Josephson effects in one-dimensional supersolids

We demonstrate that superflow past an obstacle is possible in a solid phase in the one-dimensional Gross-Pitaevskii equation with a finite-range two-body interaction. The phenomenon we find is analogous to the dc Josephson effect in superconductors, and we deduce the “Josephson relation” between the current and phase difference of the condensates separated by the obstacle. We also discuss persistent currents and nonclassical rotational inertia in annular containers with a penetrable potential barrier. The phase diagram in the plane of the current and the interaction strength is given. Our result provides a simple theoretical example of supersolidity in the presence of an obstacle.

Figure 1

Phase diagram in the case of $U\left(x\right)=0$ in the plane of $J$ and $g$. SF (SS) denotes the superfluid (supersolid) phase. In the region with $J$ above the critical current (open square), no steady state exists (NS). The solid curve represents the Landau critical current in the liquid phase. Open triangles represent the liquid-solid phase boundary determined from the Bragg peak.

Figure 2

Density profile for $U_0/{\epsilon }_0=0.01$ and $g=20(<g_{\mathrm{c}})$, where $J_{\mathrm{c}}/J_0=0.668$.

Figure 3

Upper panel: density profile for $(g,J/J_0)=(22,0.0921),(25,0.0782),(30,0.0598)$. For all curves, we set $U_0/{\epsilon }_0=10$. Lower panel: the profile of the phase of the condensate wave function for $(g,J/J_0)=(25,0.0782)$ and $U_0/{\epsilon }_0={10}^{-5},U_0/{\epsilon }_0=10$. $\Delta \varphi$ denotes the phase shift induced by the barrier.

Figure 4

$J$-$\Delta \varphi$ characteristic (the Josephson relation) for various values of $U_0$ and for $g=25>g_{\mathrm{c}}$. For a technical reason, we take $A\left(x\right)$ for $U_0/{\epsilon }_0={10}^{-5}$ as $A_0\left(x\right)$ in (22).

Figure 5

$g$ dependence of the critical current $J_{\mathrm{c}}$ for various strengths of potential barrier. The solid curve represents the Landau critical current.

Figure 6

$v$ dependence of $J\left(v\right)$ for $g=25$ and $d=a$, $U_0={\epsilon }_0$. Here, $v_0\equiv \hslash /mL$.

Figure 7

$v$ dependence of total momentum $P$ for $g=25$ and $d=a$, $U_0={\epsilon }_0$. Alternatively, this curve can be regarded as the relation between the angular momentum $L_z=RP$ of the condensate and the angular velocity $\omega =v/R$ of the container with annular geometry with radius $R=L/\left(2\pi \right)$. The unit of angular velocity is ${\omega }_0=\hslash /\left(mLR\right)$.

Figure 8

$d$ dependence of the Josephson relation for $U_0/{\epsilon }_0=10$ and $g=25$.

Figure 9

Density profile for $g=25,J=0,U_0=10{\epsilon }_0$, and $d=2a$.