Phase-jump nucleation in a one-dimensional model of a supersolid

We consider the activation of phase jumps in a one-dimensional Bose-Einstein condensate. Our system includes a nonlocal interaction term which mimics, for example, a dipole-blockaded Rydberg system. In the mean-field limit the condensate can form superfluid droplets, arranged periodically on a line, thus displaying a supersolidlike ground state. Under an imposed velocity, phase jumps will develop. We study these phase jumps numerically and analytically, and are able to write down a relationship between the velocity, the width of the density peaks, the number of phase jumps, and $\Lambda$, a parameter that determines the number of peaks in the condensate.

Figure 1

The density modulated ground state displaying 14 peaks for $\Lambda =101.66$ and $v=0$.

Figure 2

Free energy–velocity plots for the families of phase jumps for $\Lambda =101.66$ when $n=0$ (solid line) and $n=1$ (dashed line). The intersection between the curves occurs at $v=0.049$.

Figure 3

The density (solid line) and phase (dashed line) against $x$ when $\Lambda =101.66$, $n=0$, and $v=0.15$. Panel (a) is over the whole $x$ domain whereas panel (b) takes $-6.49<x<2.65$. We observe a large region (slightly larger than the peak size $a_w$) in which ${\varphi }_x\approx v$; here ${\varphi }_x=0.15$ over the region $-4.06<x<-0.47$. Included are visual definitions of $\lambda$ and $\delta l$.

Figure 4

(a) Density profile (log-linear plot) for different velocities (indicated on the figure) with $\Lambda =101.66$ held fixed. The curves have been shifted so that the minimum of the density is located at $x=0$. The density minimum and the variation length $\delta l$ decrease as $v$ increases. (b) Evolution of the density minimum (log-linear plot) with $v$ for different values of the parameter $\Lambda$ (indicated on the figure). In all cases we have $\lambda =64/14$.

Figure 5

The phase as a function of $x$ when $\Lambda =101.66$ and (a) $v=0.78$ ($n=1$, solid line) and (b) $v=0.79$ ($n=2$, dashed line). The phase is calculated modulo $2\pi$.

Figure 6

A plot of the NCRIF against velocity calculated through Eq. (26) (solid line) and Eq. (25) (dashed line). Here $\Lambda =101.66$ and $n=0$.