Spontaneous crystallization and filamentation of solitons in dipolar condensates

Intersite interactions play a crucial role in polar gases in optical lattices even in the absence of hopping. We show that due to these long-range interactions a destabilized stack of quasi-one-dimensional Bose-Einstein condensates develops a correlated modulational instability in the nonoverlapping sites. Interestingly, this density pattern may evolve spontaneously into soliton filaments or into a checkerboard soliton crystal, offering a unique possibility to observe such two-dimensional arrangements in ultracold gases. These self-assembled structures may be observed under realistic conditions within current experimental feasibilities.

Figure 1

Scheme of the stack of disjoint quasi-1D dipolar BECs.

Figure 2

Bogoliubov spectrum for a ${}^{52}$Cr BEC ($\mu =6{\mu }_B$, where ${\mu }_B$ is the Bohr magneton) with a density ${10}^{14}$ cm${}^{-3}$ and $a=-8.5a_0$ ($a_0$ is the Bohr radius), occupying $N_m=10$ sites of a lattice with the intersite spacing $\Delta =512$ nm and a lattice depth of $13.3$$E_R$ (recoil energy), which results in ${\omega }_{\perp }=2\pi \times 26.7$ kHz and $l_{\perp }=85.3$ nm. Here, $q_c=0.07/l_{\perp }$.

Figure 3

(Top) BEC wave function's density distribution (arb. units) after 200 ms of time evolution for the same parameters as in Fig. 2. For plotting purposes the $y$ width of the tubes has been magnified. (Bottom) Dynamics of the Fourier transform of the associated column density $\Sigma (z,t)$ (arb. units). The dominating $q=0$ peak has been removed for clarity and the remaining distribution has been normalized to the maximum. The arrows indicate the harmonics of $q_c$.

Figure 4

Filamentation of solitons. Here, a snapshot of time evolution of the BEC density distribution (arb. units) after 500 ms for, in particular, $N_m=20$ lattice sites, $\mu =18$${\mu }_B$ and $a=-41.7a_0$. The remaining parameters are chosen as in the case of Fig. 2.

Figure 5

(Top) Function $\chi \left(\tau \right)$ for a typical case within the filamentation regime ($g=-0.019$, $g_d=0.0034$, $\Delta =6l_{\perp }$). In particular, for the parameters that we employed in Fig. 2 the time ($t=\tau /{\omega }_{\perp }$) that we here consider equals $t=700$ ms. (Inset) Time-averaged values of $\chi$ for long times, for different values of $g_d$ and constant $g=-0.019$. (Bottom) Phase diagram of the possible regimes for $g<0$, $g_d>0$.

Figure 6

Spontaneous crystallization of solitons in the case of negative $g_d$. Here, BEC density distribution (arb. units) for $a=306a_0$, $\mu =36{\mu }_B$, and the remaining parameters such as in Fig. 2.

Figure 7

(Top) The inset shows a typical example of time evolution of ${\chi }_{NN}\left(\tau \right)$ and ${\chi }_{NNN}\left(\tau \right)$ within the soliton crystal regime ($g=0.069$, $gd=-0.032$, and $\Delta =6l_{\perp }$). We average ${\chi }_{\alpha }\left(\tau \right)$ within three different time intervals $\Delta {\tau }_{1,2,3}$, and we depict in the figure the corresponding averaged ${\chi }_{\alpha }$ (red circles, green squares, blue triangles) for different values of $g_d$ and constant $g=0.069$. Note that for $|g_d|/g>0.47$, ${\chi }_{NNN}$ decreases in time, indicating destruction of the checkerboard crystal. In particular, for the parameters we employed in Fig. 2, the time ($t=\tau /{\omega }_{\perp }$) that we here consider equals $t=1200$ ms. (Bottom) Phase diagram of the possible regimes for $g>0$, $g_d<0$.