Supersolid edge and bulk phases of a dipolar quantum gas in a box

We investigate the novel density distributions acquired by a dipolar Bose-Einstein condensed gas confined in a box potential, with special focus on the effects of supersolidity. Different from the case of harmonic trapping, the ground-state density reveals a strong depletion in the bulk region and an accumulation of atoms near the walls, well separated from the bulk, as a consequence of the competition between the attractive and the repulsive nature of the dipolar force. In a quasi-two-dimensional geometry characterized by cylindrical box trapping, we observe the emergence of a ringlike configuration near the boundary of the box, revealing peculiar supersolid and crystal effects in a useful range of parameters. In the case of square box trapping, the density oscillations along the edges, caused by the enhanced accumulation of atoms near the vertices, exhibit interesting analogies with the case of box-trapped quasi-one-dimensional configurations. For sufficiently large values of the atom number, the bulk region can also exhibit supersolidity, the resulting geometry reflecting the symmetry of the confining potential even for large systems.

Figure 1

Ground-state integrated density profiles $n(x,y)={\int dz\left|\mathrm{\Psi }(x,y,z)\right|}^2$ for a gas of ${10}^5$ atoms of ${}^{164}\mathrm{Dy}$ confined in the polarization direction by a harmonic potential of frequency ${\omega }_z=\left(2\pi \right)100\mathrm{Hz}$, and by a box potential in the x-y plane, with the shape of a circle of radius $\mathrm{R}=10.185\mu \mathrm{m}$. The value of ${\epsilon }_{dd}$ for panels (a–c) is, respectively, 1.32, 1.404, and 1.467. The height of the box is fixed to $V_0=100\hslash {\omega }_z$

Figure 2

Estimate of the superfluid fraction of the edge region as a function of ${\epsilon }_{dd}$, based on Leggett's variational formula (4) (blue squares), applied to the configuration described in Fig. 1. Red circles represent the ratio between the number of atoms that settle on the edge and the total number of atoms in the system

Figure 3

Ground-state integrated density profiles $n(x,y)={\int dz\left|\mathrm{\Psi }(x,y,z)\right|}^2$ for a gas of ${10}^5$ atoms of ${}^{164}\mathrm{Dy}$ confined in the polarization direction by a harmonic potential of frequency ${\omega }_z=\left(2\pi \right)100\mathrm{Hz}$ and by a box potential in the x-y plane, with the shape of a square of side $L=16\mu \mathrm{m}$. The value of ${\epsilon }_{dd}$ for (a)–(c) is, respectively, 1.32, 1.404, and 1.467. The height of the box is fixed to $V_0=100\hslash {\omega }_z$

Figure 4

Ground-state integrated density profiles $n\left(x\right)={\int dydz\left|\mathrm{\Psi }(x,y,z)\right|}^2$ of $\mathrm{N}=4\times {10}^4$ atoms of ${}^{164}\mathrm{Dy}$ in a transverse harmonic confinement of frequencies ${\omega }_y={\omega }_z=\left(2\pi \right)100\mathrm{Hz}$ confined by a one-dimensional box potential of height $V_0=100\hslash {\omega }_z$ at positions $\mathrm{x}=\pm 12$ $\mu \mathrm{m}$ [(a)–(c)] or with periodic boundary conditions at $\mathrm{x}=\pm 12$ $\mu \mathrm{m}$ [(d)–(f)]. Insets (g) and (h) show the excitation spectrum calculated by solving the Bogolyubov-de Gennes equations for the configurations of panels (d) and (e). The profiles reported in panels (a) and (d) [respectively, (b), (e), and (c), (f)] are calculated by fixing ${\epsilon }_{dd}$=1.32 (respectively, 1.39 and 1.46).

Figure 5

Ground-state integrated density profiles $n(x,y)={\int dz\left|\mathrm{\Psi }(x,y,z)\right|}^2$ for a gas of $2\times {10}^6$ atoms of ${}^{164}\mathrm{Dy}$ confined in the polarization direction by a harmonic potential of frequency ${\omega }_z=\left(2\pi \right)100\mathrm{Hz}$ and by a box potential in the x-y plane with the shape of a triangle of side $L=58.58\mu \mathrm{m}$ (a), a square of side $L=45.84\mu \mathrm{m}$ (b), a pentagon of side $L=38.33\mu \mathrm{m}$ (c), a hexagon of side $L=33.82\mu \mathrm{m}$ (d), and a circle of radius $R=21.75\mu \mathrm{m}$ (e). The value of ${\epsilon }_{dd}=1.36$ is the same for all the configurations, which also have the same area.

Figure 6

Roton energy of a quasi-one-dimensional dipolar Bose gas with transverse harmonic confinement of frequencies ${\omega }_y={\omega }_z=\left(2\pi \right)100\mathrm{Hz}$, with periodic boundary conditions along the $x$ axis, as function of the linear density $n=N/L$, where N is the number of atoms and L is the length of the simulation cell along $x$ and of ${\epsilon }_{dd}$. The blue dots correspond, from bottom to top, to the configurations of Figs. 4, while the white empty square corresponds to the ring edge of the configuration shown in Fig. 5 (circular transverse confinement). The dark area corresponds to configurations in which the roton mode, calculated starting from a uniform configuration, is unstable and where spontaneous density modulations of supersolid or crystal nature are formed. The green, horizontal dashed line is a guide to the eye showing that, for fixed ${\epsilon }_{dd}$, starting from a low-density superfluid, increasing the density results in a phase transition to a supersolid (and eventually to a droplet crystal), while at even higher densities, the system turns superfluid again.