Effects of atom numbers on the miscibility-immiscibility transition of a binary Bose-Einstein condensate

The effects of atom numbers on the miscibility-immiscibility transition of a harmonically trapped population imbalanced binary Bose-Einstein condensate are investigated. In two-dimensional parameter space expanded by the strength of interspecies interaction and the ratio of atom numbers between two components, we have mapped out the ground-state phase diagrams for the ideal model without intraspecies interactions, and for the ${}^{23}\mathrm{Na}\text{−}{}^{23}\mathrm{Na}$ mixture and the ${}^{87}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ mixture with experimentally accessible intraspecies scattering lengths, respectively. It is found that all the phase diagrams include the miscible, symmetric immiscible, and asymmetric immiscible phases, and there exists a tricritical point where those three different phases merge. Furthermore, the phase boundaries among those three phases show a strong dependence on the atom numbers of two components. These results can be observed in current experiments.

Figure 1

The ground-state phase diagram of a binary BEC without intraspecies interactions in the $g_{12}\text{−}\gamma$ plane. The red, green, and blue solid lines represent the phase boundaries given by numerical calculations, the green dot-dashed line is the critical value of $g_{12}$ for the asymmetric immiscible ground state given by the variational method, and the solid black circle denotes the tricritical point. The insets filled by red and blue represent the density distributions of two components given by numerical calculations, and the dashed black lines in the insets of the miscible (M) and asymmetric immiscible (AI) phases are the density distributions of Gaussian variational approximation, while the dashed black line in the inset of the symmetric immiscible (SI) phase is ${\varphi }_2={\pi }^{-1/4}{e}^{-z^{\prime 2}/2}$. The system parameters of the insets in the miscible, asymmetric immiscible, and symmetric immiscible phases are $\left(\gamma ,g_{12}\right)=\left(2,1\right)$, $\left(\gamma ,g_{12}\right)=\left(2,4\right)$, and $\left(\gamma ,g_{12}\right)=\left(7,3\right)$, respectively.

Figure 2

Parts (a) and (b) show the locations $z_{1,m}^{\prime }$ and $z_{2,m}^{\prime }$ of the maximum values of two wave functions in the $\gamma \text{−}{g}_{12}$ plane, respectively. Parts (c) and (d) show the dependencies of ${\mathrm{\Delta }}_j$ and $\delta$ on $g_{12}$ for fixed $\gamma$, where $\gamma =2$ and 10 in (c) and (d), respectively. The vertical dot-dashed lines in (c) and (d) indicate the phase boundaries. The inset in (d) magnifies the curve of $\delta$.

Figure 3

The ground-state phase diagram for the ${}^{23}\mathrm{Na}$ mixture with $a_{11}=a_{22}=a=54.54a_B$ in different trap geometries, where the insets in each phase are the representative density profiles of two wave functions for $\gamma =8$. The trapping frequencies are $\left({\omega }_{\perp },{\omega }_z\right)=\left(100,10\right)\mathrm{Hz}$ in (a), $\left({\omega }_{\perp },{\omega }_z\right)=\left(10,100\right)\mathrm{Hz}$ in (b), and $\left({\omega }_{\perp },{\omega }_z\right)=\left(10,10\right)\mathrm{Hz}$ in (c), respectively. For M, SI, and AI phases, the interspecies scattering lengths of the insets are $a_{12}/a=1.01,1.03,1.04$ in (a), $a_{12}/a=1.04,1.2,1.3$ in (b), and $a_{12}/a=1.1,2.2,2.6$ in (c), respectively. In each inset, the green outside surface represents the isosurface of the density profile, the cross section depicts the density distribution in the $x^{\prime }{y}^{\prime }$ plane at $z^{\prime }=0$ (for the oblate and spherical traps) or in the $y^{\prime }{z}^{\prime }$ plane at $x^{\prime }=0$ (for the prolate trap), and the spatial scales in both axial and transverse directions are in $[-6\xi ,6\xi ]$, where $\xi =\sqrt{\hslash /m{\omega }_m}$, with ${\omega }_m=\text{min}\{{\omega }_{\perp },{\omega }_z\}$, is the unit of space.

Figure 4

The ground-state phase diagram in the $a_{12}/\sqrt{a_{11}{a}_{22}}\text{−}\gamma$ plane for the ${}^{87}\mathrm{Rb}$ mixture with $a_{11}\ne a_{22}$, where $(a_{11},a_{22})=(98.98,100.4)a_B$ in (a) and (b), $(a_{11},a_{22})=(95,100.4)a_B$ in (c) and (d), and the insets in each phase diagram are the representative density profiles in three different phases. The trapping frequencies are $({\omega }_{\perp },{\omega }_z)=(10,10)\mathrm{Hz}$ in (a) and (c), and $({\omega }_{\perp },{\omega }_z)=(100,100)\mathrm{Hz}$ in (b) and (d). The parameters of the insets in (a) and (c) are $\gamma =5$ and $a_{12}/\sqrt{a_{1}1a_{22}}=1.05,1.25,1.4$ for the M, SI, and AI regions, respectively. In (b), the parameters of the insets in the M, SI, and AI regions are $\left(\gamma ,a_{12}/\sqrt{a_{11}{a}_{22}}\right)=\left(5,1.02\right),\left(5,1.08\right),\left(5,1.14\right)$, respectively. In (d), the parameters of the insets in the left SI region are $\gamma =1.1$ and $a_{12}/\sqrt{a_{11}{a}_{22}}=1.04$, and the parameters of the insets in AI and the right SI regions are $\left(\gamma ,a_{12}/\sqrt{a_{11}{a}_{22}}\right)=\left(5,1.08\right),\left(5,1.14\right)$, respectively. In each inset, the green outside surface represents the isosurface of the density profile, the cross section depicts the density distribution in the $x^{\prime }{y}^{\prime }$ plane at $z^{\prime }=0$, and the spatial scales in both axial and transverse directions are in $[-6\xi ,6\xi ]$, where $\xi =\sqrt{\hslash /m{\omega }_m}$ with ${\omega }_m=\text{min}\{{\omega }_{\perp },{\omega }_z\}$ is the unit of space.