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arXiv:2202.02230 · Physical Review A

Vortices in quantum droplets of heteronuclear Bose mixtures

Matteo Caldara, Francesco Ancilotto

Published Jun 30, 2022• 16 citations
arXivHTMLPDF

Abstract

We have theoretically investigated the structure of spinning self-bound droplets made of a $^{41}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ Bose mixture by solving the Gross-Pitaevskii equation including beyond-mean-field correction in the Lee-Huang-Yang form. The structure and energetics of vortex formation in the self-bound mixture have been elucidated, showing that the formation of linear vortices in the heavier species is energetically favored over other configurations. A fake (partially filled) core develops as a consequence in the other species, resulting in a hole which might be imaged in experiments. We computed the minimal size of $^{41}\mathrm{K}\ensuremath{-}^{87}\mathrm{Rb}$ droplets which can host stable vortex lines in their interior, which is important information for experiments aimed at the observation of vortices in such systems. The different role of quantized vortices and capillary waves, which are the two ways angular momentum can be stored in a swirling superfluid, is addressed in detail by computing the relation between angular momentum and rotational frequency. The results show intriguing similarities with the case of a prototypical superfluid, i.e., $^{4}\mathrm{He}$ droplets when set into rotation. A two-branches curve in the stability diagram, qualitatively similar to the one expected for classical (incompressible and viscous) rotating liquid droplets, is obtained when vortices are present in the droplets, while prolate (i.e., nonaxisymmetric) shapes are only permitted in vortex-free droplets.

