Three-dimensional droplets of swirling superfluids

A method for the creation of three-dimensional (3D) solitary topological modes, corresponding to vortical droplets of a two-component dilute superfluid, is presented. We use the recently introduced system of nonlinearly coupled Gross-Pitaevskii equations, which include contact attraction between the components, and quartic repulsion stemming from the Lee-Huang-Yang correction to the mean-field energy. Self-trapped vortex tori, carrying the topological charges $m_1=m_2=1$ or $m_1=m_2=2$ in their components, are constructed by means of numerical and approximate analytical methods. The analysis reveals stability regions for the vortex droplets (in broad and relatively narrow parameter regions for $m_{1,2}=1$ and $m_{1,2}=2$, respectively). The results provide a scenario for the creation of stable 3D self-trapped states with the double vorticity ($m_{1,2}=2$). The stable modes are shaped as flat-top ones, with the space between the inner hole, induced by the vorticity, and the outer boundary filled by a nearly constant density. On the other hand, all modes with hidden vorticity, i.e., topological charges of the two components $m_1=-m_2=1$, are unstable. The stability of the droplets with $m_{1,2}=1$ against splitting (which is the main scenario of possible instability) is explained by estimating analytically the energy of the split and unsplit states. The predicted results may be implemented, exploiting dilute quantum droplets in mixtures of Bose-Einstein condensates.

Figure 1

The norm versus the chemical potential (a), the energy versus the norm (b), and the amplitude of the wave function versus the width (c) for droplet modes with indicated values of the vorticity. Red and black branches correspond to unstable and stable states, respectively, with red dots separating stable and unstable segments. For $m_{1,2}=2$ a stability segment exists, too, as shown in (c), but it is located at large values of $N$ outside of the regions displayed in (a), (b). The dashed line in (a) designates the value of ${\mu }_{\mathrm{co}}$, given by Eq. (5), below which localized modes do not exist. The results are displayed, here and in other figures, for $g_{\mathrm{LHY}}=0.50$ and $g=1.75$. Here and in the other figures, all quantities are plotted in dimensionless units (see main text).

Figure 2

Typical profiles of unstable sharp (a) and stable flat-top (b) vortex droplets with $m_{1,2}=1$ for values of $\mu$, indicated in the panels. Here $g_{\mathrm{LHY}}=0.50$ and $g=1.75$.

Figure 3

The real part of the instability growth rate versus the chemical potential of the stationary droplet $\mu$, at values of the perturbation azimuthal index $k$ indicated in the panels, for $m_{1,2}=1$ (a) and $m_{1,2}=2$ (b) at $g_{\mathrm{LHY}}=0.50$ and $g=1.75$. The vertical dashed line designates the cutoff value of $\mu$ given by Eq. (5). Thus, stability windows are clearly observed in both panels (a) and (b), even if the window is rather narrow in (b). Panel (c) shows the cutoff value ${\mu }_{\mathrm{co}}$, and the stability boundary ${\mu }_{\mathrm{st}}$, versus the relative strength $g$ of the intercomponent attraction, for the vortex droplets with $m_{1,2}=1$. They are stable in the region of ${\mu }_{\mathrm{co}}<\mu <{\mu }_{\mathrm{st}}$. Panel (d) shows minimal norm $N_{\min }$ of vortex droplets and norm $N_{\mathrm{st}}$ above which such states become stable as functions of $g$. In (c,d) $g_{\mathrm{LHY}}=0.50$.

Figure 4

Isosurface density plots, at levels $|{\psi }_1{|}^2=0.1$ in the first three rows, and $|{\psi }_1{|}^2=0.5$ in the bottom one, showing unstable evolution of the perturbed vortex droplet with $m_{1.2}=1$ and ${\mu }_{1,2}=-0.04$ (first to third rows), and stable evolution for ${\mu }_{1,2}=-0.16$ (the fourth row). Only the ${\psi }_1$ component is displayed. The breakup of the droplets in the first, second, and third rows is induced by small perturbations proportional to unstable eigenmodes produced by the solution of Eq. (7) with azimuthal indices $k=1,2,3$, respectively. A broadband random perturbation is applied to the stable droplet in the bottom row. In all cases $g_{\mathrm{LHY}}=0.50$ and $g=1.75$.

Figure 5

The same as in Fig. 4, but for droplets with intrinsic vorticity $m_{1.2}=2$ and chemical potentials ${\mu }_{1,2}=-0.04$ and ${\mu }_{1,2}=-0.183$ for the unstable state in the first three rows, and the stable one in the bottom row, respectively. All the isosurfaces are plotted at $|{\psi }_1{|}^2=0.05$. The breakup of the droplet in the first, second, and third rows is initiated by perturbation eigenmodes corresponding to azimuthal indices $k=3,4,5$, respectively. In all cases $g_{\mathrm{LHY}}=0.50$ and $g=1.75$.

Figure 6

Isosurface density plots, at levels $|{\psi }_{1,2}{|}^2=0.1$ showing unstable evolution of the perturbed vortex droplet with $m_1=+1,m_2=-1$ and ${\mu }_{1,2}=-0.04$. Top row shows ${\psi }_1$ component, bottom row shows ${\psi }_2$ component. The breakup is induced by small perturbation with azimuthal index $k=3$. In all cases $g_{\mathrm{LHY}}=0.05$ and $g=1.75$.