Supersolidity in an elongated dipolar condensate

We present a theory for the emergence of a supersolid state in a cigar-shaped dipolar quantum Bose gas. Our approach is based on a reduced three-dimensional (3D) theory, where the condensate wave function is decomposed into an axial field and a transverse part described variationally. This provides an accurate fully 3D description that is specific to the regime of current experiments and efficient to compute. We apply this theory to understand the phase diagram for a gas in an infinite tube potential. We find that the supersolid transition has continuous and discontinuous regions as the averaged density varies. We develop two simplified analytic models to characterize the phase diagram and elucidate the roles of quantum droplets and of the roton excitation.

Figure 1

(a) Schematic of the tube-dipolar system with confinement in the $xy$ plane and the dipoles oriented along the $y$ axis. (b) Contrast of density modulation and (c) superfluid fraction as a function of the average linear density and the $s$-wave scattering length. The $s$-wave scattering length for the transition to the modulated state ($a^{*}$, white line) and for the roton instability of the uniform BEC ($a_{\text{rot}}^{*}$, red line) are shown. The dashed line (where $a^{*}=a_{\text{rot}}^{*}$) between the circle markers at the densities $n_{\text{low}}=1.3\times {10}^3/\mu \mathrm{m}$ and $n_{\text{high}}=4.6\times {10}^3/\mu \mathrm{m}$ indicates a continuous transition. Results for ${}^{164}\mathrm{Dy}$ using $a_{dd}=130.8a_0$, with ${\omega }_{x,y}=2\pi \times 150\mathrm{Hz}$.

Figure 2

Continuous transition to density modulated state. (a) Schematic showing a density isosurface of $|{\mathrm{\Psi }}_{\text{CM}}{|}^2$. (b) Contrast, (c) transverse anisotropy $\eta$, and the (d) transverse width $l$ and unit cell size $L$, as $a_s$ varies. In (b)–(d) we compare the results of the full theory for $\mathrm{\Psi }$ (black line) to the model ${\mathrm{\Psi }}_{\text{CM}}$ (red line). In the uniform BEC state both theories are identical (gray line). The vertical dashed line indicates the transition point $a^{*}$. In (d) the length $L$ is only uniquely defined in the modulated state and the critical roton wavelength $2\pi /k_{\text{rot}}^{*}$ [see (e)] is shown for reference. (e) Excitation spectrum of uniform BEC state. Parameters as in Fig. 1 with $n=2.5\times {10}^3/\mu \mathrm{m}$.

Figure 3

Low-density discontinuous transition. (a) Schematic showing a density isosurface of $|{\mathrm{\Psi }}_{\text{DA}}{|}^2$. Comparison of $\mathrm{\Psi }$ (black line and black dotted line) and ${\mathrm{\Psi }}_{\text{DA}}$ (blue line and blue dotted line) results for the (b) energy per particle, (c) transverse anisotropy $\eta$, and [(d) and (e)] length scales as $a_s$ varies. Solid (dotted) lines are used for each theory when its energy is lower (higher) than ${\mathrm{\Psi }}_{\text{BEC}}$. In (b) the CM ansatz results are also shown (red line). In (d) the critical roton wavelength is indicated at $a_{\text{rot}}^{*}$. In (e) ${\sigma }_z$ is the $1/e$ halfwidth of ${\left|\mathrm{\Psi }\right|}^2$ along the $z$ axis. Parameters as in Fig. 1 with $n=0.7\times {10}^3/\mu \mathrm{m}$.

Figure 4

High-density discontinuous transition. Comparison of $\mathrm{\Psi }$ (black line and black dotted line) and ${\mathrm{\Psi }}_{\text{CM}}$ (red line and red dotted line) results for the (a) contrast, (b) transverse anisotropy $\eta$, and the (c) length scales $l$ and $L$ as $a_s$ varies. In (c) we also indicate the roton wavelength at $a_{\text{rot}}^{*}$ for reference. (d) Energy per particle comparison of the theories. The solid lines are used for each theory when its energy is lower than ${\mathrm{\Psi }}_{\text{BEC}}$. Parameters as in Fig. 1 with $n=6.25\times {10}^3/\mu \mathrm{m}$.

Figure 5

(a) Phase diagram summarizing the results of Fig. 1 with $n_{max}=3.25\times {10}^3/\mu \mathrm{m}$ being where $a_{\text{rot}}^{*}$ is maximized. (b) The value of $n_{max}$ versus ${\omega }_{x,y}$ for ${}^{166}\mathrm{Er}$ and ${}^{164}\mathrm{Dy}$ systems. The vertical bars show the continuous transition range $[n_{\text{low}},n_{\text{high}}]$ for several cases. The cross indicates the parameters of the calculation in Ref. [18]. The inset shows the rescaled data.