\xii$; the QPA reproduces the TPAexactly in the symmetric limit $\xii=1$ and obeys an exact species-exchange covariance, $\si(K,\xii)=\xii\,\si(K,1/\xii)$, both verified numerically. Benchmarked against an independent GP boundary-value solver over
K∈{1.2,1.5,2,3,5} and $\xii\in\{0.5,0.7,1,1/0.7,2\}$, the QPA systematically lowers the variational interfacial tension relative to the TPA in every asymmetric case tested -- removing up to
∼30% of the TPA's residual tension error in the weakest-segregation, most-asymmetric regime, though typically only a few percent elsewhere -- while the DPA over-estimates the tension by
9--
42% against
6.0--
9.6% for the TPA and QPA. This energy improvement does not translate into a smaller pointwise profile error: with a properly
L-independent
L2 metric, the QPA profile error exceeds the TPA's in the majority of (species,
K, $\xii$) combinations tested, but the TPA is not universally superior -- in
8 of
40 such combinations, concentrated at
K≥2 and related in mirrored pairs by the species-exchange covariance, the QPA is marginally better instead. Within the parameter window, optimisation protocol, and admissible domain explored here, neither the TPA nor the QPA reaches sub-percent tension accuracy. The results clarify both the usefulness and the intrinsic limitations of piecewise-quadratic interfacial models for binary quantum fluids: alarger variational trial family can be a genuinely better energetic approximation without being a better pointwise one.
Within Gross-Pitaevskii (GP) theory we derive the interface potential $V(\ensuremath{\ell})$ which describes the interaction between the interface separating two demixed Bose-condensed gases and an optical hard wall at a distance $\ensuremath{\ell}$. Previous work revealed that this interaction gives rise to extraordinary wetting and prewetting phenomena. Calculations that explore nonequilibrium properties by using $\ensuremath{\ell}$ as a constraint provide a thorough explanation for this behavior. We find that at bulk two-phase coexistence, $V(\ensuremath{\ell})$ for both complete wetting and partial wetting is monotonic with exponential decay. Remarkably, at the first-order wetting phase transition, $V(\ensuremath{\ell})$ is independent of $\ensuremath{\ell}$. This anomaly explains the infinite continuous degeneracy of the grand potential reported earlier. As a physical application, using $V(\ensuremath{\ell})$ we study the three-phase contact line where the interface meets the wall under a contact angle $\ensuremath{\theta}$. Employing an interface displacement model we calculate the structure of this inhomogeneity and its line tension $\ensuremath{\tau}$. Contrary to what happens at a usual first-order wetting transition in systems with short-range forces, $\ensuremath{\tau}$ does not approach a nonzero positive constant for $\ensuremath{\theta}\ensuremath{\rightarrow}0$, but instead approaches zero (from below) in the manner $\ensuremath{\tau}\ensuremath{\propto}\ensuremath{-}\ensuremath{\theta}$ as would be expected for a critical wetting transition. This hybrid character of $\ensuremath{\tau}$ is a consequence of the absence of a barrier in $V(\ensuremath{\ell})$ at wetting. For a typical $V(\ensuremath{\ell})=\mathcal{S}\phantom{\rule{0.16em}{0ex}}exp(\ensuremath{-}\ensuremath{\ell}/\ensuremath{\xi})$, with
S the spreading coefficient and $\ensuremath{\xi}$ a decay length, we conjecture that $\ensuremath{\tau}=\ensuremath{-}2\phantom{\rule{0.16em}{0ex}}(1\ensuremath{-}ln2)\phantom{\rule{0.16em}{0ex}}\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\ensuremath{\xi}\phantom{\rule{0.16em}{0ex}}sin\ensuremath{\theta}$ is exact within GP theory, with $\ensuremath{\gamma}$ the interfacial tension and $0\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}\ensuremath{\pi}$.
An ultralow-temperature binary mixture of Bose-Einstein condensates adsorbed at an optical wall can undergo a wetting phase transition in which one of the species excludes the other from contact with the wall. Interestingly, while hard-wall boundary conditions entail the wetting transition to be of first order, using Gross-Pitaevskii theory we show that first-order wetting as well as critical wetting can occur when a realistic exponential optical wall potential (evanescent wave) with a finite turn-on length $\ensuremath{\lambda}$ is assumed. The relevant surface excess energies are computed in an expansion in $\ensuremath{\lambda}/{\ensuremath{\xi}}_{i}$, where ${\ensuremath{\xi}}_{i}$ is the healing length of condensate
i. Experimentally, the wetting transition may best be approached by varying the interspecies scattering length
a12 using Feshbach resonances. In the hard-wall limit, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, exact results are derived for the prewetting and first-order wetting phase boundaries.
In previous work within Gross-Pitaevskii (GP) theory for ultracold gases, wetting phase transitions were predicted for a phase-segregated two-component Bose-Einstein condensate (BEC) adsorbed at an optical wall. The wetting phase diagram was found to depend on intrinsic atomic parameters, being the masses and the scattering lengths, and on the extrinsic wall boundary condition. Here we study wetting transitions in GP theory without an optical wall in a setting with three phase-segregated BEC components instead of two. The boundary condition is removed by replacing the wall with the third component and treating the three phases on an equal footing. This leads to an unequivocal wetting phase diagram that depends only on intrinsic atomic parameters. It features first-order and critical wetting transitions, and prewetting phenomena. The phase boundaries are computed by numerical solution of the GP equations. In addition, useful analytic results are obtained by extending the established double-parabola approximation to three components.
In this study, we determine the interface position of a binary Bose-Einstein condensate mixture in the prewetting phase, confined by a hard wall, at absolute zero temperature. An approximate analytical result is obtained by employing the double-parabola approximation (DPA) to the quartic potential energy density within the Gross-Pitaevskii (GP) theory. The interface position depends on the control parameter and the chemical potential ratio. Our analytical result shows good agreement with numerical results, which indicates that the DPA is a reliable method. This study provides insights for experiments focused on observing the wetting transition of gases in the Bose-Einstein condensate phase.