Impurity in a heteronuclear two-component Bose mixture

We study the fate of an impurity in an ultracold heteronuclear Bose mixture, focusing on the experimentally relevant case of a ${}^{41}\mathrm{K}\text{−}{}^{87}\mathrm{Rb}$ mixture, with the impurity in a ${}^{41}\mathrm{K}$ hyperfine state. Our paper provides a comprehensive description of an impurity in a BEC mixture with contact interactions across its phase diagram. We present results for the miscible and immiscible regimes, as well as for the impurity in a self-bound quantum droplet. Here, varying the interactions, we find exotic states where the impurity localizes either at the center or at the surface of the droplet.

Figure 1

Mobile impurity in a heteronuclear ${}^{87}\mathrm{Rb}\text{−}{}^{41}\mathrm{K}$ mixture. (a) Quantum phases of the mixture. In the self-bound phase (pink region), the mixture forms a droplet, stabilized by quantum fluctuations, on which the impurity can reside. In the miscible phase (green region), the impurity lives in a bosonic mixture, whereas the yellow region corresponds to the immiscible, phase-separated regime, in which the impurity occupies the most energetically favorable of the two domains. (b) Tunable scattering lengths $a_{\text{I-Rb}}$ between the impurity and ${}^{87}\mathrm{Rb}$ (red) and the intercomponent scattering length $a_{\text{K-Rb}}$ (orange) across the magnetic field interval $B\in [60,105]$ G; note the Feshbach resonance at $B=78.9$ G. In this magnetic field range, $a_{\text{I-K}}$, $a_{\text{Rb-Rb}}$, and $a_{\text{K-K}}$ are approximately constant (see main text). Dashed blue: Effective scaled mean-field coupling $\delta \tilde{g}=\delta g\sqrt{m_1{m}_2}/4\pi {\hslash }^2$ of the two-component mixture. When $\delta g<0$, the system is unstable toward collapse and is stabilized by quantum fluctuations into a self-bound droplet phase (pink region). Full gray lines denote the transitions of impurities localized at the center (light pink), at the surface (pink) $63.5\text{G}<B<66.1\text{G}$, or expelled from the droplet (dark pink) for $N_D=4\times {10}^4$ particles in the droplet [see also Figs. 3].

Figure 2

Impurity properties in the miscible and immiscible phase. (a) Polaron spectral function $A\left(E\right)$ in the miscible phase of the mixture, as a function of external magnetic field $B$ and the polaron energy. The dotted white line shows the mean-field solution of the equations of motion, whereas the dashed blue line in the immiscible region shows the energy of the polaron coupled to a single component, see main text. Energies are scaled by the peak density $n$ of the mixture. (b) Quasiparticle weight $Z$ in the immiscible region, as a function of magnetic field $B$.

Figure 3

Localized impurity in a self-bound droplet. (a), (b) Representative impurity-droplet Feynman diagrams at second-order perturbation theory in the small parameters $a_{Ii}/{\xi }_i$, $i=A,B$. The red and green dashed lines refer to the first and second normal modes of the condensate, respectively, each one being a different superposition of an excitation in the first and second components; the blue lines represent impurity propagators. (c)–(f) Effective impurity potential $V_{\text{eff}}\left(r\right)$ (red curves) for an impurity in a self-bound droplet (yellow shaded region) with $N_D=4\times {10}^4$. The impurity density (blue shaded) and its ground-state energy (blue dashed lines) are shown for several magnetic fields. (c) $B=63.5$ G. The potential does not support bound states in three dimensions. (d) $B=65.1$ G and (e) $B=66.0$ G. The impurity is localized at the surface of the droplet at a distance $r\approx 1\mu \mathrm{m}$ from the center. (f) $B=66.6$ G. The impurity is centered in the self-bound droplet.

Figure 4

Densities and excited states of the impurity. (a)–(c) Impurity density of the ground and excited states for $N_D=4\times {10}^4$. Main figures: Isosurfaces of constant density. Planes display integrated densities along the axis orthogonal to the plane. Densities (color bar) are scaled by the maximum density of each configuration. (a) Ground state of an impurity centered in the droplet at $B=66.6$ G. (b) Ground state of an impurity localized at the droplet surface at $B=65.8$ G. (c) Excited state of an impurity at $B=65.8$ G for $\ell =10$ and $m=10$. (d) Effective potential $V_{\text{eff}}\left(r\right)$ and density of the impurity $n_I\left(r\right)$ for the $n=0,\cdots ,3$ $s$-wave bound states for a spherical droplet with density $n_D\left(r\right)$ (yellow). The eigenstates are localized on the surface of the droplet at $r\approx 1.1\mu \mathrm{m}$ from the droplet center. e) Spectrum of the impurity eigenstates in the presence of the effective potential $V_{\text{eff}}$(r) of (a). The black dashed line is the analytical prediction $E_I\left(\ell \right)-E_I\left(0\right)\propto \ell (\ell +1)$ discussed in the text. Eigenstates with $n=1$ and $\ell =0$ and $\ell =10$, $m=10$ are shown in (b) and (c) respectively. The parameters for (d) and (e) are $N_D=4\times {10}^4$ and $B=65.8$ G.

Figure 5

Feynman diagrams of the interaction between the two-component BEC (red and green lines) and the impurity (blue lines). The arrows denote normal and anomalous propagators following the convention in Ref. [57].

Figure 6

Energies $E_I$ of the impurity (blue) and $E_D$ of the droplet (red) in units of ${\hslash }^2/m_1{a}_{11}^2$ as a function of (a) the number of particles in the droplet $N_D$ and (b) the magnetic field $B$(G). In (a), we set $B=65.8$ G and in (b) $N_D=4\times {10}^4$.

Figure 7

Effective mass in the droplet phase. $m^{*}/m-1$ for the first three vibrational states in Fig. 4.