DOI
10.1103/physreva.105.063328

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We have theoretically investigated the structure of spinning self-bound droplets made of a 41K\ensuremath-87Rb^{41}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}41K\ensuremath-87Rb Bose mixture by solving the Gross-Pitaevskii equation including beyond-mean-field correction in the Lee-Huang-Yang form. The structure and energetics of vortex formation in the self-bound mixture have been elucidated, showing that the formation of linear vortices in the heavier species is energetically favored over other configurations. A fake (partially filled) core develops as a consequence in the other species, resulting in a hole which might be imaged in experiments. We computed the minimal size of 41K\ensuremath−87Rb^{41}\mathrm{K}\ensuremath{-}^{87}\mathrm{Rb}41K\ensuremath−87Rb droplets which can host stable vortex lines in their interior, which is important information for experiments aimed at the observation of vortices in such systems. The different role of quantized vortices and capillary waves, which are the two ways angular momentum can be stored in a swirling superfluid, is addressed in detail by computing the relation between angular momentum and rotational frequency. The results show intriguing similarities with the case of a prototypical superfluid, i.e., 4He^{4}\mathrm{He}4He droplets when set into rotation. A two-branches curve in the stability diagram, qualitatively similar to the one expected for classical (incompressible and viscous) rotating liquid droplets, is obtained when vortices are present in the droplets, while prolate (i.e., nonaxisymmetric) shapes are only permitted in vortex-free droplets.
Using density functional theory, we have theoretically studied the formation and the stability of vortices in quantum liquid droplets composed of a mixture of hyperfine states of potassium. Following the experimental setup that produced quantum droplets for the first time, we work with squeezed drops that are compressed in one direction. By squeezing the drops even more, toward a quasi-two-dimensional geometry, we study the minimum atom number able to show a stable vortex and obtain that this number is significantly smaller than previous predictions for spherical droplets. The reduction of the critical atom number for forming a stable vortex could make their experimental observation in these droplets, which is still lacking, more feasible. Contrary to results obtained in heteronuclear mixtures, where the energetically preferred vortices are partially filled with the species not participating in the rotation, our results show a relevant stability island of fully empty vortices. Increasing the number of particles in the drop and the speed of rotation, we estimate the transition line between empty and filled vortices.
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 m1=m2=1{m}_{1}={m}_{2}=1m1​=m2​=1 or m1=m2=2{m}_{1}={m}_{2}=2m1​=m2​=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 m1,2=1{m}_{1,2}=1m1,2​=1 and m1,2=2{m}_{1,2}=2m1,2​=2, respectively). The results provide a scenario for the creation of stable 3D self-trapped states with the double vorticity (m1,2=2{m}_{1,2}=2m1,2​=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 ${m}_{1}=\ensuremath{-}{m}_{2}=1$, are unstable. The stability of the droplets with m1,2=1{m}_{1,2}=1m1,2​=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.
The angular momentum of rotating superfluid droplets originates from quantized vortices and capillary waves, the interplay between which remains to be uncovered. Here, the rotation of isolated submicrometer superfluid ^{4}He droplets is studied by ultrafast x-ray diffraction using a free electron laser. The diffraction patterns provide simultaneous access to the morphology of the droplets and the vortex arrays they host. In capsule-shaped droplets, vortices form a distorted triangular lattice, whereas they arrange along elliptical contours in ellipsoidal droplets. The combined action of vortices and capillary waves results in droplet shapes close to those of classical droplets rotating with the same angular velocity. The findings are corroborated by density functional theory calculations describing the velocity fields and shape deformations of a rotating superfluid cylinder.
Abstract We systematically explore isotropic self-bound liquid states and topological vortex excitations in two-dimensional binary Bose-Einstein condensates with tunable Rydberg soft-core interactions, modeled by the Gross-Pitaevskii equation incorporating the two-dimensional Lee-Huang-Yang quantum correction. Combining analytical Thomas-Fermi treatment with high-precision numerical simulations, we obtain stable fundamental and vortex-carrying quantum droplet solutions and investigate their collisional dynamics. The particle number N and Rydberg coupling strength κ are shown to systematically modulate the spatial and thermodynamic properties of both vortex-free and ring-shaped vortex liquids. In the repulsive regime κ > 0, our results agree quantitatively with those of magnetic quadrupolar condensates, confirming a common stabilization scenario for two-dimensional quantum fluids. More importantly, the Rydberg system distinguishes itself by offering fully isotropic, sign-switchable interactions and sustaining stable vortex droplets with winding numbers up to m = 4 over broad parameter ranges—features that are severely constrained in anisotropic quadrupolar counterparts.Collisions between moving zero-vorticity and vortex self-bound droplets are briefly discussed, demonstrate that their collisional dynamics are universal under the competing nonlocal repulsion and the logarithmic Lee-Huang-Yang effect.
Helium nanodroplets are considered ideal model systems to explore quantum hydrodynamics in self-contained, isolated superfluids. However, exploring the dynamic properties of individual droplets is experimentally challenging. In this work, we used single-shot femtosecond x-ray coherent diffractive imaging to investigate the rotation of single, isolated superfluid helium-4 droplets containing ~10(8) to 10(11) atoms. The formation of quantum vortex lattices inside the droplets is confirmed by observing characteristic Bragg patterns from xenon clusters trapped in the vortex cores. The vortex densities are up to five orders of magnitude larger than those observed in bulk liquid helium. The droplets exhibit large centrifugal deformations but retain axially symmetric shapes at angular velocities well beyond the stability range of viscous classical droplets.
Within density functional theory, we have studied the interplay between vortex arrays and capillary waves in spinning prolate 4He^{4}\mathrm{He}4He droplets made of several thousand helium atoms. Surface capillary waves are ubiquitous in prolate superfluid 4He^{4}\mathrm{He}4He droplets, and depending on the size and angular momentum of the droplet, they may coexist with vortex arrays. We have found that the equilibrium configuration of small prolate droplets is vortex free, evolving towards vortex hosting as the droplet size increases. This result is in agreement with a recent experiment [O'Connell et al., Phys. Rev. Lett. 124, 215301 (2020)] that disclosed that vortex arrays and capillary waves coexist in the equilibrium configuration of very large drops. In contrast to viscous droplets executing rigid-body rotation, the stability phase diagram of spinning 4He^{4}\mathrm{He}4He droplets cannot be universally described in terms of dimensionless angular momentum and angular velocity variables: Instead, the rotational properties of superfluid helium droplets display a clear dependence on the droplet size and the number of vortices they host.