Supersolid behavior of a dipolar Bose-Einstein condensate confined in a tube

Motivated by a recent experiment [L. Chomaz et al., Nat. Phys. 14, 442 (2018)], we perform numerical simulations of a dipolar Bose-Einstein condensate (BEC) in a tubular, periodic confinement at $T=0$ within density functional theory, where the beyond-mean-field correction to the ground-state energy is included in the local density approximation. We study the excitation spectrum of the system by solving the corresponding Bogoliubov–de Gennes equations. The calculated spectrum shows a roton minimum, and the roton gap decreases by reducing the effective scattering length. As the roton gap disappears, the system spontaneously develops a periodic linear structure formed by denser clusters of atomic dipoles immersed in a dilute superfluid background. This structure shows the hallmarks of a supersolid system, i.e., (i) a finite nonclassical translational inertia along the tube axis and (ii) the appearance of two gapless modes, i.e., a phonon mode associated with density fluctuations and resulting from the translational discrete symmetry of the system, and a Nambu-Goldstone gapless mode corresponding to phase fluctuations, resulting from the spontaneous breaking of the gauge symmetry. A further decrease in the scattering length eventually leads to the formation of a periodic linear array of self-bound droplets.

Figure 1

Upper panel: Dispersion relation of excitations propagating along the tube axis in the homogeneous system. Energies are in atomic units. Lower panel: Integrated density $n_y(x,z)$ just below and at the critical value of ${\epsilon }_{dd}$ where the roton gap vanishes. The total number of atoms is $N=6\times {10}^4$. The lowest plot shows the density $n$ along the tube axis for different values of ${\epsilon }_{dd}$.

Figure 2

Upper panel: Superfluid fraction as function of ${\epsilon }_{dd}$. Lower panel: Excitation spectrum along the tube axis, calculated for ${\epsilon }_{dd}=1.45$. The rightmost value of $k_x$ corresponds to the first Brillouin zone boundary along the $x$ axis, i.e., $k_x=\pi /d, d=L/11$ being the length of the unit cell containing exactly one droplet in Fig. 1.

Figure 3

Energy per particle (in atomic units) as a function of ${\epsilon }_{dd}$. The inset shows the array of self-bound droplets.