Approximate theories for binary magnetic quantum droplets

We develop two approximate theories to describe the miscible and immiscible droplets that can occur in a binary mixture of highly magnetic bosonic atoms. In addition to allowing simpler calculations, the approximate theories provide insight into the role of quantum fluctuations in the two regimes. Results are validated by comparison to those from the extended Gross-Pitaevskii equation. As an application, we solve for the ground-state droplets crossing the miscible-immiscible transition as a function of the short-ranged interspecies interaction parameter. We consider regimes where the transition occurs suddenly or as a smooth cross-over. Using dynamical calculations, we show that the character of the transition is revealed in the number of domains produced when ramping the droplet into the immiscible regime.

Figure 1

Real part of the quantum fluctuation energy integrand. $I_{E+}^{5/2}$ (solid red, upper) and $I_{E-}^{5/2}$ (solid blue, lower) from Eq. (5) and IA of $I_{E+}^{5/2}$ (dashed black) from Eq. (14) for $\{a_{11},a_{22},a_{12}\}=\{70,120,50\}{a}_0$ and, $\{a_{11}^{\mathrm{dd}}, a_{22}^{\mathrm{dd}}, a_{12}^{\mathrm{dd}}\}=130.1a_0\equiv a^{\mathrm{dd}}$ and $E_0={\hslash }^2/\left[m{\left(a^{\mathrm{dd}}\right)}^2\right]$. In (a) ${\theta }_k=0$, (b) ${\theta }_k=\pi /2$ and the relative density ratio $n_1/n_2$ is varied. In (c) $n_1/n_2=1$ and ${\theta }_k$ is varied.

Figure 2

(a) Chemical potential correction $\mathrm{\Delta }{\mu }_1/n_2^{3/2}$ (red, lower) and $\mathrm{\Delta }{\mu }_2/n_2^{3/2}$ (blue, upper) from Eq. (31) in terms of the ratio $n_1/n_2$ and (b) similarly $\mathrm{\Delta }{\mu }_i/n_1^{3/2}$ in terms of the ratio $n_2/n_1$. Also shown are the single component limit (23) ($\times$) and the impurity limit (24) ($⊚$). $\{a_{11},a_{22},a_{12}\}/a_0=\{50,120,100\}$ with $a_{ij}^{\mathrm{dd}}$ as in Fig. 1.

Figure 3

Immiscible BMG droplets. (a1), (b1) Axial density of component 1 (red) and 2 (blue), (a2), (b2) corresponding chemical potential corrections, (a3), (b3) isodensity surfaces at $n_i=2\times {10}^{21}/{\mathrm{m}}^3$, and (a4), (b4) ${\varepsilon }_{\text{IA}}$ from (37) (red, upper) and overlap of EGPE states ${\chi }_{12}$ (black, lower). Showing the EGPE (solid), and IA (dashed). Parameters are $a_{11}=70a_0, N_1={10}^4, N_2=5\times {10}^3$, and (a) $a_{22}=70a_0, a_{12}=75a_0$, (b) $a_{22}=85a_0, a_{12}=84.5a_0$.

Figure 4

Miscible BMG droplets. (a)–(d) axial density profiles of component 1 (red), component 2 (blue) and total density (black), using the EGPE (solid), and SMA (dashed). (b1), (d1) show the $n_i=7.5\times {10}^{20}/{\mathrm{m}}^3$ isodensity surfaces for (b) and (d), respectively. (e), (f) Comparison of total (black), component 1 (red), and component 2 (blue) peak densities as $N_1/N$ varies, using the EGPE ($\times$) and SMA ($◯$). Scattering lengths are (a)–(f) $a_{11}=70a_0, a_{12}=65a_0$, (a), (b), (e) $a_{22}=70a_0$, and (c), (d), (f) $a_{22}=85a_0$. The total number of atoms in (a)–(f) $N=2.7\times {10}^4$, with (a), (d) $N_1=1.5\times {10}^4$, (b,c) $N_1=2.2\times {10}^4$. (g) plot of ${\varepsilon }_{\mathrm{sma}}$ from Eq. (39) ($▴$) and ${\varepsilon }_{\mathrm{shape}}$ from Eq. (38) ($•$) with $N_{\mathrm{total}}=27\times {10}^3$, except for (yellow) for which $N_{\mathrm{total}}=5\times {10}^3$, and {$a_{11}, a_{22}, a_{12}\}/a_0$ (blue): {85, 70, 30}, (light-green): {70, 70, 65}, (light-blue): {85, 70, 60}, (yellow): {85, 70, 65}, (green): {85, 70, 65} (in order from top at the left of the subfigure).

Figure 5

(a), (c), (e) Miscible to asymmetric immiscible transition with $a_{11}=a_{22}=70a_0, N_1={10}^4$, and $N_2=5\times {10}^3$. (b), (d), (f) Miscible to symmetric immiscible transition with $a_{11}=70a_0, a_{22}=85a_0, N_1={10}^4$, and $N_2=5\times {10}^3$. (a), (b) Pseudospin density $m_z$, (c), (d) component overlap ${\chi }_{12}$, and (e), (f) the energy per particle $E/Nh$, as $a_{12}$ varies. In (e), (f) we compare EGPE ($\times$) with the SMA ($◯$) and IA ($\square$).

Figure 6

Preparation of immiscible droplets from an initial miscible droplet. (a), (b) $a_{12}$ from 65 to $75a_0$ with other parameters as in Figs. 5, 5 and 5; (c), (d) $a_{12}$ from 70 to $80a_0$ with other parameters as in Figs. 5, 5 and 5; (a), (c) using a ${\tau }_r=10\mathrm{ms}$ and (b), (d) a 40-ms ramp.