Long-lifetime supersolid in a two-component dipolar Bose-Einstein condensate

Recent studies on supersolidity in a single-component Bose-Einstein condensate (BEC) have relied on the Lee-Huang-Yang (LHY) correction for stabilization of self-bound droplets, which, however, involves a high density inside the droplets, limiting the lifetime of the supersolid. Here we propose a two-component mixture of dipolar and nondipolar BECs, such as an ${}^{166}\mathrm{Er}\text{−}{}^{87}\mathrm{Rb}$ mixture, to create and stabilize a supersolid without the LHY correction, which can suppress the atomic loss and may allow observation of the long-time dynamics of the supersolid. In such a system, supersolidity can be controlled by the difference in the trap centers between the two components.

Figure 1

(a) Integrated density distributions $n_{jz}(y,x)=\int dz{\left|{\psi }_j\left(\mathit{r}\right)\right|}^2$ and $n_{jx}(y,z)=\int dx{\left|{\psi }_j\left(\mathit{r}\right)\right|}^2$ of the ground state for $N_1={10}^5$, $N_2=2\times {10}^5$, $a_{12}=100a_B$, and $\delta =0$. (b) Time evolution of the number $N_j$ of atoms in component $j$ with the three-body recombination loss, where the state in (a) is used as the initial state. The insets show $n_{1z}(y,x)$ and $n_{2z}(y,x)$ at $t=2$ s. (c) Dependence of the contrast $C$ on the sag $\delta$. The insets show $n_{1z}(y,x)$ and $n_{1x}(y,z)$ for $\delta =1.7\mu \mathrm{m}$. (d) Dependence of the contrast $C$ on $a_{12}$. The insets show $n_{1z}(y,x)$ for $a_{12}=70a_B$ and $120a_B$. In (c) and (d), other parameters are the same as those in (a). The images are $77\mu \mathrm{m}\times 19.3\mu \mathrm{m}$ in size. The color bar scales from 0 to $1.3\times {10}^3\mu {\mathrm{m}}^{-2}$.

Figure 2

Out-of-phase Goldstone mode in a one-dimensional supersolid. (a)–(c) Sag is absent, with $\delta =0$ and (d)–(f) sag is present, with $\delta =2.2\mu \mathrm{m}$. Integrated density profiles (a), (d) $n_{1z}(y,x)=\int dz{\left|{\psi }_1\left(\mathit{r}\right)\right|}^2$ and (b), (e) $n_{1x}(y,z)=\int dx{\left|{\psi }_1\left(\mathit{r}\right)\right|}^2$ of component 1 of the ground states, where the images are $77\mu \mathrm{m}\times 19.3\mu \mathrm{m}$. (c), (f) Time evolution of the integrated density profile $n_1(y,t)=\int dxdz{\left|{\psi }_1(\mathit{r},t)\right|}^2$ of component 1, where the three-body loss is included. The initial phase is imprinted as in Eq. (5) with $\varphi =0.2\pi$. The color bar scales from 0 to $1.3\times {10}^3\mu {\mathrm{m}}^{-2}$ for $n_{1z}(y,x)$ and $n_{1x}(y,z)$, and to $5.6\times {10}^3\mu {\mathrm{m}}^{-1}$ for $n_1(y,t)$. (g) Time evolution of the center-of-mass position of component 1 in the $y$ direction, $\langle y_1\left(t\right)\rangle =\int d\mathit{r}y{\left|{\psi }_1(\mathit{r},t)\right|}^2$, without and with sag. ${\omega }_{2x,y,z}=1.6{\omega }_{1x,y,z}$ in (d)–(f). Other parameters are the same as those in Fig. 1.

Figure 3

Two-dimensional supersolid in an oblate trap for $a_{12}=100a_B$. (a) Integrated density profiles $n_{jz}(x,y)=\int dz{\left|{\psi }_j\left(\mathit{r}\right)\right|}^2$ and $n_{jy}(x,z)=\int dy{\left|{\psi }_j\left(\mathit{r}\right)\right|}^2$ of components 1 and 2. (b) Normalized moment of inertia $\mathrm{\Theta }/{\mathrm{\Theta }}_{\mathrm{rig}}$ as a function of the sag $\delta$. The insets show $n_{1z}(x,y)$ and $n_{jy}(x,z)$. The sizes of the images are $55.2\mu \mathrm{m}\times 55.2\mu \mathrm{m}$ for $n_{jz}(x,y)$ and $55.2\mu \mathrm{m}\times 13.8\mu \mathrm{m}$ for $n_{jy}(x,z)$. The color bar scales from 0 to $6.4\times {10}^2\mu {\mathrm{m}}^{-2}$ for $n_{jz}(x,y)$ and to $1.1\times {10}^3\mu {\mathrm{m}}^{-2}$ for $n_{jy}(x,z)$.