Supersolidity of cnoidal waves in an ultracold Bose gas

A one-dimensional Bose-Einstein condensate may experience nonlinear periodic modulations known as cnoidal waves. We argue that such structures represent promising candidates for the study of supersolidity-related phenomena in a nonequilibrium state. A mean-field treatment makes it possible to rederive Leggett's formula for the superfluid fraction of the system and to estimate it analytically. We determine the excitation spectrum, for which we obtain analytical results in the two opposite limiting cases of (i) a linearly modulated background and (ii) a train of dark solitons. The presence of two Goldstone (gapless) modes, associated with the spontaneous breaking of $\mathrm{U}\left(1\right)$ symmetry and of continuous translational invariance, at long wavelength is verified. We also calculate the static structure factor and the compressibility of cnoidal waves, which show a divergent behavior at the edges of each Brillouin zone.

Figure 1

Schematic behavior of the potential $W$ as a function of $n$. The value of $W$ at the local minimum (maximum) is $W_{\min }$ ($W_{\max }$). Here $n_1\le n_2\le n_3$ are the roots of $W\left(n\right)=\mathcal{E}$.

Figure 2

Density profiles $n\left(x\right)$ of cnoidal waves for (a) $\eta =0.6$ and (b) $\eta =1.6$ and for $m_e=0.1$ (blue solid lines) and $m_e=0.5$ (red dashed lines). The value of $m_e$ for the two yellow dash-dotted curves was chosen so as to minimize the superfluid density (23) [see also the red dashed lines of Figs. 3 and 3]. It is equal to 0.993 in (a) and to $m_e^{\max }=0.943$ in (b).

Figure 3

(a) and (b) contrast of the fringes and (c) and (d) corresponding superfluid density as functions of $m_e$ for (a) and (c) $\eta =0.3$ (blue solid line), 0.6 (red dashed line), and 0.9 (yellow dash-dotted line) and (b) and (d) $\eta =1.3$ (blue solid line), 1.6 (red dashed line), and 2.0 (yellow dash-dotted line). For the curves in (b) and (d) the corresponding values of $m_e^{\max }$ are 0.997, 0.943, and 0.826, respectively.

Figure 4

Lowest three bands of the excitation spectrum as functions of $q$ for (a) the same parameters $\eta =0.6$ and $m_e=0.5$ as the red dashed curve of Fig. 2 and (b) $\eta =2.0$ and a value $m_e=0.8$ very close to $m_e^{\max }=0.826$ [see yellow dash-dotted curves of Figs. 3 and 3]. In both cases the sign of the current density $\mathcal{J}$ is taken to be positive. The black dashed lines show the spectrum of a uniform system having the same average density $\overline{n}$ and velocity (a) $\overline{v}/c=1.18$ and (b) $\overline{v}/c=1.10$ as the cnoidal waves under consideration.

Figure 5

Static structure factor as a function of $q$ (black solid lines). The blue dashed and red dash-dotted curves show the contributions of the lower and upper gapless bands, respectively, for cases corresponding to the same values of $m_e$ and $\eta$ as in Fig. 4: (a) $\eta =0.6$ and $m_e=0.5$ and (b) $\eta =2.0$ and $m_e=0.8$.

Figure 6

Compressibility as a function of $q$ (black solid lines). The blue dashed and red dash-dotted lines show the contributions of the lower and upper gapless bands, respectively, for cases corresponding to the same values of $m_e$ and $\eta$ as in Fig. 4: (a) $\eta =0.6$ and $m_e=0.5$ and (b) $\eta =2.0$ and $m_e=0.8$.

Figure 7

Lowest band of the excitation spectrum as a function of $q$. In each panel we compare the numerical results (blue solid curve) with the analytic prediction (30) (black dashed curve) at $m_e=0.99$, i.e., close to the dark-soliton limit. We take $\eta$ equal to (a) 0.1, (b) 0.3, (c) 0.6, and (d) 0.9.

Figure 8

Density profiles of the cnoidal wave for $m_e=0.99$ and different values of $\eta$, corresponding to the same values of $\eta$ as in Fig. 7: (a) 0.1, (b) 0.3, (c) 0.6, and (d) 0.9.