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A Quadruple-Parabola Approximation for the Static Interfacial Properties of Asymmetric Phase-Separated Binary Bose--Einstein Condensates

Hoàng Văn Quyết

Published Sep 29, 2026• 0 citations
Publisher

Abstract

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 $K\in\{1.2,1.5,2,3,5\}$ 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 $\sim30\%$ 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 $L^2$ 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 $K\ge2$ 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.

DOI
10.21203/rs.3.rs-11029284/v1

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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, \xii1≠\xii2\xii_1\neq\xii_2\xii1​=\xii2​. 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 \siDPA≥\siTPA≥\siQPA≥\siGP\si_{\rm DPA}\ge\si_{\rm TPA}\ge\si_{\rm QPA}\ge\si_{\rm GP}\siDPA​≥\siTPA​≥\siQPA​≥\siGP​ for arbitrary \xii\xii\xii; the QPA reproduces the TPAexactly in the symmetric limit \xii=1\xii=1\xii=1 and obeys an exact species-exchange covariance, \si(K,\xii)=\xii \si(K,1/\xii)\si(K,\xii)=\xii\,\si(K,1/\xii)\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}K\in\{1.2,1.5,2,3,5\}K∈{1.2,1.5,2,3,5} and \xii∈{0.5,0.7,1,1/0.7,2}\xii\in\{0.5,0.7,1,1/0.7,2\}\xii∈{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%\sim30\%∼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 999--42%42\%42% against 6.06.06.0--9.6%9.6\%9.6% for the TPA and QPA. This energy improvement does not translate into a smaller pointwise profile error: with a properly LLL-independent L2L^2L2 metric, the QPA profile error exceeds the TPA's in the majority of (species, KKK, \xii\xii\xii) combinations tested, but the TPA is not universally superior -- in 888 of 404040 such combinations, concentrated at K≥2K\ge2K≥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.
The interfacial profiles and interfacial tensions of phase-separated binary mixtures of Bose-Einstein condensates are studied theoretically. The two condensates are characterized by their respective healing lengths ${\ensuremath{\xi}}_{1}$ and ${\ensuremath{\xi}}_{2}$ and by the interspecies repulsive interaction KKK. An exact solution to the Gross-Pitaevskii (GP) equations is obtained for the special case ${\ensuremath{\xi}}_{2}/{\ensuremath{\xi}}_{1}=1/2$ and K=3/2K=3/2K=3/2. Furthermore, applying a double-parabola approximation (DPA) to the energy density featured in GP theory allows us to define a DPA model, which is much simpler to handle than GP theory but nevertheless still captures the main physics. In particular, a compact analytic expression for the interfacial tension is derived that is useful for all ${\ensuremath{\xi}}_{1},\phantom{\rule{0.28em}{0ex}}{\ensuremath{\xi}}_{2}$, and KKK. An application to wetting phenomena is presented for condensates adsorbed at an optical wall. The wetting phase boundary obtained within the DPA model nearly coincides with the exact one in GP theory.
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.
Based on an analogy between the Gross-Pitaevskii (GP) equations for binary mixtures of Bose-Einstein condensates (BECs) at zero temperature and Newton’s equations of motion for a particle in a conservative field, we derive exact analytical solutions for the coupled GP equations in several phase-segregated BEC scenarios. In a straightforward manner, exact analytical solutions are obtained for both the symmetric and anti-symmetric cases. Furthermore, our approach enables us to propose approximate analytical solutions as an alternative to those derived from the widely used double-parabola approximation (DPA). Numerical computations demonstrate that our approximate solutions closely match the exact solutions within the GP framework. This finding is of particular significance for determining the wavefunctions of the condensates, a fundamental task in investigating the static properties of BECs.
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.