Droplet Crystal Ground States of a Dipolar Bose Gas

We show that the ground state of a dipolar Bose gas in a cylindrically symmetric harmonic trap has a rich phase diagram, including droplet crystal states in which a set of droplets arrange into a lattice pattern that breaks the rotational symmetry. An analytic model for small droplet crystals is developed and used to obtain a zero temperature phase diagram that is numerically validated. We show that in certain regimes a coherent low-density halo surrounds the droplet crystal, giving rise to a novel phase with localized and extended features.

Figure 1

Stationary solutions of the extended GPE (1) for a ${}^{164}\mathrm{Dy}$ gas with $a_s=70a_0$ in a $(\omega ,{\omega }_z)=2\pi \times (60,300)\mathrm{Hz}$ trap, atom numbers and $\nu$ values (see text) as indicated. Density isosurfaces are at $2\times {10}^{19}{\mathrm{m}}^{-3}$ (blue) and $2\times {10}^{20}{\mathrm{m}}^{-3}$ (red). Density slices are at $z=0$.

Figure 2

(a) Energy as a function of total atom number for $a_s=70a_0$ using the GPE solutions obtained the trap $(\omega ,{\omega }_z)=2\pi (60,300)\mathrm{Hz}$. Filled circle markers indicate the first nine states appearing in Fig. 1. (Inset) The predictions of the variational solution. (b) As above, but for $a_s=65a_0$. Horizontal colored lines indicate the ground state at each $N$ value with arbitrary scale.

Figure 3

Ground state phase diagram for $\nu$-droplet crystal states of ${}^{164}\mathrm{Dy}$ atoms in a $(\omega ,{\omega }_z)=2\pi (60,300)\mathrm{Hz}$ trap. The solid lines and colored regions show the extended GPE results for where the $\nu$-droplet states with $\nu \le 6$ are the lowest energy state. The gray region is where the ground state has $\nu >6$. Transition lines between states found using the variational model are indicated by dashed lines. The horizontal lines indicate the results of Fig. 2 and the filled circles correspond to states in Figs. 1. Hatched regions are where $N_{\mathrm{halo}}/N>1\%$.

Figure 4

(a) Number of atoms in the halo $N_{\mathrm{halo}}$ as $N$ increases and (b) corresponding chemical potentials. Filled circles indicate the states in Figs. 1. Filled square indicates state (c) below. (c),(d) $\nu =6$ droplet crystal states with $N=97.7\times {10}^3$. Same parameters as Fig. 2, except for the higher trap frequencies in (d).