Symmetry and Self-Bound Droplets in Dipolar Molecular Gases

Abstract

Recent experiments with degenerate molecular gases dressed by elliptically polarized microwave fields have enabled new control of dipolar interactions via engineered anisotropy. We reveal a symmetry structure of the dipolar interaction that generates degeneracies among the interaction parameters, enabling a classification of spatial symmetries and equilibrium shapes of the gases. Exploiting these symmetries, we analyze solutions including beyond-meanfield quantum fluctuations, and develop a complementary variational theory. We map out the phase diagram of self-bound droplets and characterize their widths, energies, and densities.

Figures (5)

• Figure 1: Symmetry of interaction strengths, and symmetry and relative lengths. The blue dots and lines show equivalent states under permutations of the coordinates. There is also simple reflection symmetry about the line $\epsilon_2=0$: the $\epsilon_2<0$ results and their connection to the dots are not shown (if three dots are linked above, there are another three for $\epsilon_2<0$, and if two are shown there is another one for $\epsilon_2<0$). The three colored regions (six including $\epsilon_2<0$) demarcate domains with different relative widths. The notation $x=y>z$ means that the cloud has $x,y$ symmetry, and the $x,y$ width is greater than the $z$ width, unless broken by a confining potential. The dashed lines indicate $\epsilon_0=0$ and other members of its sextet.
• Figure 2: Phase diagram for $N=1000$ using the eGPE (red circles and thick curve) and variational (thin red curve) with $E=0$, and for $N=5000$ using the eGPE (blue circles and thick curve) and variational (thin blue curve). Also shown is the variational energy for $N=5000$ (background color), the stability boundary in the thermodynamic limit, Eq. \ref{['e:tlbound']} (dashed lines), and contours where $\operatorname{Im}\{\mathcal{Q}_5\} = \operatorname{Re}\{\mathcal{Q}_5\}/10$ (thin black curve) and $\operatorname{Im}\{\mathcal{Q}_5\} = \operatorname{Re}\{\mathcal{Q}_5\}/4$ (thin white curve), and the symmetry axes from Fig. \ref{['f:sym']} (black lines at $\epsilon_0=\pm\epsilon_2/\sqrt3$). The cases from Fig. \ref{['f:profile']} are marked with $\times$ and the case from Fig. \ref{['f:props']}(b) is marked with a dash-dotted line.
• Figure 3: Profiles for $N=5000$ and (a) $\epsilon_0=3$, $\epsilon_2=0$, (b) $\epsilon_0=-1.5$, $\epsilon_2=0$ (c) $\epsilon_0=0$, $\epsilon_2=\sqrt3$ for the $x$ axis (solid), $y$ axis (dash-dotted), and $z$ axis (dashed). On the right (not to scale) are isodensity surfaces at $90\%$ (red) and $5\%$ (blue) of peak density.
• Figure 4: Properties of droplets for $N=1000$ (red) and $N=5000$ (blue), thick curves are eGPE and thin curves are variational. (a,b) RMS widths along the $x$ (solid), $y$ (dash-dotted), and $z$ (dashed) axes [in (a) $\sqrt{\langle y^2\rangle}$ is equal to and obscured by $\sqrt{\langle x^2\rangle}$; for variational $\sqrt{\langle r_i^2\rangle}=l_i/\sqrt2$]. (c,d) Energy per particle, also showing Eq. \ref{['e:Escale']} (black dashed). (e,f) Peak density, also showing Eq. \ref{['e:nscale']} (black dashed). In (a,c,e) results are for $\epsilon_2=0$, and (b,d,f) for $\epsilon_0+\sqrt3\epsilon_2=3$.
• Figure 5: Quantum fluctuations coefficient along $\epsilon_2=0$ (solid) and $\epsilon_0=0$ (dashed). Shown are the real part (blue), the imaginary part (red), and the small parameter approximation \ref{['e:Q5small']} (thin black).