Nucleation and wetting transitions in three-component Bose-Einstein condensates in Gross-Pitaevskii theory: exact results

TL;DR

This work offers exact results for nucleation and wetting transitions in a three-component BEC mixture within Gross-Pitaevskii theory. By solving the GP equations in carefully chosen solvable regimes, it derives exact nucleation conditions and, in several cases, identifies degenerate first-order wetting transitions coinciding with nucleation boundaries. It provides both symmetric and asymmetric intermediate-segregation analyses and a strong-segregation treatment, including a Schrödinger-type formulation and hypergeometric solutions. The results are contrasted with the double-parabola approximation, showing that DP captures transition locations well but can mispredict the order of wetting transitions. Overall, the findings illuminate a close interplay between nucleation and wetting in multi-component BECs and point to experimental tunability via interaction strengths.

Abstract

Nucleation and wetting transitions are studied in a three-component Bose-Einstein condensate mixture within Gross-Pitaevskii theory. For special cases of intermediate segregation between components 1 and 2, the nucleation phase transition of a surfactant film of component 3 is obtained by exact solution. Additional exact results for the nucleation transition are derived in the limit of strong segregation between components 1 and 2. In this limit the exact first-order wetting phase boundary is obtained using analytical and numerical methods, and is contrasted with the exact nucleation and wetting phase boundary derived previously for a two-component Bose-Einstein condensate mixture at a hard optical wall. Exact results for the three-component mixture are compared with results from the double-parabola approximation used in an earlier work.

Figures (9)

• Figure 1: Wetting phase diagram for $K_{12} = 3$, $\xi_1 = \xi_2$ and $K \equiv K_{13}=K_{23}$. Nucleation transition and critical wetting transition at three-phase coexistence ($\mu_3 = \bar{\mu}_3$) in the plane of healing length ratio $\bar{\xi}_3/(\xi_2+\bar{\xi}_3)$ vs inverse coupling $1/K$. The solid red line is the exact nucleation line given by Eq. \ref{['exactnuclsym']}. Open circles denote numerically determined critical wetting transition points. These do, in general, not coincide with the nucleation transition. See Fig. \ref{['fig:interface_tensions_Malomed_total']} for examples. The point $D$ (star) denotes the exact location of a degenerate first-order wetting transition which is coincident with the nucleation transition.
• Figure 2: Reduced grand potential $\tilde{\Omega} = \Omega/(4P\xi_2)$ vs inverse coupling $1/K$, with $K\equiv K_{13}=K_{23}$ at three-phase coexistence ($\mu_3 = \bar{\mu}_3$). Fixed parameters are $K_{12} = 3$ and $\xi_1 = \xi_2$. (a) A nucleation transition takes place at $N$ and a critical wetting transition at $W$, e.g., for healing length ratio $\bar{\xi}_3/\xi_2 = 1/9$. (b) The nucleation transition at $N$ and the (degenerate) first-order wetting transition at $W$ exactly coincide for $\bar{\xi}_3/\xi_2 = 1/2$ and $K=3/2$. (c) A nucleation transition takes place at $N$ and a critical wetting transition at $W$, e.g., for healing length ratio $\bar{\xi}_3/\xi_2 = 1$.
• Figure 3: Wetting phase diagram for $K_{12}=3/2$, $\xi_2 = 2 \,\xi_1$ and $K \equiv K_{13} = K_{23}$. Nucleation transition and wetting transition at three-phase coexistence ($\mu_3 = \bar{\mu}_3$) in the plane of healing length ratio $\bar{\xi}_3/(\xi_2+\bar{\xi}_3)$ vs inverse coupling $1/K$. The solid red line is the exact nucleation condition given by Eq. \ref{['exactnuclsym3halfs']}. Open circles denote numerically determined first-order wetting transition points. These do, in general, not coincide with the nucleation transition. See Fig. \ref{['fig:interface_tensions_Indekeu']} for examples.
• Figure 4: Reduced grand potential $\tilde{\Omega} = \Omega/(4P\xi_2)$ vs inverse coupling $1/K$, with $K\equiv K_{13}=K_{23}$ at three-phase coexistence ($\mu_3 = \bar{\mu}_3$). Fixed parameters are $K_{12} = 3/2$ and $\xi_2 = 2\,\xi_1$. (a) A nucleation transition takes place at $N$ and a first-order wetting transition at $W$, e.g., for healing length ratio $\bar{\xi}_3/\xi_2 = 1/9$; (b) The exact nucleation transition at $N$ and the numerically determined (non-degenerate) first-order wetting transition at $W$ coincide for $\bar{\xi}_3/\xi_2 \approx 1/\sqrt{10}$ and $K \approx 6/5$; (c) A nucleation transition takes place at $N$ and a first-order wetting transition at $W$, e.g., for healing length ratio $\bar{\xi}_3/\xi_2 = 1$.
• Figure 5: Numerically computed wave function profiles at bulk three-phase coexistence ($\mu_3 = \bar{\mu}_3$) evaluated at the first-order wetting transition $W$ in Fig. \ref{['fig:interface_tensions_Indekeu']}(a), with $K_{13} = K_{23} = K_{\rm wet} \approx 1.2547$, $K_{12} = 3/2$ and $\xi_2 = 2\,\xi_1, \bar{\xi}_3/\xi_2=1/9$. (a) Nonwet state. Thin surfactant film of component 3 adsorbed at the 1-2 interface. (b) Wet state (infinitely thick layer of component 3), with the 1-3 and 3-2 interfaces located about $\tilde{x}^-$ and $\tilde{x}^+$, respectively. Both states, (a) and (b), have the same grand potential and therefore "coexist" at first-order wetting.