Analysis beyond the Thomas-Fermi approximation of the density profiles of a miscible two-component Bose-Einstein condensate

We investigate a harmonically trapped two-component Bose-Einstein condensate within the miscible regime, close to its boundaries, for different ratios of effective intra- and interspecies interactions. We derive analytically a universal equation for the density around the different boundaries in one, two, and three dimensions, for both the coexisting and spatially separated regimes. We also present a general procedure to solve the Thomas-Fermi approximation in all three spatial dimensionalities, reducing the complexity of the Thomas-Fermi problem for the spatially separated case in one and three dimensions to a single numerical inversion. Finally, we analytically determine the frontier between the two different regimes of the system.

Figure 1

Different density distributions obtained within the TF approximation (solid lines) from Eqs. (4) and (5) and by numerically integrating (dotted lines) the TCGPEs [Eq. (2)] for the coexisting (a,b) and spatially separated (c) regimes. The different boundaries obtained in the TF approximation [Eqs. (6)] are highlighted. In this plot we assume $g_s>g_{3-s}$ with $\mathrm{\Pi }=1$.

Figure 2

Numerical solution of the universal equation [Eqs. (12) and (19)] (black solid line) and the two asymptotic behaviors given by the Hastings-McLeod solution [Eq. (20)]: the Airy function (green dot-dashed line) and $\sqrt{-\xi }$ (red dashed line). We also plot the asymptotic behavior of the Airy function [Eq. (21)] for $\xi \to +\infty$ (blue dotted line).

Figure 3

Comparison between the results of the density profiles obtained through the universal equation [Eqs. (12) and (19)], the TF approximation (Sec. 4), and the numerical solution of the TCGPEs [Eq. (2)]. The first row shows the TF density profile (blue dotted line) and the numerical solution of the TCGPEs (solid lines) of a TCBEC in the spatially separated regime for the 1D (first column), 2D (second column), and 3D (third column) cases for two different values of $g_2$ in each plot. We also plot a magnification around $R_1$ (second row), $R_2$ (third row), and ${\tilde{R}}_1$ (fourth row) where we include the asymptotic approximation of the universal equation $\sqrt{2}\text{Ai}\left(r\right)$ (dot-dashed line) derived in Sec. 3. The parameter values used are (i) for the 1D case $\chi =0.985, g_2=200$ (black lines), and $g_2=1000$ (red or gray lines); (ii) for the 2D case $\chi =0.9, g_2=1000$ (black lines), and $g_2=10000$ (red or gray lines); and (iii) for the 3D case $\chi =0.9, g_2=1000$ (black lines), and $g_2=10000$ (red or gray lines). Note that the magnifications only include the component under study. For a complete description of the dimensionless parameters and scalings see the text.