Supersolidity and crystallization of a dipolar Bose gas in an infinite tube

We calculate the ground states of a dipolar Bose gas confined in an infinite tube potential. We use the extended Gross-Pitaevskii equation theory and present a numerical method to efficiently obtain solutions. A key feature of this method is an analytic result for a truncated dipole-dipole interaction potential that enables the long-ranged interactions to be accurately evaluated within a unit cell. Our focus is on the transition of the ground state to a crystal driven by dipole-dipole interactions as the short-ranged interaction strength is varied. We find that the transition is continuous or discontinuous depending upon the average system density. These results give deeper insight into the supersolid phase transition observed in recent experiments and validate the utility of the reduced three-dimensional theory developed in [Phys. Rev. Res. 2, 043318 (2020)] for making qualitatively accurate predictions.

Figure 1

Plot of the relative error in the DDI energy computed using the bare interaction kernel Eq. (10) (red) and the cutoff kernel in Eq. (12) (blue) using ${\psi }_{\mathrm{tr}}$ with $l_x=x_0/2$ and $l_y=2x_0$. The black line shows the relative error in the integration of the density. We fixed $L_{\rho }=R$ and the grid point density to eight per unit length.

Figure 2

(a) Example of the superfluid fraction calculated by nonclassical translational inertia [Eq. (22)] (red dots) and the upper and lower bounds (blue lines) given by Eqs. (23) and (24), respectively. The difference between the results is barely visible. (b) The differences $f_s^u-f_s$ and $f_s^l-f_s$ (blue lines). Other parameters are as follows: ${}^{164}\mathrm{Dy}$ Bose gas with $n=2500/\mu \mathrm{m}$ and ${\omega }_{x,y}/2\pi =150\mathrm{Hz}$.

Figure 3

(a) Phase diagram for a ${}^{164}\mathrm{Dy}$ Bose gas confined by a radially symmetric infinite tube trap with ${\omega }_{x,y}/2\pi =150\mathrm{Hz}$. The black line indicates the crystallization transition boundary $a^{*}$. This line is solid where the transition is discontinuous and is dashed where it is continuous. The circles demarcate the start and the end of the continuous region at $n_{\mathrm{low}}$ and $n_{\mathrm{high}}$, respectively. The parameters where a roton excitation in the uniform state softens to zero energy are shown with the red line. The crystallization transition boundary of the reduced 3D theory is shown in gray and the roton boundary from that theory in magenta. (b1)–(b3), (c1)–(d1) Example ground states for parameters indicated in subplot (a). Isodensity surfaces of ${\left|\psi \right|}^2$ (repeated over three unit cells for the modulated cases) taken at $75\%$ (red) and $1\%$ (blue) of the peak density. The gray isosurface indicates a cutaway isoenergy surface of the trapping potential. Other parameters are as follows: (b1) $a_s=86a_0$, (b2) $a_s=92.27a_0$, (b3) $a_s=93a_0$, (b1)–(b3) $n=2500/\mu \mathrm{m}$, (c1) $a_s=86a_0, n=820/\mu \mathrm{m}$, (d1) $a_s=91a_0, n=5000/\mu \mathrm{m}$.

Figure 4

Superfluid fraction (blue) and density contrast (red) as a function of the average linear density of the crystalline state along the transition boundary $a^{*}$ (see Fig. 3). The vertical lines identify the low-density ($n_{\mathrm{low}}$) and high-density ($n_{\mathrm{high}}$) end points of the region in which the transition is continuous. The gray-shaded region indicates contrast values which we take to be zero (see text). Other parameters as in Fig. 3.

Figure 5

Behavior of the ground-state density contrast (red) and superfluid fraction (blue) as $a_s$ varies across the transition for three different cases: (a) $n=710/\mu \mathrm{m}$, (b) $n=2500/\mu \mathrm{m}$, and (c) $n=7000/\mu \mathrm{m}$. The vertical gray dashed lines indicate $a_{\mathrm{rot}}^{*}$ for each case. Other parameters as in Fig. 3.

Figure 6

(a) Dispersion relation of the lowest Bogoliubov excitation band for a uniform BEC state for $a_s$ values as indicated. A roton excitation manifests as a local minimum in this dispersion relation (marked with $\times$). The roton softens to zero energy as $a_s\to a_{\mathrm{rot}}^{*}$ from above. (b) Energy per particle of the uniform (blue line) and modulated (red line) states; the black dot indicates where the energies cross. (c) Length $L$ of the uc (undefined out of the crystalline phase). The red dot has coordinates $(a_{\mathrm{rot}}^{*},{\lambda }_{\mathrm{rot}}$) where ${\lambda }_{\mathrm{rot}}=2\pi /k_{\mathrm{rot}}$ is the roton wavelength and $k_{\mathrm{rot}}>0$ is the wave vector where the dispersion touches zero. Results for $n=2500/\mu \mathrm{m}$ and other parameters as in Fig. 3.

Figure 7

[(a), (c)] Energy per particle of modulated state (red) and uniform state (blue); the black dot indicates where the energies cross. [(b), (d)] Length of the uc for the crystalline state. The red dot has coordinates $(a_{\mathrm{rot}}^{*},{\lambda }_{\mathrm{rot}}$). [(a), (b)] $n=710/\mu \mathrm{m}$ and [(c), (d)] $n=7000/\mu \mathrm{m}$, with other parameters as in Fig. 3.