Miscibility of binary Bose-Einstein condensates with $p$-wave interaction

We investigate the ground-state phase diagram of a binary mixture of Bose-Einstein condensates (BECs) with competing interspecies $s$- and $p$-wave interactions. Exploiting a pseudopotential model for the $l=1$ partial wave, we derive an extended Gross-Pitaevskii (GP) equation for the BEC mixture that incorporates both $s$- and $p$-wave interactions. Based on it, we study the miscible-immiscible transition of a binary BEC mixture in the presence of interspecies $p$-wave interaction, by combining numerical solution of the GP equation and Gaussian variational analysis. Our paper uncovers a dual effect—to either enhance or reduce miscibility—of positive interspecies $p$-wave interaction, which can be precisely controlled by adjusting relevant experimental parameters. By completely characterizing the miscibility phase diagram, we establish a promising avenue towards experimental control of the miscibility of binary BEC mixtures via high partial-wave interactions.

Figure 1

(a), (b) Ground-state phase diagram of a binary mixture of BECs in one dimension, as a function of the dimensionless interspecies $s$-wave interaction ${\beta }_{12}$ and $p$-wave interaction ${\beta }_p$. The diagram is calculated via numerical imaginary-time evolution of the GP equation. The schematic filled with red and green represents the density distributions of the BECs in the miscible (M) and immiscible (IM) phases. We consider in (a) ${\beta }_1={\beta }_2=0$, i.e., without intraspecies interactions; in (b) ${\beta }_1={\beta }_2=2$. Since the boundaries of order parameters $d$ and $\eta$ are identical, only the diagrams of $\eta$ are shown. (c), (d) Dependence of the order parameter $\eta$ on the interspecies interaction ${\beta }_{12}$ for different ${\beta }_p$. We consider in (c) ${\beta }_1={\beta }_2=0$ and in (d) ${\beta }_1={\beta }_2=2$.

Figure 2

(a) Ground-state phase diagram of a binary mixture of BECs in one dimension, obtained via Gaussian variational analysis (see Sec. 3b). Several points $({\beta }_{12},\beta p)$ near the phase transition point $P_2=(4.52,1.5)$, including $P_1=(4.4,1.5), P_3=(4.6,1.5)$, and $P_4=(4.6,1.7)$, are marked. These points are used to illustrate the physical mechanism underlying the abrupt jumps of the order parameter across the phase boundary (see Sec. 3b). The dashed line corresponds to ${\beta }_p=1.5$. (b), (c) Gaussian variational energy of the BEC mixture, $E/N$, as a function of the order parameter $d$ [panel (b)] and $\eta$ [panel (c)], at points $P_1$ (blue dashed line), $P_3$ (orange solid line), and $P_4$ (purple dotted line). Light gray arrows indicate two possible metastable states with nonvanishing order parameter.

Figure 3

Ground-state phase diagram of a binary mixture of BECs in (a) one dimension, (b) two dimensions, and (c) three dimensions, as a function of the dimensionless interspecies $s$-wave interaction ${\beta }_{12}$ and $p$-wave interaction ${\beta }_p$. As the boundaries of order parameters $d$ and $\eta$ are identical, only the distribution of $\eta$ is plotted. In the pure yellow region, the system is in a completely M phase ($\eta =1$ and $d=0$), while in other regions, it is in an AIM phase ($\eta <1$ and $d>0$). In (c), along the line ${\beta }_{12}=43$, several points from $P_5=(43,{10}^3)–P_{10}=(43,0)$ are marked. They offer insights into the dual effects of $p$-wave interaction for phase separation, as analyzed in Sec. 4a. The inset of panel (c) zooms in the region inside the red dashed circle of panel (c).

Figure 4

Dependence of the order parameters $\eta$ (a) and $d$ (b) on the interspecies $s$-wave interaction strength ${\beta }_{12}$, for various interspecies $p$-wave interaction strength ${\beta }_p$ and spatial dimension. Solid line, ${\beta }_p=0.1$; dashed lines, ${\beta }_p=1$; purple dotted line, ${\beta }_p=10.0$ and in three dimensions.

Figure 5

Ground-state phase diagram of a ${}^{87}\mathrm{Rb}\text{−}{}^{23}\mathrm{Na}$ mixture in the $N\text{−}{a}_{12}$ plane. The interspecies scattering lengths are $a_{11}=113.4\mathrm{a}.\mathrm{u}.$ and $a_{22}=57.0\mathrm{a}.\mathrm{u}.$. The $p$-wave interspecies interactions ${\beta }_p/{\beta }_{12}^{\mathrm{BG}}$ are zero in (a), 0.2 in (b), 0.8 in (c), and 3.6 in (d), and the background scattering length $a_{12}^{\mathrm{BG}}=67\mathrm{a}.\mathrm{u}.$ (a1–d1) Distribution of the order parameter $\eta$. (a2–d2) Distribution the order parameter $d$. The red and black dashed lines represent the phase boundaries determined by numerical calculation, and the light gray solid lines in (a1)–(d1) represent the critical value of $a_{12}=80\mathrm{a}.\mathrm{u}.$ The density profiles of the two BEC components at locations $P_{11}=(5\times {10}^2,100\mathrm{a}.\mathrm{u}.), P_{12}=(1\times {10}^4,100\mathrm{a}.\mathrm{u}.), P_{13}=(1.8\times {10}^4,100\mathrm{a}.\mathrm{u}.)$, and $P_{14}=(3\times {10}^4,100\mathrm{a}.\mathrm{u}.)$ in panel (a2) are shown in (e1)–(e4) respectively.

Figure 6

Ground-state phase diagram for the ${}^{87}\mathrm{Rb}\text{−}{}^{23}\mathrm{Na}$ mixture with fixed $a_{11}=113.4\mathrm{a}.\mathrm{u}.$ and $a_{22}=57.0\mathrm{a}.\mathrm{u}.$ in the $N\text{−}{({\beta }_p/{\beta }_{12}^{\mathrm{BG}})}^{1/3}$ plane. The value of $s$-wave interspecies interaction in (a) is the background scattering length $a_{12}=67\mathrm{a}.\mathrm{u}.$, while in (b), it is set to $a_{12}=100\mathrm{a}.\mathrm{u}.$ (a1) and (b1) correspond to the order parameter $\eta$, while (a2) and (b2) correspond to the order parameter $d$. The red, black, and yellow dashed lines represent the phase boundaries given by numerical imaginary-time evolution of the GP equation.