Impurity Crystal in a Bose-Einstein Condensate

We investigate the behavior of impurity fields immersed in a larger condensate field in various dimensions. We discuss the localization of a single impurity field within a condensate and note the effects of surface energy. We derive the functional form of the attractive condensate-mediated interaction between two impurities. Generalizing the analysis to $N$ impurity fields, we show that within various parameter regimes a crystal of impurity fields can form spontaneously in the condensate. Finally, the system of condensate and crystallized impurity structure is shown to have nonclassical rotational inertia, which is characteristic of superfluidity; i.e., the system can be seen to exhibit supersolid behavior.

Figure 1

Phase diagram. The vertical line represents $\gamma ={\gamma }_0$ while the curve $i$ represents $\sqrt{(N-1)\gamma +{\gamma }_0}/\sqrt{N}$. All numerical simulations of Eqs. (2, 3) are in two space dimensions in a $64\times 64$ periodic plane. The initial conditions were uniform miscible states plus small fluctuations. In each sector the left image plots the condensate density $|\psi {|}^2$ while the right plots $\sum _{k=1}^N|{\chi }_k{|}^2$. The images I, II, and IV correspond to $N=36$ impurity fields, while image III corresponds to $N=6$. In III the initial masses are $\int |\psi {|}^2{d}^2\mathit{x}=4096$ and $\int |{\chi }_k{|}^2{d}^2\mathit{x}=40.96$, $\lambda =0.2$, $\gamma =1$, and ${\gamma }_0=0$, while in IV the initial masses are $\int |\psi {|}^2{d}^2\mathit{x}=4096$ and $\int |{\chi }_k{|}^2{d}^2\mathit{x}=40.96$, $\lambda =2$, $\gamma =1.5$. and ${\gamma }_0=0$. Note that, as discussed in the text, the surface-tension effects will shift the boundaries. See a movie of crystallization in supplementary material Ref. 9.

Figure 2

Numerical simulation of Eqs. (2, 3) for $N=36$ impurity fields in a $128\times 128$ periodic plane. Here the initial masses are $\int |\psi {|}^2{d}^2\mathit{x}=16384$ and $\int |{\chi }_k{|}^2{d}^2\mathit{x}=200$, $\lambda =0.4$, $\gamma =1$, and ${\gamma }_0=0$. (a) plots the condensate density $|\psi {|}^2$ while (b) plots $\sum _{k=1}^N|{\chi }_k{|}^2$. Inhomogeneous regions, i.e., where miscible states coexist with the crystalline state, indicate that the system has not yet reached an equilibrium state.