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Journal of Science Natural Science

PHENOMENOLOGICAL ANALOGY BETWEEN GROSS-PITAEVSKII THEORY FOR BOSE-EINSTEIN CONDENSATE MIXTURES IN INFINITE SPACE AND CLASSICAL MECHANICS

Pham Duy Thanh, Tran Ky Vi, Nguyễn Văn Thụ

Published Mar 31, 2025• 0 citations

Abstract

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.

DOI: 10.18173/2354-1059.2025-0004

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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.

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

K

. 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=3/2

. 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

K

. 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.

We present a study of binary mixtures of Bose–Einstein condensates confined in a double-well potential within the framework of the mean field Gross–Pitaevskii (GP) equation. We re-examine both the single component and the binary mixture cases for such a potential, and we investigate what are the situations in which a simpler two-mode approach leads to an accurate description of their dynamics. We also estimate the validity of the most usual dimensionality reductions used to solve the GP equations. To this end, we compare both the semi-analytical two-mode approaches and the numerical simulations of the one-dimensional (1D) reductions with the full 3D numerical solutions of the GP equation. Our analysis provides a guide to clarify the validity of several simplified models that describe mean-field nonlinear dynamics, using an experimentally feasible binary mixture of an F = 1 spinor condensate with two of its Zeeman manifolds populated, m = ±1.

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