Self-Bound Doubly Dipolar Bose-Einstein Condensates

We analyze the physics of self-bound droplets in a doubly dipolar Bose-Einstein condensate composed by particles with both electric and magnetic dipole moments. Using the particularly relevant case of dysprosium, we show that the anisotropy of the doubly dipolar interaction potential is highly versatile and nontrivial, depending critically on the relative orientation and strength between the two dipole moments. This opens novel possibilities for exploring intriguing quantum many-body physics. Interestingly, by varying the angle between the two dipoles we find a dimensional crossover from quasi-one-dimensional to quasi-two-dimensional self-bound droplets. This opens a so far unique scenario in condensate physics, in which a dimensional crossover is solely driven by interactions in the absence of any confinement.

Figure 1

(a) The dipole moments $d_m$ and $d_e$ of a Dy atom in the stretched state $|S⟩$ as a function of $\alpha$ for $\mathcal{B}=100\mathrm{G}$ and $\mathcal{E}=2.68\mathrm{kV}/\mathrm{cm}$. With these field strengths the maximum values of $d_m$ and $d_e$ attained are $13{\mu }_B$ and 0.12D, respectively. The solid line shows $\gamma =g_e/g_m$. The inset shows the lifetime of the state $|S⟩$. The 2D potential, $V_d^{y=0}(\mathit{r})$ (in arbitrary units), taking $d_m$ and $d_e$ from (a) for (b) $\alpha =0$, (c) $\alpha =1\mathrm{rad}$, $\alpha =1.3\mathrm{rad}$ and $\alpha =\pi /2\mathrm{rad}$. For $\alpha >{\alpha }_a=1.239\mathrm{rad}$, $V_d^{y=0}(\mathit{r})$ is purely attractive, see (d) and (e). The thick arrow shows the effective polarization direction. (f) Polarization angle ${\theta }_p$ vs $\alpha$ for different $\gamma$; the inset shows ${\theta }_p$ as a function of $\gamma$ for $\alpha =\pi /2$.

Figure 2

(a) Stable and unstable region for a self-bound (Gaussian) droplet of Dy atoms in the state $|S⟩$, for $\mathcal{B}=100\mathrm{G}$, $\mathcal{E}=2.68\mathrm{kV}/\mathrm{cm}$, and $N=2000$, as a function of the scattering length $a$ and $\alpha$. The stable regime is partitioned using $L_y^0/L_x^0$. (b) Equilibrium widths from the Gaussian calculations (solid lines), and the 3D numerical simulations of Eq. (8) (filled circles) as a function of $\alpha$ for $a=110a_0$, where $a_0$ is the Bohr radius and other parameters as in (a). The inset shows the peak density of the droplet. The angle ${\alpha }_a$ is shown by a dashed vertical line. The widths are scaled by the length $a_m={\mu }_{0}m{d}_m^2/12\pi {\hslash }^2\simeq 222a_0$. (c)–(e) The density plots for the droplet ground state from 3D numerics for different $\alpha$, indicating a structural crossover from cigar- to pancake-shaped droplets.

Figure 3

(a) Excitation frequencies of the Dy droplet obtained from the Gaussian variational calculations as a function of $\alpha$ for the same parameters as in Fig. 2. The inset shows ${\omega }_2/|\mu |$ as a function of $\alpha$. (b) The eigenvector ${\sum }_{j=x^{\prime },y,z^{\prime }}{b}_j{\overrightarrow{e}}_j$ corresponding to ${\omega }_1$ as a function of $\alpha$. It changes from being a breathing mode on the $xy$ plane for $\alpha =0$ to a pure $y$ oscillation as $\alpha$ approaches $\pi /2$. The $N$ dependence of the droplet widths at $\alpha =0$ and $\alpha =\pi /2$ is shown in (c) and (d), respectively.